Entanglement transformation with no classical communication

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青春变形记读后感英语

青春变形记读后感英语Metamorphosis of Youth is a thought-provoking novel that delves into the complex and often turbulent journey of adolescence. As I immersed myself in the narrative, I found myself captivated by the profound insights it offered into the universal experiences of growing up.At the heart of the story lies the protagonist, a young individual navigating the challenges and transformations that define the teenage years. The author masterfully captures the rawness of emotion, the internal struggles, and the profound changes that shape the protagonist's sense of self. Through vivid descriptions and poignant dialogues, the reader is taken on a journey that resonates with the universal human experience.One of the most striking aspects of the novel is its ability to delve into the nuances of identity formation. The protagonist's journey is marked by a constant search for self-understanding, as they grapple with the ever-evolving complexities of their own personality. The author skillfully portrays the protagonist's journey of self-discovery,exploring the delicate balance between the individual's innate qualities and the external influences that shape their development.The narrative's exploration of the protagonist's relationships is equally compelling. The author masterfully weaves the intricate web of familial dynamics, friendships, and romantic entanglements, each serving as a mirror to the protagonist's own growth and transformation. The portrayal of these relationships is both authentic and insightful, reflecting the profound impact they have on the protagonist's emotional and psychological development.Throughout the novel, the author navigates the delicate balance between the protagonist's internal world and the external pressures they face. The protagonist's struggles with societal expectations, peer influence, and the ever-evolving landscape of adolescence are powerfully depicted. The reader is invited to empathize with the protagonist's challenges, as they grapple with the complexities of finding their place in the world.One of the most remarkable aspects of Metamorphosis of Youth is its ability to capture the universal themes of adolescence while maintaining a deeply personal and relatable narrative. The author's skillful use of language and imagery transports the reader into the protagonist's world, allowing them to viscerally experience the emotional rollercoaster that defines the teenage years.The novel's exploration of themes such as identity, belonging, and personal growth resonates profoundly with the reader. The protagonist's journey serves as a mirror to the reader's own experiences, inviting them to reflect on their own transformations and the unique challenges they have faced.Moreover, the novel's exploration of the protagonist's mental health and emotional well-being is particularly poignant. The author delves into the complexities of navigating the turbulent waters of adolescence, addressing issues such as anxiety, depression, and the profound impact they can have on an individual's sense of self. This sensitive and nuanced portrayal serves as a powerful reminder of the importance of understanding and supporting young people during this crucial stage of development.One of the most compelling aspects of Metamorphosis of Youth is its ability to transcend the boundaries of a single narrative. The protagonist's journey becomes a universal metaphor for the human experience, inviting the reader to reflect on their own personal growth and transformation. The novel's themes of resilience, self-acceptance, and the power of personal agency resonate deeply, leaving a lasting impact on the reader.In conclusion, Metamorphosis of Youth is a masterful work ofliterature that captivates the reader from the very first page. The author's ability to weave a compelling narrative that speaks to the universal experiences of adolescence is truly remarkable. Through its exploration of identity, relationships, and personal growth, the novel invites the reader to embark on a journey of self-reflection and empathy, ultimately leaving a lasting impression on the reader's heart and mind.。

到公园清理垃圾英语作文

到公园清理垃圾英语作文Title: Cleaning Up Garbage in the Park。

Cleaning up garbage in the park is not just a responsibility but also a duty we owe to our community and environment. Recently, I had the opportunity to participate in a park cleanup initiative, and the experience left a profound impact on me. In this essay, I will delve into the importance of such activities, the challenges faced, and the satisfaction derived from contributing to a cleaner environment.First and foremost, the significance of cleaning up garbage in the park cannot be overstated. Parks serve as vital green spaces within urban landscapes, providing a sanctuary for relaxation, recreation, and connection with nature. However, the accumulation of litter detracts from their beauty and poses environmental hazards. Plastic bags, bottles, and wrappers not only mar the aesthetic appeal of the park but also endanger wildlife through ingestion orentanglement. Moreover, decomposing organic waste produces methane, a potent greenhouse gas that contributes to climate change. By participating in park cleanup efforts, we mitigate these adverse effects and preserve the natural beauty of our surroundings.Nevertheless, cleaning up garbage in the park presents its own set of challenges. One such challenge is the sheer volume of waste accumulated over time. Despite regular maintenance by park authorities, littering remains a persistent issue, exacerbated by the lack of awareness and irresponsible behavior of some visitors. Sorting through heaps of trash can be physically demanding and time-consuming, requiring patience and perseverance. Additionally, ensuring proper disposal of hazardous materials such as broken glass or sharp objects necessitates caution to avoid injuries. Despite these challenges, the collective effort of volunteers can make a significant difference in restoring the cleanliness of the park.Participating in a park cleanup initiative was a deeplygratifying experience. As I worked alongside fellow volunteers, armed with gloves and trash bags, there was a sense of camaraderie and purpose that transcendedindividual motivations. Each piece of litter collected symbolized a small victory in our collective endeavor to reclaim the park from neglect. Moreover, witnessing the transformation from a litter-strewn environment to apristine landscape filled me with a sense of accomplishment and pride. The satisfaction derived from knowing that our efforts would benefit both present and future generations was immeasurable.Furthermore, park cleanup activities foster a sense of environmental stewardship and civic responsibility among participants. By actively engaging in the preservation of public spaces, individuals develop a deeper appreciationfor the natural world and recognize their role in its protection. Through education and outreach initiatives, we can raise awareness about the importance of waste reduction, recycling, and sustainable practices. Empowering communities to take ownership of their surroundings fosters a culture of environmental consciousness and collectiveaction.In conclusion, cleaning up garbage in the park is not merely a chore but a meaningful endeavor that contributes to the well-being of our community and environment. Despite the challenges posed by litter accumulation, the satisfaction derived from restoring the cleanliness of public spaces and fostering a sense of environmental stewardship makes the effort worthwhile. By working together, we can create cleaner, healthier, and more sustainable communities for generations to come.。

傲慢与偏见黑布林英语阅读读后感高二

傲慢与偏见黑布林英语阅读读后感高二Pride and Prejudice by Jane Austen is a timeless classic that captivates readers with its witty dialogue, complex characters, and intricate plot. Set in the 19th century English countryside, this novel skillfully examines societal norms, love, and the human condition. As a high school sophomore, reading this novel has been an enlightening experience that has broadened my understanding of literature and society.One of the most compelling aspects of Pride and Prejudice is Austen's ability to craft multidimensional characters. Elizabeth Bennet, the novel's protagonist, immediately caught my attention with her intelligence, independence, and strong-willed nature. It was refreshing to witness her refusal to conform to societal expectations and her determination to marry for love rather than for financial security. In asociety where marriage was often seen as the sole means of survival for women, Elizabeth's defiance challengestraditional gender roles and encourages readers to question societal norms.Through Elizabeth's interactions with Mr. Darcy, Austen explores the theme of pride and prejudice, both in oneself and others. Initially perceiving Darcy as arrogant and aloof, Elizabeth's judgment is clouded by her own prejudices. However, as the novel progresses, she begins to peel back the layers of his character and discover his genuine nature. This gradual transformation in Elizabeth's perception not only teaches the importance of introspection but also the dangers of relying solely on first impressions.Furthermore, Pride and Prejudice offers a vivid portrayal of the class-conscious society prevalent in the 19th century. The Bennet family's lower social standing serves as a backdrop against which the prejudices and expectations of theupper class are highlighted. The interactions between the Bennet sisters and the landed gentry expose the stark differences in values, financial security, and social status. Austen's exploration of these class dynamics underscores the inequality and snobbery that existed during this era, prompting readers to reflect upon the relevancy of such societal divisions today.The novel's intricate plot, filled with romantic entanglements, misunderstandings, and humorous encounters, keeps readers engrossed from beginning to end. Austen's keen observations of human nature are masterfully woven into the story, resulting in a narrative that is both entertaining and thought-provoking. The balance between romance and social commentary is striking, allowing readers to delve into the lives and motivations of the characters while simultaneously pondering the broader issues raised by the novel.Reading Pride and Prejudice as a high school student has been a transformative experience. Austen's portrayal of relationships, society, and individual growth has deepened my understanding of human nature and the historical context in which this novel was written. I have come to appreciate the power of literature in prompting critical thinking and fostering empathy for characters that exist beyond the confines of the pages.In conclusion, Pride and Prejudice is a literary masterpiece that continues to resonate with readers across generations. Austen's ability to create relatable characters, explore social themes, and craft a compelling plot make this novel an enduring classic. As a high school sophomore, reading this novel has broadened my perspective on love, societal expectations, and the importance of challengingone's own biases. Pride and Prejudice serves as testament to Austen's extraordinary talent and her invaluablecontributions to English literature.。

红发会英语读后感

红发会英语读后感After reading "Red Hair Association", I was deeply moved by the story and the characters. The novel is a powerful narrative of friendship, love, and resilience, set against the backdrop of a society struggling to reconcile its past and present. The novel follows the story of three childhood friends, who find themselves caught up in the chaos of political and social upheaval in their homeland. The characters undergo significant transformation throughout the story, and their personal journeys resonate with the larger themes of the novel.One of the central themes of the novel is the power of friendship and its ability to sustain and uplift individuals in the face of adversity. The bond between the three friends, despite the challenges they face, remains unbreakable. Their loyalty and support for each other help them navigate thedifficult terrain of their lives, and it is this solidarity that becomes a source of strength for them.In addition to the theme of friendship, the novel also explores the complexities of love and relationships. The characters experience romantic entanglements, heartbreak, and the pursuit of love in various forms. These relationships are intricately woven into the fabric of the narrative, adding layers of depth and emotion to the story.Furthermore, the novel delves into the larger socio-political context of the characters' lives, shedding light on the historical and cultural forces shaping their world. The backdrop of societal change and upheaval serves as a powerful catalyst for the characters' personal growth and transformation. The novel skillfully weaves together the personal and the political, offering a nuanced portrayal of the ways in which individual lives intersect with broader historical forces.The prose itself is evocative and deeply moving, conveying the vivid emotions and inner turmoil of the characters. The author's vivid descriptions and lyrical language bring the world of the novel to life, immersing the reader in the rich tapestry of the characters' experiences.Overall, "Red Hair Association" is a compelling and thought-provoking novel that resonates with its exploration of friendship, love, and resilience in the face of adversity. The characters' personal journeys, set against the backdrop of societal change, offer a poignant and insightfulreflection on the human experience. This is a novel that lingers in the mind long after the final page is turned, leaving a lasting impression on the reader.。

SCI写作句型汇总

SCI写作句型汇总S C I论文写作中一些常用的句型总结(一)很多文献已经讨论过了一、在Introduction里面经常会使用到的一个句子：很多文献已经讨论过了。

它的可能的说法有很多很多，这里列举几种我很久以前搜集的：A.??Solar energy conversion by photoelectrochemical cells?has been intensively investigated.?(Nature 1991, 353, 737 - 740?)B.?This was demonstrated in a number of studies that?showed that composite plasmonic-metal/semiconductor photocatalysts achieved significantly higher rates in various photocatalytic reactions compared with their pure semiconductor counterparts.C.?Several excellent reviews describing?these applications are available, and we do not discuss these topicsD.?Much work so far has focused on?wide band gap semiconductors for water splitting for the sake of chemical stability.（DOI:10.1038/NMAT3151）E.?Recent developments of?Lewis acids and water-soluble organometalliccatalysts?have attracted much attention.（Chem. Rev. 2002, 102, 3641?3666）F.?An interesting approach?in the use of zeolite as a water-tolerant solid acid?was described by?Ogawa et al（Chem.Rev. 2002, 102, 3641?3666）G.?Considerable research efforts have been devoted to?the direct transition metal-catalyzed conversion of aryl halides toaryl nitriles. (J. Org. Chem. 2000, 65, 7984-7989) H.?There are manyexcellent reviews in the literature dealing with the basic concepts of?the photocatalytic processand the reader is referred in particular to those by Hoffmann and coworkers,Mills and coworkers, and Kamat.(Metal oxide catalysis,19,P755)I. Nishimiya and Tsutsumi?have reported on（proposed）the influence of the Si/Al ratio of various zeolites on the acid strength, which were estimated by calorimetry using ammonia. (Chem.Rev. 2002, 102, 3641?3666)二、在results and discussion中经常会用到的：如图所示A. GIXRD patterns in?Figure 1A show?the bulk structural information on as-deposited films.?B.?As shown in Figure 7B,?the steady-state current density decreases after cycling between 0.35 and 0.7 V, which is probably due to the dissolution of FeOx.?C.?As can be seen from?parts a and b of Figure 7, the reaction cycles start with the thermodynamically most favorable VOx structures（J. Phys. Chem. C 2014, 118, 24950?24958）这与XX能够相互印证：A.?This is supported by?the appearance in the Ni-doped compounds of an ultraviolet–visible absorption band at 420–520nm (see Fig. 3 inset), corresponding to an energy range of about 2.9 to 2.3 eV.B. ?This?is consistent with the observation from?SEM–EDS. (Z.Zou et al. / Chemical Physics Letters 332 (2000) 271–277)C.?This indicates a good agreement between?the observed and calculated intensities in monoclinic with space groupP2/c when the O atoms are included in the model.D. The results?are in good consistent with?the observed photocatalytic activity...E. Identical conclusions were obtained in studies?where the SPR intensity and wavelength were modulated by manipulating the composition, shape,or size of plasmonic nanostructures.?F.??It was also found that areas of persistent divergent surfaceflow?coincide?with?regions where convection appears to be consistently suppressed even when SSTs are above 27.5°C.(二)1. 值得注意的是...A.?It must also be mentioned that?the recycling of aqueous organic solvent is less desirable than that of pure organic liquid.B.?Another interesting finding is that?zeolites with 10-membered ring pores showed high selectivities (>99%) to cyclohexanol, whereas those with 12-membered ring pores, such as mordenite, produced large amounts of dicyclohexyl ether. (Chem. Rev. 2002, 102,3641?3666)C.?It should be pointed out that?the nanometer-scale distribution of electrocatalyst centers on the electrode surface is also a predominant factor for high ORR electrocatalytic activity.D.?Notably,?the Ru II and Rh I complexes possessing the same BINAP chirality form antipodal amino acids as the predominant products.?(Angew. Chem. Int. Ed., 2002, 41: 2008–2022)E. Given the multitude of various transformations published,?it is noteworthy that?only very few distinct?activation?methods have been identified.?(Chem. Soc. Rev., 2009,?38, 2178-2189)F.?It is important to highlight that?these two directing effects will lead to different enantiomers of the products even if both the“H-bond-catalyst” and the?cata lyst?acting by steric shielding have the same absolute stereochemistry. (Chem. Soc. Rev.,?2009,?38, 2178-2189)G.?It is worthwhile mentioning that?these PPNDs can be very stable for several months without the observations of any floating or precipitated dots, which is attributed to the electrostatic repulsions between the positively charge PPNDs resulting in electrosteric stabilization.(Adv. Mater., 2012, 24: 2037–2041)2.?...仍然是个挑战A.?There is thereby an urgent need but it is still a significant challenge to?rationally design and delicately tail or the electroactive MTMOs for advanced LIBs, ECs, MOBs, and FCs.?（Angew. Chem. Int. Ed.2 014, 53, 1488 – 1504）B.?However, systems that are?sufficiently stable and efficient for practical use?have not yet been realized.C.??It?remains?challenging?to?develop highly active HER catalysts based on materials that are more abundant at lower costs. (J. Am. Chem.Soc.,?2011,?133, ?7296–7299)D.?One of the?great?challenges?in the twenty-first century?is?unquestionably energy storage. (Nature Materials?2005, 4, 366 - 377?)众所周知A.?It is well established (accepted) / It is known to all / It is commonlyknown?that?many characteristics of functional materials, such as composition, crystalline phase, structural and morphological features, and the sur-/interface properties between the electrode and electrolyte, would greatly influencethe performance of these unique MTMOs in electrochemical energy storage/conversion applications.（Angew. Chem. Int. Ed.2014,53, 1488 – 1504）B.?It is generally accepted (believed) that?for a-Fe2O3-based sensors the change in resistance is mainly caused by the adsorption and desorption of gases on the surface of the sensor structure. (Adv. Mater. 2005, 17, 582)C.?As we all know,?soybean abounds with carbon,?nitrogen?and oxygen elements owing to the existence of sugar,?proteins?and?lipids. (Chem. Commun., 2012,?48, 9367-9369)D.?There is no denying that?their presence may mediate spin moments to align parallel without acting alone to show d0-FM. (Nanoscale, 2013,?5, 3918-3930)(三)1. 正如下文将提到的...A.?As will be described below（也可以是As we shall see below）,?as the Si/Al ratio increases, the surface of the zeolite becomes more hydrophobic and possesses stronger affinity for ethyl acetate and the number of acid sites decreases.（Chem. Rev. 2002, 102, 3641?3666）B. This behavior is to be expected and?will?be?further?discussed?below. (J. Am. Chem. Soc.,?1955,?77, 3701–3707)C.?There are also some small deviations with respect to the flow direction,?whichwe?will?discuss?below.（Science, 2001, 291, 630-633）D.?Below,?we?will?see?what this implies.E.?Complete details of this case?will?be provided at a?later?time.E.?很多论文中，也经常直接用see below来表示，比如：The observation of nanocluster spheres at the ends of the nanowires is suggestive of a VLS growth process (see?below). (Science, 1998, ?279, 208-211)2. 这与XX能够相互印证...A.?This is supported by?the appearance in the Ni-doped compounds of an ultraviolet–visible absorption band at 420–520 nm (see Fig. 3 inset), corresponding to an energy range of about2.9 to 2.3 eVB.This is consistent with the observation from?SEM–EDS. (Chem. Phys. Lett. 2000, 332, 271–277)C.?Identical conclusions were obtained?in studies where the SPR intensity and wavelength were modulated by manipulating the composition, shape, or size of plasmonic nanostructures.?（Nat. Mater. 2011, DOI: 10.1038/NMAT3151）D. In addition, the shape of the titration curve versus the PPi/1 ratio,?coinciding withthat?obtained by fluorescent titration studies, suggested that both 2:1 and 1:1 host-to-guest complexes are formed. (J. Am. Chem. Soc. 1999, 121, 9463-9464)E.?This unusual luminescence behavior is?in accord with?a recent theoretical prediction; MoS2, an indirect bandgap material in its bulk form, becomes a direct bandgapsemiconductor when thinned to a monolayer.?(Nano Lett.,?2010,?10, 1271–1275)3.?我们的研究可能在哪些方面得到应用A.?Our ?ndings suggest that?the use of solar energy for photocatalytic watersplitting?might provide a viable source for?‘clean’ hydrogen fuel, once the catalyticef?ciency of the semiconductor system has been improved by increasing its surface area and suitable modi?cations of the surface sites.B. Along with this green and cost-effective protocol of synthesis,?we expect that?these novel carbon nanodots?have potential applications in?bioimaging andelectrocatalysis.(Chem. Commun., 2012,?48, 9367-9369)C.?This system could potentially be applied as?the gain medium of solid-state organic-based lasers or as a component of high value photovoltaic (PV) materials, where destructive high energy UV radiation would be converted to useful low energy NIR radiation. (Chem. Soc. Rev., 2013,?42, 29-43)D.?Since the use of?graphene?may enhance the photocatalytic properties of TiO2?under UV and visible-light irradiation,?graphene–TiO2?composites?may potentially be used to?enhance the bactericidal activity.?(Chem. Soc. Rev., 2012,?41, 782-796)E.??It is the first report that CQDs are both amino-functionalized and highly fluorescent,?which suggests their promising applications in?chemical sensing.(Carbon, 2012,?50,?2810–2815)(四)1. 什么东西还尚未发现/系统研究A. However,systems that are sufficiently stable and efficient for practical use?have not yet been realized.B. Nevertheless,for conventional nanostructured MTMOs as mentioned above,?some problematic disadvantages cannot be overlooked.（Angew. Chem. Int. Ed.2014,53, 1488 – 1504）C.?There are relatively few studies devoted to?determination of cmc values for block copolymer micelles. (Macromolecules1991, 24, 1033-1040)D. This might be the reason why, despite of the great influence of the preparation on the catalytic activity of gold catalysts,?no systematic study concerning?the synthesis conditions?has been published yet.?(Applied Catalysis A: General 2002, 226, ?1–13)E.?These possibilities remain to be?explored.F.??Further effort is required to?understand and better control the parameters dominating the particle surface passivation and resulting properties for carbon dots of brighter photoluminescence. (J. Am. Chem. Soc.,?2006,?128?, 7756–7757)2.?由于/因为...A.?Liquid ammonia?is particularly attractive as?an alternative to water?due to?its stability in the presence of strong reducing agents such as alkali metals that are used to access lower oxidation states.B.?The unique nature of?the cyanide ligand?results from?its ability to act both as a σdonor and a π acceptor combined with its negativecharge and ambidentate nature.C.?Qdots are also excellent probes for two-photon confocalmicroscopy?because?they are characterized by a very large absorption cross section?(Science ?2005,?307, 538-544).D.?As a result of?the reductive strategy we used and of the strong bonding between the surface and the aryl groups, low residual currents (similar to those observed at a bare electrode) were obtained over a large window of potentials, the same as for the unmodified parent GC electrode. (J. Am. Chem. Soc. 1992, 114, 5883-5884)E.?The small Tafel slope of the defect-rich MoS2 ultrathin nanosheets is advantageous for practical?applications,?since?itwill lead to a faster increment of HER rate with increasing overpotential.(Adv. Mater., 2013, 25: 5807–5813)F. Fluorescent carbon-based materials have drawn increasing attention in recent years?owing to?exceptional advantages such as high optical absorptivity, chemical stability, biocompatibility, and low toxicity.(Angew. Chem. Int. Ed., 2013, 52: 3953–3957)G.??On the basis of?measurements of the heat of immersion of water on zeolites, Tsutsumi etal. claimed that the surface consists of siloxane bondings and is hydrophobicin the region of low Al content. (Chem. Rev. 2002, 102, 3641?3666)H.?Nanoparticle spatial distributions might have a large significance for catalyst stability,?given that?metal particle growth is a relevant deactivation mechanism for commercial catalysts.?3. ...很重要A.?The inhibition of additional nucleation during growth, in other words, the complete separation?of nucleation and growth,?is?critical(essential, important)?for?the successful synthesis of monodisperse nanocrystals. (Nature Materials?3, 891 - 895 (2004))B.??In the current study,?Cys,?homocysteine?(Hcy) and?glutathione?(GSH) were chosen as model?thiol?compounds since they?play important (significant, vital, critical) roles?in many biological processes and monitoring of these?thiol?compounds?is of great importance for?diagnosis of diseases.(Chem. Commun., 2012,?48, 1147-1149)C.?This is because according to nucleation theory,?what really matters?in addition to the change in temperature ΔT?(or supersaturation) is the cooling rate.(Chem. Soc. Rev., 2014,?43, 2013-2026)(五)1. 相反/不同于A.?On the contrary,?mononuclear complexes, called single-ion magnets (SIM), have shown hysteresis loops of butterfly/phonon bottleneck type, with negligiblecoercivity, and therefore with much shorter relaxation times of magnetization. (Angew. Chem. Int. Ed., 2014, 53: 4413–4417)B.?In contrast,?the Dy compound has significantly larger value of the transversal magnetic moment already in the ground state (ca. 10?1?μB), therefore allowing a fast Q TM. （Angew. Chem. Int. Ed., 2014, 53: 4413–4417）C.?In contrast to?the structural similarity of these complexes, their magnetic behavior exhibits strong divergence.?(Angew. Chem. Int. Ed., 2014, 53: 4413–4417)D.?Contrary to?other conducting polymer semiconductors, carbon nitride ischemically and thermally stable and does not rely on complicated device manufacturing. (Nature materials, 2009, 8(1): 76-80.)E.?Unlike?the spherical particles they are derived from that Rayleigh light-scatter in the blue, these nanoprisms exhibit scattering in the red, which could be useful in developing multicolor diagnostic labels on the basis not only of nanoparticle composition and size but also of shape. (Science 2001,? 294, 1901-1903)2. 发现，阐明，报道，证实可供选择的词包括：verify, confirm, elucidate, identify, define, characterize, clarify, establish, ascertain, explain, observe, illuminate, illustrate，demonstrate, show, indicate, exhibit, presented, reveal, display, manifest，suggest, propose, estimate, prove, imply, disclose，report, describe，facilitate theidentification of?举例：A. These stacks appear as nanorods in the two-dimensional TEM images, but tilting experiments?confirm that they are nanoprisms.?(Science 2001,? 294, 1901-1903)B. Note that TEM?shows?that about 20% of the nanoprisms are truncated.?(Science 2001,? 294, 1901-1903)C. Therefore, these calculations not only allow us to?identify?the important features in the spectrum of the nanoprisms but also the subtle relation between particle shape and the frequency of the bands that make up their spectra.?(Science 2001,? 294, 1901-1903)D. We?observed?a decrease in intensity of the characteristic surface plasmon band in the ultraviolet-visible (UV-Vis) spectroscopy for the spherical particles at λmax?= 400 nm with a concomitant grow th of three new bands of λmax?= 335 (weak), 470 (medium), and 670 nm (strong), respectively. (Science 2001,? 294, 1901-1903)E. In this article, we present data?demonstrating?that opiate and nonopiate analgesia systems can be selectively activated by different environmental manipulationsand?describe?the neural circuitry involved. (Science 1982, 216, 1185-1192)F. This?suggests?that the cobalt in CoP has a partial positive charge (δ+), while the phosphorus has a partial negative charge (δ?),?implying?a tran sfer of electron density from Co to P.?(Angew. Chem., 2014, 126: 6828–6832)3. 如何指出当前研究的不足A. Although these inorganic substructures can exhibit a high density of functional groups, such as bridging OH groups, andthe substructures contribute significantly to the adsorption properties of the material,surprisingly little attention has been devoted to?the post-synthetic functionalization of the inorganic units within MOFs. (Chem. Eur. J., 2013, 19: 5533–5536.)B.?Little is known,?however, about the microstructure of this material. (Nature Materials 2013,12, 554–561)C.?So far, very little information is available, and only in?the absorber film, not in the whole operational devices. （Nano Lett.,?2014,?14?(2), pp 888–893）D.?In fact it should be noted that very little optimisation work has been carried out on?these devices. (Chem. Commun., 2013,?49, 7893-7895)E. By far the most architectures have been prepared using a solution processed perovskite material,?yet a few examples have been reported that?have used an evaporated perovskite layer. (Adv. Mater., 2014, 27: 1837–1841.)F. Water balance issues have been effectively addressed in PEMFC technology through a large body of work encompassing imaging, detailed water content and water balance measurements, materials optimization and modeling,?but very few of these activities have been undertaken for?anion exchange membrane fuel cells,? primarily due to limited materials availability and device lifetime. (J. Polym. Sci. Part B: Polym. Phys., 2013, 51: 1727–1735)G. However,?none of these studies?tested for Th17 memory, a recently identified T cell that specializes in controlling extracellular bacterial infections at mucosal surfaces. (PNAS, 2013,?111, 787–792)H. However,?uncertainty still remains as to?the mechanism by which Li salt addition results in an extension of the cathodicreduction limit. (Energy Environ. Sci., 2014,?7, 232-250)I.?There have been a number of high profile cases where failure to?identify the most stable crystal form of a drug has led to severe formulation problems in manufacture. (Chem. Soc. Rev., 2014,?43, 2080-2088)J. However,?these measurements systematically underestimate?the amount of ordered material. ( Nature Materials 2013, 12, 1038–1044)(六)1.?取决于a.?This is an important distinction, as the overall activity of a catalyst will?depend on?the material properties, synthesis method, and other possible species that can be formed during activation.?(Nat. Mater.?2017,16,225–229)b.?This quantitative partitioning?was determined by?growing crystals of the 1:1 host–guest complex between?ExBox4+?and corannulene. (Nat. Chem.?2014,?6177–178)c.?They suggested that the Au particle size may?be the decisive factor for?achieving highly active Au catalysts.(Acc. Chem. Res.,?2014,?47, 740–749)d.?Low-valent late transition-metal catalysis has?become indispensable to?chemical synthesis, but homogeneous high-valent transition-metal catalysis is underdeveloped, mainly owing to the reactivity of high-valent transition-metal complexes and the challenges associated with synthesizing them.?(Nature2015,?517,449–454)e.?The polar effect?is a remarkable property that enables?considerably endergonic C–H abstractions?that would not be possible otherwise.?(Nature?2015, 525, 87–90)f.?Advances in heterogeneous catalysis?must rely on?the rational design of new catalysts. (Nat. Nanotechnol.?2017, 12, 100–101)g.?Likely, the origin of the chemoselectivity may?be also closely related to?the H?bonding with the N or O?atom of the nitroso moiety, a similar H-bonding effect is known in enamine-based nitroso chemistry. (Angew. Chem. Int. Ed.?2014, 53: 4149–4153)2.?有很大潜力a.?The quest for new methodologies to assemble complex organic molecules?continues to be a great impetus to?research efforts to discover or to optimize new catalytic transformations. (Nat. Chem.?2015,?7, 477–482)b.?Nanosized faujasite (FAU) crystals?have great potential as?catalysts or adsorbents to more efficiently process present and forthcoming synthetic and renewablefeedstocks in oil refining, petrochemistry and fine chemistry. (Nat. Mater.?2015, 14, 447–451)c.?For this purpose, vibrational spectroscopy?has proved promising?and very useful.?(Acc Chem Res. 2015, 48, 407–413.)d.?While a detailed mechanism remains to be elucidated and?there is room for improvement?in the yields and selectivities, it should be remarked that chirality transfer upon trifluoromethylation of enantioenriched allylsilanes was shown. (Top Catal.?2014,?57: 967.?)e.?The future looks bright for?the use of PGMs as catalysts, both on laboratory and industrial scales, because the preparation of most kinds of single-atom metal catalyst is likely to be straightforward, and because characterization of such catalysts has become easier with the advent of techniques that readilydiscriminate single atoms from small clusters and nanoparticles. (Nature?2015, 525, 325–326)f.?The unique mesostructure of the 3D-dendritic MSNSs with mesopore channels of short length and large diameter?is supposed to be the key role in?immobilization of active and robust heterogeneous catalysts, and?it would have more hopeful prospects in?catalytic applications. (ACS Appl. Mater. Interfaces,?2015,?7, 17450–17459)g.?Visible-light photoredox catalysis?offers exciting opportunities to?achieve challenging carbon–carbon bond formations under mild and ecologically benign conditions. (Acc. Chem. Res.,?2016, 49, 1990–1996)3. 因此同义词：Therefore, thus, consequently, hence, accordingly, so, as a result这一条比较简单，这里主要讲一下这些词的副词词性和灵活运用。

XVII. The Star

XVII. The StarInstinct Devotion to the moment.Goal Spiritual introspection,insight into the perfectionof the cosmos, renewal,and rebirth.Guiding Star of Bethlehem (tmst inprinciple a higher order).Creative hope (maintaining Ia conception of what is |possible). |Destructive hope (denial of|the truth), illusion. |Carefreeness, trust in the |future. |FoundationThe cardThe Star is the card touching the longing in the depths of our unconscious, and this longing is equivalent to the longing of life for itself. For inthe water of the unconscious that Nut or Isis pours over herself we recognize the spiral-formed energy around the seven-pointed star, which is an exact representation of the even-pointed star in the great firmament.' We therefore see the creative impulse of the eternal in the waters of life, responsible for the births of humanity as well as the births of the universe. This creative impulse is not only the orig- inal symbol of life, but also the hope that out ofthe rubble (the Tower) new life will grow time and again. The entire card is a vision of indescribable power, which we call the principle of eternal renewal, a return to the fundamentals in which we recognize ourselves as part of the greater whole, and this greater whole is the impulse of life itself.Analysis and Descriptioni. Motif (the star fairy)This seductive trump shows us a naked, feminine figure in a lapis-lazuli blue and pink-colored light. She has descended from the universe to carry long- ing into the world, the longing for life fulfilled inthe womb and at the mother's breast. With her descent, she penetrates into the light of our con- sciousness in order to connect us with our inner femininity.Lady of the stars |The picture portrays Nut, the lady of the stars.2! She carries two chalices, a golden one from which-j she pours out the water high above her head, and'| a silver one from which the immortal tincture of|her life flows; both are similar to female breasts.|j Her nakedness does not just simply haveattractive power of the unveiled: she is the vision o|| the indescribable in which the melancholy ofuluH mate inaccessibility resonates. This longing, expressed in the newborn by the desire formother's breast, and therefore carrying the sublim-H mat wish of merging with the cosmos within it (to|| be one with the mother), cannot be fulfilled in life|| The star fairy, who has her home in the depths o|| the cosmos, is an expression of the unconscious^ longing shining through the beauty of awoman. She is not the fire, she is just the longin|§| for the light; this is why she bears the projectionour search. We desire her, but what we desire is archetype of the eternal feminine and not thetal we encounter in real life.In the long hair winding around her bodycloak, we do not find the indication of livingbut rather of dreaming of living the longing dreamed life and, so to speak, surrendering witiout living the inner yearnings. On the one han|this symbolizes the entanglement with the unlib^ated longings, yet on the other hand we there! |J approach the cosmic connection in this card: |the hail is the connection between the cosmos;|^^Bthe material.1. Crowley speaks here of the seven-pointed star of Venus or of the star of Babylon, for Babylon is yet a further materialization of the original idea of Nuith; she is the Scarlet Woman, the sacred Harlot who is the lady ofAtu XI. •2. The figure of the goddess is shown in manifestagjthat is, not as the surrounding space of heaven sho\||Atu XX, where she is the pure philosophical idea^ tinuous and omniform. In this card she is definitel'y| sonified as a human-seeming figure. (Crowley) 1I The StarPosture (chalices)That which we dream through the goddess of thestars, that is, those inner longings she calls forthwithin us, we see united in her form; but that' which has been dreamed into her by the cosmosi we see in the contents of the chalices the Star| Goddess pours out upon herself. It is the water of, life that she pours out of the two large chalices,I and she not only waters the ground, but also her-' self in the mirror of the elements. We then recog-nize in the water of life the cosmic urge expressingitself in our striving for renewal. Crowley writes:"From this golden cup she pours this etherealwater, which is also milk and oil and blood, uponher own head, indicating the eternal renewal of the categories, the inexhaustible possibilities of exis- tence."Water of life (microcosmos)That which was just nebulous feeling and vision-ary longing now takes on form and shape through the contact with the life crystals. The Star Goddess perceives herself (and we perceive ourselves in her) not as the actual goddess of love, but rather as the instinctive drive with which people fall in lovewith love, and therefore with the power of hope;the foundation of renewal or the will of creationfrom out of itself. Through the small universe inthe energy spirials, reflecting the great one, we rec- ognize the unlimited possibilites of self-develop- ment present in this card. We recognize in the energy crystals the psychical ability of bringingforth our cosmic self and placing it on the level of that universal plane, of which the small blue star in the waters of life is only a reflection.2. Background (cosmos)Star (macrocosmos)Conversely, on the universal plane the large star reflecting onto the purifying water in the self- renewal of the goddess (which Crowley calls a veil before the face of the immortal goddess) is the principle of renewal that covers the entire networkof the universe. If the small star embodies the vis- ible love of people, then the large one is the invis- ible love of God. It symbolizes that boundless por- tion of us that, completely recognized by ourselves, rules the longing for the eternal. The star standsfor the outpouring of highest inspiration, within i which the spiral haze of cosmic energies flow. Tills j is shown in the image of the goddess who places ; herself under the rain of stars (the virgin sprinkles herself with the water of life).GlobeThe mighty globe in the background is a symbolof the Earth that constantly renews itself; at thesame time it also shows the perspective of longino that leaves all the boundaries of the material, and looks at the mother of all becoming (earth) through the eyes of the daughter from the distance (star world). In this we recognize unlimited possi- bilities, not only of hope, but also of illusions, for the foam of rising visions lets all concepts become reality. It is as if the virgin, quickened by the con- cept of her divinity and possessed by the ideal divine love, does not come closer to the earth (matter), but instead strives for fusion with the universe.3. Foreground (matter)Chalices, crystals (crystalization)In the connection between the two goblets, one can see the interaction between the moon and stars. The energy that initially emanates from the sun is reflected by the moon onto the earth, and thereby receives its crystalline structure. The silver chalice defines the physical renewal of the earth and thereby retrieves a piece of the matter, which has removed itself into the firmament, back to reality. The water flowing upon the earth lets this become fertile. The water that is again poured into water shows the excess and the abundance of ener- gy flowing to us from the infinity of the heavens. The golden chalice expresses cosmic wholeness and the blossoming of hope after destruction (the Tower). That this hope is not in vain is expressedin the crystals at the bottom edge of the card. Roses, butterflies (transformation)Renewal and hope rise in dreams with these and link with visions of the eternal into a mighty starof desire. Yet the Star is not only the principle of hope coming forth out of itself from the wish for renewal, but it is also the enticing call of the godsthat has been dreamed into the "impression of the soul" by the "matrix of longing." Life then begins in flowing and it will also end there when every- thing that we are streams back to the yearnings of nature. (Crowley: "The milk of the stars from her paps"). The principle of love is hinred at in the roses whirling upwards and the butterflies as the sign of renewal or symbol for transformation.InterpretationsBackground (the revelation of Nut)If under the night stars in the desert thou presently burnest mine incense before me, invok- ing me with a pure heart, and the Serpent flame therein, thou shah come a little to lie in my bosom. For one kiss wilt thou then be willing to give all; but whoso gives one particle of- dust shall lose all in that hour. Ye shall gather goods and store of women and spices; ye shall wear rich jewels; ye shall exceed the nations ot the earth in splendour and pride; but always in the love of me, and so shall ye come to my joy. I charge you earnestly to come before me in a single robe, and covered with a rich headdress. I love you! I yearn to you! Pale or purple, veiled or voluptuous, Iwho am all pleasure and purple, and drunken- ness of the innermost sense, desire you. Put on the wings, and arouse the coiled splendour with- in you: come unto me! At all my meetings with vou shall hear the priestess say—and her eyes shall burn with desire as she stands bare and rejoicing in my secret temple—calling rorth the flame of the hearts of all in her love-chant. Sing the rapturous love-song unto me! Burn to me perfumes! Wear to me jewels! Drink to me, for Ilove you! I love you! I am the blue-lidded daugh- ter of Sunset: I am the naked brilliance of the voluptuous night-sky. To me! To me!Aleister Crowley, "The Book ot the Law"GeneralThe Star is not only the card of hope, the uncon- scious will of life, and the insight into higher cor- relations; it also stands for the unconscious desire to abandon oneself in the streams of flowing delight and cheerful irresponsibility: "What do I dream about? What is it that dreams of me? What is dreamed about within me? If a person knows what he dreams about it does not really make him into a person or truly human, nor does he move anywhere with it—he then instead becomes a psy- chologist; a dreamer who knows his own dreams." (From a letter to one of the authors) ConsciousnessThe Star represents the intuitive understanding of cosmic correlations and trust in the laws of the absolute. Just as the cross-sum of "Adjustment" (17 =1+7=8) that is connected with this card expresses the conscious and more intellectually- related understanding of the powers of order, the Star embodies the emotional understanding, inner insight, and related trust in the lawfulness of cre- ation. This card can then characterize the begin- ning of a time when we discover with relief that we have escaped the tower ot worries and experience the carefree lightness of which the Bible says: "Look at the birds of the air: they neither sow nor reap nor gather into barns, and yet your heavenly Father feeds them" (Matthew 6.26).ProfessionThe card indicates that we plan or start things reaching far into the future, and in the positivecourse of which we may set our legitimate hopes.It often occurs that in this early stage we are not at all aware of the far-reaching effects of our actions. Only in retrospect does the decisive course charac- terized by the Star, which has been set for thefuture during these times, become clear. These are often profound insights into the greater correla- tions through which we grow beyond the narrow- ness of our immediate perspective. It is as if we would take on a bird's-eye view from which thehigh tower of obstacles can be surveyed and broad g vistas of a welcome future open up. i]Relationships jIn the area of partnership the Star stands for a rela-1 tionship with a promising future. It shows fresh,far-reaching perspectives in an existing relation- ship, or it can signify the longing, hope, and prospect of a new, lasting partnership. This card, however, warns of the danger of becoming entan-; gled in infantility, since many of the goals we strive; for in partnerhsip are frequently only the attemptto revive the unfulfilled yearnings we developed at the mother's nourishing breast. On this level, it teaches us that earthly love is a power going beyond itself and real partnership necessarily embodies a promise incapable of being fulfilled (in this manner), but in animating us to transcenden-'tal striving can find its fulfillment there. ' Analogous Correlations fi;i. Archetypes and Symbols s^.Archetype lThe water of life. iLetter u -|He = H ("window"); numerical value of five. Tq Hebrew letter stands for window: symbol of tlljlonging for the angels and for the world of stars that they have dreamed.NumberSeventeen is the number of new hope. Calculated from the last full moon, it is the seventeenth night in which the crescent of the new moon (as a silver stripe) can be glimpsed on the western horizon after three dark, moonless nights.Tree of LifeChokmah-Tiphareth: Chokmah's indescribable vision shimmers through the face ofTiphareth, in which in turn the reflection of the indescribable is expressed, but this time as an inner vision. Chokmah (wisdom) stands for the first manifesta- tion of the divine in contact with the psychical reality; Tiphareth (harmony) describes the condi- tion of balance and is at the same time a reflection of the star plane (Kether sphere)./ Ching61 Chung Fu—"Inmost Sincerity."Rune ® (3Wunjo ("joy") and Laguz ("water," "vital energy"). Wunjo symbolizes the "good star" and Laguz/Laf the "flowing harmony," quenching the inner long- ing for God through flowing and growth (the longing of life for itself fulfilled).ZodiacAquarius. Crowley compares the card with the "water-bearer," guardian of the water ot life, which Nut, Lady of the Stars, pours our other chalices. DeitiesAphrodite/Venus (the "naked goddess"); Asherah,Astarte, Ischtar, or Isis, and the seven priestesses of the Oracle, the seven wise men of Arabia (also called the seven pillars of wisdom).MythologyIndian myth of the eternal becoming and dying ofall ages, world, and creations; in the Greek mythology, the seven daughters of Atlas and the Pleiades, who were pursued by Orion until Zeus turned them into constellations in the firmament. Cult sitesGarden of the Eumenides (on the Kolonos hillnear Athens); the Gardens of the Hesperides; the Islands of the Blessed (mysteries), or the Rhine Cliffs of the Lorelei.Sabbath"The Festival of the Liberation of the Spring Waters." Tibetan celebration at the first full moon after the rising ofSirius (Dog Star).RitualRitualistic purification (bath) as symbol of renew-al of life and contemplation.2. Associations and PerceptionsPicture"Birth of Venus" by Sandro Botticelli. Even for the ancient Greeks, beauty was identical with truth.The spirit is nothing without harmony is whatthey taught.Writing"The Song of Solomon."MusicThe "Song of the Rhine Daughters" in "Rhinegold" by Richard Wagner or "Is it a Dream, Can it be Real?" terzet and closing ensemble from the "Rosenkavalier" by Richard Strauss. The celes- tial beauty of the high female voices give an impression of the enchanting fascination of thiscard.FragrancesJasmine, rose, ophire, almond oil.GemsAquamarine, rose quartz, star sapphire, or lapis lazuli, reminiscent of the starry skies.AstrologyJupiter in Aquarius (eleventh house) in the sense of confidence and farsightedness, or Venus in Pisces (twelfth house) as an expression of longing for the unfathomable and the striving for God.AlchemyWhitening (albedo). After the blackening (nigre- do) there follows the whitening through washing (ablucio, baptisma), by which the soul (anima) having escaped through death is reunited with the body.Quotation"We, the water of life! Source which arises out of us and flows that much richer [he more you drink from us ... Drink! Drink! Do what you want!" Michael Ende "The Never-Ending Story"。

Quantum Imaging

a r X i v :q u a n t -p h /0203046v 1 11 M a r 2002Quantum imagingL.A.Lugiato,A.Gatti and E.BrambillaIstituto Nazionale per la Fisica della Materia,Dipartimento di Scienze Chimiche Fisiche e Matematiche,Universit`adell’Insubria,Via Valleggio 11,22100Como,ItalyWe provide a brief overview of the newly born ﬁeld of quantum imaging,and discuss someconcepts that lie at the root of this ﬁeld.I.PREAMBLE This article is based on the plenary talk that one of us (L.A.L.)gave at the ICSSUR Conference in Boston in June 2001.The starting point is provided by the general topic of the spatial aspects of quantum optical ﬂuctuations.This has been the object of several studies in the past,but only recently there has been a constant focus of attention,which is mainly due to the fact that the spatial features may open new possibilities,e.g.in the direction of parallel processing and multichannel operation by quantum optical procedures.Once realized that these studies may have interesting perspectives,it is natural to coin a new name as quantum imaging to designate this ﬁeld.This article includes an introductory part,in which we discuss few concepts that play a key role in this ﬁeld,such as the intrinsic connection between quantum entanglement and squeezing,the spatial squeezing and the near ﬁeld/far ﬁeld duality.Next,we discuss several topics in the ﬁeld of quantum imaging and illustrate recent ideas and results.The topics include -Detection of weak phase and amplitude objects beyond the standard quantum limit,-Ampliﬁcation of weak optical images preserving the signal-to-noise ratio (noiseless ampliﬁcation),-Entangled two-photon microscopy,-Quantum limits in the detection of small displacements and in image reconstruction,-Quantum lithography,-Quantum teleportation of optical images.Several of the results and approaches that we will illustrate have been pursued by the participants in the European Project QUANTIM (Quantum Imaging),that started in January 2001.II.INTRODUCTIONA.Intrinsic connection between squeezing and quantum entanglementOne might consider that squeezing is a rather old-fashioned ﬁeld,whereas for entanglement one immediately thinks of such topics as quantum computing,or quantum teleportation,or cryptography,and concludes that it is a new and exciting ﬁeld.However one must remark that,in the framework of the continuous variable approach,squeezing and quantum entanglement are intrinsically linked,they are basically two faces of the same phenomenon.This circumstance explains by the way why a series of meetings entitled Squeezed States and Uncertainty Relations is still gathering so many people.In order to demonstrate this point in details,let us consider two radiation beams and the associated annihilation operators of photons a 1and a 2,and let us focus on the simple linear transformation to another couple of beams b 1and b 2:b 1=a 1+a 22,b 2=a 1−a 22(1)This transformation can be implemented very easily.If,for example,a1and a2have the same frequency and the same polarization,it is realized by a50/50beam splitter.If they have the same frequency but horizontal and vertical polarization,respectively,one can use a polarizing beam splitter in which the two output beams are polarized at450 and1350,respectively.Now,the general result for transformation(1)is that–if a1and a2are EPR(Einstein-Podolski-Rosen[1])entangled beams with respect to quadrature components,then beams b1and b2are squeezed with respect to two orthogonal quadrature components and,vice versa,–if a1and a2are squeezed beams with respect to two orthogonal quadrature components,b1and b2are EPR entangled beams with respect to quadrature components.To prove this result,let us consider the interaction Hamiltonian for parametric down-conversion in the nondegenerate conﬁguration and in the approximation of classical undepleted pump[2]H=igα a†1a†2−a1a2 (2)where g is the coupling constant andαis the classical amplitude of the pump beam.As it is well known,in the nondegenerate conﬁguration the two beams a1and a2are entangled both with respect to photon number and with respect to two quadrature components[2–4].The entanglement can be explicitly shown by applying the time evolution operator corresponding to the Hamiltonian(2)to the uncorrelated vacuum state,for an interaction timeτinte−icosh(gτint/¯h).(3)Clearly the state described by Eq.(3)is not factorizable,and implies perfect correlation between the photon number in the two beams.If we now introduce into Eq.(2)the expression of a1and a2as a function of b1and b2a1=b1+b22,a2=b1−b22,(4)we obtain immediately the alternative expressionH=i gα2b†2 2−b22 ,(5)which corresponds to the sum of two interaction hamiltonians for parametric down-conversion in the degenerate conﬁguration.Hence the beams b1and b2are squeezed,and,because of the minus sign in front of the second term, the squeezing is in orthogonal quadrature components.For example,whenαis real b1is squeezed with respect to the Y quadrature(imaginary part of the annihilation operator)and b2is squeezed with respect to the X quadrature (real part of the annihilation operator).B.Spatially multimode versus singlemode squeezingIn almost all literature on squeezing one considers singlemode squeezing.If one wants to detect a good level of squeezing,the local oscillator must be matched to the squeezed spatial mode and,in addition,it is necessary to detect the whole beam.If one detects only part of the beam the squeezing is immediately degraded,because a portion of a mode necessarily involves higher order modes,in which squeezing is absent.What we can call local squeezing,i.e. squeezing in small regions of the transverse plane,can be obtained only in presence of spatially multimode squeezing, i.e.squeezing in a band of spatial modes.This has been predicted by Sokolov and Kolobov for the traveling wave optical parametric ampliﬁer(OPA)[5,6]and by our group for the optical parametric oscillator(OPO)[7,8].Let us dwell a moment,for example,on the case of the OPA(Fig.1a),in which one has a slab ofχ(2)material which is pumped by a coherent plane wave of frequency2ωs.A fraction of the pump photons are down-converted into signal-idler photon pairs,which are distributed over a broad band of temporal frequencies around the degeneratefrequencyωs.For eachﬁxed temporal frequency,the photon pairs are distributed over a band of spatial frequencieslabeled by the transverse component q of the wave vector.If,in addition to the pumpﬁeld,we inject a coherent plane wave with frequencyωs and transverse wave vector q(Fig.1b),in the output we have a signal wave which corresponds to an ampliﬁed version of the input wave and for this reason the system is called optical parametric ampliﬁer.Because of the pairwise emission of photons,there isalso an idler wave which,close to degeneracy,is symmetrical with respect to the signal wave.Referring to the case in which only the pump is injected,two regimes can be distinguished.One is that of purespontaneous parametric down-conversion,as in the case of a very thin crystal.In this case coincidences between partners of single photon pairs are detected.The other is that of dominant stimulated parametric down-conversion,inwhich a large number of photon pairs at a time is detected.In the following we will consider both cases alternatively.C.Nearﬁeld/Farﬁeld dualityWe want to illustrate the key spatial quantum properties of theﬁeld emitted by an OPA,in the linear regime of negligible pump depletion,or by an OPO below threshold.In the OPA case,we consider the conﬁguration of a largephoton number.In the nearﬁeld(see Fig.2)one has the phenomenon of spatially multimode squeezing or local squeezing discussedin Sec.II B.A good level of squeezing is found,provided the region which is detected has a linear size not smaller than the inverse of the spatial bandwidth of emission in the Fourier plane.If,on the other hand,one looks at thefarﬁeld(which can be reached,typically,by using a lens as shown in Fig.2)oneﬁnds the phenomenon of spatialentanglement between small regions located symmetrically with respect to the center.Precisely,if one considers two symmetrical pixels1and2(Fig.3),the intensityﬂuctuations in the two pixels are very well correlated or,equivalently,theﬂuctuations in the intensity diﬀerence between the two pixels are very much below the shot noise[9,10].Because this phenomenon arises for any pair of symmetrical pixels,we call it spatial entanglement.The same eﬀect occursalso for quadrature components,because in the two pixels theﬂuctuations of the quadrature component X are almost exactly correlated,and those of the quadrature component Y are almost exactly anticorrelated[11].The minimumsize of the symmetrical small regions,among which oneﬁnds spatial entanglement,is determined by theﬁnal aperture of optical elements,and is given,in the paraxial approximation,byλf/a,whereλis the wavelength,f is the focallength of the lens and a is the aperture of optical elements(e.g.the lens aperture)(Fig.2).In a more realistic model of the OPA,theﬁnite waist of the pumpﬁeld should be taken into account.In this case the minimum size of theregions where entanglement is detectable in the farﬁeld is mostly determined by the pump waist.The spatial entanglement of intensityﬂuctuations in the farﬁeld is quite evident even in single shots(the pumpﬁeldis typically pulsed).Fig.4a shows a numerical simulation in a case of non collinear phase matching at degenerated frequency.One observes the presence of symmetrical intensity peaks,which become broader and broader as onereduces the waist of the pumpﬁeld.A similar situation is observed in an experiment performed using a LBO crystal [12].We observeﬁnally that the nearﬁeld/farﬁeld duality can be understood on the basis of the intrinsic connectionbetween squeezing and quantum entanglement.The spatial entanglement in the farﬁeld arises from the correlation between the modes a q∼exp[i q· x]and a− q∼exp[−i q· x],,which in the farﬁeld give rise to two separated and opposite spots in the transverse plane( q is the transverse wave vector and x is the position vector in the transverse plane).On the other hand,in the nearﬁeld there is no squeezing in modes a q and a− q separately,whereas there is large√2,which annihilate photons squeezing in the combination modes and b q=(a q+a− q)/on spatial modes∼cos( q· x)and∼sin( q· x).In the nearﬁeld it is possible to observe this squeezing by using a localoscillator with a cos( q· x)or sin( q· x)spatial conﬁguration[7].One notices immediately that the relation betweenmodes a q,a− q and modes b q,b− q coincides with Eq.(1),which,as we have seen,transforms entangled beams into squeezed beams,and viceversa.III.TOPICS IN QUANTUM IMAGINGNext we focus speciﬁcally on studying the quantum aspects of the very classicalﬁeld of imaging.We will discuss several topics in order.A.Detection of weak amplitude or phase objects beyond the standard quantum limitLet us considerﬁrst the case of a weak amplitude object which is located,say,in the signal part of theﬁeld emitted by an OPA(Fig.5).Both signal and idler are very noisy and therefore,in the case of large photon number,if the object is weak and we detect only the signalﬁeld,the signal-to-noise ratio for the object is low.But,because of the spatial entanglement,theﬂuctuations in the intensity diﬀerence between signal and idler are small.Hence if we detect the intensity diﬀerence,the signal-to-noise ratio for the object becomes much better.This scheme is conceptually related to well known´’ghost image”experiments[13,14]in the regime of detection of single photon pairs.Next,let us pass to the case of a weak phase object in which one can exploit,instead,the property of spatially multimode squeezing.The conﬁguration is the standard one of a Mach-Zender interferometer in which,as it is well known,one can detect a small phase shift with a sensitivity beyond the standard quantum limit by injecting a squeezed beam in the port through which usually normal vacuum enters.If we have a weak phase image(Fig.6)we can obtain the same result by injecting a spatially multimode squeezed light[15].B.Quantum imaging with entangled photon pairsThis approach has been formulated by Abouraddy,Saleh,Sergienko,and Teich for the regime in which single photon pairs are detected.They have shown that using entangled photons in an imaging system oﬀers possibilities that cannot be attained when entanglement is absent[16].In a classical imaging conﬁguration one has a source,an illumination system,an object,an imaging system and a detector.Theﬁeld at the detection plane is related to that at the object plane by a linear integral transformation with a kernel h.Let us focus on the interesting case illustrated in Fig.7,in which the imaging is performed using photon pairs,in which case the topography of the system allows for two branches with diﬀerent kernels to be simultaneously illuminated,a system that was heuristically considered by Belinskii and Klyshko[17].The two-photon wave function is denotedψ( x, x′)( x and x′are source position vectors in the transverse plane)and the two kernels are denoted h1 and h2for the signal and idler beams1and2,respectively.Next,let us introduce two assumptions.Theﬁrst is that the two photons are entangled.This is expressed by the fact that the wave functionψincludes a factorδ( x− x′), which expresses the fact that the two photons are generated at the same point.The second assumption is that the ﬁrst photon is detected by a bucket detector,which can reveal its presence,but not at all its location.On the other hand,the other detector is capable of scanning the position of the photon,and one detects the coincidences between photon pairs.The key point is that in this situation the probability distribution for the position of the second photon depends not only on h2but also on h1,and this is true even though the position of photon1is not detected.Thus, suppose that,for example,there is nothing in the path of photon2,whereas in the path of photon1there is an object which may be transmissive/reﬂective,diﬀusive or scattering(see Fig.7).By using a bucket detector for the ﬁrst photon,and by scanning the position of the second photon and detecting coincidences one can obtain an image of the object which is in the path of theﬁrst photon despite the fact that the position of this photon is not scanned.The essential point is that,when the two photons are entangled,such distributed quantum imaging is in general partially coherent,and can possibly be fully coherent in which case phase information about the object is preserved,whereas for classically correlated but unentangled photons the imaging would be incoherent.These authors have also applied the principle of entangled-photon imaging to quantum holography[16].C.Entangled two-photon microscopyIn recent years there has been a lively interest in entangled-photon microscopy[18].The surge in the development ofﬂuorescence microscopy based on two-photon excitation using laser light has been driven by the principal advantages of this technique over single-photon excitation:a pair of low-energy photons can deposit as much energy as a single ultraviolet photon thereby exciting aﬂuorescent molecule within a sample with greater penetration depth,better resolution,and less risk of damage upon absorption along the optical path.However, in order to obtain two-photon absorption with a classical light source such as a laser,a very large photon-ﬂux density is necessary to place two photons within a small enough volume and time window so that theﬂuorescent molecule can absorb them.In this latter case,a femtosecond-pulsed high-power laser is used directly as the source of light, which can produce undesired photodamage of the specimen.On the other hand,entangled photons generated by the process of spontaneous parametric downconversion in a nonlinear crystal comprise intrinsically paired photons within a small volume and short time window.In principle, therefore,far smaller photon-ﬂux densities can be used to eﬀect absorption so that the risk of photodamage to thespecimen is reduced(see Fig.8).Yet other possible advantages arise from the direct proportionality of absorption to photon-ﬂux density(rather than the quadratic relation that holds for ordinary two-photon absorption)and the factthat the sum of the photon energies of each entangled pair is a constant and equal to the energy of the downconverted pump photon whereas the sum energy is far broader for photons from a femtosecond laser).However,one of theprincipal challenges in implementing this form of microscopy is obtaining an entangled-photonﬂux that is suﬃcient to excite two-photon transitions,which have a limited cross section[19].D.Image ampliﬁcation by parametric down conversionLet us come back to the conﬁguration of Fig.1b in the case of a large number of photon pairs.Let us assume thatnow,instead of a plane wave at frequency close toωs,we inject a coherent monochromatic image(Fig.9)of frequency ωs.Parametric image ampliﬁcation has been extensively studied from a classical viewpoint(see e.g.[12]).A basicpoint in Fig.9is that,if the image is injected oﬀaxis,one obtains in the output a signal image,that represents an ampliﬁed version of the input image,and also a symmetrical idler image.An interesting situation arises if one has,inaddition to the ampliﬁer,a pair of lenses located at focal distances with respect to the object plane,to the ampliﬁer and to the image plane(Fig.10).As it was shown by our group[9,11,20],in the limit of large ampliﬁcation the twooutput images can be considered twin of each other even from a quantum mechanical viewpoint.As a matter of fact, they do not only display the same intensity distribution but also the same local quantumﬂuctuations.Precisely,letus consider two symmetrical pixels in the two images(Fig.11).It turns out that the intensityﬂuctuations in the two pixels are identical,i.e.exactly correlated/synchronized.On the other hand,the phaseﬂuctuations are exactly anticorrelated.Hence the situation with respect to phase and intensityﬂuctuations is similar to what one obtains by breaking a fossil in two parts(Fig.12).In fact the two parts have the same´’intensity”,if one establishes an analogybetween intensity and thickness,but they have opposite phase,since one is clockwise and the other anticlockwise, one is convex and the other concave.So in this way,from one image one obtains twin images in a state of spatialentanglement which involves also the quadrature components X and Y,as it was already described in the case of pure parametricﬂuorescence without any signal injection in Sec.II C.There is,however,a negative point that concerns the signal-to-noise ratio.When the input image is injected oﬀaxis, this mechanism of ampliﬁcation is phase insensitive and therefore,as it is well known,it adds3dB of quantum noisein the output[21].In order to have noiseless ampliﬁcation,i.e.ampliﬁcation which preserves the signal-to-noise ratio, one must inject two coherent images symmetrically(Fig.13)[20].In this case one has in the input two identical,butuncorrelated images and in the output two ampliﬁed images in a state of spatial quantum entanglement.One can prove that this symmetrical conﬁguration is phase sensitive and,in fact,as it was shown in[22,23]the ampliﬁcationcan become noiseless.A couple of years ago there was a landmark experiment by Kumar and collaborators[24]which demonstrated the noiseless ampliﬁcation of a simple test pattern.E.Measurement of small displacementsThis topic has been recently studied from a quantum mechanical standpoint by Fabre and his group[25].Theyanalyze the displacement of a light beam.According to the Rayleigh diﬀraction limit,the position of the beam can be measured with an error on the order of the beam section.However,one can use,rather,a split detector which measures the intensities i1and i2from the two halves of the beam cross section(Fig.14).If one displaces gradually the beam with respect to the detector and plots the intensity diﬀerence,one obtains a curve like that shown in Fig.14and the precision in the measurement of the displacement is limited only by noise.Thestandard quantum limit is given[26]by the ratio of the Rayleigh limit to the square root of the photon number.In this way one can measure shifts in the sub-nanometer range.In the case of the microscopic observation of biologicalsamples,one cannot raise the photon number and therefore it is important to have the possibility of going beyond the standard quantum limit by reducing the noise in the intensity diﬀerence i1−i2below the shot noise level.To achieve this,a possibility is to use a beam in a state of spatial entanglement between its upper and its lower part, because in this way the number of photons in the two parts is basically the same and theﬂuctuations in the intensitydiﬀerence are very small.Another possibility,which was proposed in[25],is to synthesize a special two-mode state which arises from the superposition of a Gaussian mode in a squeezed vacuum state and a spatially antisymmetrical mode which lies in a coherent state.The antisymmetrical mode is obtained from a Gaussian mode byﬂipping upside down one of the two halves.One shows that in this way one can measure displacements beyond the standard quantum limit.Experiments with this conﬁguration are presently conducted in the laboratory of Bachor at the University of Canberra in collaboration with the group of Fabre[27].F.Image reconstructionKolobov and Fabre have recently studied the quantum limits in the process of image reconstruction[28].The very well known scheme they analyzed is shown in Fig.15.The object is contained in aﬁnite region of size X.The imaging system is composed by two lenses and an aperture.Because of the transverseﬁniteness of the imaging system,diﬀraction comes into play,with a consequence that some details of the object are blurred.In order to better reconstruct the object,one can proceed as follows.One considers the linear operator H which transforms the object into the image,its eigenvalues and its eigenfunctions,which are called prolate spheroidal functions:Hf n( x)=µn f n( x)(6) If the imaging were perfect,H would be the identity operator and all eigenvalues would be equal to unity.An imperfect imaging introduces a sort of loss so thatµn≤1.Now one expands the image e( x)on the basis of theeigenfunctions:e( x)= n c n f n( x)(7) and obtains the coeﬃcients c n.The object a( x)is reconstructed by the following expressione( x)= n c nµn f n( x)(8) where the coeﬃcients c n have been replaced by c n√whereζis a c-number and|0 is the vacuum state.One can generalize the argument above by considering a state of the formexp ζa N−ζ∗ a† N |0 (10)which can be obtained,for example,by a material with aχN nonlinearity,in which a single pump photon is down-converted into N entangled photons.One can consider the interference of two beams,one of which is in state(10), while the other lies in the stateexp ζ ae iχ N−ζ∗ a†e−iχ N |0 (11)obtained from(10)by a rotation of the angleχwhich corresponds to the phase diﬀerence between the two paths plus, possibly,a constant.Clearly,a rotation of2π[1]Einstein A.,Podolsky B.and Rosen N.,Phys.Rev.47,777(1935)[2]Walls D.F.and Milburn G.,Quantum Optics(Springer-Verlag,Berlin1994);Scully M.O.and Zubairy M.S.,QuantumOptics(Cambridge University Press1997);Schleich W.Quantum Optics in Phase Space(Wiley VCH,2001);Barnett S.M.and Radmore P.M.,Methods in Theoretical Quantum Optics(Clarendon Press Oxford1997)[3]Reid M.D.and Drummond P.D.,Phys.Rev.Lett.60,2731(1988);Reid M.D.,Phys.Rev.A40,913(1989)[4]Ou Z.Y.,Pereira S.F.,Kimble H.J.and Peng K.C.,Phys.Rev.Lett.68,3663(1992);Ou Z.Y.,Pereira S.F.and KimbleH.J.,Appl.Phys.B55,265(1992)[5]Kolobov M.I.and Sokolov I.V.,Sov.Phys JETP69,1097(1989);Phys.Lett.A140,101(1989);Europhys.Lett.15,271(1991)[6]Kolobov 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A.F.,Sergienko A.V.,and Teich M.C.,Phys.Rev.A62,043816(2000);Abouraddy A.F.,SalehB.E.A.,Sergienko A.V.,and Teich M.C.,Phys.Rev.Lett.87,123602(2001);Abouraddy A.F.,Saleh B.E.A.,SergienkoA.V.,and Teich M.C.,Optics Exp.9,498(2001);Abouraddy A.F.,SalehB.E.A.,Sergienko A.V.,and Teich M.C.,Entangled-photon Fourier optics,J.Opt.Soc.Am.B19,in press(2002)[17]Belinskii A.V.and Klyshko D.N.,Sov.Phys.JETP78,259(1994)[18]Teich M.C.and Saleh B.E.A.,ˇCeskoslovensk´yˇc asopis pro fyziku(Prague)47,3(1997)[19]H.-B.Fei,B.M.Jost,S.Popescu,B.E.A.Saleh,and M.C.Teich,Phys.Rev.Lett.78,1679(1997)[20]Gatti A.,Brambilla E.,Lugiato L.A.and Kolobov M.I.,J.Opt.B Quantum Semiclass.Opt.2,196(2000)[21]Caves C.M.,Phys.Rev.D26,1817(1982)[22]Kolobov M.I.and Lugiato L.A.,Phys Rev.A52,4930(1995)[23]Sokolov I.V.,Kolobov M.I.and Lugiato L.A.,Phys.Rev.A60,2420(1998)[24]Choi S.-K.,Vasilyev M.and Kumar P.,Phys.Rev.Lett.83,1938(1999)[25]Fabre C.,Fouet J.B.,and Maitre A.,Opt.Lett.25,76(2000)[26]Treps N.,Eﬀets quantiques dans les 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lens.Fig.3.Illustration of the concept of spatial entanglement.I i,X i,Y i (i=1,2)denote the intensities and the quadrature components measured in the two pixels1and2,respectively.Fig.4(a)Intensity distribution in the farﬁeld for a single shot of the pulsed pumpﬁeld.a)Numerical simulations.The waist of the pump beam is1000µm,300µm,150µm in the three frames from top to bottom,respectively.。

Quantum entanglements and entangled mutual entropy

a rXiv:q uant-ph/981282v218A pr2QUANTUM ENTANGLEMENTS AND ENTANGLED MUTUAL ENTROPY VIACHESLAV P BELAVKIN AND MASANORI OHYA Abstract.The mathematical structure of quantum entanglement is studied and classiﬁed from the point of view of quantum compound states.We show that the classical-quantum correspondences such as encodings can be treated as diagonal (d-)entanglements.The mutual entropy of the d-compound and entangled states lead to two diﬀerent types of entropies for a given quantum state:the von Neumann entropy,which is achieved as the supremum of the information over all d-entanglements,and the dimensional entropy,which is achieved at the standard entanglement,the true quantum entanglement,co-inciding with a d-entanglement only in the case of pure marginal states.The q-capacity of a quantum noiseless channel,deﬁned as the supremum over all entanglements,is given by the logarithm of the dimensionality of the input algebra.It doubles the classical capacity,achieved as the supremum over all d-entanglements (encodings),which is bounded by the logarithm of the dimen-sionality of a maximal Abelian subalgebra.1.Introduction Recently,the speciﬁcally quantum correlations,called in quantum physics entan-glements,are used to study quantum information processes,in particular,quantum computation,quantum teleportation,quantum cryptography [19,21,22].There have been mathematical studeis of the entanglements in [20,17,18],in which the entangled state is deﬁned by a state not written as a form k λk ρk ⊗σk with any states ρk and σk .However it is obvious that there exist several correlated states written as separable forms above.Such correlated,or entangled states have been also discussed in several contexts in quantum probability such as quantum mea-surement and ﬁltering [3,4],quantum compound state[1,14]and lifting [2].In thispaper,we study the mathematical structure of quantum entangled states to provide a ﬁner classiﬁcation of quantum sates,and we discuss the informational degree of entanglement and entangled quantum mutual entropy.We show that the entangled states can be treated as generalized compound states,the nonseparable states of quantum compound systems which are not repre-setable by convex combinations of the product states.The compound states,called o-entangled,are deﬁned by orthogonal decompositions of their marginal states.This2VIACHESLA V P BELA VKIN AND MASANORI OHYAis a particular case of so called separable state of a compound system,the convex combination of the product states which we call c-entangled.The o-entangled com-pound states are most informative among c-entangled states in the sense that the maximum of mutual entropy over all c-entanglements to the quantum system A is achieved on the extreme o-entangled states as the von Neumann entropy S(̺)of a given normal state̺on A.Thus the maximum of mutual entropy over all classical couplings,described by c-entanglements of(quantum)probe systems B to the system A,is bounded by ln rank A,the logarithm of the rank of the vonNeumann algebra A,deﬁned as the dimensionality of the maximal Abelian sub-algebra A◦⊆A.Due to dim A≤(rank A)2,it is achieved on the normal tracial ρ=(rank A)−1I only in the case ofﬁnite dimensional A.More general than o-entangled states,the d-entangled states,are deﬁned asc-entangled states by orthogonal decomposition of only one marginal state on the probe algebra B.They can give bigger mutual entropy for a quantum noisy channel than the o-entangled state which gains the same information as d-entangled extremestates in the case of a deterministic channel.We prove that the truly(strongest)entangled states are most informative in thesense that the maximum of mutual entropy over all entanglements to the quantum system A is achieved on the quasi-compound state,given by an extreme entangle-ment of the probe system B=A with coinciding marginals,called standard for agiven̺.The standard entangled state is o-entangled only in the case of Abelian A or pure marginal state̺.The gained information for such extreme q-compound state deﬁnes another type of entropy,the quasi-entropy S(̺)which is bigger than the von Neumann entropy S(̺)in the case of non-Abelian A(and mixed̺.)The maximum of mutual entropy over all quantum couplings,described by true quan-tum entanglements of probe systems B to the system A is bounded by ln dim A,the logarithm of the dimensionality of the von Neumann algebra A,which is achieved on a normal tracialρin the case ofﬁnite dimensional A.Thus the q-entropy S(̺), which can be called the dimensional entropy,is the true quantum entropy,in con-trast to the von Neumann entropy S(̺),which is semi-classical entropy as it can be achieved as a supremum over all couplings with the classical probe systems B.These entropies coincide in the classical case of Abelian A when rank A=dim A.In the case of non-Abelianﬁnite-dimensional A the q-capacity C q=ln dim A is achieved as the supremum of mutual entropy over all q-encodings(correspondences),described by entanglements.It is strictly bigger then the classical capacity C=ln rank A of the identity channel,which is achieved as the supremum over usual encodings, described by the classical-quantum correspondences A◦→A.In this short paper we consider the case of a simple algebra A=L(H)for whichsome results are rather obvious and given without proofs.The proofs are given in the complete paper[5]for a more general case of decomposable algebra A to include the classical discrete systems as a particular quantum case,and will be published elsewhere.pound States and EntanglementsLet H denote the(separable)Hilbert space of a quantum system,and A=L(H) be the algebra of all linear bounded operators on H.A bounded linear functional ̺:A→C is called a state on A if it is positive(i.e.,̺(A)≥0for any positive operator A in A)and normalized̺(I)=1for the identity operator I in A.AQUANTUM ENTANGLEMENTS AND ENTANGLED MUTUAL ENTROPY3normal state can be expressed as(1)̺(A)=tr Gκ†Aκ=tr Aρ,A∈A.In(2.1),G is another separable Hilbert space,κis a linear Hilbert-Schmidt operator from G to H andκ†is the adjoint operator ofκfrom H to G.Thisκis called the amplitude operator,and it is called just the amplitude if G is one dimensional space C,corresponding to the pure state̺(A)=κ†Aκfor aκ∈H withκ†κ= κ 2=1, in which caseκ†is the adjoint functional from H to C.Moreover the density operatorρin(2.1)isκκ†uniquely deﬁned as a positive trace class operator P A∈A .Thus the predual space A∗can be identiﬁed with the Banach space T(H)of all trace class operators in H(the density operators P A∈A∗,P B∈B∗of the states ̺,ςon diﬀerent algebras A,B will be usually denoted by diﬀerent lettersρ,σcorresponding to their Greek variations̺,ς.)In general,G is not one dimensional,the dimensionality dim G must be not less than rankρ,the dimensionality of the range ranρ⊆H of the density operatorρ.We shall equip it with an isometric involution J=J†,J2=I,having the properties of complex conjugation on G,J λjζj= ¯λj Jζj,∀λj∈C,ζj∈Gwith respect to which Jσ=σJ for the positive and so self-adjoint operatorσ=κ†κ=σ†on G.The latter can also be expressed as the symmetricity property ˜ς=ςof the stateς(B)=tr Bσgiven by the real and so symmetric density operator ¯σ=σ=˜σon G with respect to the complex conjugation¯B=JBJ and the tilda operation(G-transponation)˜B=JB†J on the algebra B=L(G).For example,G can be realized as a subspace of l2(N)of complex sequences N∋n→ζ(n)∈C,with n|ζ(n)|2<+∞in the diagonal representation σ=[µ(n)δm n].The involution J can be identiﬁed with the complex conjugation Cζ(n)=¯ζ(n),i.e.,C:ζ= n|n ζ(n)→Cζ= n|n ¯ζ(n)in the standard basis{|n }⊂G of l2(N).In this caseκ= κn n|is given by orthogonal eigen-amplitudesκn∈H,κ†mκn=0,m=n,normalized to the eigen-valuesλ(n)=κ†nκn=µ(n)of the density operatorρsuch thatρ= κnκ†n is a Schatten decomposition,i.e.the spectral decomposition ofρinto one-dimensional orthogonal projectors.In any other basis the operator J is deﬁned then by J= U†CU,where U is the corresponding unitary transformation.One can also identify G with H by Uκn=λ(n)1/2|n such that the operatorρis real and symmetric, JρJ=ρ=Jρ†J in G=H with respect to the involution J deﬁned in H by Jκn=κn.Here U is an isometric operator H→l2(N)diagonalizing the operator ρ:UρU†= |n λ(n) n|.The amplitude operatorκ=ρ1/2corresponding to B=A,σ=ρis called standard.Given the amplitude operatorκ,one can deﬁne not only the states̺ ρ=κκ† and ς σ=κ†κ on the algebras A=L(H)and B=L(G)but also a pure entanglement state̟on the algebra B⊗A of all bounded operators on the tensor product Hilbert space G⊗H by̟(B⊗A)=tr G˜Bκ†Aκ=tr H Aκ˜Bκ†.4VIACHESLA V P BELA VKIN AND MASANORI OHYAIndeed,thus deﬁned ̟is uniquely extended by linearity to a normal state on the algebra B ⊗A generated by all linear combinations C = λj B j ⊗A j due to̟(I ⊗I )=tr κ†κ=1and ̟ C †C = i,k ¯λi λk tr G ˜B k ˜B †i κ†A †iA k κ= i,k ¯λi λk tr G ˜B †i κ†A †i A k κ˜B k =tr G χ†χ≥0,where χ= j A j κ˜Bj .This state is pure on L (G ⊗H )as it is given by an amplitude ϑ∈G ⊗H deﬁned as(ζ⊗η)†ϑ=η†κJζ,∀ζ∈G ,η∈H ,and it has the states ̺and ςas the marginals of ̟:̟(I ⊗A )=tr H Aρ,̟(B ⊗I )=tr G Bσ.(2)As follows from the next theorem for the case F =C ,any pure state ̟(B ⊗A )=ϑ†(B ⊗A )ϑ,B ∈B ,A ∈A given on L (G ⊗H )by an amplitude ϑ∈G ⊗H with ϑ†ϑ=1,can be achieved by a unique entanglement of its marginal states ςand ̺.Theorem 2.1.Let ̟:B ⊗A →C be a compound state̟(B ⊗A )=tr F υ†(B ⊗A )υ,(3)deﬁned by an amplitude operator υ:F →G ⊗H on a separable Hilbert space F into the tensor product Hilbert space G ⊗H with tr υ†υ=1.Then this state can be achieved as an entanglement̟(B ⊗A )=tr G ˜Bκ†(I ⊗A )κ=tr F⊗H (I ⊗A )κ˜Bκ†(4)of the states (2)with σ=κ†κand ρ=tr F κκ†,where κis an amplitude operator G →F ⊗H .The entangling operator κis uniquely deﬁned by ˜κU =υup to a unitary transformation U of the minimal domain F =dom υ.Note that the entangled state (4)is written as̟(B ⊗A )=tr G ˜Bπ(A )=tr H Aπ∗ ˜B ,(5)where π(A )=κ†(I ⊗A )κ,bounded by A σ∈B ∗for any A ∈L (H ),is in the predual space B ∗⊂B of all trace-class operators in G ,and π∗(B )=tr F κBκ†,bounded by B ρ∈A ∗,is in A ∗⊂A .The map πis the Steinspring form [9]of the general completely positive map A →B ∗,written in the eigen-basis {|k }⊂F of the density operator υ†υas π(A )= m,n|m κ†m (I ⊗A )κn n |,A ∈A (6)while the dual operation π∗is the Kraus form [10]of the general completely positive map A →A ∗,given in this basis as π∗(B )= n,mn |B |m tr F κn κ†m =tr G ˜Bω.(7)QUANTUM ENTANGLEMENTS AND ENTANGLED MUTUAL ENTROPY5 It corresponds to the general form(8)ω= m,n|n m|⊗tr Fκnκ†mof the density operatorω=υυ†for the entangled state̟(B⊗A)=tr(B⊗A)ωin this basis,characterized by the weak orthogonality property(9)tr Fψ(m)†ψ(n)=µ(n)δm nin terms of the amplitude operatorsψ(n)=(I⊗ n|)˜κ=˜κn.Deﬁnition2.1.The dual mapπ∗:B→A∗to a completely positive mapπ:A→B∗,normalized as tr Gπ(I)=1,is called the quantum entanglement of the state ς=π(I)on B to the state̺=π∗(I)on A.The entanglement by(10)π◦∗(A)=ρ1/2Aρ1/2=π◦(A)of the stateς=̺on the algebra B=A is called standard for the system(A,̺).The standard entanglement deﬁnes the standard compound state̟0(B⊗A)=tr H˜Bρ1/2Aρ1/2=tr H Aρ1/2˜Bρ1/2on the algebra A⊗A,which is pure,given by the amplitudeϑ0associated with̟0 is˜κ0,whereκ0=ρ1/2.Example2.1.In quantum physics the entangled states are usually obtained by a unitary transformation U of an initial disentangled state,described by the density operatorσ0⊗ρ0⊗τ0on the tensor product Hilbert space G⊗H⊗K,that is,̟(B⊗A)=tr U†(B⊗A⊗I)U(σ0⊗ρ0⊗τ0).In the simple case,when K=C,τ0=1,the joint amplitude operatorυis deﬁned on the tensor product F=G⊗H0with H0=ranρ0asυ=U1(σ0⊗ρ0)1/2.The entangling operatorκ,describing the entangled state̟,is constructed as it was done in the proof of Theorem2.1by transponation of the operatorυU†,where U is arbitrary isometric operator F→G⊗H0.The dynamical procedure of such entanglement in terms of the completely positive mapπ∗:A→B∗is the subject of Belavkin quantumﬁltering theory[8].The quantumﬁltering dilation theorem[8] proves that any entanglementπcan be obtained the unitary entanglement as the result of quantumﬁltering by tracing out some degrees of freedom of a quantum environment,described by the density operatorτ0on the Hilbert space K,even in the continuous time case.3.C-and D-Entanglements and EncodingsThe compound states play the role of joint input-output probability measures in classical information channels,and can be pure in quantum case even if the marginal states are mixed.The pure compound states achieved by an entanglement of mixed input and output states exhibit new,non-classical type of correlations which are responsible for the EPR type paradoxes in the interpretation of quantum theory. The mixed compound states on B⊗A which are given as the convex combinations ̟= nςn⊗̺nµ(n),µ(n)≥0, nµ(n)=16VIACHESLA V P BELA VKIN AND MASANORI OHYAof tensor products of pure or mixed normalized states̺n∈A∗,ςn∈B∗as in classical case,do not exhibit such paradoxical behavior,and are usually considered as the proper candidates for the input-output states in the communication chan-nels.Such separable compound states are achieved by c-entanglements,the convex combinations of the primitive entanglements B→tr G Bωn,given by the density operatorsωn=σn⊗ρn of the product states̟n=ςn⊗̺n:(11)π∗(B)= nρn tr G Bσnµ(n),A compound state of this sort was introduced by Ohya[1,15]in order to deﬁne the quantum mutual entropy expressing the amount of information transmitted from an input quantum system to an output quantum system through a quantum channel,using a Schatten decompositionσ= nσnµ(n),σn=|n n|of the input density operatorσ.It corresponds to a particular,diagonal type(12)π(A)= n|n κ†n(I⊗A)κn n|of the entangling map(6)in an eigen-basis{|n }∈G of the density operatorσ, and is discussed in this section.Let us consider aﬁnite or inﬁnite input system indexed by the natural numbers n∈N.The associated space G⊆l2(N)is the Hilbert space of the input system described by a quantum projection-valued measure n→|n n|on N,given an orthogonal partition of unity I= |n n|∈B of theﬁnite or inﬁnite dimensional input Hilbert space G.Each input pure state,identiﬁed with the one-dimensional density operator|n n|∈B corresponding to the elementary symbol n∈N,deﬁnes the elementary output state̺n on A.If the elementary states̺n are pure,they are described by output amplitudesηn∈H satisfyingη†nηn=1=trρn,where ρn=ηnη†n are the corresponding output one-dimensional density operators.If these amplitudes are non-orthogonalη†mηn=δm n,they cannot be identiﬁed with the input amplitudes|n .The elementary joint input-output states are given by the density operators |n n|⊗ρn in G⊗H.Their mixtures(13)ω= nµ(n)|n n|⊗ρn,deﬁne the compound states on B⊗A,given by the quantum correspondences n→|n n|with the probabilitiesµ(n).Here we note that the quantum correspondence is described by a classical-quantum channel,and the general d-compound state for a quantum-quantum channel in quantum communication can be obtained in this way due to the orthogonality of the decomposition(13),corresponding to the orthogonality of the Schatten decompositionσ= n|n µ(n) n|forσ=tr Hω.The comparison of the general compound state(8)with(13)suggests that the quantum correspondences are described as the diagonal entanglements(14)π∗(B)= nµ(n) n|B|n ρn,They are dual to the orthogonal decompositions(12):π(A)= nµ(n)|n η†n Aηn n|= n|n η(n)†Aη(n) n|,QUANTUM ENTANGLEMENTS AND ENTANGLED MUTUAL ENTROPY7 whereη(n)=µ(n)1/2ηn.These are the entanglements with the stronger orthogo-nality(15)ψ(m)ψ(n)†=µ(n)δm n,for the amplitude operatorsψ(n):F→H of the decomposition of the amplitude operatorυ= n|n ⊗ψ(n)in comparison with the orthogonality(9).The orthog-onality(15)can be achieved in the following manner:Take in(6)κn=|n ⊗η(n) with m|n =δm n so thatκ†m(I⊗A)κn=µ(n)η†n Aηnδm nfor any A∈A.Then the strong orthogonality condition(15)is fulﬁlled by the amplitude operatorsψ(n)=η(n) n|=˜κn,andκ†κ= nµ(n)|n n|=σ,κκ†= nη(n)η(n)†=ρ.It corresponds to the amplitude operator for the compound state(13)of the form (16)υ= n|n ⊗ψ(n)U,where U is arbitrary unitary operator from F onto G,i.e.υis unitary equivalent to the diagonal amplitude operatorκ= n|n n|⊗η(n)on F=G into G⊗H.Thus,we have proved the following theorem in the case of pure output statesρn=ηnη†n.Theorem3.1.Letπbe the operator(13),deﬁning a d-compound state of the form (17)̟(B⊗A)= n n|B|n tr F nψ†n Aψnµ(n)Then it corresponds to the entanglement by the orthogonal decomposition(12)map-ping the algebra A into a diagonal subalgebra of B.Note that(2.9)deﬁnes the general form of a positive map on A with values in the simultaneously diagonal trace-class operators in A.Deﬁnition 3.1.The completely positive convex combination(11)is called c-entanglement,and is called d-entanglement,or quantum encoding if it has the diag-onal form(14)on B.The d-entanglement is called o-entanglement and compound state is called o-compound if all density operatorsρn are orthogonal:ρmρn=ρnρm for all m and n.Note that due to the commutativity of the operators B⊗I with I⊗A on G⊗H, one can treat the correspondences as the nondemolition measurements[4]in B with respect to A.So,the compound state is the state prepared for such measurements on the input G.It coincides with the mixture of the states,corresponding to those after the measurement without reading the sent message.The set of all d-entanglements corresponding to a given Schatten decomposition of the input stateσon B is obviously convex with the extreme points given by the pure output statesρn8VIACHESLA V P BELA VKIN AND MASANORI OHYAon A,corresponding to a not necessarily orthogonal decompositionsρ= nρ(n) into one-dimensional density operatorsρ(n)=µ(n)ρn.The Schatten decompositionsρ= nλ(n)ρn correspond to the extreme d-entanglements,ρn=ηnη†n,µ(n)=λ(n),characterized by orthogonalityρmρn=0, m=n.They form a convex set of d-entanglements with mixed commutingρn for each Schatten decomposition ofρ.The orthogonal d-entanglements were used in[7] to construct a particular type of Accardi’s transitional expectations[6]and to deﬁne the entropy in a quantum dynamical system via such transitional expectations.The established structure of the general q-compound states suggests also the general formΦ∗(B,̺0)=tr FX†(B⊗ρ0)X=tr G ˜B⊗I Y(I⊗ρ0)Y†1of transitional expectationsΦ∗:B×A◦∗→A∗,describing the entanglementsπ∗=Φ∗(̺0)of the statesς=π(I)to̺=π∗(I)for each initial state̺0∈A◦∗with the density operatorρ0∈A◦⊆L(H0)byπ∗(B)=tr Fκ(B⊗I)κ†,whereκ=X†(I⊗ρ0)1/2.It is given by an entangling transition operator X:F⊗H→G⊗H0, which is deﬁned by a transitional amplitude operator Y:H0⊗F→G⊗H up to a unitary operator U in F as(ζ⊗η0)†X(Uξ⊗η)=(η0⊗Jξ)†Y†(Jζ⊗η).The dual mapΦ:A→B∗⊗A◦is obviously normal and completely positive, (18)Φ(A)=X(I⊗A)X†∈B∗⊗A◦,∀A∈A,with tr GΦ(I)=I◦,and is calledﬁltering map with the output statesΦ(I)(I⊗ρ0)ς=tr Hin the theory of CPﬂows[8]over A=A◦.The operators Y normalized as tr F Y†Y=I◦describe A-valued q-compound statesE(B⊗A)=tr F Y†(B⊗A)Y=tr G ˜B⊗I Φ(A),deﬁned as the normal completely positive maps B⊗A→A◦with E(I⊗I)=I◦.If the A-valued compound state has the diagonal form given by the orthogonal decomposition(19)Φ(A)= n|n tr FΨ(n)†AΨ(n) n|,corresponding to Y= n|n ⊗Ψ(n),whereΨ(n):H0⊗F→H,it is achieved by the d-transitional expectationsΦ∗(B,̺0)= n n|B|n Ψ(n)(ρ0⊗I)Ψ(n)†.The d-transitional expectations correspond to the instruments[11]of the dynamical theory of quantum measurements.The elementaryﬁlters1Θn(A)=QUANTUM ENTANGLEMENTS AND ENTANGLED MUTUAL ENTROPY9 deﬁne posterior states̺n=̺0Θn on A for quantum nondemolition measurements in B,which are called indirect if the corresponding density operatorsρn are non-orthogonal.They describe the posterior states with orthogonalρn=Ψn(ρ0⊗I)Ψ†n,Ψn=Ψ(n)/µ(n)1/2for allρ0iﬀΨ(n)†Ψ(n)=δm n M(n).4.Quantum Entropy via EntanglementsAs it was shown in the previous section,the diagonal entanglements describe the classical-quantum encodingsκ:B→A∗,i.e.correspondences of classical symbols to quantum,in general not orthogonal and pure,states.As we have seen in contrast to the classical case,not every entanglement can be achieved in this way.The general entangled states̟are described by the density operators ω=υυ†of the form(8)which are not necessarily block-diagonal in the eigen-representation of the density operatorσ,and they cannot be achieved even by a more general c-entanglement(11).Such nonseparable entangled states are called in[15]the quasicompound(q-compound)states,so we can call also the quantum nonseparable correspondences the quasi-encodings(q-encodings)in contrast to the d-correspondences,described by the diagonal entanglements.As we shall prove in this section,the most informative for a quantum system (A,̺)is the standard entanglementπ◦∗=π0of the probe system(B◦,ς0)=(A,̺), described in(10).The other extreme cases of the self-dual input entanglementsπ∗(A)= nρ(n)1/2Aρ(n)1/2=π(A),are the pure c-entanglements,given by the decompositionsρ= ρ(n)into pure statesρ(n)=ηnη†nµ(n).We shall see that these c-entanglements,corresponding to the separable states(20)ω= nηnη†n⊗ηnη†nµ(n),are in general less informative then the pure d-entanglements,given in an orthonor-mal basis{η◦n}⊂H byπ◦(A)= nη◦nη†n Aηnη◦†nµ(n)=π◦∗(A).Now,let us consider the entangled mutual entropy and quantum entropies of states by means of the above three types of compound states.To deﬁne the quantum mutual entropy,we need the relative entropy[12,13,23]of the compound state ̟with respect to a reference stateϕon the algebra A⊗B.It is deﬁned by the density operatorsω,φ∈B⊗A of these states as(21)S(̟,ϕ)=trω(lnω−lnφ).It has a positive value S(̟,ϕ)∈[0,∞]if the states are equally normalized,say (as usually)trω=1=trφ,and it can beﬁnite only if the state̟is absolutely continuous with respect to the reference stateϕ,i.e.iﬀ̟(E)=0for the maximal null-orthoprojector Eφ=0.10VIACHESLA V P BELA VKIN AND MASANORI OHYAThe mutual entropy Iω(A,B)of a compound state̟achieved by an entangle-mentπ∗:B→A∗with the marginalsς(B)=̟(B⊗I)=tr G Bσ,̺(A)=̟(I⊗A)=tr H Aρis deﬁned as the relative entropy(21)with respect to the product stateϕ=ς⊗̺:(22)I A,B(̟)=trω(lnω−ln(σ⊗I)−ln(I⊗ρ)).Here the operatorωis uniquely deﬁned by the entanglementπ∗as its density in (7),or the G-transposed to the operator˜ωinπ(A)=κ†(I⊗A)κ=tr H A˜ω.This quantity describes an information gain in a quantum system(A,̺)via an entanglementπ∗of another system(B,ς).It is naturally treated as a measure of the strength of an entanglement,having zero value only for completely disentangled states,corresponding to̟=ς⊗̺.The following proposition follows from the monotonicity property[24,16] (23)̟=K∗̟0,ϕ=K∗ϕ0⇒S(̟,ϕ)≤S(̟0,ϕ0).of the general relative entropy on a von Neuman algebra M with respect to the predual K∗to any normal completely positive unital map K:M→M◦. Proposition4.1.Letπ◦∗:B◦→A∗be an entanglementπ◦∗of a stateς0=π◦(I) on a discrete decomposable algebra B◦⊆L(G0)to the state̺=π◦∗(I)on A,and π∗=π◦∗K be an entanglement deﬁned as the composition with a normal completely positive unital map K:B→B◦.Then I A,B(̟)≤I A,B◦(̟0),where̟,̟0 are the compound states achieved byπ◦∗,π∗respectively.In particular,for any c-entanglementπ∗to(A,ς)there exists a not less informative d-entanglementπ◦∗=κwith an Abelian B◦,and the standard entanglementπ0(A)=ρ1/2Aρ1/2ofς0=̺on B◦=A is the maximal one in this sense.Note that any extreme d-entanglementπ◦∗(B)= n n|B|n ρ◦nµ(n),B∈B◦,withρ= nρ◦nµ(n)decomposed into pure normalized statesρ◦n=ηnη†n,is maxi-mal among all c-entanglements in the sense I A,B(̟0)≥I A,B(̟).This is because trρ◦n lnρ◦n=0,and therefore the information gainI A,B(̟)= nµ(n)trρn(lnρn−lnρ).with aﬁxedπ∗(I)=ρachieves its supremum−tr Hρlnρat any such extreme d-entanglementπ◦∗.Thus the supremum of the information gain(22)over all c-entanglements to the system(A,̺)is the von Neumann entropy(24)S A(̺)=−tr Hρlnρ.It is achieved on any extremeπ◦∗,for example given by the maximal Abelian subal-gebra B◦⊆A,with the measureµ=λ,corresponding to a Schatten decomposition ρ= nη◦nη◦†nλ(n),η◦†mη◦n=δm n.The maximal value ln rank A of the von Neumann entropy is deﬁned by the dimensionality rank A=dim B◦of the maximal Abelian subalgebra of the decomposable algebra A,i.e.by dim H.QUANTUM ENTANGLEMENTS AND ENTANGLED MUTUAL ENTROPY11Deﬁnition4.1.The maximal mutual entropyS A(̺)=supπ∗(I)=ρI A,B(̟)=I A,B◦(̟0),(25)achieved on B◦=A by the standard q-entanglementπ◦∗(A)=ρ1/2Aρ1/2for aﬁxed state̺(A)=tr H Aρ,is called q-entropy of the state̺.The diﬀerencesS B|A(̟)= S B(ς)−I A,B(̟)S B|A(̟)=S B(ς)−I A,B(̟)are respectively called the q-conditional entropy on B with respect to A and the degree of disentanglement for the compound state̟. Obviously, S B|A(̟)is positive in contrast to the disentanglement S B|A(̟), having the positive maximal value S B|A(̟)=S B(ς)in the case̟=ς⊗̺of complete disentanglement,but which can achieve also a negative valueinf π∗(I)=ρD B|A(̟)=S A(ς)− S A(̺)=trρlnρ(26)for the entangled states as the following theorem states.Obviously S A(̺)= S A(̺) if the algebra A is completely decomposable,i.e.Abelian,and the maximal value ln rank A of S A(̺)can be written as ln dim A in this case.The disentanglement S B|A(̟)is always positive in this case,as well as in the case of Abelian B when S B|A(̟)= S B|A(̟).Theorem4.2.The q-entropy for the simple algebra A=L(H)is given by the formulaS A(̺)=−2tr Hρlnρ=2S A(ρ),(27)It is positive, S A(̺)∈[0,∞],and if A isﬁnite dimensional,it is bounded,with the maximal value S A(̺◦)=ln dim A which is achieved on the tracialρ◦=(dim H)−1I, where dim A=(dim H)2.5.Quantum Channel and its Q-CapacityLet H0be a Hilbert space describing a quantum input system and H describe its output Hilbert space.A quantum channel is an aﬃne operation sending each input state deﬁned on H0to an output state deﬁned on H such that the mixtures of states are preserved.A deterministic quantum channel is given by a linear isometry Y:H0→H with Y†Y=I◦(I◦is the identify operator in H0)such that each input state vectorη∈H0, η =1is transmitted into an output state vector Yη∈H, Yη =1.The orthogonal mixturesρ0= nµ(n)ρ◦n of the pure input statesρ◦n=η◦nη◦†n are sent into the orthogonal mixturesρ= nµ(n)ρn of the corresponding pure statesρn=Yρ◦n Y†.A noisy quantum channel sends pure input states̺0into mixed ones̺=Λ∗(̺0) given by the dualΛ∗to a normal completely positive unital mapΛ:A→A0,Λ(A)=tr F1Y†AY,A∈Awhere Y is a linear operator from H0⊗F+to H with tr F+Y†Y=I◦,and F+is a separable Hilbert space of quantum noise in the channel.Each input mixed state12VIACHESLA V P BELA VKIN AND MASANORI OHYA̺0on A ◦⊆L (H 0)is transmitted into an output state ̺=̺0Λgiven by the density operator Λ∗(ρ0)=Y ρ0⊗I + Y †∈A ∗for each density operator ρ0∈A ◦∗,where I+is the identity operator in F +.With-out loss of generality we can assume that the input algebra A ◦is the smallest decomposable algebra,generated by the range Λ(A )of the given map Λ.The input entanglements κ:B →A ◦∗described as normal CP maps with κ(I )=̺0,deﬁne the quantum correspondences (q-encodings)of probe systems (B ,ς),ς=κ∗(I ),to (A ◦,̺0).As it was proven in the previous section,the most informative is the standard entanglement κ=π◦∗,at least in the case of the trivial channel Λ=I.This extreme input q-entanglement π◦(A ◦)=ρ1/20A ◦ρ1/20=π◦∗(A ◦),A ◦∈A ◦,corresponding to the choice (B ,ς)=(A ◦,̺0),deﬁnes the following density operator ω=(I ⊗Λ)∗ ω◦q ,ω◦q =ϑ0ϑ†0(28)of the input-output compound state ̟◦q Λon A ◦⊗A .It is given by the amplitudeϑ0∈H ⊗20deﬁned as ˜ϑ0=ρ1/20.The other extreme cases of the self-dual input entanglements,the pure c-entanglements corresponding to (20),can be less infor-mative then the d-entanglements,given by the decompositions ρ0= ρ0(n )into pure states ρ0(n )=ηn η†n µ(n ).They deﬁne the density operators ω=(I ⊗Λ)∗(ω◦d ),ω◦d = n η◦n η◦†n ⊗ηn η†n µ0(n ),(29)of the A ◦⊗A -compound state ̟◦d Λ,which are known as the Ohya compound states ̟◦o Λ[1]in the caseρ0(n )=η◦n η◦†n λ0(n ),η◦†m η◦n =δm n ,of orthogonality of the density operators ρ0(n )normalized to the eigen-values λ0(n )of ρ0.They are described by the input-output density operators ω=(I ⊗Λ)∗(ω◦o ),ω◦o = nη◦n η◦†n ⊗η◦n η◦†n λ0(n ),(30)coinciding with (28)in the case of Abelian A ◦.These input-output compound states ̟are achieved by compositions λ=π◦Λ,describing the entanglements λ∗of the extreme probe system (B ◦,ς0)=(A ◦,̺0)to the output (A ,̺)of the channel.If K :B →B ◦is a normal completely positive unital mapK (B )=tr F −X †BX,B ∈B ,where X is a bounded operator F −⊗G 0→G with tr F −X †X =I ◦,the compositions κ=π◦∗K,π∗=Λ∗κare the entanglements of the probe system (B ,ς)to the channel input (A ◦,̺0)and to the output (A ,̺)via this channel.The state ς=ς0K is given by K ∗(σ0)=X I −⊗σ0 X †∈B ∗for each density operator σ0∈B ◦∗,where I −is the identity operator in F −.Theresulting entanglement π∗=λ∗K deﬁnes the compound state ̟=̟0(K ⊗Λ)on B ⊗A with̟0(B ◦⊗A ◦)=tr ˜B ◦π◦(A ◦)=tr υ†0(B ◦⊗A ◦)υ0.。

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(the Schmidt projection of Bennett, et al. [2] and its development [8]), a classical communication channel is not necessary. In the case of single copy, however, it was an open question to ﬁnd the optimal strategy without communication to concentrate a general bipartite pure state to an ME state [5]. In this work we present an approach to transforming a bipartite entangled state only through the local operations of Alice or Bob (without the need for a classical communication channel), which is also equivalent to only transforming the reduced density matrices ρA = T rB |Ψ1 Ψ1 | or ρB = T rA |Ψ1 Ψ1 |. By introducing the coeﬃcient matrix Λ = (λi,j ) [9, 10], where 2 i,j |λi,j | = 1, we rewrite the input state as λ1,1 · · · λ1,N |1 B . . . .. . |Ψ1 = (|1 A , · · · , |M A ) . . . . . . . (1) λM,1 · · · λM,N |N
B
It is straightforward that any operation to transform |Ψ1 by the individual eﬀort of Alice or Bob is equivalent to the local manipulation of their basis vectors α or β . One example of local manipulation on their bases is that Alice and Bob independently apply the operations αT → αT U † and β → V β with the unitary operators obtained from the singular value decomposition Λ = U † Λd V and transform the coeﬃcient matrix of |Ψ1 to a diagonal form Λd = diag (λ1 , λ2 , · · · , λmin(M,N ) ), (2) where min(M, N ) is the Schmidt rank of |Ψ1 . Then the essential step in transforming it to another entangled state |Ψ2 is reduced to connecting their diﬀerent Λd . Here K † , K T , K ∗ represent the Hermitian conjugate, the transpose and the entry-wise complex conjugate of an arbitrary matrix K , respectively. The Schmidt coeﬃcients λi can be arranged in a standard form with λi ≥ λi+1 by permuting basis vectors. With the coeﬃcient matrix Λ, the reduced density matrices of an
arXiv:0708.4184v7 [quanoptimal scheme to realize the transformations between single copies of two bipartite entangled states without classical communication between the sharing parties. The scheme achieves the upper bound for the success probabilities [Phys. Rev. A 63, 022301 (2001), Phys. Rev. Lett. 83, 1455 (1999)] of generating maximally entangled states if applied to entanglement concentration. Such strategy also dispenses with the interaction with an ancilla system in the implementation. We also show that classical communications are indispensable in realizing the deterministic transformations of a single bipartite entangled state. With a ﬁnite number of identical pairs of two entangled bosons, on the other hand, we can realize the deterministic transformation to any target entangled state of equal or less Schmidt rank through an extension of the scheme.
Entanglement transformation with no classical communication
Bing He1, ∗ and J´ anos A. Bergou1
1 Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10065, USA (Dated: July 3, 2008)
PACS numbers: 03.67.Bg, 03.65.Ud, 03.67.Mn
Entanglement is a key resource in realizing various quantum information protocols, as well as manifests the non-classical nature of physics. An entangled quantum state produced in laboratory and/or transported over a distance, inevitably, shows some degradation (the ﬁdelity less than one) due to decoherence and loss in the process. To faithfully teleport a quantum state to a remote place, it is necessary to have a pair of particles in the maximally entangled (ME) state [1]. This problem gives rise to the study of entanglement concentration [2], which is the procedure to generate an ME state out of copies of partially entangled states. One of the generalized problems of this type is how to realize the transformation between two entangled pure states with the possible supply of multiple copies or even a single copy [3, 4] of the input state. An arbitrary input state of this type shared by two parties, say Alice and Bob, is given as |Ψ1 = i,j λi,j |i A |j B (normalized), where the vector αT = (|1 A , · · · , |M A ) forms an orthonormal basis of Alice’s space HA and β T = (|1 B , · · · , |N B ) forms that of Bob’s space HB (M and N can be diﬀerent). A routine strategy to transform a single copy of this state involves a generalized measurement of Alice followed by one-way communication of its result to Bob. Classical communications coordinate the operations of Alice and Bob. It has been concluded that strategies with at least one-way classical communication are more powerful than those with no communication in manipulating one bipartite entangled state [5]. Against intuitions, classical communication is not cheap in some situations [6]. A typical example is the superdense coding [7]: Alice and Bob can use n qubits to transmit 2n bits of classical information, if they share n pairs of Bell states. However, if the entangled pairs they share are not perfect and it costs them more than n bits of classical communication to transform the pairs to the ME states, it would totally destroy the purpose of superdense coding. In multi-copy concentration of a bipartite pure state