Quantum Lower Bounds by Entropy Numbers

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A Note on Quantum Hamming Bound

a r X i v :0711.4603v 1 [q u a n t -p h ] 29 N o v 2007A Note on Quantum Hamming BoundSalah A.AlyDepartment of Computer Science,Texas A&M University,College Station,TX 77843-3112,USAEmail:salah@Proving the quantum Hamming bound for degenerate nonbinary stabilizer codes has been an open problem for a decade.In this note,I prove this bound for double error-correcting degenerate stabilizer codes.Also,I compute the maximum length of single and double error-correcting MDS stabilizer codes over ﬁnite ﬁelds.1Bounds on Quantum CodesQuantum stabilizer codes are a known class of quantum codes that can protect quantum information against noise and decoherence.Stabilizer codes can be constructed from self-orthogonal or dual-containing classical codes,see for example [3,8,11]and references therein.It is desirable to study upper and lower bounds on the minimum distance of classical and quantum codes,so the computer search on the code parameters can be minimized.It is a well-known fact that Singleton and Hamming bounds hold for classical codes [10].Also,upper and lower bounds on the achievable minimum distance of quantum stabilizer codes are needed.Perhaps the simplest upper bound is the quantum Singleton bound,also known as the Knill-Laﬂamme bound.The binary version of the quantum Singleton bound was ﬁrst proved by Knill and Laﬂamme in [12],see also [1,2],and later generalized by Rains using weight enumerators in [16].Theorem 1(Quantum Singleton Bound).An ((n,K,d ))q stabilizer code with K >1satisﬁesK ≤q n −2d +2.(1)Codes which meet the quantum Singleton bound with equality are called quantum MDS codes.If we assume that K =q k ,then this bound can be stated as k ≤n −2d +2.In [11]It has been shown that these codes cannot be indeﬁnitely long and showed that the maximal length of a q -ary quantum MDS codes is upper bounded by 2q 2−2.This could probably be tightened to q 2+2.It would be interesting to ﬁnd quantum MDS codes of length greater than q 2+2since it would disprove the MDS conjecture for classical codes [10].A related open question is regarding the construction of codes with lengths between q and q 2−1.At the moment there are no analytical methods for constructing a quantum MDS code of arbitrary length in this range (see [9]for some numerical results).Another important bound for quantum codes is the quantum Hamming bound.The quantum Hamming bound states (see [6,7])that:1Theorem2(Quantum Hamming Bound).Any pure((n,K,d))q stabilizer code satisﬁes⌊(d−1)/2⌋i=0 n i (q2−1)i≤q n/K.(2)The previous quantum Hamming bound holds only for nondegenerate(pure)quantum codes. However,the degenerate(impure)quantum codes are particularly interesting class of quantum codes because they can pack more quantum information.In addition,the errors of small weights do not need active error correction strategies.So far no degenerate quantum code has been found that beats this bound.Gottesman showed that impure single and double error-correcting binary quantum codes cannot beat the quantum Hamming bound[8].It is proved in[11]that Hamming bound holds for quantum stabilizer codes with distance d=3.In general,does Hamming bound exist for any distance d in((n,K,d))q stabilizer codes?This has been an open question for a decade.In this note we prove Hamming bound for double error-correcting stabilizer codes with distance d=5and also give a sketch to prove it for general distance d.2Quantum Hamming Bound Holds for Distance d=5There have been several approaches to prove bounds on the quantum code parameters.In[1] Ashikhmin and Litsyn derived many bounds for quantum codes by extending a novel method originally introduced by Delsarte[5]for classical ing this method they proved the binary versions of Theorems1,2.We use this method to show that the Hamming bound holds for all double error-correcting quantum codes.See[11]for a similar result for single error-correcting codes.But ﬁrst we need Theorem3and the Krawtchouk polynomial of degree j in the variable x,K j(x)=js=0(−1)s(q2−1)j−s x s n−x j−s .(3)Theorem 3.Let Q be an((n,K,d))q stabilizer code of dimension K>1.Suppose that S is a nonempty subset of{0,...,d−1}and N={0,...,n}.Letf(x)=ni=0f i K i(x)(4)be a polynomial satisfying the conditionsi)f x>0for all x in S,and f x≥0otherwise; ii)f(x)≤0for all x in N\S.ThenK≤1f x.(5) 2Proof.See[11].We demonstrate usefulness of the previous Theorem by showing that quantum Hamming bound holds for impure codes when d=5.Lemma4(Quantum Hamming Bound).An((n,K,5))q stabilizer code with K>1satisﬁesK≤q n n(n−1)(q2−1)2/2+n(q2−1)+1 .(6) Proof.Let f(x)= n j=0f j K j(x),where f x=( e j=0K j(x))2,S={0,1,...,4}and N={0,1,...,n}. Calculating f(x)and f x gives usf0=(1+n(q2−1)+n(n−1)(q2−1)2/2)2f1=12(n−3)(n−2)(q2−1)2−(n−2)(q2−1))2f3=(1−2(n−3)(q2−1)+12(n−5)(n−4)(q2−1)2)2and,f(0)=q2n(1+n(q2−1)+1f x(7)So,there are four diﬀerent comparisons where f(0)/f0≥f(x)/f x,for x=1,2,3,4.Weﬁnd a lower bound for n that holds for all values of q.From Lemmas5,6,7,and8,shown below,for n≥7 it follows thatmax{f(0)/f0,f(1)/f1,f(2)/f2,f(3)/f3,f(4)/f4}=f(0)/f0(8)While the above method is a general method to prove Hamming bound for impure quantum codes,the number of terms increases with a large minimum distance.It becomes diﬃcult toﬁnd the true bound using this method.However,one can derive more consequences from Theorem3; see,for instance,[1,2,13,15].3Lemma5.The inequality f(0)/f0≥f(1)/f1holds for n≥6and q≥2.Proof.Let f(0)/f0≥f(1)/f1then1(n−1)2(n−2)2(q2−1)4(n−1)2(n−2)2(q2−1)4≥(1+n(q2−1)n(n−1)+(q2−1)2)(q2(q2−1)(n−1)+1)2divide both sides by(q2−1)2(q2−1)2and approximate1)(n−1)2by approximating both sides,theﬁnal result is(n−2)≥4orn≥6Lemma6.The inequality f(0)/f0≥f(2)/f2holds for n≥7and q≥2.Proof.Letq2n(−(n−2)(q2−1)+(n−3)(n−2)(q2−1)2/2)2 by simplifying both sides(−(n−2)(q2−1)+(n−3)(n−2)(q2−1)2/2)2≥(q4+2(n−2)(q2−1))(1+n(q2−1)+n(n−1)(q2−1)2/2)Simplifying L.H.S,(n−2)to(n−3)then(q2−1)4((n−3)2/2−(n−2))2≥(q4+2(n−2)(q2−1))(1+n(q2−1)+n(n−1)(q2−1)2/2)4by simplifying both sides((n−3)2/2−(n−2))2≥(q2q2−1)(1+n+n(n−1)/2)((n−3)2/2−(n−2))2≥2(2n+1)(n2+2n+2)(n−3)2((n−3)/2−1)2≥2(2n+1)((n+1)2+1)(n−5)2/4≥2(2n+1)n≥7Lemma7.The inequality f(0)/f0≥f(3)/f3holds for n≥7and q≥2.Proof.Letq2n(1−2(n−3)(q2−1)+(n−3)(n−4)(q2−1)2/2)2 by simpliﬁcation(1−2(n−3)(q2−1)+(n−3)(n−4)(q2−1)2/2)2≥6(q2−1)(1+n(q2−1)+n(n−1)(q2−1)2/2)((n−4)2(q2−1)2−4(n−4)(q2−1)+2)24((n−5)4≥3(q2−1)3(2+2n+n2)(n−5)4≥4(2+2n+n2)(n−5)2≥2n≥7Lemma8.The inequality f(0)/f0≥f(4)/f4holds for n≥7and q≥2.Proof.Letq2n(3−3(n−4)(q2−1)+(n−4)(n−5)(q2−1)2/2)2divide by q2n and simplifying(3−3(n−4)(q2−1)+(n−4)(n−5)(q2−1)2/2)2≥6(1+n(q2−1)+n(n−1)(q2−1)2/2)5then by approximating(n−4)to(n−5)in L.H.S and(n−1)to4n in R.H.S,we canﬁnd that (−3(n−4)(q2−1)+(n−5)2(q2−1)2/2)2≥6(1+n(q2−1)+n2(q2−1)2/2)((n−5)2(q2−1)+(n−5)2(q2−1)2/2)2≥6(1+n(q2−1)+n2(q2−1)2/2) (q2−1)4((n−5)2+(n−5)2/2)2≥6(1+n(q2−1)+n2(q2−1)2/2)dividing both sides by(q2−1)4and simplifying6(1+n(q2−1)+n2(q2−1)2/2)(9/4)(n−5)4≥(9)1+n(q2−1)If the Hamming bound is tighter than the Singleton bound for any((n,K,3))q quantum code,then it means that MDS codes cannot exist for that set of n,K.This occurs whenq nq n−2d+2=q n−4≥3.2Upper Bound on the Maximal Length of Double Error-correctingMDS CodesLemma10.The maximal length of double error-correcting quantum MDS codes is upper bounded by:(q2−3)+ 2(q2−1)(11)n≤Proof.It is known that the Hamming bound for d=5is given by:q nK≤1+n(q2−1)+n(n−1)(q2−3)2+8(q8−1)(q2−3)2+8(q8−1)(q2−3)2+8(q8−1)[3]A.R.Calderbank,E.M.Rains,P.W.Shor,and N.J.A.Sloane.Quantum error correction viacodes over GF(4).IEEE rm.Theory,44:1369–1387,1998.[4]P.Delsarte.Four fundamental parameters of a code and their combinatorial signiﬁcance.Information and Control,23(5):407-438,December1973.[5]P.Delsarte.Bounds for unrestricted codes by linear programming.Philips Res.Reports,27:272–289,1972.[6]K.Feng and Z.Ma.Aﬁnite Gilbert-Varshamov bound for pure stabilizer quantum codes.IEEE rm.Theory,50(12):3323–3325,2004.[7]D.Gottesman.A class of quantum error-correcting codes saturating the quantum Hammingbound.Phys.Rev.A,54:1862–1868,1996.[8]D.Gottesman.Stabilizer codes and quantum error correction.Caltech Ph.D.Thesis,eprint:quant-ph/9705052,1997.[9]M.Grassl,T.Beth,and M.R¨o tteler.On optimal quantum codes.Internat.J.QuantumInformation,2(1):757–775,2004.[10]W.C.Huﬀman and V.Pless.Fundamentals of Error-Correcting Codes.University Press,Cambridge,2003.[11]A.Ketkar,A.Klappenecker,S.Kumar,and P.K.Sarvepalli.Nonbinary stabilizer codes overﬁniteﬁelds.IEEE rm.Theory,52(11):4892–4914,2006.[12]E.Knill and ﬂamme.A theory of quantum error–correcting codes.Physical Review A,55(2):900–911,1997.[13]V.I.Levenshtein.Krawtchouk polynomials and universal bounds for codes and designs inHamming spaces.IEEE rm.Theory,41(5):1303–1321,1995.[14]J.H Van Lint.Introduction to coding theory.Third Edition,Springer-Verlag1999.[15]R.J.McEliece,E.R.Rodemich,jr.H.Rumsey,and L.R.Welch.New upper bounds on the rateof a code via the Delsarte-MacWilliams inequalities.IEEE rm.Theory,23(2):157, 1977.[16]E.M.Rains.Nonbinary quantum codes.IEEE rm.Theory,45:1827–1832,1999.84AppendixAn Approach (Sketch)to Prove Hamming Bound for Degen-erate Nonbinary Stabilizer Codes with Minimum Distance dOne way to prove the quantum Hamming bound for impure nonbinary stabilizer codes with d ≤(n −k +2)/2is to expand f (x )/f x in terms of Krawtchouk polynomials.Let f (x )= n j =0f j K j (x )and f x =( ei =0K i (x ))2.The Krawtchouk polynomial of degree e in the variables x and q is given byK e (q,x )=e j =0(−1)j (q 2−1)e −jx j n −xe −j(17)Theorem 11.Let Q be an ((n,K,d ))q stabilizer code of dimension K >1.Suppose that S is anonempty subset of {0,1,...,d −1}and N ={0,1,...,n }.Letf (x )=n i =0f i K i (x )be a polynomial satisfying the conditions:i)f x >0for all x ∈S ,and f x ≥0otherwise;ii)f (x )≤0for all x ∈N \S .ThenK ≤1f x.Notice that f (x )=ni =0f i K i (x )can be written as f i =q−2nnx =0f (x )K x (i ).Lemma 12(Sketch).Let Q be an ((n,K,d ))q stabilizer code of dimension k ≥1.Suppose that S is a non-empty subset of {0,1,2,....,2e },where e =⌊d −1f x equals toK ≤q nf(x)f x(19) And our goal is toﬁnd max{f(0)/f0,f(1)/f1,...,f(d−1)/f d−1}that may equal to f(0)/f0.Now,for x=0,weﬁnd thatf(0)f0=K0(q,0)+f1K1(q,0)f0(20)orf(0)e i=0K i(0) 2and for any other value of y∈{1,2,...,d−1},weﬁnd thatf(y)f y=f0K0(q,y)f y+......+f n K n(q,y) f y= n j=0 e i=0K i(j) 2K j(y)f0−f(y)f0−f(y)e i=0K i(0) 2− n j=0 e i=0K i(j) 2K j(y)e i=0K i(0) 2−K j(y)f0−f j K j(q,y)in the previous equation,f j>0and f y>0,so,if we prove thatf j K j(q,0)≥0,(24)f ythen the claim holds.As shown in[4],[14],we seek a constant value for the left side in24,so, multiplying both sides by K e(i)K e(i)K i(q,0)≥0(25)f yand take n i=0ni=0 K e(i)K i(q,0)f y ≥0n i=0K e(i)K i(q,0)f y≥0(26) from[14],given that n i=0K e(i)K i(q,j)=q nδej,by substitution,q nδe0≥0f yδe0≥0(27)f yNow,δe0=1,andδey=1or0;and obviously f y≥f0.So,if y=e=⇒δe0/f0≥0,and similarly,δey=1=⇒f y−f0≥0.11。

量子力学07

For the next several lectures we will be discussing the von Neumann entropy and various concepts relating to it.This lecture is intended to introduce the notion of entropy and its connection to compression.7.1Shannon entropyBefore we discuss the von Neumann entropy,we will take a few moments to discuss the Shannon entropy.This is a purely classical notion,but it is appropriate to start here.The Shannon entropy of a probability vector p∈RΣis deﬁned as follows:p(a)log(p(a)).H(p)=−∑a∈Σp(a)>0Here,and always in this course,the base of the logarithm is2.(We will write ln(α)if we wish to refer to the natural logarithm of a real numberα.)It is typical to express the Shannon entropy slightly more concisely asp(a)log(p(a)),H(p)=−∑a∈Σwhich is meaningful if we make the interpretation0log(0)=0.This is sensible given thatαlog(α)=0.limα↓0There is no reason why we cannot extend the deﬁnition of the Shannon entropy to arbitrary vectors with nonnegative entries if it is useful to do this—but mostly we will focus on probability vectors.There are standard ways to interpret the Shannon entropy.For instance,the quantity H(p)can be viewed as a measure of the amount of uncertainty in a random experiment described by the probability vector p,or as a measure of the amount of information one gains by learning the value of such an experiment.Indeed,it is possible to start with simple axioms for what a measure of uncertainty or information should satisfy,and to derive from these axioms that such a measure must be equivalent to the Shannon entropy.Something to keep in mind,however,when using these interpretations as a guide,is that the Shannon entropy is usually only a meaningful measure of uncertainty in an asymptotic sense—as the number of experiments becomes large.When a small number of samples from some experi-ment is considered,the Shannon entropy may not conform to your intuition about uncertainty,as the following example is meant to demonstrate.Example7.1.LetΣ={0,1,...,2m2},and deﬁne a probability vector p∈RΣas follows:p(a)= 1−1m2−m21≤a≤2m2.56nn∑j=1Y j−E[Y j] ≥ε =0,which is true by the(weak)law of large numbers.uu∗, Ξ⊗1L(Z) (uu∗)represents one way of measuring the quality with whichΦacts trivially on the state uu∗.Now,any puriﬁcation u∈W⊗Z ofσmust take the formu=vec √k∑j=1√σB 2=k ∑j=1 σ,A j 2.Another expression for the channelﬁdelity isF channel(Ξ,σ)=ρ⊗n,B j A i。

量子力学09

−α T α 1− α T = λ ξ α ⊗ ( ρ1 1 ) + ( 1 − λ ) ξ 2 ⊗ ( ρ2 ) .
The identity vec(1 )∗ ( X ⊗ Y T ) vec(1 ) = Tr( XY ) then gives the desired result. Now, toward the proof of Theorem 9.4, we require the following lemma. Lemma 9.5. Let P1 , P2 , Q1 , Q2 , R1 , R2 ∈ Pd (X ) be positive deﬁnite operators that satisfy these conditions: 1. [ P1 , P2 ] = [Q1 , Q2 ] = [ R1 , R2 ] = 0,
′ S(ρ ξ ) = − log(e) f ρ ,ξ (0) = − log( e) lim
α ↓0
Tr ξ α ρ1−α − 1 , α
where the second equality follows by substituting f ρ,ξ (0) = 1 into the deﬁnition of the derivative. Now consider the following lemma that concerns the relationship among the functions f ρ,ξ for various choices of ρ and ξ . Lemma 9.3. Let ξ 1 , ξ 2 , ρ1 , ρ2 ∈ Pd (X ) be positive deﬁnite operators. Then for every choice of α, λ ∈ [0, 1] we have

Note on Bound States and the Bekenstein Bound

a r X i v :h e p -t h /0406037v 1 3 J u n 2004Preprint typeset in JHEP style -HYPER VERSIONDonald Marolf Physics Department,UCSB,Santa Barbara,CA 93106.Radu Roiban Physics Department,UCSB,Santa Barbara,CA 93106.Abstract:In this brief note we draw attention to examples of quantum ﬁeld theories which may hold interesting lessons for attempts to devise a precise formulation of the Bekenstein bound.Our comments mirror the recent results of Bousso (hep-th/03110223)indicating that the species problem remains an issue for precise formulations of this bound.Keywords:Bekenstein Bound.1.IntroductionThere has been much discussion in the literature of the idea that quantum systems may be subject to certain fundamental bounds relating their entropy(S)to their size(measured in terms of a radius R or an enveloping area),and perhaps to their energy(E).Such proposed bounds include the Bekenstein bound S<αRE[1,2],the holographic bound S<A/4ℓ2p [3,4],and the more subtle Causal[5]and Covariant[6]Entropy Bounds.Such bounds were originally motivated by considerations of black hole thermodynamics[1,2,3,4].Though this motivation has been criticized by various authors[7,8,9,10],the proposed bounds remain interesting topics of discussion and investigation.The Covariant Entropy Bound represents a reﬁnement of the holographic bound as, at least when spacetime can be treated classically,it gives a precise deﬁnition of what is meant by the area A.Similarly,the parameters playing the role of size and energy for the Causal Entropy Bound are well-deﬁned in this context,though the same is not true of the original holographic bound.It is also of interest to study whether a more precise conjecture can be found to replace the Bekenstein bound S<αRE.This was explored in two recent papers[11,12]by Bousso.The Bekenstein bound is unique among those above in that it does not involve the Planck length.It may therefore be conjectured to hold in ordinary ﬁeld theories,without considering coupling to gravity.This is advantageous for testing the bound,as we have more knowledge as to which such theories exist than we do when gravity is considered.An alternate interpretation of the Bekenstein bound is that,although it does not explicitly refer to the Planck length,it should apply only toﬁeld theories which can in principle be consistently coupled to the gravitationalﬁeld.We shall have little to say here about this more restrictive conjecture.In[11](following Bekenstein[13,14,15])it was argued that a precise version of this conjecture might apply to arbitrary quantumﬁeld theories.In particular,it was argued that a more precise formulation might be able to handle the so-called‘species problem’, referring to the fact that naive interpretations of the bound S<αRE(whereαis aﬁxed constant of order1)are readily violated in any theory containing a large number ofﬁelds.A simple example arises from a one-particle wavepacket state in a theory of N masslessscalarﬁelds.Such a state has RE∼1,but S∼ln N.Thus,the most naive interpretation of the Bekenstein bound is violated.Bekenstein has long argued that the bound should not apply to such wavepacket states (which will eventually spread out in space),but only to‘complete systems’[13,14,15] which are truly conﬁned to aﬁnite region and that one should include contributions from the energy of any‘walls’used to hold the system together.It is here that some cleverness is needed to make this statement precise since inﬂat spacetime,even if walls are introduced, the full system(including the walls)will necessarily possess an overall center of mass degree of freedom which will be unconﬁned and will eventually spread out across all of space.Thus, it is not clear in what sense any sub-system of the universe is truly‘complete’in this sense.Theﬁnal section of[11]suggested that one should simply disregard the overall center of mass degree of freedom and instead consider‘bound states,’with the size R being the width of the bound state.Following[11],we shall not yet be too precise about how this width is deﬁned.We also note that another proposal was explored in[12],in which the conjecture was made precise in the context of discrete light cone quantumﬁeld theory, where the size is controlled by the size of a compact direction in the spacetime.However, it was noted in[12]that a large number of species can violate the bound as easily in this second context as in the naive example above.Here we consider the‘bound state’proposal of[11]to remove the center of mass degree of freedom and deﬁne the resulting quotient to be the set of bound states.One may then test bounds of the form S<αRM,where M is the mass of the bound states.Even in this context one may quickly construct counter-examples.First,consider again a theory of N free massless scalar particles.The above quotient of the one-particle Hilbert space leaves an N-dimensional vector space corresponding to particleﬂavor.But S=ln N and M=0clearly violates the bound,and atﬁnite M the bound is violated for large enough N.This counter-example was also pointed out in[12].A similar trivial violation may be constructed from any theory having a clear‘bound state’(say,QCD with its hadrons)and then considering a Lagrangian built from a large number N of mutually non-interacting copies of this system.Now,while the examples above are counter-examples in the technical sense,they ap-pear to be somewhat trivial.This triviality might be taken as an indication that,with a bit of reﬁnement,the bound state version of the Bekenstein bound could be made more robust. Our purpose below is to point out less trivial counter-examples in which the states appear at some intuitive level to all be‘bound states held together by the same force’–though the existence of dual formulations again raises the question of to what extent‘bound states’are fundamentally diﬀerent from any other sort of state.Our counter-examples concern N=1and N=2supersymmetric SU(N c)gauge theory,where the degrees of freedom are under some control(see for example[16])in the infrared limit.2.Examples:N=1and N=2SU(N c)gauge theory with fundamentalmatterLet us consider an SU(N c)gauge theory in3+1dimensions with matter in the fundamentalrepresentation.The infrared behavior depends on the number N f of matter multiplets Q and˜Q(see for example[16]).Quite generally,if N f<3N c among the low energy degrees of freedom oneﬁnds,in some description,N2f mesons M=Q˜Q and baryons,which are composites of the high energy matterﬁelds.Of particular interest to our discussion is the situation32R(M)=3N f−N c2N c<N f<2N c).Furthermore,even if one changes the rules and requires that the mesons be placed in wavepackets with E∼1/R,the bound is readily violated at large N c,N f.In the limit N fց31The cases with no conformal invariance are quite hard to analyze because the masses of the low energy degrees of freedom depend on the K¨a hler potential and are not under control.2The other conformally invariant answer would be inﬁnite size,but conformal invariance is broken away from theﬁxed point so that the mesons will have someﬁnite size in the full theory(which can then be neglected in the long wavelength limit).option(1)would be ruled out if a convincing theory of the above models coupled to gravity could be found.We mention in passing that similar results hold for SU(n)gauge theory with m<n massless fundamental fermions and N=2supersymmetry in3+1dimensions which are conﬁning rather than conformal.Here the mass of mesons can be tuned to be arbitrarily small while taking the conﬁnement energy scale to inﬁnity as fast as one likes, and thus presumably keeping the size of the meson bound states small.Let us return brieﬂy to the actual context discussed in[11],in which Bousso attempted to study the conﬁnement of degrees of freedom to aﬁxed region of space through the use of an external potential.The argument in[12]was that an attempt to use a single potential to conﬁne a large number of species inevitably leads to large radiative corrections,over which one has little control.The bound states in the models above are of a similar nature,as the gluons couple equally to each of the N fﬂavors of Q and˜Q,though now supersymmetry does allow one to retain some control over the analysis.In the context of such bound states oneﬁnds that degrees of freedom(the relative motion of the constituents)can in fact be conﬁned without great cost in energy.This suggests that if tools could be found to make the analysis tractable,external potential problems of the sort studied in[11]could also lead to large numbers of states localized in a region ofﬁxed size.As usual,we expect that strongly coupled quantumﬁeld theories are capable of all manner of surprising behaviors not immediately obvious from their perturbative description.Acknowledgments:We are grateful to Tom Banks,Jan de Boer,and especially Raphael Bousso for several interesting discussions on these issues.D.M.was supported in part by NSF grant PHY03-54978,and by funds from the University of California and the Kavli Institute of Theoretical Physics.R.R.was supported in part by the National Science Foundation under Grant No.PHY00-98395as well as by the Department of Energy under Grant No.DE-FG02-91ER40618.References[1]J.D.Bekenstein,‘Black Holes And Entropy,’Phys.Rev.D7,2333(1973);J.D.Bekenstein,‘Generalized Second Law Of Thermodynamics In Black Hole Physics,’Phys.Rev.D9,3292 (1974).[2]J.D.Bekenstein,‘Quantum information and quantum black holes,’arXiv:gr-qc/0107049.[3]L.Susskind,‘The World as a hologram,’J.Math.Phys.36,6377(1995)[arXiv:hep-th/9409089].[4]G.’t Hooft,‘Dimensional Reduction In Quantum Gravity,’arXiv:gr-qc/9310026.[5]R.Brustein and G.Veneziano,‘A Causal Entropy Bound,’Phys.Rev.Lett.84,5695(2000)[arXiv:hep-th/9912055].[6]R.Bousso,‘A covariant entropy conjecture,’JHEP07,004(1999),[arxiv:hep-th/9905177].[7]W.G.Unruh and R.M.Wald,‘Acceleration Radiation And Generalized Second Law OfThermodynamics,’Phys.Rev.D25,942(1982).[8]D.Marolf and R.Sorkin,‘Perfect mirrors and the self-accelerating box paradox,’Phys.Rev.D66,104004(2002)[arXiv:hep-th/0201255].[9]D.Marolf and R.D.Sorkin,‘On the status of highly entropic objects,’arXiv:hep-th/0309218.[10]D.Marolf,D.Minic and S.F.Ross,‘Notes on spacetime thermodynamics and theobserver-dependence of entropy,’arXiv:hep-th/0310022.[11]R.Bousso,‘Bound states and the Bekenstein bound,’arXiv:hep-th/0310148.[12]R.Bousso,‘Harmonic resolution as a holographic quantum number,’JHEP0403,054(2004)[arXiv:hep-th/0310223].[13]J.D.Bekenstein,‘Speciﬁc Entropy And The Sign Of The Energy,’Phys.Rev.D26,950(1982).[14]J.D.Bekenstein,‘Entropy Bounds And The Second Law For Black Holes,’Phys.Rev.D27,2262(1983).[15]J.D.Bekenstein,‘On Page’s examples challenging the entropy bound,’arXiv:gr-qc/0006003.[16]K.A.Intriligator and N.Seiberg,‘Lectures on supersymmetric gauge theories andelectric-magnetic duality,’Nucl.Phys.Proc.Suppl.45BC,1(1996)[arXiv:hep-th/9509066].。

Asymptotic Quantum Search and a Quantum Algorithm for Calculation of a Lower Bound of the P

ASYMPTOTIC QUANTUM SEARCH AND A QUANTUM ALGORITHM FOR CALCULATION OF A LOWER BOUND OF THE PROBABILITY OF FINDING A DIOPHANTINE EQUATIONTHAT ACCEPTS INTEGER SOLUTIONSR. V. Ramos and J. L. de Oliveirarubens@deti.ufc.br jluzeilton@deti.ufc.brDepartment of Teleinformatic Engineering, Federal University of CearaCampus do Pici, C. P. 6007, 60455-740, Fortaleza, Brazil.Several mathematical problems can be modeled as a search in a database. An example is the problem of finding the minimum of a function. Quantum algorithms for solving this problem have been proposed and all of them use the quantum search algorithm as a subroutine and several intermediate measurements are realized. In this work, it is proposed a new quantum algorithm for finding the minimum of a function in which quantum search is not used as a subroutine and only one measurement is needed. This is also named asymptotic quantum search. As an example, we propose a quantum algorithm based on asymptotic quantum search and quantum counting able to calculate a lower bound of the probability of finding a Diophantine equation with integer solution.Keywords:Quantum algorithms, quantum search, optimization, Diophantine equations.1. IntroductionThe Grover’s quantum search algorithm is an important result in quantum computation that proves that quantum superposition can speed-up the task of finding a specific value within an unordered database. The quantum search is proved to use O(N1/2) operations of the oracle (in comparison with the O(N) operations of the best classical algorithm), indicating a quadratic speed-up [1-3]. Several mathematical problems can be modeled as a search, like the problem of finding the minimum (or the maximum) of a function. Thus, some algorithms for finding the minimum or maximum using quantum search have been proposed [4,5], using the generalization of the quantum search proposed in [6] as a subroutine that is called several times. Every time the quantum search is called, at the end a measurement is realized. The number of measurements in the algorithm proposed in [4] is (log2N), where N is the number of elements in the database [7]. Aiming to reduce the number of measurements, Kowada at al [7] proposed a new quantum algorithm for finding the minimum of a function that realizes O(log N) measurements. Here, we go beyond, proposing a quantum algorithm for finding the minimum that realizes only one measurement, at the final of the algorithm. Furthermore, conversely the already proposed quantum algorithms for finding the minimum, the quantum search in the proposed algorithm is not used as a subroutine, indeed it is a part of the algorithm as any other quantum gate. In this work, as an example, we apply asymptotic quantum search together with quantum counting algorithm [3,8] in order to create a quantum algorithm able to calculate a lower bound of the probability of finding a Diophantine equation that accepts integer solutions. We are going to call this number by D.(1)(2) (3)2. Quantum algorithm for finding the minimum of a functionThe circuit for the proposed quantum algorithm is shown in Fig. 1.Fig. 1- Circuit for asymptotic quantum search.In Fig. 1, U f is the quantum gate that implements the function f , U f x 0 = x f (x ) , the gate QBSC is a quantum bit string comparator [9], QBSC x y 0 = x y b , where b =0 if x >y and b =1 if x y . At last, the oracle in the amplitude amplification recognizes if the quantum registers 4 and 5 are in the total state 0 N 0 . In order to understand the quantum circuit in Fig. 1, we firstly show the operation of the quantum circuit U , following the states in the marked positions, as shown below:1152432151452243331451452233000000000000NNNx xNN NNNx f x x xy Nx f x xy x y c xc U x Hc xf x c xf x yc x f x f y(4.a)(4.b)(5)(6.a)41451452323, ,415243 212431101 N x y x yf x f y f x f y n x x jj xj f x f y n x x jj j xf x yf y f x yf y c x f x f y c x f x f y50xf x f y.In (1)-(4.b), N is the number of qubits, hence, there are 2N elements in the database. In (4.b), 1 n (x ) 2N is the number of elements that obey f (x ) f (y ) for a given x and all y ’s. Thus, the better the solution the larger is the value of n (x ). At last, y i can assume any y value.min 515243 215243 16min min 101 00Nn x N x j x j f x f y n x N x j xj f x f y x c x f x c x f x c x f xminmin 1522443321423 10001 0Nn x N N N N x j xj x x f x f y n x N Nx kxk x x f x f y c x fx y c xf x H y5(6.b)(7)(8)min minmin 6min min 115224343 154231 021421002001001N j NNNN NNx x xx x n x NN Nx j x j x x H y f x f y n x N x kk c x f x c n x xf x c x y c x f x H ymin53 00N x x x f x f y.Finallymin 71515243243 15423 1 0214210000200000000N N j NN NN NNNx x xxf x f y n xNNNx j xj x x H y f x f y NN x k k U c xc n x x c x y c x y53 01Nn x N xf x f yThe worse the solution the lower is the value of n (x ) and the faster is the decay. Now, using k -times the gate U together with the multi-controlled CNOTs, the quantum state just before the quantum amplification is:min 15243 1154231 1 021142120000200020N j N k N N N Nx x f x f y n x k r N N NN x j r x j x x H y f x f y n r N N Nx k k c n x x c n x x H y c n x x H y31 501x k N r x f x f y(9)At the amplitude amplification, the oracle recognizes the state 0 N 0 in the fourth and fifth quantum registers. Therefore, only the first term in (8) will have their amplitudes amplified. Looking closer the first term in (8), one sees that the amplitude of the searched answer is min x c , since n (x )=2N for x =x min . The second term with largest amplitude has n (x )=2N -1, hence, after the k -th application of U its amplitude will be c x /2k . Thus, if k is large enough only the term corresponding to the searched answer will have considerable amplitude and, after amplitude amplification, the searched answer will be obtained with high probability ina single set of measurements. For example, choosing x c for all x , the largest amplitude after kusage of U(before amplitude amplification) is min x c while the second largest amplitude will bemin 2k x x c c . The oracle in the amplitude amplification is applied /(4 ) times, where [9]sin k.If k is large enough, then 2Ngood p . Obviously, the efficiency of the proposed algorithm is equal to theefficiency of the amplitude amplification.3. Quantum algorithm for finding a lower bound of DA Diophantine equation is a polynomial equation with any number of unknowns and with integercoefficients. For example, (x +1)n +(y +2)n +(z +3)n -Cxyz =0, where C and n . Since there is not a universal process to determine whether any Diophantine equation accepts or not integer solutions, the best that one can do is to use a brute-force method. In this case, one can not determine the probability of finding a Diophantine equation that accepts integer solutions, D , exactly, but it is possible to determine a lower bound for it. In this direction, the quantum algorithm proposed can be used to implement a brute-force attack (using the intrinsic quantum parallelism) and the accuracy of the answer obtained is limited by the amount of qubits used, the larger the amount of qubits used the closer to D is the obtained answer. Since only a finite amount of qubits can be used, the answer obtained is simply a lower bound of D . Since the (infinite) set of Diophantine equations considered is enumerable, we will use integer numbers to enumerate the Diophantine equations. The quantum circuit for the proposed quantum algorithm is shown in Fig. 2. In this quantum circuit, we assume that the set of Diophantine equations considered has only three unknowns (x ,y ,z ), however, the extension to a larger number of unknowns is straightforward.(10)(11)(12)(13)(15)(16)(14)(17)31126354,,32126543,,3125643,,,,,,0000,,,,000,,,,,,00lM lxyz px y z Ml xyz p px y z lxyz p px y zx y z p c x y z c x y z D x y z p c x y z D x y z y z4125634,,,,51126534,, ,,,,,,,,,,,,,,0,,,,,,,,1p p i j mxyz p p px y z x y z xyzp i j mp i j mpx y zi j mD x y z D x y z c x y z D x y z y z D x y z cx y zD x y z y z D x y z !62126543,, ,,,,,,352317132,,,,,,01,,,,0,,2,,,,p p i j mpp s s s s s s opts lxyzp i j mpx y zi j mD x y z D x y z t p p p p p pM opt opt opt p opt opt opt xyz s M xyz p pcx y zD x y z y z c x y z D x y z c n x y z x y z D x y !4634353,,,,,,010p p popt opt opt l Mpx y z x y z x y z z !!Finally38126354,,2133163542 ,,,,,,,,0000,,0000,,2,,pps s s opt sp p popt opt opt lM lxyz px y zt p p popt opt opt xyz s l MlM pxyz x y zx y z x y z U c x y z c x y z p c n x y z x y z !In (17), 1 n (x ,y ,z ) 23M is the number of elements that obey |D p (x,y,z )| |D p (x’,y’,z’)| for a given x ,y ,z and all x’,y’,z’. Thus, the better the solution the larger is the value of n (x ,y ,z ). We assume that, for the Diophantine equation number p there are t p minimums. Moreover3,,2s s sp p pMopt opt opt n x y z for s =1,2,…,t p . Thestates !i i =1,2,3,4 and ! are states with undesired terms. Now, using k -times the gate U together with the multi-controlled CNOT gates, the quantum state just before the quantum amplification is:(20)(19)(18)3312635421 ,,,,,,,,,,2,,0000p p s s s opt sp p popt opt optt kp p p M l Mlopt opt optxyz xyz ps x y zx y z x y z p c x y z c n x y z x y z"The state " represents the uninteresting terms. At the amplitude amplification, the oracle recognizes the state 0 3M 0 in the fifth and sixth quantum registers. Therefore, only the first term in (18) will have their amplitudes amplified. Looking closer the first term in (18), one sees that the amplitude of one of the searched answers is p opt sxyz c (comparing to the single variable case in Section 2, we have n (x,y,z )=23M for,,,,p p popt opt opt x y z x y z ). The second term with largest amplitude has n (x ,y ,z )=23M -1, hence, after the k -thapplication of U its amplitude will be c xyz /2k . Thus, if k is large enough only the term corresponding to the searched answer will have considerable amplitude and, after amplitude amplification, it will be obtained with high probability in a single measurement. For example,choosing xyz c for all xyz , thelargest amplitude after k usage of U (before amplitude amplification)is p opt sxyz c thesecond largest amplitude will be 2popt sk xyz xyz c c . The oracle in the amplitude amplification is applied/(4 ) times, wheresin k.If k is large enough, then 32Mgood p p t . The efficiency of the proposed algorithm is equal to theefficiency of the amplitude amplification. Thus, the final state after the amplitude amplification is1,,,,ps s ss s st p p pp p pend opt opt opt p opt opt opt ps y z D x y z errIn (20) one can see that, for each one of the 2N Diophantine equations, the algorithm calculate the minimums values testing 2M different values for each unknown. Moreover, the term err represents the remained unwanted states due to the non ideal amplitude amplification.In order to calculate the lower bound of D , we use the state (20) as input of a quantum countingalgorithm whose oracle recognizes a zero in the last register of (20), that is,,,0p p pp opt opt optD x y z .The quantum counting algorithm returns the number of elements in the database that pass in the oracle test. Let us name this value by r (N ,M ). Hence,(21),lim ,2N D N M r N M #$and a lower bound for D is simply ,2estND r N M .4. DiscussionsThe problem to decide if a given Diophantine equation has or not integer solutions is, according toChurch-Turing computability thesis, an undecidable problem. In fact, this problem is related to the halting problem for Turing machine: The Diophantine equation problem could be solved if and only if the Turing halting problem could be [10]. Since there is not a universal process to decide if a given program will halt or not a Turing machine, the best that one can do is to run the program and wait for a halt. The point is: one can never know if he/she waited enough time in order to observe the halt. The Diophantine equation problem has a similar statement. Since there is not a universal procedure to decide if a given Diophantine equation accepts an integer solution, the best that one can do in order to check if it accepts or not an integer solution, is to test the integer numbers. The point here is: one can never know the answer because there are infinite integers to be tested. Associated to the halting problem is the Chaitin number , the probability of a given program to halt the Turing machine [11-17]. The number is an incompressible number and it can not be calculated. Similarly, one can define the probability of finding a Diophantine equation with integer solutions, D . Obviously, D is also a number that can not be calculated, because the knowledge of D implies in the knowledge of a way to decide if any given Diophantine equation accepts or not any integer solution that, by its turn, implies in the knowledge of and in the solution of the halting problem. As happens with , it is possible to calculate classically a lower bound for D , however, it must be a brute-force algorithm. Classical algorithms for calculation of D will differ basically in the method that the zeros of the Diophantine equation are obtained. The quantum algorithm here proposed is also a brute-force algorithm but its advantage is the quantum parallelism, since several Diophantine equations are tested simultaneously.4. ConclusionsIt was proposed a new quantum algorithm for finding the minimum of a function that requires only one measurement. This is important since the complete circuit is simplified and measurements of qubits are not noise free. This algorithm is an asymptotic quantum search. As an example of its application, we constructed a quantum algorithm for finding a lower bound for the probability of finding a Diophantine equation that accepts integer solutions. A lower bound can also be calculated using a classical computer,however, our algorithm is more efficient since it can test 2N(for N qubits) Diophantine equations simultaneously. At last, the algorithm proposed requires only one measurement, after the quantum counting stage.AcknowledgeThe authors thank the anonymous reviewer for useful comments.References1.Grover, L. V.: A fast quantum mechanical algorithm for database search, in Proc., 28th Annual ACM Symposium on the Theory of Computing, New York, pp. 212 (1996).2. Grover, L. V.: Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett., 79, pp. 325 (1997).3. M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge, ch. 4 and 6 (2000).4. Dürr C., and Høyer, P.: A quantum algorithm for finding the minimum, arXive:quant-ph/9607014 (1999).5. Ahuja, A., and Kapoor, S.: A quantum algorithm for finding the maximum, arXive:quant-ph/9911082 (1999).6. Boyer, M., Brassard, G., Høyer, P., and Tapp, A.: Tight bounds on quantum searching, arXive:quant-ph/9605034 (1996).7. Kowada, L. A. B., Lavor, C., Portugal, R., and de Figueiredo, C. M. H.: A new quantum algorithm to solve the minimum searching problem, International Journal of Quantum Information, Vol. 6, no. 3, 427 - 436, 2008.8. Kaye, P., Laflamme, R., and Mosca, M.: An introduction to Quantum Computing, 1st Ed., Oxford University Press, 2007.9. Oliveira, D. S., and Ramos, R. V.: Quantum Bit String Comparator: Circuits and Applications, Quantum Computers and Computing, vol. 7, no. 1, 17-26, 2008.10. T. D. Kieu, Quantum algorithms for Hilbert’s tenth problem, – quant-ph/0110136, 2003.11. C. Calude, Theories of Computational Complexity, North-Holland, Amsterdam, 1998.12. G. Chaitin, Meta Math!: the quest for omega, Pantheon Books, 2005.13. G. Chaitin, Irreducible Complexity in Pure Mathematics, Preprint at (/math.HO/0411091), 2004.14. G. Chaitin, Information-theoretic computational complexity, IEEE Trans. on Inf. Theory, IT-20, 10, 1974.15. C. Calude, Information and Randomness, Spring-Verlag, Berlin, 1994.16. G. Chaitin, Algorithmic Information Theory,1a Ed. Cambridge University Press, 1987.17. G. Markowsky, Introduction to algorithmic information theory, J. of Universal Comp. Science, 2, no. 5, 245, 1996.。

量子信息和量子纠缠理论

Multipartite Schmidt-correlated State
Fully separable
PPT
Fully separable (maximally entangled)
~ 1 (N)
M.J. Zhao, S.M. Fei and Z.X. Wang, Phys. Lett. A 372(2008)2552
S. Albeverio, S.M. Fei, Phys. Lett. A 276(2000)8 S. Albeverio, S.M. Fei and W.L. Yang, Comm. Theor. Phys. 38 (2002) 301
S. Albeverio, S.M. Fei and W.L. Yang, Phys. Rev. A 66 (2002) 012301 M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. A 60, 1888 (1999)
Separable! Separable！
Separability of mixed states: no general criteria a) Peres (PPT) criterion:
Peres PRL 77, 1413 (1996)
2x2, 2x3:
PPT
Separable
Horodeckis, Phys. Lett. A 223,1 (1996)
Caltech (Kimble et al)
/~qoptics/teleport.html
Nature 390(1997) 575
Science 282(1998) 706
Wigner functions before &after

量子相变中的量子纠缠度量研究-英文版

RG flow
• c-theorem:
cUV cIR
fixed point
(Zamolodchikov, 1986)
UV
fixed point
IR
• Massive theory (off critical) Block entropy saturation
fixed point
RG flow
UV
Irreversibility of RG trajectories
Loss of entanglement
Open Systems & Quantum Information Milano, 10 Marzo 2006
• Local Entropy: when the subsystem A is a single site.
• The concurrence measures the entanglement between two sites after having traced out the remaining sites. • The transition is signaled by the first derivative of the concurrence, which diverges logarithmically (specific heat).

h 1 lnh 1
S1 has the same singularity as
Local (single site) entropy: 1 ρ1 = I + σ z σ z S1 = Tr ρ1lnρ1 2
σz

Local measures of entanglement based on the 2-site density matrix depend on 2-point functions

物理化学英语词汇

道尔顿定律Dalton law
道尔顿分压定律Dalton partial pressure law
德拜效应Debye effect
德拜立方公式Debye cubic formula
德拜－休克尔极限公式Debye-Huckel’s limiting equation
等焓过程isenthalpic process
电池电动势electromotive force of cells
电池反应cell reaction
电导conductance
电导率conductivity
电动势的温度系数temperature coefficient of electromotive force
电动电势zeta potential
标准摩尔反应焓standard molar reaction enthalpy
标准摩尔反应吉布斯函数standard Gibbs function of molar reaction
标准摩尔反应熵standard molar reaction entropy
标准摩尔焓函数standard molar enthalpy function
表面活性剂surfactants
表面吸附量surface excess
表面张力surface tension
表面质量作用定律surface mass action law
波义尔定律Boyle law
波义尔温度Boyle temperature
波义尔点Boyle point
玻尔兹曼常数Boltzmann constant
物理化学英语词汇
BET公式BET formula
pVT性质pVT property
ζ电势zeta potential

Kagome Spin Liquid戈薇的自旋液体

Triangular
Second Renormalization
Kagome
h (corr) 12 0.081
S 1S 2l1b ˆl2c ˆ
0.005 S1S2l1l2
- 0.039 S1 S2 ly1ly2
0.112 S1 S2
0.1 S 1S 2l 2b ˆ l 1c ˆ
n,n ~n
~n Z 1n P n n'n ~n' ~n' n
Step I: Divide lattice to disjoint blocks. Diagonalize H on each Block.
block excitations are the ''atoms'' (composite particles)
Truncate: M lowest states per block
• The triangular lattice flows to chiral symmetry breaking, probably the 3 sublattice Neel phase.
• Future: Investigations of the quantum phase transition in the effective Hamiltonian by following the RG flow.
Effective Bond Interactions
h( corr) 12

0.953
S 1S 2l1b ˆl2c ˆ

0.2111 S1S2l1l2
0.053 S1 S2 ly1ly2
0.1079 S1 S2

量子力学索引英汉对照

21-centimeter line, 21厘米线AAbsorption, 吸收Addition of angular momenta, 角动量叠加Adiabatic approximation, 绝热近似Adiabatic process, 绝热过程Adjoint, 自伴的Agnostic position, 不可知论立场Aharonov-Bohm effect, 阿哈罗诺夫-玻姆效应Airy equation, 艾里方程；Airy function, 艾里函数Allowed energy, 允许能量Allowed transition, 允许跃迁Alpha decay, 衰变；Alpha particle, 粒子Angular equation, 角向方程Angular momentum, 角动量Anomalous magnetic moment, 反常磁矩Antibonding, 反键Anti-hermitian operator, 反厄米算符Associated Laguerre polynomial, 连带拉盖尔多项式Associated Legendre function, 连带勒让德多项式Atoms, 原子Average value, 平均值Azimuthal angle, 方位角Azimuthal quantum number, 角量子数BBalmer series, 巴尔末线系Band structure, 能带结构Baryon, 重子Berry's phase, 贝利相位Bessel functions, 贝塞尔函数Binding energy, 束缚能Binomial coefficient, 二项式系数Biot-Savart law, 毕奥-沙法尔定律Blackbody spectrum, 黑体谱Bloch's theorem, 布洛赫定理Bohr energies, 玻尔能量；Bohr magneton, 玻尔磁子；Bohr radius, 玻尔半径Boltzmann constant, 玻尔兹曼常数Bond, 化学键Born approximation, 玻恩近似Born's statistical interpretation, 玻恩统计诠释Bose condensation, 玻色凝聚Bose-Einstein distribution, 玻色-爱因斯坦分布Boson, 玻色子Bound state, 束缚态Boundary conditions, 边界条件Bra, 左矢Bulk modulus, 体积模量CCanonical commutation relations, 正则对易关系Canonical momentum, 正则动量Cauchy's integral formula, 柯西积分公式Centrifugal term, 离心项Chandrasekhar limit, 钱德拉赛卡极限Chemical potential, 化学势Classical electron radius, 经典电子半径Clebsch-Gordan coefficients, 克-高系数Coherent States, 相干态Collapse of wave function, 波函数塌缩Commutator, 对易子Compatible observables, 对易的可观测量Complete inner product space, 完备内积空间Completeness, 完备性Conductor, 导体Configuration, 位形Connection formulas, 连接公式Conservation, 守恒Conservative systems, 保守系Continuity equation, 连续性方程Continuous spectrum, 连续谱Continuous variables, 连续变量Contour integral, 围道积分Copenhagen interpretation, 哥本哈根诠释Coulomb barrier, 库仑势垒Coulomb potential, 库仑势Covalent bond, 共价键Critical temperature, 临界温度Cross-section, 截面Crystal, 晶体Cubic symmetry, 立方对称性Cyclotron motion, 螺旋运动DDarwin term, 达尔文项de Broglie formula, 德布罗意公式de Broglie wavelength, 德布罗意波长Decay mode, 衰变模式Degeneracy, 简并度Degeneracy pressure, 简并压Degenerate perturbation theory, 简并微扰论Degenerate states, 简并态Degrees of freedom, 自由度Delta-function barrier, 势垒Delta-function well, 势阱Derivative operator, 求导算符Determinant, 行列式Determinate state, 确定的态Deuterium, 氘Deuteron, 氘核Diagonal matrix, 对角矩阵Diagonalizable matrix, 对角化Differential cross-section, 微分截面Dipole moment, 偶极矩Dirac delta function, 狄拉克函数Dirac equation, 狄拉克方程Dirac notation, 狄拉克记号Dirac orthonormality, 狄拉克正交归一性Direct integral, 直接积分Discrete spectrum, 分立谱Discrete variable, 离散变量Dispersion relation, 色散关系Displacement operator, 位移算符Distinguishable particles, 可分辨粒子Distribution, 分布Doping, 掺杂Double well, 双势阱Dual space, 对偶空间Dynamic phase, 动力学相位EEffective nuclear charge, 有效核电荷Effective potential, 有效势Ehrenfest's theorem, 厄伦费斯特定理Eigenfunction, 本征函数Eigenvalue, 本征值Eigenvector, 本征矢Einstein's A and B coefficients, 爱因斯坦A,B系数；Einstein's mass-energy formula, 爱因斯坦质能公式Electric dipole, 电偶极Electric dipole moment, 电偶极矩Electric dipole radiation, 电偶极辐射Electric dipole transition, 电偶极跃迁Electric quadrupole transition, 电四极跃迁Electric field, 电场Electromagnetic wave, 电磁波Electron, 电子Emission, 发射Energy, 能量Energy-time uncertainty principle, 能量-时间不确定性关系Ensemble, 系综Equilibrium, 平衡Equipartition theorem, 配分函数Euler's formula, 欧拉公式Even function, 偶函数Exchange force, 交换力Exchange integral, 交换积分Exchange operator, 交换算符Excited state, 激发态Exclusion principle, 不相容原理Expectation value, 期待值FFermi-Dirac distribution, 费米-狄拉克分布Fermi energy, 费米能Fermi surface, 费米面Fermi temperature, 费米温度Fermi's golden rule, 费米黄金规则Fermion, 费米子Feynman diagram, 费曼图Feynman-Hellman theorem, 费曼-海尔曼定理Fine structure, 精细结构Fine structure constant, 精细结构常数Finite square well, 有限深方势阱First-order correction, 一级修正Flux quantization, 磁通量子化Forbidden transition, 禁戒跃迁Foucault pendulum, 傅科摆Fourier series, 傅里叶级数Fourier transform, 傅里叶变换Free electron, 自由电子Free electron density, 自由电子密度Free electron gas, 自由电子气Free particle, 自由粒子Function space, 函数空间Fusion, 聚变Gg-factor, g-因子Gamma function, 函数Gap, 能隙Gauge invariance, 规范不变性Gauge transformation, 规范变换Gaussian wave packet, 高斯波包Generalized function, 广义函数Generating function, 生成函数Generator, 生成元Geometric phase, 几何相位Geometric series, 几何级数Golden rule, 黄金规则"Good" quantum number, "好"量子数"Good" states, "好"的态Gradient, 梯度Gram-Schmidt orthogonalization, 格莱姆-施密特正交化法Graphical solution, 图解法Green's function, 格林函数Ground state, 基态Group theory, 群论Group velocity, 群速Gyromagnetic railo, 回转磁比值HHalf-integer angular momentum, 半整数角动量Half-life, 半衰期Hamiltonian, 哈密顿量Hankel functions, 汉克尔函数Hannay's angle, 哈内角Hard-sphere scattering, 硬球散射Harmonic oscillator, 谐振子Heisenberg picture, 海森堡绘景Heisenberg uncertainty principle, 海森堡不确定性关系Helium, 氦Helmholtz equation, 亥姆霍兹方程Hermite polynomials, 厄米多项式Hermitian conjugate, 厄米共轭Hermitian matrix, 厄米矩阵Hidden variables, 隐变量Hilbert space, 希尔伯特空间Hole, 空穴Hooke's law, 胡克定律Hund's rules, 洪特规则Hydrogen atom, 氢原子Hydrogen ion, 氢离子Hydrogen molecule, 氢分子Hydrogen molecule ion, 氢分子离子Hydrogenic atom, 类氢原子Hyperfine splitting, 超精细分裂IIdea gas, 理想气体Idempotent operaror, 幂等算符Identical particles, 全同粒子Identity operator, 恒等算符Impact parameter, 碰撞参数Impulse approximation, 脉冲近似Incident wave, 入射波Incoherent perturbation, 非相干微扰Incompatible observables, 不对易的可观测量Incompleteness, 不完备性Indeterminacy, 非确定性Indistinguishable particles, 不可分辨粒子Infinite spherical well, 无限深球势阱Infinite square well, 无限深方势阱Inner product, 内积Insulator, 绝缘体Integration by parts, 分部积分Intrinsic angular momentum, 内禀角动量Inverse beta decay, 逆衰变Inverse Fourier transform, 傅里叶逆变换KKet, 右矢Kinetic energy, 动能Kramers' relation, 克莱默斯关系Kronecker delta, 克劳尼克LLCAO technique, 原子轨道线性组合法Ladder operators, 阶梯算符Lagrange multiplier, 拉格朗日乘子Laguerre polynomial, 拉盖尔多项式Lamb shift, 兰姆移动Lande g-factor, 朗德g-因子Laplacian, 拉普拉斯的Larmor formula, 拉摩公式Larmor frequency, 拉摩频率Larmor precession, 拉摩进动Laser, 激光Legendre polynomial, 勒让德多项式Levi-Civita symbol, 列维-西维塔符号Lifetime, 寿命Linear algebra, 线性代数Linear combination, 线性组合Linear combination of atomic orbitals, 原子轨道的线性组合Linear operator, 线性算符Linear transformation, 线性变换Lorentz force law, 洛伦兹力定律Lowering operator, 下降算符Luminoscity, 照度Lyman series, 赖曼线系MMagnetic dipole, 磁偶极Magnetic dipole moment, 磁偶极矩Magnetic dipole transition, 磁偶极跃迁Magnetic field, 磁场Magnetic flux, 磁通量Magnetic quantum number, 磁量子数Magnetic resonance, 磁共振Many worlds interpretation, 多世界诠释Matrix, 矩阵；Matrix element, 矩阵元Maxwell-Boltzmann distribution, 麦克斯韦-玻尔兹曼分布Maxwell's equations, 麦克斯韦方程Mean value, 平均值Measurement, 测量Median value, 中位值Meson, 介子Metastable state, 亚稳态Minimum-uncertainty wave packet, 最小不确定度波包Molecule, 分子Momentum, 动量Momentum operator, 动量算符Momentum space wave function, 动量空间波函数Momentum transfer, 动量转移Most probable value, 最可几值Muon, 子Muon-catalysed fusion, 子催化的聚变Muonic hydrogen, 原子Muonium, 子素NNeumann function, 纽曼函数Neutrino oscillations, 中微子振荡Neutron star, 中子星Node, 节点Nomenclature, 术语Nondegenerate perturbationtheory, 非简并微扰论Non-normalizable function, 不可归一化的函数Normalization, 归一化Nuclear lifetime, 核寿命Nuclear magnetic resonance, 核磁共振Null vector, 零矢量OObservable, 可观测量Observer, 观测者Occupation number, 占有数Odd function, 奇函数Operator, 算符Optical theorem, 光学定理Orbital, 轨道的Orbital angular momentum, 轨道角动量Orthodox position, 正统立场Orthogonality, 正交性Orthogonalization, 正交化Orthohelium, 正氦Orthonormality, 正交归一性Orthorhombic symmetry, 斜方对称Overlap integral, 交叠积分PParahelium, 仲氦Partial wave amplitude, 分波幅Partial wave analysis, 分波法Paschen series, 帕邢线系Pauli exclusion principle, 泡利不相容原理Pauli spin matrices, 泡利自旋矩阵Periodic table, 周期表Perturbation theory, 微扰论Phase, 相位Phase shift, 相移Phase velocity, 相速Photon, 光子Planck's blackbody formula, 普朗克黑体辐射公式Planck's constant, 普朗克常数Polar angle, 极角Polarization, 极化Population inversion, 粒子数反转Position, 位置；Position operator, 位置算符Position-momentum uncertainty principles, 位置-动量不确定性关系Position space wave function, 坐标空间波函数Positronium, 电子偶素Potential energy, 势能Potential well, 势阱Power law potential, 幂律势Power series expansion, 幂级数展开Principal quantum number, 主量子数Probability, 几率Probability current, 几率流Probability density, 几率密度Projection operator, 投影算符Propagator, 传播子Proton, 质子QQuantum dynamics, 量子动力学Quantum electrodynamics, 量子电动力学Quantum number, 量子数Quantum statics, 量子统计Quantum statistical mechanics, 量子统计力学Quark, 夸克RRabi flopping frequency, 拉比翻转频率Radial equation, 径向方程Radial wave function, 径向波函数Radiation, 辐射Radius, 半径Raising operator, 上升算符Rayleigh's formula, 瑞利公式Realist position, 实在论立场Recursion formula, 递推公式Reduced mass, 约化质量Reflected wave, 反射波Reflection coefficient, 反射系数Relativistic correction, 相对论修正Rigid rotor, 刚性转子Rodrigues formula, 罗德里格斯公式Rotating wave approximation, 旋转波近似Rutherford scattering, 卢瑟福散射Rydberg constant, 里德堡常数Rydberg formula, 里德堡公式SScalar potential, 标势Scattering, 散射Scattering amplitude, 散射幅Scattering angle, 散射角Scattering matrix, 散射矩阵Scattering state, 散射态Schrodinger equation, 薛定谔方程Schrodinger picture, 薛定谔绘景Schwarz inequality, 施瓦兹不等式Screening, 屏蔽Second-order correction, 二级修正Selection rules, 选择定则Semiconductor, 半导体Separable solutions, 分离变量解Separation of variables, 变量分离Shell, 壳Simple harmonic oscillator, 简谐振子Simultaneous diagonalization, 同时对角化Singlet state, 单态Slater determinant, 斯拉特行列式Soft-sphere scattering, 软球散射Solenoid, 螺线管Solids, 固体Spectral decomposition, 谱分解Spectrum, 谱Spherical Bessel functions, 球贝塞尔函数Spherical coordinates, 球坐标Spherical Hankel functions, 球汉克尔函数Spherical harmonics, 球谐函数Spherical Neumann functions, 球纽曼函数Spin, 自旋Spin matrices, 自旋矩阵Spin-orbit coupling, 自旋-轨道耦合Spin-orbit interaction, 自旋-轨道相互作用Spinor, 旋量Spin-spin coupling, 自旋-自旋耦合Spontaneous emission, 自发辐射Square-integrable function, 平方可积函数Square well, 方势阱Standard deviation, 标准偏差Stark effect, 斯塔克效应Stationary state, 定态Statistical interpretation, 统计诠释Statistical mechanics, 统计力学Stefan-Boltzmann law, 斯特番-玻尔兹曼定律Step function, 阶跃函数Stem-Gerlach experiment, 斯特恩-盖拉赫实验Stimulated emission, 受激辐射Stirling's approximation, 斯特林近似Superconductor, 超导体Symmetrization, 对称化Symmetry, 对称TTaylor series, 泰勒级数Temperature, 温度Tetragonal symmetry, 正方对称Thermal equilibrium, 热平衡Thomas precession, 托马斯进动Time-dependent perturbation theory, 含时微扰论Time-dependent Schrodinger equation, 含时薛定谔方程Time-independent perturbation theory, 定态微扰论Time-independent Schrodinger equation, 定态薛定谔方程Total cross-section, 总截面Transfer matrix, 转移矩阵Transformation, 变换Transition, 跃迁；Transition probability, 跃迁几率Transition rate, 跃迁速率Translation,平移Transmission coefficient, 透射系数Transmitted wave, 透射波Trial wave function, 试探波函数Triplet state, 三重态Tunneling, 隧穿Turning points, 回转点Two-fold degeneracy , 二重简并Two-level systems, 二能级体系UUncertainty principle, 不确定性关系Unstable particles, 不稳定粒子VValence electron, 价电子Van der Waals interaction, 范德瓦尔斯相互作用Variables, 变量Variance, 方差Variational principle, 变分原理Vector, 矢量Vector potential, 矢势Velocity, 速度Vertex factor, 顶角因子Virial theorem, 维里定理WWave function, 波函数Wavelength, 波长Wave number, 波数Wave packet, 波包Wave vector, 波矢White dwarf, 白矮星Wien's displacement law, 维恩位移定律YYukawa potential, 汤川势ZZeeman effect, 塞曼效应。

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a r X i v :q u a n t -p h /0611294v13N ov26Quantum Lower Bounds by Entropy Numbers Stefan Heinrich Department of Computer Science University of Kaiserslautern D-67653Kaiserslautern,Germany e-mail:heinrich@informatik.uni-kl.de homepage:http://www.uni-kl.de/AG-Heinrich Abstract We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the n -th minimal error in the quantum setting of information-based complexity.As an application,we improve some lower bounds on quantum approximation of embed-dings between ﬁnite dimensional L p spaces and of Sobolev embeddings.1Introduction There is one major technique for proving lower bounds in the quantum set-ting of IBC (information-based complexity)as introduced in [5].It uses the polynomial method [1]together with a result on approximation by polyno-mials from [14].This method has been applied in [5],[9],[19].Other papers on the quantum complexity of continuous problems use this implicitly by reducing mean computation to the problem in consideration and then using the lower bound for mean computation of [14]directly ([15],[18],[11],[16]).This approach,however,does not work for the case of approximation of embedding operators in spaces with norms diﬀerent from the inﬁnity norm.To settle such situations,a more sophisticated way of reduction to known bounds was developed in [6],based on a multiplicativity property of the n -th minimal quantum error.In this paper we introduce an approach which is new for the IBC quan-tum setting.We again use the polynomial method of [1],but combine it with methods related to entropy [4].We derive lower bounds for the n -th mini-mal quantum error in terms of certain entropy numbers.Similar ideas have been applied before in [17],the model and methods however being diﬀerent,1see also related work[13].As an application,we improve the lower bounds [6,7]on approximation as well as those of[8]by removing the logarithmic factors.Let us also mention that a modiﬁcation of the polynomial method based on trigonometric polynomials was used in[2,3]for proving lower bounds for a type of query diﬀerent from that introduced in[5],the so-called power-query[16].Our method can also be applied in this setting and simpliﬁes the analysis from[2,3].We comment on this at the end of the paper.2Lower bounds by entropyWe work in the quantum setting of IBC as introduced in[5].We refer to this paper for the needed notions.Let D and K be nonempty sets, let F(D,K)denote the set of all functions on D with values in K,let F⊆F(D,K)be nonempty,and let G be a normed linear space.Let A be a quantum algorithm from F to G.For any subset C⊆G deﬁne the function p C:F→R byp C(f)=P{A(f)∈C}(f∈F)–the probability that the output of algorithm A at input f belongs to C. This quantity is well-deﬁned for all subsets C since the output of A takes onlyﬁnitely many values,see[5].Furthermore,deﬁneP A,F=span{p C:C⊆G}⊆F(F,R)to be the linear span of the functions p C.We need some notions related to entropy.We refer to[4]for the deﬁni-tions.For a nonempty subset W of a normed space G and k∈N(we use the notation N={1,2,...}and N0={0,1,2,...})deﬁne the k-th inner entropy number asϕk(W,G)=sup{ε:there exist u1,...,u k+1∈W such thatu i−u j ≥2εfor all1≤i=j≤k+1}.(1) It is worth while mentioning a related notion.The k-th entropy number is deﬁned to beεk(W,G)=inf ε:there exist g1,...,g k∈G such thatg−g i G≤εfor all g∈W .(2)min1≤i≤kThenϕk(W,G)≤εk(W,G)≤2ϕk(W,G),(3)2see[4],relations(1.1.3)and(1.1.4).Also observe that theﬁrst numbers of both types are related to the radius and diameter of W as follows:1ϕ1(W,G)=2 (i=1,...,k+1)(11) are disjoint.It follows from(10)and(11)that for i=1,...,k+13P{A(f i)∈V i}≥Let usﬁrst complete the proof of(ii):If A has no queries,its output does not depend on f∈F,and in particular,the distribution of the random variables A(f1)and A(f2)is the same.But then(12)impliesP{A(f1)∈V1∩V2}≥1/2,thus V1∩V2=∅,a contradiction,which proves(9)in case(ii).Now we deal with case(i).Let C be the set of all C⊂G of the formC= i∈I V i,with I being any subset of{1,...,k+1}.Clearly,|C|=2k+1.(13) Let P A,F be endowed with the supremum normp ∞=supf∈F|p(f)|.We havep C ∞≤1(C∈C).(14) Moreover,p C1−p C2∞≥14, whilep C2(f i)=P{A(f i)∈C2}≤P{A(f i)∈G\V i}≤12,implying(15).For p∈P A,F let B(p,r)be the closed ball of radius r around p in P A,F.By(15)the balls B(p C,1/4)have disjoint interior for C∈C. Moreover,by(14), C∈C B(p C,1/4)⊆B(0,5/4).4A volume comparison gives2k+1=|C|≤5dim P A,F,hence,taking logarithms,we get a contradiction to(5),which completes the proof.Let e q n(S,F)denote the n-th minimal quantum error,that is,the inﬁmum of e(S,A,F)taken over all quantum algorithms A from F to G with at most n queries(see[5]).As an immediate consequence of Lemma1,and also for later use,we note the following.Corollary1.1Now we are ready to state the new lower bound on the n -th minimal quantum error.Proposition 1.Let D,K be nonempty sets,let F ⊆F (D,K )be a nonempty set of functions,G a normed space,S :F →G a mapping,and L ∈N .Suppose L =(f u )u ∈{0,1}L ⊆F (D,K )is a system of functions satisfying condition (I).Thene q n (S,F )≥ϕk (S (F ∩L ),G )whenever k,n ∈N satisfy 2n ≤L andk +1>(log 25) eL2n 2n (19)(see,e.g.,[12],(4.7)on p.122,for the inequality).SetU ={u ∈{0,1}L :f u ∈F }and let M L,2n (U )denote the space of all restrictions of functions from M L,2n to U .Clearly,dim M L,2n (U )≤dim M L,2n .(20)DeﬁneΨ:P A,F ∩L →F (U,R )by setting for p ∈P A,F ∩L and u ∈U(Ψp )(u )=p (f u ).Obviously,Ψis linear,moreover,for C ⊆G(Ψp C )(u )=p C (f u )(u ∈U ).6By Lemma2,p C(f u),as a function of u∈U,is the restriction of an element of M L,2n to U.Hence,Ψp C∈M L,2n(U),and by linearity and the deﬁnition of P A,F∩L as the linear span of functions p C,we getΨ(P A,F∩L)⊆M L,2n(U).Furthermore,Ψis one-to-one,since{f u:u∈U}=F∩ing(19)and (20)it follows thatdim P A,F∩L≤dim M L,2n(U)≤ eLNNi=1|f(i)|p 1/pif p<∞,f L N∞=max1≤i≤N|f(i)|,and let B(L N p)be its unit ball.Deﬁne J N pq:L N p→L N q to be the identity operator J N pq f=f(f∈L N p).As already mentioned,the lower bound for approximation of J N pq was ob-tained using a multiplicativity property of the n-th minimal quantum error ([6],Proposition1).The result involved some logarithmic factors of negative power([6],Proposition6).Based on Proposition1above we improve this bound by removing the logarithmic factors.Proposition2.Let1≤p,q≤∞.There is a constant c>0such that for all n∈N0,N∈N with n≤cNe q n(J N p,q,B(L N p))≥1Proof.It suﬃces to prove the case p=∞,q=1.We put L=N and f u=u for u∈{0,1}N.Clearly,the system L=(f u)u∈{0,1}N satisﬁes condition(I) andL⊂B(L N∞).(21) Let{f ui:1≤i≤k+1}be a maximal system withf ui −f ujL N1≥14 .On the other hand,2N≤k+1i=1 u∈{0,1}N: f u−f u i L N1<14log2(48.(24) Consider the function g:(0,1]→R,g(x)=x log2e+log212(0<x≤c2).(25) Now putc=min c12,1and assumen ≤cN.(27)If n =0,Corollary 1gives e q 0(J N∞,1,B(L N ∞))= J N ∞,1 =1.(28)Hence we can suppose that n ≥1,which,by (27),implies N ≥c −1.Conse-quently,from (26),log 2log 252.(29)Since by (26)and (27),2n/N ≤2c ≤c 2,we get from (25)2n 2n<c 1N +2n2n <c 1.(31)This implies,using also (23),(log 25)eN 8.Using Proposition 2we can also remove the logarithmic factors in anotherlower bound –for Sobolev embeddings J pq :W r p ([0,1]d )→L q([0,1]d ),see [7]for the notation and Proposition 2of that paper for the previous result.The following can be derived from Proposition 2using the same argument as in [7],p.43,relations (87)and (88).Corollary 2.Let 1≤p,q ≤∞,r,d ∈N ,and assume r p ,2q .Then there is a constant c >0such that for all n ∈Ne q n (J pq ,B (W r p ([0,1]d )))≥cn−r/d .9Furthermore,the lower bounds from[6]were also used in[8],Proposition 3and ing Proposition2,these results can be improved in the respective way,too.We omit the details.Let usﬁnally comment on lower bounds for power queries introduced in [16].An inspection of the proof of Lemma1shows that the type of query is not used at all in the argument,so the statement also holds for power queries.One part of the argument in both[2,3]consists of proving that for a quantum algorithm with at most n power queries and for a suitable subset F0⊆F,which can be identiﬁed with the interval[0,1],the respective space is contained in the(complex)linear span of functions e2πiαt(t∈[0,1]), P A,Fwith frequenciesαfrom a set of cardinality not greater than c n for some≤2c n.Moreover,also 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