QI_problem_set_1

Problem Set 1

Due 10th Oct.&12th Oct.

1求二体系统量子态|ψ?12=a |H 1?|H 2?+b |V 1?|V 2?的约化密度矩阵ρ1。

2计算二元对称信道的信道容量。

3空间H 中存在两组正交归一化态{|ψi ?}，{ ?ψi

?},则存在幺正变换U ，使得U |ψi ?= ?ψi ?，试构造出该U 变换。

4空间H 中存在两组归一化态{|ψi ?}、{ ?ψi ?}，它们满足:?i,j,有?ψi |ψj ?=??ψi ?ψj ?.请证明，则存在U ，使得U |ψi ?= ?ψi ?，并构造出该U 变换。

5对两比特态

|??=1√2

|0?A (12|0?B +√32|1?B )+1√2|1?A (√32|0?B +12|1?B )i)求约化密度矩阵ρA ,ρB ;ii)求|??的Schmidt 分解形式。

6对三粒子系统纯态|?ABC ?，在空间H A ?H B ?H C 中是否存在H A ,H B ,H C 中的正交基{|i A ?},{|i B ?},{|i C ?}，使得|?ABC ?=∑i

√p i |i A ??|i B ??|i C ?一定成立?给出理由。7设|ψ?为量子比特态，在Bloch 球面上均匀随机分布。i)随机地猜想一个态|??，求猜测态相对于|ψ?的平均保真度ˉF =?|??|ψ?|2?ii)对此量子态做正交测量ˉF =?|??|ψ?|2?。测量后系统被制备到:ρ=P ↑?ψ|P ↑|ψ?+P ↓?ψ|P ↓|ψ?，求ρ与原来的态|ψ?的平均保真度。(ˉF

=)

补充题:

8Consider the state

|ψ?AB=1

2(√

2|00?+|01?+|11?

)

FindρB=T r A|ψ??ψ|andρA=T r B|ψ??ψ|.Show that these matrices have the same eigenvalues by showing T rρ2A=T rρ2B.What are these eigenvalues?

9a The quantum state

cos θ

2

|0?+e i?sinθ

2

|1?

corresponds to the point on the Bloch sphere

j=(sinθcos?,sinθsin?,cosθ)

Show that the operator

σj=j xσx+j yσy+j zσz

has eigenvectors which correspond to the points±j on the Bloch sphere.

9b Show that antipodal points on the Bloch sphere are orthogonal quantum states.

9c Show that if the vectors j and k are perpendicular,thenσj andσk anticommute.

9d Show that applyingσj to a quantum state|ψ?rotates|ψ?by the angleπaround the j-axis on the Bloch sphere.

Hint:one way to do it is use1a,1b,1c.(There are more straightforward,although calculation-intensive,ways to do it,so this hint may not be of much use).