Refining upper bounds of the measurement-induced nonbilocal correlation in the Swapping experiment

ZHANG Ying; GUO Zhihua; HE Kan

doi:10.15983/j.cnki.jsnu.2023025

Journal of Shaanxi Normal University(Natural Science Edition) >

2023 , Vol. 51 >Issue 5: 43 - 48

DOI: https://doi.org/10.15983/j.cnki.jsnu.2023025

Refining upper bounds of the measurement-induced nonbilocal correlation in the Swapping experiment

ZHANG Ying ¹ ,
GUO Zhihua ² ,
HE Kan ^,¹^,^*

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¹ College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, Shanxi, China
² School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, Shaanxi, China

Received date: 2022-02-18

Online published: 2023-09-28

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Abstract

Measurement-induced nonbilocal correlation is a measure which quantifies the nonlocality in the entanglement-swapping experiment under local measurements. The measure can be calculated for the pure state, however not for the mixed state. It is obtained that a refined upper bound of the measurement-induced nonbilocal correlation for a special class of mixed states, that is, the marginal state ρ_BC of every state in this class is a product state ρ_BC=ρ_B$\otimes$ρ_C, ρ_B is non-degenerate and ρ_C is degenerate.

Key words： measurement-induced nonbilocal correlation; upper bounds of measures; quantum nonlocality; quantum correlations

Cite this article

ZHANG Ying , GUO Zhihua , HE Kan . Refining upper bounds of the measurement-induced nonbilocal correlation in the Swapping experiment[J]. Journal of Shaanxi Normal University(Natural Science Edition), 2023 , 51(5) : 43 -48 . DOI: 10.15983/j.cnki.jsnu.2023025

量子非局域性是一个量子力学的基本概念,它被广泛应用于量子信息处理任务中,如减少通信复杂度、量子密钥分配、私密随机数生成以及与设备无关的纠缠见证^{[1⇓⇓⇓⇓-6]}。从概念上讲,非局域性是一种不能被任何局域隐变量理论所描述的关联,它与纠缠和量子特性等其他现象密切相关,但又有所区别^[7]。学者们通过贝尔不等式的方法对非局域性进行了广泛研究^{[8⇓⇓⇓⇓-13]}。但仅通过贝尔不等式的角度去研究非局域性显然是不够的,这是因为存在着无量子特性的非局域性^[14]和无纠缠的非局域性^{[15⇓⇓-18]}。Luo等^[19]提出了测量诱导的非局域(measurement-induced nonlocality, MIN),并得到了任意维纯态和2×n维混合态的解析解。Mirafzali等^[20]研究了任意m×n维两体态的MIN值,获得了MIN的上界计算公式,该上界比文献[19]中的上界更便于计算且更精细。

之后,量子非局域性的研究延伸到量子网络实验上。在量子网络实验中有多个观察者和多个源^{[21⇓⇓⇓⇓⇓⇓⇓⇓⇓-31]},最简单的例子是涉及3个观察者和2个源的量子Swapping实验。如图1所示,Alice、Bob、Charles是观测者,且具有观测算子组{A_x}_x_=0,1、{B_y}_y_=0,1、{C_z}_z_=0,1, 源S₁发送一个两体系统ρ_AB,源S₂发送一个两体系统ρ_CD,注意Bob的观测算子是作用在ρ_BC这个两体系统上的。

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图1 双局域场景

Fig.1 Scenario of bilocality

为了有效捕获该实验中的非局域关联,Zhang等^[32-33]量化了该场景下的非局域性,提出了测量诱导的非双局域关联(measurement-induced non-bilocal correlation)这一度量概念,给出了纯态情形下该度量的解析解和混合态情形下的一个上界。但是,该上界不便于计算,对于某些混合态,可以继续细化上界的计算公式以减少计算量。本文将对一类混合态给出比文献[32]中测量诱导的非双局域关联的上界更便于计算的一个上界公式。这类特殊态,就是当边缘态ρ_BC=ρ_B$\otimes$ρ_C中ρ_B非简并且ρ_C简并的混合态。

1 测量诱导的非双局域关联

考虑一个有限维的四体系统,对应复希尔伯特空间H=H_A$\otimes$H_B$\otimes$H_C$\otimes$H_D,其中H_A(H_B、H_C、H_D)是粒子A(B、C、D)的希尔伯特空间,且具有维数m(n、u、v)。令D(H)是H上有界、迹为1的半正定算子的集合。给定由A、B共享的量子态ρ_AB∈D(H_A$\otimes$H_B),和由C、D共享的量子态ρ_CD∈D(H_C$\otimes$H_D),定义测量诱导的非双局域关联^[32]为

N_C(ρ_AB$\otimes$ρ_CD)=

(1)$\begin{array}{l}N_{C}\left(\rho_{A B} \otimes \rho_{C D}\right)= \\\quad \max _{\Pi^{B C}}\left\|\rho_{A B} \otimes \rho_{C D}-\Pi^{B C}\left(\rho_{A B} \otimes \rho_{C D}\right)\right\|^{2},\end{array}$

其中最大值取遍不扰动局域态ρ_BC的冯诺依曼测量Π^BC={

Π h B C

},即∑_h

Π h B C

ρ_BC

Π h B C

=ρ_BC,Π^BC(ρ_AB$\otimes$ρ_CD)=∑_h(I^A$\otimes$

Π h B C

$\otimes$I^D)(ρ_AB$\otimes$ρ_CD)(I^A$\otimes$

Π h B C

$\otimes$I^D)。这里,‖·‖是希尔伯特施密特范数,定义为‖X‖≡

t r (X † X)

。令ρ_A和ρ_B是ρ_AB对于A方和B方的约化密度矩阵,ρ_C和ρ_D也同样定义。

记L(H_A)为线性算子的希尔伯特空间,其上的内积定义为<X|Y>≡tr(X^†Y)。在这个空间里,若<X_i|X_j>=δ_ij,则算子族{X_i:i=0,1,…,m²-1}被称为一个正交厄米算子基。令{X_i:i=0,1,…,m²-1}、{Y_j:j=0,1,…,n²-1}、{Z_k:k=0,1,…,u²-1}和{W_l:l=0,1,…,v²-1}分别是L(H_A)、L(H_B)、L(H_C)和L(H_D)上的正交厄米算子基,其中X₀=I^A/

m

,Y₀=I^B/

n

,Z₀=I^C/

u

和W₀=I^D/

v

,则一般的两体态ρ_AB和ρ_CD可被表示为

(2)$\begin{aligned}\rho_{A B}= & \frac{1}{\sqrt{m n}} \frac{I^{A}}{\sqrt{m}} \otimes \frac{I^{B}}{\sqrt{n}}+\sum_{i=1}^{m^{2}-1} x_{i} X_{i} \otimes \frac{I^{B}}{\sqrt{n}}+\frac{I^{A}}{\sqrt{m}} \otimes \\& \sum_{j=1}^{n^{2}-1} y_{j} Y_{j}+\sum_{i=1}^{m^{2}-1} \sum_{j=1}^{n^{2}-1} t_{i j}^{A B} X_{i} \otimes Y_{j}, \\\rho_{C D}= & \frac{1}{\sqrt{u v}} \frac{I^{C}}{\sqrt{u}} \otimes \frac{I^{D}}{\sqrt{v}}+\sum_{k=1}^{u^{2}-1} z_{k} Z_{k} \otimes \frac{I^{D}}{\sqrt{v}}+\frac{I^{C}}{\sqrt{u}} \otimes \\& \sum_{l=1}^{v^{2}-1} w_{l} W_{l}+\sum_{k=1}^{u^{2}-1} \sum_{l=1}^{v^{2}-1} t_{k l}^{C D} Z_{k} \otimes W_{l 。}\end{aligned}$

式中:ρ_AB的关联矩阵是T_AB≡(

t i j A B

);ρ_CD的关联矩阵是T_CD≡(

t k l C D

)。现在假设

T A B ¯

1 m n y T x T A B

和

T C D ¯

1 u v w T z T C D

这里x=(x₁,x₂,…,

x m 2 - 1

)^T。类似地,对y、z和w进行假设,于是有

ρ_AB$\otimes$ρ_CD=

∑ i = 0 m 2 - 1 ∑ j = 0 n 2 - 1 ∑ k = 0 u 2 - 1 ∑ l = 0 v 2 - 1 t i j A B t k l C D

X_i$\otimes$Y_j$\otimes$

Z_k$\otimes$W_l,

ρ_BC_,_AD=

∑ j = 0 n 2 - 1 ∑ k = 0 u 2 - 1 ∑ i = 0 m 2 - 1 ∑ l = 0 v 2 - 1 t i j A B t k l C D

Y_j$\otimes$Z_k$\otimes$

X_i$\otimes$W_l,

式中矩阵

T B C, A D ¯

t i j A B t k l C D

)_jk_,_il=

T A B ¯ T

$\otimes$

T C D ¯

,且ρ_BC_,_AD的关联矩阵是

T B C, A D ¯

去掉第一行和第一列的子矩阵,记为T_BC_,_AD。

对于(2)式中表示的ρ_AB和ρ_CD,文献[32]推导了测量诱导的非双局域关联的上界

N_c(ρ_AB$\otimes$ρ_CD)=tr(T_BC_,_AD

T T B C, A D

m i n G

(tr(GT_BC_,_AD

T T B C, A D

G^T))≤

∑ s = 1 n 2 u 2 - n u

λ_s。

式中:G≡(g_ho)是一个nu×(n²u²-1)维矩阵,且元素为g_ho≡tr

Π h B C

Y_j$\otimes$Z_k(o=ju²+k;j=0,1,…,n²-1;k=0,1,…,u²-1;jk≠0);T_BC_,_AD=(t_jk_,_il)是一个(n²u²-1)×(m²v²-1)维矩阵;{λ_s:s=1,2,…,n²u²-1}是T_BC_,_AD

T T B C, A D

按照降序排列的特征值。

对于Swapping网络中的局域量子态ρ_BC,由于它是一个直积态ρ_BC=ρ_B$\otimes$ρ_C,不失一般性,若ρ_B是非简并且有谱投影(这里指谱分解下的投影算子){|e_B><e_B|},而ρ_C是简并的,则Π^BC={

Π e B

$\otimes$

Π f C

}={|e_B><e_B|$\otimes$

Π f C

},且

(3)$\begin{array}{l}N_{C}\left(\rho_{A B} \otimes \rho_{C D}\right)=\left(\operatorname{tr} \mid\left(\boldsymbol{T}_{A B} \boldsymbol{T}_{A B}^{\mathrm{T}}\right)-\right. \\\left.\operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right)\left(\frac{1}{u v}+\|\boldsymbol{z}\|^{2}+\|\boldsymbol{w}\|^{2}+\right. \\\left.\operatorname{tr}\left(\boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}}\right)\right)+\left(\frac{1}{m n}+\|\boldsymbol{x}\|^{2}+\|\boldsymbol{y}\|^{2}+\right. \\\left.\operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right)\left(\operatorname{tr} \boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}}\right)- \\\left.\min _{C} \operatorname{tr}\left(\boldsymbol{C}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}} \boldsymbol{C}^{\mathrm{T}}\right)\right) \leqslant\left(\operatorname{tr}\left(\boldsymbol{T}_{A B} \boldsymbol{T}_{A B}^{\mathrm{T}}\right)-\right. \\\left.\operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right)\left(\frac{1}{u v}+\|\boldsymbol{z}\|^{2}+\|\boldsymbol{w}\|^{2}+\right. \\\left.\operatorname{tr}\left(\boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}}\right)\right)+\left(\frac{1}{m n}+\|\boldsymbol{x}\|^{2}+\|\boldsymbol{y}\|^{2}+\right. \\\left.\operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right)\left(\sum_{s^{\prime}=1}^{u^{2}-u} \lambda_{s}^{\prime}\right) 。\end{array}$

式中:B≡(b_ej)是一个n×(n²-1)维矩阵,元素为b_ej≡tr(|e_B><e_B|Y_j)=<e_B|Y_j|e_B>(j=1,2…,n²-1);C≡(c_fk)是一个u×(u²-1)维矩阵,其元素为c_jk≡tr(

Π f C

Z_k)(k=1,2,…,u²-1);{λ'_s':s'=1,2,…,u²-1}是矩阵T_CD

T T C D

按降序排列的特征值。

最后,若ρ_B和ρ_C都是非简并且有谱投影{|e_B><e_B|}和{|f_C><f_C|},则

$\begin{aligned}N_{c}\left(\rho_{A B} \otimes \rho_{C D}\right)=\left(\operatorname{tr}\left(\boldsymbol{T}_{A B} \boldsymbol{T}_{A B}^{\mathrm{T}}\right)-\right. \\\left.\quad \operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right)\left(\frac{1}{u v}+\|\boldsymbol{z}\|^{2}+\|\boldsymbol{w}\|^{2}+\right. \\\left.\quad \operatorname{tr}\left(\boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}}\right)\right)+\left(\frac{1}{m n}+\|\boldsymbol{x}\|^{2}+\|\boldsymbol{y}\|^{2}+\right. \\\left.\quad \operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right)\left(\operatorname{tr}\left(\boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}}\right)-\right. \\\left.\quad \operatorname{tr}\left(\boldsymbol{C} \boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}} \boldsymbol{C}^{\mathrm{T}}\right)\right),\end{aligned}$

式中C≡(c_fk)是一个u×(u²-1)维矩阵,且c_fk≡tr(|f_C><f_C|Z_k)=<f_C|Z_k|f_C>(k=1,2,…,u²-1)。

后面,针对(3)式中混合态情况下的上界进行进一步细化,并得到了一个更便于计算的上界公式。

2 一类混合态测量诱导的

非双局域关联上界的优化

对于(3)式中假定的一类局域量子态ρ_BC=ρ_B$\otimes$ρ_C,即其中ρ_B是非简并且有谱投影{|e_B><e_B|},而ρ_C是简并的情况,文献[32]给出了态ρ_AB$\otimes$ρ_CD测量诱导的非双局域关联上界。接下来,应用文献[20]中的方法对这类态的上界进行优化。

由于(3)式中只需要对tr(T_CD

T T C D

m i n C

tr(CT_CD

T T C D

C^T)进行上界的计算,下面研究边缘态ρ_C有d个维度为u_r的简并子空间情况下量子态非双局域关联的计算问题。

我们重新展开(2)式中的量子态ρ_CD:

(4)ρ_CD=

∑ k = 1 u 2 ∑ l = 0 v 2 - 1

c_klZ'_k$\otimes$W_l。

式中:集合{W_l}与(2)式中所用的集合一样;集合{Z'_k}是由ρ_C的特征向量所构造。给出ρ_C的谱分解ρ_C=

∑ r = 1 d

e_r

∑ s = 1 u r

|x_rs><x_rs|+

∑ r = 1 d'

e'_r|x_r><x_r|,其中{e_r:r=1,2,…,d}({e'_r:r=1,2,…,d'})是ρ_C的简并(非简并)特征值。在这个关系中,假定ρ_C有d+d'(1≤d+d'≤u)个不同的特征值,且每个特征值e_r被重复了u_r次(2≤u_r≤u,d'+

∑ r = 1 d

u_r=u)。同样,ρ_C的简并特征向量被表示为|x_rs>,其中第一个指标确定了相应的特征值,第二个指标与指标“r”指定的特征值的简并度有关。ρ_C的非简并特征向量被表示为|x_r>。

首先,集合{|x_rs>}是不唯一的,同一特征向量|x_rs>的任意线性组合仍然是ρ_C的一个特征向量。为了对集合{Z'_k}进行构造,本文选择了其中一个集合,并表示为{|x'_rs>}。用{|x'_rs><x'_rs|:s,s'=1,2,…,u_r}的线性组合构造算子{Z'_k:k=B_r_-1+2,B_r_-1+1,…,B_r},其中B_r=

u 12

u 22

,…+

u r 2

,B₀=0,且设

' 1 + B r - 1

1 u r ∑ s = 1 u r

|x'_rs><x'_rs|,

这样就形成了一组满足条件tr(Z'_kZ'_k')=δ_kk'的厄米算子集合。进而令{Z

' B d + r

=|x_r><x_r|},并用|x'_rs><x_r'|和|x'_rs><x'_r's'|(r≠r')的线性组合构造集合{Z'_k:k=D,…,u²}(B_d+d'+1=D),且满足条件Z'_k=

Z k' †

,tr(Z'_kZ'_k')=δ_kk', tr(|x'_rs><x'_rs|Z'_k)=0和tr(|x_r><x_r|Z'_k)=0。

用ρ_C在新基下的展开来计算tr(T_CD

T T C D

m i n C

tr(CT_CD

T T C D

C^T)=

m a x Π C

‖ρ_CD-Π^C(ρ_CD)‖²的上界。如前所述,等式(3)中所用的Π^C={

Π f C

}一定是ρ_C的一个谱投影,即

Π^C={

Π f C

}={

Π x r s C

=|x_rs><x_rs|}+

{

Π x r C

=|x_r><x_r|}。

在施加保持ρ_C不变的冯诺依曼测量Π^C之后,态ρ_CD变为

Π^C(ρ_CD)=

∑ r = 1 d ∑ s = 1 u r

(

Π x r s C

$\otimes$I^D)ρ_CD(

Π x r s C

$\otimes$I^D)+

∑ r = 1 d'

(

Π x r C

$\otimes$I^D)ρ_CD(

Π x r C

$\otimes$I^D)。

运用等式(4),有

Π^C(ρ_CD)=

∑ r = 1 d ∑ s = 1 u r ∑ k = 1 u 2 ∑ l = 0 v 2 - 1

c_kl

Π x r s C

Z'_k

Π x r s C

$\otimes$W_t+

∑ r = 1 d' ∑ k = 1 u 2 ∑ l = 0 v 2 - 1

c_kl

Π k r C

Z'_k

Π k r C

$\otimes$W_l。

由上述对于{Z'_k}的构造可得,{

Π x r s C

Z'_k

Π x r s C

=0:k≠F_r,…,B_r}和{

Π x r C

Z'_k

Π x r C

=0:k≠B_d+1,…,B_d+d'},其中F_r=B_r_-1+1。于是有

Π^C(ρ_CD)=

∑ r = 1 d ∑ s = 1 u r ∑ k = F r B r ∑ l = 0 v 2 - 1

c_kl

Π x r s C

Z'_k

Π x r s C

$\otimes$W_l+

∑ k = B d + 1 B d + d' ∑ l = 0 v 2 - 1

c_klZ'_k$\otimes$W_l,

经计算,

ρ_CD-Π^C(ρ_CD)=

∑ r = 1 d ∑ k = F r B r ∑ l = 0 v 2 - 1

c_kl(Z'_k-

∑ s = 1 u r Π x r s C

Z'_k

Π x r s C

)W_l+

∑ k = D u 2 ∑ l = 0 v 2 - 1

c_klZ'_k$\otimes$W_l。

另一方面由等式(4)和在冯诺依曼测量下ρ_C的不变性可知

ρ_C=

∑ k = 1 u 2 c k 0

Z'_k

v

ρ_C=

∑ r = 1 d ∑ s = 1 u r Π x r s C

ρ_C

Π x r s C

∑ r = 1 d' Π x r C

ρ_C

Π x r C

故

∑ k = 1 u 2 c k 0

Z'_k=

∑ r = 1 d ∑ s = 1 u r ∑ k = F r B r c k 0 Π x r s C

Z'_k

Π x r s C

∑ k = B d + 1 B d + d c k 0

Z'_k,

移项可得

∑ r = 1 d ∑ k = F r B d c k 0

(Z'_k-

∑ s = 1 u r Π x r s C

Z'_k

Π x r s C

∑ k = D u 2 c k 0

Z'_k=0。

所以对于前面计算的ρ_CD-Π^C(ρ_CD),l=0时的项为0。若定义X_rsk≡tr

Π x r s C

Z'_k,则

(5)$\begin{array}{c}\left\|\rho_{C D}-\Pi^{C}\left(\rho_{C D}\right)\right\|^{2}=\sum_{l=1}^{v^{2}-1}\left(\sum_{k=D}^{u^{2}} c_{k l}^{2}+\right. \\\left.\sum_{r=1}^{d} \sum_{k=F_{r}}^{B_{r}}\left(c_{k l}^{2}-\sum_{k^{\prime}=F_{r}}^{B_{r}} \sum_{s=1}^{u_{r}} X_{r s k} X_{r s k^{\prime}} c_{k l} c_{k^{\prime} l}\right)\right)\end{array}$

由{Z'_k}的构造可知

X r s F r

=1/

u r

,而对于不扰动局域态的冯诺依曼测量Π^C={

Π f C

}={

Π x r s C

=|x_rs><x_rs|}+{

Π x r C

=|x_r><x_r|}来说,它必须满足2个条件:①

Π f C Π f' C

=δ_ff'

Π f C

;②

∑ f = 1 u Π f C

=I^C。对于条件①,

Π f C Π f' C

=δ_ff'

Π f C

⇒

∑ k, k' = 0 u 2 - 1

c_fkc_f'k'Z_kZ_k'=δ_ff'

∑ k = 0 u 2 - 1

c_fkZ_k⇒

∑ k, k' = F r B r

X_rskX_rs'k'Z_kZ_k'=δ_ss'

∑ k = F r B r

X_rskZ_k;

对于条件②,由于

X r s F r

=1/

u r

,故

∑ f = 1 u Π f C

=I^C⇒

∑ f = 1 u

c_fk=0(k=1,2,…,u²-1)⇒

∑ s = 1 u r

X_rsk=0(k=B_r_-1+2,…,B_r)。

所以,等式(5)中k=F_r或k'=F_r时的项为0,且可最终写为

(6)$\begin{array}{r}\left\|\rho_{C D}-\Pi^{C}\left(\rho_{C D}\right)\right\|=\sum_{k=D}^{u_{2}} \sum_{l=1}^{v^{2}-1} c_{k l}^{2}+ \\\sum_{r=1}^{d}\left(\operatorname{tr}\left(\boldsymbol{C}_{r} \boldsymbol{C}_{r}^{\mathrm{T}}\right)-\operatorname{tr} \mathbf{X}_{r} \boldsymbol{C}_{r} \boldsymbol{C}_{r}^{\mathrm{T}} \boldsymbol{X}_{r}^{\mathrm{T}}\right),\end{array}$

式中:C_r是一个(

u r 2

-1)×(v²-1)维矩阵,其元素为{c_kl:k=B_r_-1+2,…,B_r};X_r是一个u_r×(

u r 2

-1)维矩阵,其元素为{X_rsk:s=1,2,…,u_r:k=B_r_-1+2,…,B_r}。经计算可得,

(7)$\begin{array}{c}\operatorname{tr}\left(\boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}}\right)-\min _{C} \operatorname{tr}\left(\boldsymbol{C}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}} \boldsymbol{C}^{\mathrm{T}}\right)= \\\max _{I^{C}}\left\|\rho_{C D}-\Pi^{C}\left(\rho_{C D}\right)\right\|^{2}= \\\sum_{k=D}^{u^{2}} \sum_{l=1}^{v^{2}-1} c_{k l}^{2}+\sum_{r=1}^{d} \max \left(\operatorname{tr}\left(\boldsymbol{C}_{r} \boldsymbol{C}_{r}^{\mathrm{T}}\right)-\right. \\\left.\operatorname{tr}\left(\boldsymbol{X}_{r} \boldsymbol{C}_{r} \boldsymbol{C}_{r}^{\mathrm{T}} \boldsymbol{X}_{r}^{\mathrm{T}}\right)\right)= \\\sum_{k=D}^{u^{2}} \sum_{l=1}^{v^{2}-1} c_{k l}^{2}+\sum_{r=1}^{d}\left(\operatorname{tr}\left(\boldsymbol{C}_{r} \boldsymbol{C}_{r}^{\mathrm{T}}\right)-\right. \\\left.\min _{k_{r}} \operatorname{tr}\left(\boldsymbol{X}_{r} \boldsymbol{C}_{r} \boldsymbol{C}_{r}^{\mathrm{T}} \boldsymbol{X}_{r}^{\mathrm{T}}\right)\right) 。\end{array}$

根据文献[32],可从等式(7)中得出一个上界,即

(8)$\begin{array}{c}\operatorname{tr}\left(\boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}}\right)-\min _{C} \operatorname{tr}\left(\boldsymbol{C} \boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}} \boldsymbol{C}^{\mathrm{T}}\right) \leqslant \\\sum_{r=1}^{d} \sum_{k=1}^{u_{r}^{2}-u_{r}} \lambda_{r k}+\sum_{k=D}^{u^{2}} \sum_{l=1}^{v^{2}-1} c_{k l}^{2},\end{array}$

式中{λ_rk:k=1,2,…,

u r 2

-1}是矩阵C_r

C T r

按降序排列的特征值。

于是,对于(3)式可以得到一个上界计算公式

(9)$\begin{array}{l}N_{C}\left(\rho_{A B} \otimes \rho_{C D}\right)= \\\left(\operatorname{tr}\left(\boldsymbol{T}_{A B} \boldsymbol{T}_{A B}^{\mathrm{T}}\right)-\operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right)\left(\frac{1}{u v}+\right. \\\left.\|\boldsymbol{z}\|^{2}+\|\boldsymbol{w}\|^{2}+\operatorname{tr}\left(\boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}}\right)\right)+ \\\left(\frac{1}{m n}+\|\boldsymbol{x}\|^{2}+\|\boldsymbol{y}\|^{2}+\right. \\\left.\operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right)\left(\sum_{k=D}^{u^{2}} \sum_{l=1}^{v^{2}-1} c_{k l}^{2}+\right. \\\left.\sum_{r=1}^{d}\left(\operatorname{tr}\left(\boldsymbol{C}_{r} \boldsymbol{C}_{r}^{\mathrm{T}}\right)-\min _{K_{r}} \operatorname{tr}\left(\boldsymbol{X}_{r} \boldsymbol{C}_{r} \boldsymbol{C}_{r}^{\mathrm{T}} \boldsymbol{X}_{r}^{\mathrm{T}}\right)\right)\right) \leqslant \\\left(\operatorname{tr}\left(\boldsymbol{T}_{A B} \boldsymbol{T}_{A B}^{\mathrm{T}}\right)-\operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right)\left(\frac{1}{u v}+\right. \\\left.\|\boldsymbol{z}\|^{2}+\|\boldsymbol{w}\|^{2}+\operatorname{tr}\left(\boldsymbol{T}_{C D} \boldsymbol{T}_{C D}^{\mathrm{T}}\right)\right)+\left(\frac{1}{m n}+\right. \\\left.\|\boldsymbol{x}\|^{2}+\|\boldsymbol{y}\|^{2}+\operatorname{tr}\left(\boldsymbol{B} \boldsymbol{T}_{A B}^{\mathrm{T}} \boldsymbol{T}_{A B} \boldsymbol{B}^{\mathrm{T}}\right)\right) \cdot \\\left(\sum_{r=1}^{d} \sum_{k=1}^{u_{r}^{2}-u_{r}} \lambda_{r k}+\sum_{k=D}^{u^{2}} \sum_{l=1}^{v^{2}-1} c_{k l}^{2}\right) 。 \\\end{array}$

由于这个上界可在每个简并特征子空间下独立计算得出,所以减少了以前上界(在整个空间下计算)的计算量,优化了最终的结果。

3 结语

测量诱导的非双局域关联通过测量的方式,对量子Swapping实验中的非局域性进行了有效捕获。本文针对一类量子态,即边缘态ρ_BC,这个直积态ρ_BC=ρ_B$\otimes$ρ_C中ρ_B是非简并且有谱投影{|e_B><e_B|}},和ρ_C是简并的情况进行了重新计算。通过重新构造局域态ρ_C的厄米算子基,获得了一种新的计算方法,并给出了一个更优化且更便于计算的上界公式。进一步需要研究的课题是,如何获得最一般量子态的上界计算公式。

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模态框（Modal）标题

Abstract

Cite this article

图1 双局域场景

1 测量诱导的非双局域关联

2 一类混合态测量诱导的

3 结语

References