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Characterizing nonbilocal correlation: a geometric perspective

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Abstract

Exploiting the notion of measurement-induced nonlocality (Luo and Fu in Phys Rev Lett 106:120401, 2011), we introduce a new measure to quantify the nonbilocal correlation. We establish a simple relation between the nonlocal and nonbilocal measures for the arbitrary pure input states. Considering the mixed states as inputs, we derive two upper bounds of affinity-based nonbilocal measure. Finally, we have studied the nonbilocality of a different combinations of input states.

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Acknowledgements

This work was financially supported by the Council of Scientific and Industrial Research (CSIR), Government of India, under Grant No. 03(1444)/18/EMR-II.

Author information

Authors and Affiliations

  1. Department of Physics, Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur, Tamil Nadu, 613 401, India

    R. Muthuganesan & V. K. Chandrasekar

  2. Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Bu rehová 7, Praha 1-Staré Mĕsto, 115 19, Czech Republic

    R. Muthuganesan

  3. Department of Physics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632014, India

    S. Balakrishnan

Authors
  1. R. Muthuganesan
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  2. S. Balakrishnan
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  3. V. K. Chandrasekar
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Correspondence to R. Muthuganesan.

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Appendix

Appendix

Let \(|\varPsi _{ab}\rangle =\sum _is_i|i_ai_b\rangle \) and \(|\varPsi _{cd}\rangle =\sum _jr_j|j_cj_d\rangle \) are the two pure input states with \(s_i\) and \(r_j\) are the respective Schmidt coefficients of input states.

Noting that

$$\begin{aligned} \begin{aligned} \rho _{ab}\otimes \rho _{cd}&=|\varPsi _{ab}\rangle \langle \varPsi _{ab}|\otimes |\varPsi _{cd}\rangle \langle \varPsi _{cd}|\\&=\sum _{ii^{'}jj^{'}}s_is_{i^{'}}r_jr_{j^{'}}|i_a\rangle \langle i^{'}_a|\otimes |i_b\rangle \langle i^{'}_b|\otimes |j_c\rangle \langle j^{'}_c|\otimes |j_d\rangle \langle j^{'}_d|. \end{aligned} \end{aligned}$$
(30)

One can compute the marginal state

$$\begin{aligned} \begin{aligned} \rho ^{bc}&=\text {Tr}_{ad}(|\varPsi _{ab}\rangle \langle \varPsi _{ab}|\otimes |\varPsi _{cd}\rangle \langle \varPsi _{cd}|)=\sum _{ij}s_i^2r_j^2|i_bj_c\rangle \langle i_bj_c|. \end{aligned} \end{aligned}$$
(31)

For pure state \(\sqrt{\rho }=\rho \). From the above equation, the post-measurement state \(\varPi ^{bc}(\sqrt{\rho _{ab}\otimes \rho _{cd}})\) can be rewritten as

$$\begin{aligned} \begin{aligned}&\varPi ^{bc}\left( \sqrt{\rho _{ab}\otimes \rho _{cd}}\right) =\varPi ^{bc}\left( \rho _{ab}\otimes \rho _{cd}\right) \\&\quad =\sum _{hk}\left( \mathbbm {1}^{a}\otimes \varPi _{hk}^{bc}\otimes \mathbbm {1}^{d}\right) \left( |\varPsi _{ab}\rangle \langle \varPsi _{ab}|\otimes |\varPsi _{cd}\rangle \langle \varPsi _{cd}|\right) \left( \mathbbm {1}^{a}\otimes \varPi _{hk}^{bc}\otimes \mathbbm {1}^{d}\right) \\&\quad =\sum _{hk}\left( \mathbbm {1}^{a}\otimes \varPi _{hk}^{bc}\otimes \mathbbm {1}^{d}\right) \left( \sum _{ii^{'}jj^{'}}s_is_{i^{'}}r_jr_{j^{'}}|i_a\rangle \langle i^{'}_a|\otimes |i_b\rangle \langle i^{'}_b|\otimes |j_c\rangle \langle j^{'}_c|\otimes |j_d\rangle \langle j^{'}_d|\right) \\&\qquad \left( \mathbbm {1}^{a}\otimes \varPi _{hk}^{bc}\otimes \mathbbm {1}^{d}\right) =\sum _{hk}\sum _{ii^{'}jj^{'}}s_is_{i^{'}}r_jr_{j^{'}}|i_a\rangle \langle i^{'}_a|\otimes \varPi ^{bc}_{hk}|i_b j_{c}\rangle \langle i^{'}_b j^{'}_c|\varPi ^{bc}_{hk}\otimes |j_d\rangle \langle j^{'}_d|\\&\quad =\sum _{hk}\sum _{ii^{'}jj^{'}}s_is_{i^{'}}r_jr_{j^{'}}|i_a\rangle \langle i^{'}_a|\otimes U|h_{b}k_{c}\rangle \langle h_{b}k_{c}|U^{\dag }|i_b j_{c}\rangle \langle i^{'}_b j^{'}_c|U|h_{b}k_{c}\rangle \langle h_{b}k_{c}|U^{\dag }\otimes |j_d\rangle \langle j^{'}_d|. \end{aligned} \end{aligned}$$

Here, the von Neumann projective measurement is expressed as

$$\begin{aligned} \varPi ^{bc}=\{\varPi ^{bc}_{hk}\equiv U|h_{b}k_{c}\rangle \langle h_{b}k_{c}|U^{\dag }\} \end{aligned}$$
(32)

Consequently,

$$\begin{aligned} \begin{aligned}&\sqrt{\rho _{ab}\otimes \rho _{cd}}\varPi ^{bc}\left( \sqrt{\rho _{ab}\otimes \rho _{cd}}\right) \\&\quad =\left( \sum _{ii^{'}jj^{'}}s_is_{i^{'}}r_jr_{j^{'}}|i_a\rangle \langle i^{'}_a|\otimes |i_b\rangle \langle i^{'}_b|\otimes |j_c\rangle \langle j^{'}_c|\otimes |j_d\rangle \langle j^{'}_d|\right) \left( \sum _{hk}\sum _{uu^{'}vv^{'}}s_us_{u^{'}}r_vr_{v^{'}}\right. \\&\qquad \left. \langle h_{b}k_{c}|U^{\dag }|u_b v_{c}\rangle \langle u^{'}_b v^{'}_c|U|h_{b}k_{c}\rangle |u_a\rangle \langle u^{'}_a|\otimes U|h_{b}k_{c}\rangle \langle h_{b}f_{c}|U^{\dag }\otimes |v_d\rangle \langle v^{'}_d|\right) \\&\quad =\sum _{ii^{'}jj^{'}}\sum _{hk}\sum _{uu^{'}vv^{'}}s_is_{i^{'}}r_jr_{j^{'}}s_us_{u^{'}}r_vr_{v^{'}}\langle e_{B}f_{C}|U^{\dag }|u_B v_{C}\rangle \langle u^{'}_b v^{'}_c|U|h_{b}k_{f}\rangle |i_a\rangle \\&\qquad \langle i^{'}_a|u_a\rangle \langle u^{'}_a|\otimes |i_b j_c\rangle \langle i^{'}_b j^{'}_c|U|h_{b}k_{c}\rangle \langle h_{b}k_{c}|U^{\dag }\otimes |j_d\rangle \langle j^{'}_d|v_d\rangle \langle v^{'}_d|. \end{aligned} \end{aligned}$$

Then, the affinity between the pre- and post-measurement state is computed as

$$\begin{aligned} \begin{aligned} {\mathcal {A}}(\rho _{ab}\otimes \rho _{cd},\varPi ^{bc}(\rho _{ab}\otimes \rho _{cd}))&=\text {Tr}\sqrt{\rho _{ab}\otimes \rho _{cd}}\varPi ^{bc}(\sqrt{\rho _{ab}\otimes \rho _{cd}})\\&=\sum _{iujvhk}s_i^2s_{u}^2r_j^2r_{v}^2\langle h_{b}k_{c}|U^{\dag }|u_b v_{c}\rangle \langle i_b j_c|U|h_{b}k_{c}\rangle \langle u_b v_c|U|h_{b}k_{c}\rangle \\&\quad \langle h_{b}k_{c}|U^{\dag }|i_b j_c\rangle \\&=\sum _{hk}\left( \langle h_{b}k_{c}|U^{\dag }\rho _{bc} U|h_{b}k_{c}\rangle \right) ^{2}. \end{aligned} \end{aligned}$$

The nonbilocal measure for pure state is

$$\begin{aligned} \begin{aligned} N_{\mathcal {A}}\left( |\varPsi _{ab}\rangle \otimes |\varPsi _{cd}\rangle \right)&= \max _{\varPi ^{bc}} d_{\mathcal {A}}\left( \sqrt{\rho _{ab}\otimes \rho _{cd}},\varPi ^{bc}\left( \sqrt{\rho _{ab}\otimes \rho _{cd}}\right) \right) \\&= 1-\min _{\varPi ^{bc}} {\mathcal {A}}\left( \rho _{ab}\otimes \rho _{cd},\varPi ^{bc}\left( \rho _{ab}\otimes \rho _{cd}\right) \right) \\&=1-\min _{\varPi ^{bc}}\sum _{hk}\left( \langle h_{b}k_{c}|U^{\dag }\rho ^{bc} U|h_{b}k_{c}\rangle \right) ^{2}, \end{aligned} \end{aligned}$$

where the optimization is over all von Neumann measurements given in Eq. (32), leaving the marginal state \(\rho ^{bc}\) invariant. That is,

$$\begin{aligned} \rho ^{bc}=\sum _{hk}\langle h_{b}k_{c}|U^{\dag }\rho ^{bc}U|h_{b}k_{c}\rangle U|h_{b}k_{c}\rangle \langle h_{b}k_{c}|U^{\dag } \end{aligned}$$

is a spectral decomposition of \(\rho ^{bc}\) since \(\{U|h_{b}k_{c}\rangle \}\) is an orthonormal base. Comparing the above equation with Eq. (31), we obtained

$$\begin{aligned} N_{\mathcal {A}}\left( |\varPsi _{ab}\rangle \otimes |\varPsi _{cd}\rangle \right) =1-\sum _{i,j}s_i^4 r_j^4. \end{aligned}$$
(33)

Hence, the theorem is proved. It is worth mentioning that the affinity-based nonbilocal measure for pure state is equal to the Hellinger distance and Hilbert–Schmidt norm-based nonbilocal measures [38, 39].

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Muthuganesan, R., Balakrishnan, S. & Chandrasekar, V.K. Characterizing nonbilocal correlation: a geometric perspective. Quantum Inf Process 21, 216 (2022). https://doi.org/10.1007/s11128-022-03561-2

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