The effect on (2, N, 2) Bell tests with distributed measurement dependence

Bell tests, as primitive tools to detect nonlocality in bipartite systems, rely on an assumption, i.e., measurement independence. In practice, it is difficult to ensure measurement independence. It is necessary to investigate how Bell tests are affected by relaxing measurement independence. In the simplest (2, 2, 2) CHSH Bell test which consists of two parties, two measurements per party and two possible outcomes per measurement, the results between the maximal value of CHSH correlation function and distributed measurement dependence (DMD) are given, where DMD is a general measure of relaxing measurement independence. However, in a general Bell scenario of an arbitrary number of measurements per party, i.e., (2, N , 2), pertinent results are still missing. To solve it, we establish the relations between the maximal value of (2,  N , 2) Pearle–Braunstein–Caves (PBC) chain correlation function that maintains the locality and the degree of DMD, denoted as DMD-induced PBC chain inequalities. Furthermore, we show the tightness of these derived inequalities via constructing local hidden variable models that fake the upper bounds. Compared with the simplest CHSH Bell test, our derived inequalities need less amount of measurement dependence to fake the quantum prediction with N increasing, which is beneficial to analyze the security of device-independent quantum information processing tasks such as randomness expansion.

[1]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .

[2]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[3]  Franson,et al.  Bell inequality for position and time. , 1989, Physical review letters.

[4]  A. Zeilinger,et al.  Speakable and Unspeakable in Quantum Mechanics , 1989 .

[5]  S. Popescu,et al.  Quantum nonlocality as an axiom , 1994 .

[6]  M. Żukowski,et al.  Two-photon Franson-type experiments and local realism , 1999, quant-ph/9912064.

[7]  J. Bell,et al.  Speakable and Unspeakable in Quantum Mechanics: Preface to the first edition , 2004 .

[8]  S. Massar,et al.  Nonlocal correlations as an information-theoretic resource , 2004, quant-ph/0404097.

[9]  S. Wehner.  Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities , 2005, quant-ph/0510076.

[10]  N. Gisin,et al.  From Bell's theorem to secure quantum key distribution. , 2005, Physical review letters.

[11]  Miles P. Blencowe,et al.  How to silence a one-ton bell , 2008 .

[12]  M. Hall.  Local deterministic model of singlet state correlations based on relaxing measurement independence. , 2010, Physical review letters.

[13]  Stefano Pironio,et al.  Random numbers certified by Bell’s theorem , 2009, Nature.

[14]  Serge Fehr,et al.  Security and Composability of Randomness Expansion from Bell Inequalities , 2011, ArXiv.

[15]  Adrian Kent,et al.  Private randomness expansion with untrusted devices , 2010, 1011.4474.

[16]  M. Hall,et al.  Relaxed Bell inequalities and Kochen-Specker theorems , 2011, 1102.4467.

[17]  N. Gisin,et al.  How much measurement independence is needed to demonstrate nonlocality? , 2010, Physical review letters.

[18]  Dax Enshan Koh,et al.  Effects of reduced measurement independence on Bell-based randomness expansion. , 2012, Physical review letters.

[19]  Adrian Kent,et al.  Unconditionally secure device-independent quantum key distribution with only two devices , 2012, ArXiv.

[20]  Miguel Navascués,et al.  Operational framework for nonlocality. , 2011, Physical review letters.

[21]  Roger Colbeck,et al.  Free randomness can be amplified , 2011, Nature Physics.

[22]  Dax Enshan Koh,et al.  The effects of reduced"free will"on Bell-based randomness expansion , 2012 .

[23]  Manik Banik,et al.  Optimal free will on one side in reproducing the singlet correlation , 2012, 1204.3835.

[24]  Le Phuc Thinh,et al.  Min-entropy sources for Bell tests , 2014 .

[25]  Rodrigo Gallego,et al.  Full randomness from arbitrarily deterministic events , 2012, Nature Communications.

[26]  Stefano Pironio,et al.  Security of practical private randomness generation , 2011, 1111.6056.

[27]  Jan Bouda,et al.  Device-independent randomness extraction from an arbitrarily weak min-entropy source , 2014 .

[28]  Antonio Acín,et al.  Can observed randomness be certified to be fully intrinsic? , 2013, Physical review letters.

[29]  Nicolas Gisin,et al.  Arbitrarily small amount of measurement independence is sufficient to manifest quantum nonlocality. , 2014, Physical review letters.

[30]  Umesh Vazirani,et al.  Fully device-independent quantum key distribution. , 2012, 1210.1810.

[31]  Miguel Navascués,et al.  Robust and versatile black-box certification of quantum devices. , 2014, Physical review letters.

[32]  Nicolas Gisin,et al.  Exploring the Limits of Quantum Nonlocality with Entangled Photons , 2015, 1506.01649.

[33]  Jędrzej Kaniewski,et al.  Analytic and Nearly Optimal Self-Testing Bounds for the Clauser-Horne-Shimony-Holt and Mermin Inequalities. , 2016, Physical review letters.

[34]  Jean-Daniel Bancal,et al.  Device-independent parallel self-testing of two singlets , 2015, 1512.02074.

[35]  Antonio Acín,et al.  Certified randomness in quantum physics , 2016, Nature.

[36]  Fei Gao,et al.  Effects of measurement dependence on generalized Clauser-Horne-Shimony-Holt Bell test in the single-run and multiple-run scenarios , 2016 .

[37]  Xiaohong Huang,et al.  Effects of measurement dependence on tilted CHSH Bell tests , 2018, Quantum Inf. Process..

[38]  D. Kaiser,et al.  Relaxed Bell inequalities with arbitrary measurement dependence for each observer , 2018, Physical Review A.

[39]  Umesh V. Vazirani,et al.  Fully device independent quantum key distribution , 2012, Commun. ACM.