The membership problem for constant-sized quantum correlations is undecidable

When two spatially separated parties make measurements on an unknown entangled quantum state, what correlations can they achieve? How difficult is it to determine whether a given correlation is a quantum correlation? These questions are central to problems in quantum communication and computation. Previous work has shown that the general membership problem for quantum correlations is computationally undecidable. In the current work we show something stronger: there is a family of constant-sized correlations -- that is, correlations for which the number of measurements and number of measurement outcomes are fixed -- such that solving the quantum membership problem for this family is computationally impossible. Thus, the undecidability that arises in understanding Bell experiments is not dependent on varying the number of measurements in the experiment. This places strong constraints on the types of descriptions that can be given for quantum correlation sets. Our proof is based on a combination of techniques from quantum self-testing and from undecidability results of the third author for linear system nonlocal games.

[1]  Koon Tong Goh,et al.  Geometry of the set of quantum correlations , 2017, 1710.05892.

[2]  Zheng-Feng Ji,et al.  Compression of quantum multi-prover interactive proofs , 2016, STOC.

[3]  Pierre McKenzie,et al.  Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing , 2017, STOC.

[4]  A. Winter,et al.  Information causality as a physical principle , 2009, Nature.

[5]  Anand Natarajan,et al.  A quantum linearity test for robustly verifying entanglement , 2017, STOC.

[6]  J. S. BELLt.  Einstein-Podolsky-Rosen Paradox , 2018 .

[7]  Thomas Vidick,et al.  MIP* = RE , 2021, Communications of the ACM.

[8]  Joseph Fitzsimons,et al.  Quantum proof systems for iterated exponential time, and beyond , 2018, Electron. Colloquium Comput. Complex..

[9]  Matthew Coudron,et al.  Complexity lower bounds for computing the approximately-commuting operator value of non-local games to high precision , 2019, CCC.

[10]  Yaoyun Shi,et al.  Optimal robust quantum self-testing by binary nonlocal XOR games , 2012, 1207.1819.

[11]  George W. Polites,et al.  An introduction to the theory of groups , 1968 .

[12]  Maggie,et al.  and Beyond: , 2017 .

[13]  A. F..  Foundations of Physics , 1936, Nature.

[14]  Honghao Fu,et al.  Constant-sized correlations are sufficient to self-test maximally entangled states with unbounded dimension , 2019, Quantum.

[15]  Peter Høyer,et al.  Consequences and limits of nonlocal strategies , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[16]  Necessary and sufficient condition for quantum-generated correlations , 2003, quant-ph/0309137.

[17]  H. Buhrman,et al.  Limit on nonlocality in any world in which communication complexity is not trivial. , 2005, Physical review letters.

[18]  M. Navascués,et al.  A glance beyond the quantum model , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  O. Kharlampovich,et al.  Algorithmically complex residually finite groups , 2012, 1204.6506.

[20]  I. Ial,et al.  Nature Communications , 2010, Nature Cell Biology.

[21]  Kathleen Daly,et al.  Volume 7 , 1998 .

[22]  Andrea Coladangelo,et al.  Unconditional separation of finite and infinite-dimensional quantum correlations , 2018, 1804.05116.

[23]  M. Junge,et al.  Connes' embedding problem and Tsirelson's problem , 2010, 1008.1142.

[24]  L. J. Landau,et al.  Empirical two-point correlation functions , 1988 .

[25]  Danna Zhou,et al.  d. , 1840, Microbial pathogenesis.

[26]  Anand Natarajan,et al.  Low-Degree Testing for Quantum States, and a Quantum Entangled Games PCP for QMA , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[27]  Travis B. Russell.  Geometry of the set of synchronous quantum correlations , 2019, Journal of Mathematical Physics.

[28]  Valerio Capraro,et al.  Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture , 2013, 1309.2034.

[29]  Umesh V. Vazirani,et al.  Classical command of quantum systems , 2013, Nature.

[30]  Ursula Dresdner,et al.  Computation Finite And Infinite Machines , 2016 .

[31]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[32]  Geometry of quantum correlations , 2008, 0802.3632.

[33]  T. Fritz,et al.  Local orthogonality as a multipartite principle for quantum correlations , 2012, Nature Communications.

[34]  Anand Natarajan,et al.  NEEXP is Contained in MIP* , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[35]  A. Varvitsiotis,et al.  Geometric structure of quantum correlators via semidefinite programming , 2018, Physical Review A.

[36]  D. Pérez-García,et al.  Undecidability of the Spectral Gap in One Dimension , 2018, Physical Review X.

[37]  M. Wolf,et al.  All-multipartite Bell-correlation inequalities for two dichotomic observables per site , 2001, quant-ph/0102024.

[38]  Tsuyoshi Ito,et al.  A Multi-prover Interactive Proof for NEXP Sound against Entangled Provers , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[39]  A. Acín,et al.  A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations , 2008, 0803.4290.

[40]  G. Higman,et al.  Embedding Theorems for Groups , 1949 .

[41]  B. Tsirelson.  Quantum analogues of the Bell inequalities. The case of two spatially separated domains , 1987 .

[42]  William Slofstra,et al.  THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED , 2017, Forum of Mathematics, Pi.

[43]  William Slofstra,et al.  Perfect Commuting-Operator Strategies for Linear System Games , 2016, 1606.02278.

[44]  Physical Review , 1965, Nature.

[45]  Hamoon Mousavi,et al.  On the complexity of zero gap MIP , 2020, ICALP.

[46]  Tobias J. Hagge,et al.  Physics , 1929, Nature.

[47]  T. Fritz.  TSIRELSON'S PROBLEM AND KIRCHBERG'S CONJECTURE , 2010, 1008.1168.

[48]  B. Tsirelson.  Some results and problems on quan-tum Bell-type inequalities , 1993 .

[49]  W. B.,et al.  (1) Proceedings of the London Mathematical Society (2) Journal of the London Mathematical Society , 1927, Nature.

[50]  A. Acín,et al.  Almost-Quantum Correlations Violate the No-Restriction Hypothesis. , 2017, Physical review letters.

[51]  Francesco Pappalardi,et al.  On Artin's Conjecture for Primitive Roots , 1993 .

[52]  William Slofstra,et al.  Tsirelson’s problem and an embedding theorem for groups arising from non-local games , 2016, Journal of the American Mathematical Society.