Quantum hedging in two-round prover-verifier interactions

We consider the problem of a particular kind of quantum correlation that arises in some two-party games. In these games, one player is presented with a question they must answer, yielding an outcome of either 'win' or 'lose'. Molina and Watrous (arXiv:1104.1140) studied such a game that exhibited a perfect form of hedging, where the risk of losing a first game can completely offset the corresponding risk for a second game. This is a non-classical quantum phenomenon, and establishes the impossibility of performing strong error-reduction for quantum interactive proof systems by parallel repetition, unlike for classical interactive proof systems. We take a step in this article towards a better understanding of the hedging phenomenon by giving a complete characterization of when perfect hedging is possible for a natural generalization of the game in arXiv:1104.1140. Exploring in a different direction the subject of quantum hedging, and motivated by implementation concerns regarding loss-tolerance, we also consider a variation of the protocol where the player who receives the question can choose to restart the game rather than return an answer. We show that in this setting there is no possible hedging for any game played with state spaces corresponding to finite-dimensional complex Euclidean spaces.

[1]  Pablo A. Parrilo,et al.  Quantum algorithms for the ordered search problem via semidefinite programming , 2007 .

[2]  Uriel Feige.  On the success probability of the two provers in one-round proof systems , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[3]  Carlos Palazuelos,et al.  Rank-one quantum games , 2011, computational complexity.

[4]  A. C. Cem Say,et al.  Probabilistic and quantum finite automata with postselection , 2011, ArXiv.

[5]  Karol Horodecki,et al.  Realistic noise-tolerant randomness amplification using finite number of devices , 2016, Nature Communications.

[6]  Ronald de Wolf,et al.  Near-Optimal and Explicit Bell Inequality Violations , 2012, Theory Comput..

[7]  Ran Raz.  Quantum Information and the PCP Theorem , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[8]  Rusins Freivalds,et al.  Postselection Finite Quantum Automata , 2010, UC.

[9]  Gus Gutoski,et al.  Quantum Strategies and Local Operations , 2010, 1003.0038.

[10]  Sheng Zhang,et al.  Quantum coin flipping secure against channel noises , 2015 .

[11]  Ronald de Wolf,et al.  Rational approximations and quantum algorithms with postselection , 2015, Quantum Inf. Comput..

[12]  L. Fortnow.  Complexity-Theoretic Aspects of Interactive Proof Systems , 1989 .

[13]  B. De Moor,et al.  Optimizing completely positive maps using semidefinite programming , 2002 .

[14]  Somshubhro Bandyopadhyay,et al.  Conclusive exclusion of quantum states , 2013, 1306.4683.

[15]  Ronald de Wolf,et al.  Near-Optimal and Explicit Bell Inequality Violations , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[16]  Mark Braverman,et al.  Small Value Parallel Repetition for General Games , 2014, STOC.

[17]  Abel Molina,et al.  Hedging bets with correlated quantum strategies , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[18]  S. Massar,et al.  Family of loss-tolerant quantum coin-flipping protocols , 2010 .

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

[20]  Matthias Christandl,et al.  Highly entangled states with almost no secrecy. , 2010, Physical review letters.

[21]  Stephanie Wehner,et al.  Entanglement in Interactive Proof Systems with Binary Answers , 2005, STACS.

[22]  Hoi-Kwong Lo,et al.  Loss-tolerant quantum cryptography with imperfect sources , 2013, 1312.3514.

[23]  Uriel Feige,et al.  Proceedings of the thirty-ninth annual ACM symposium on Theory of computing , 2007, STOC 2007.

[24]  Liang Jiang,et al.  Unforgeable noise-tolerant quantum tokens , 2011, Proceedings of the National Academy of Sciences.

[25]  Thomas Holenstein,et al.  Parallel repetition: simplifications and the no-signaling case , 2007, STOC '07.

[26]  Ran Raz.  A Parallel Repetition Theorem , 1998, SIAM J. Comput..

[27]  Mermin,et al.  Simple unified form for the major no-hidden-variables theorems. , 1990, Physical review letters.

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

[29]  A. Peres.  Incompatible results of quantum measurements , 1990 .

[30]  P. Parrilo,et al.  Symmetry groups, semidefinite programs, and sums of squares , 2002, math/0211450.

[31]  J. S. BELLt,et al.  The Einstein-Podolsky-Rosen paradox , 1974, Synthese.

[32]  Mark M. Wilde,et al.  Two-Message Quantum Interactive Proofs and the Quantum Separability Problem , 2012, 2013 IEEE Conference on Computational Complexity.

[33]  Rahul Jain,et al.  Two-Message Quantum Interactive Proofs Are in PSPACE , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[34]  Scott Aaronson,et al.  Quantum computing, postselection, and probabilistic polynomial-time , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[35]  Nicolas Cerf,et al.  Optical quantum cloning , 2006 .

[36]  Matthew F Pusey,et al.  On the reality of the quantum state , 2011, Nature Physics.

[37]  Oded Regev,et al.  Bell violations through independent bases games , 2011, Quantum Inf. Comput..

[38]  S. Guha,et al.  Optimal measurements for symmetric quantum states with applications to optical communication , 2015, 1507.04737.

[39]  Gus Gutoski,et al.  Toward a general theory of quantum games , 2006, STOC '07.

[40]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.