The Power of Unentanglement

The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k)=QMA(2) for kges2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. *We give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O tilde(radicn) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on Dinur's version of the PCP Theorem and is inherently non-relativizing. *We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all kges=2. *We give evidence that QMA(2) sube PSPACE, by showing that this would follow from "strong amplification" of QMA(2) protocols. *We prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.

[1]  Avi Wigderson,et al.  Multi-prover interactive proofs: how to remove intractability assumptions , 2019, STOC '88.

[2]  M. Hastings.  Superadditivity of communication capacity using entangled inputs , 2009 .

[3]  Fernando G.S.L. Brandao,et al.  Entanglement Theory and the Quantum Simulation of Many-Body Physics , 2008, 0810.0026.

[4]  M. Hastings.  A counterexample to additivity of minimum output entropy , 2008, 0809.3972.

[5]  Alain Tapp,et al.  All Languages in NP Have Very Short Quantum Proofs , 2007, 2009 Third International Conference on Quantum, Nano and Micro Technologies.

[6]  Irit Dinur.  The PCP theorem by gap amplification , 2007, J. ACM.

[7]  Irit Dinur,et al.  The PCP theorem by gap amplification , 2006, STOC.

[8]  Matthias Christandl,et al.  Quantum computational complexity of the N-representability problem: QMA complete. , 2007, Physical review letters.

[9]  F. Verstraete,et al.  N-representability is QMA-complete , 2006, quant-ph/0609125.

[10]  Greg Kuperberg,et al.  Quantum versus Classical Proofs and Advice , 2006, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[11]  Dmitry Gavinsky,et al.  On the role of shared entanglement , 2006, Quantum Inf. Comput..

[12]  Hartmut Klauck,et al.  Interaction in Quantum Communication , 2006, IEEE Transactions on Information Theory.

[13]  Scott Aaronson,et al.  QMA/qpoly Is Contained In PSPACE/poly: De-Merlinizing Quantum Protocols , 2005, Electron. Colloquium Comput. Complex..

[14]  R. Renner,et al.  A de Finetti representation for finite symmetric quantum states , 2004, quant-ph/0410229.

[15]  Chris Marriott,et al.  Quantum Arthur–Merlin games , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[16]  A. Winter,et al.  “Squashed entanglement”: An additive entanglement measure , 2003, quant-ph/0308088.

[17]  Keiji Matsumoto,et al.  Quantum Merlin-Arthur Proof Systems: Are Multiple Merlins More Helpful to Arthur? , 2003, Chic. J. Theor. Comput. Sci..

[18]  P. Shor.  Equivalence of Additivity Questions in Quantum Information Theory , 2003, quant-ph/0305035.

[19]  Dorit Aharonov,et al.  Quantum NP - A Survey , 2002, quant-ph/0210077.

[20]  Mikhail N. Vyalyi,et al.  Classical and Quantum Computation , 2002, Graduate studies in mathematics.

[21]  Luca Trevisan,et al.  The Approximability of Constraint Satisfaction Problems , 2001, SIAM J. Comput..

[22]  Keiji Matsumoto,et al.  Quantum multi-prover interactive proof systems with limited prior entanglement , 2001, J. Comput. Syst. Sci..

[23]  John Watrous,et al.  Succinct quantum proofs for properties of finite groups , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[24]  John Watrous,et al.  Space-Bounded Quantum Complexity , 1999, J. Comput. Syst. Sci..

[25]  M. Plenio,et al.  Entanglement measures and purification procedures , 1997, quant-ph/9707035.

[26]  Charles H. Bennett,et al.  Strengths and Weaknesses of Quantum Computing , 1997, SIAM J. Comput..

[27]  E. Knill.  Quantum Randomness and Nondeterminism , 1996, quant-ph/9610012.

[28]  Charles H. Bennett,et al.  Mixed-state entanglement and quantum error correction. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[29]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[30]  Ekert,et al.  "Event-ready-detectors" Bell experiment via entanglement swapping. , 1993, Physical review letters.

[31]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[32]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and EPR channels , 1993 .

[33]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[34]  Moisés Aranda,et al.  the birthday problem , 2008 .

[35]  T. Paul,et al.  Quantum computation and quantum information , 2001, SOEN.

[36]  D. Petz,et al.  Quantum Entropy and Its Use , 1993 .