Quantum interactive proofs and the complexity of entanglement detection

We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by employing prior work on entanglement purification protocols to prove that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.

[1]  Bill Fefferman,et al.  The Power of Unentanglement , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[2]  Daniel Gooch,et al.  Communications of the ACM , 2011, XRDS.

[3]  Aram Wettroth Harrow,et al.  Quantum de Finetti Theorems Under Local Measurements with Applications , 2012, STOC '13.

[4]  Michal Horodecki,et al.  A Decoupling Approach to the Quantum Capacity , 2007, Open Syst. Inf. Dyn..

[5]  R. Cleve,et al.  SUBSTITUTING QUANTUM ENTANGLEMENT FOR COMMUNICATION , 1997, quant-ph/9704026.

[6]  Jing Chen,et al.  Short Multi-Prover Quantum Proofs for SAT without Entangled Measurements , 2010, 1011.0716.

[7]  A. Uhlmann.  The "transition probability" in the state space of a ∗-algebra , 1976 .

[8]  Yong Zhang,et al.  Fast amplification of QMA , 2009, Quantum Inf. Comput..

[9]  John Watrous,et al.  Quantum Computational Complexity , 2008, Encyclopedia of Complexity and Systems Science.

[10]  P. Parrilo,et al.  Detecting multipartite entanglement , 2004, quant-ph/0407143.

[11]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.

[12]  François Le Gall,et al.  On QMA protocols with two short quantum proofs , 2011, Quantum Inf. Comput..

[13]  Salman Beigi,et al.  NP vs QMA_log(2) , 2008, Quantum Inf. Comput..

[14]  P. Goldbart,et al.  Geometric measure of entanglement and applications to bipartite and multipartite quantum states , 2003, quant-ph/0307219.

[15]  Michael Sipser,et al.  Introduction to the Theory of Computation , 1996, SIGA.

[16]  Alexei Y. Kitaev,et al.  Parallelization, amplification, and exponential time simulation of quantum interactive proof systems , 2000, STOC '00.

[17]  Debbie W. Leung,et al.  A Communication-Efficient Nonlocal Measurement With Application to Communication Complexity and Bipartite Gate Capacities , 2008, IEEE Transactions on Information Theory.

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

[19]  Masaru Kada,et al.  The efficiency of quantum identity testing of multiple states , 2008, 0809.2037.

[20]  John Watrous.  Quantum statistical zero-knowledge , 2002 .

[21]  László Babai,et al.  Arthur-Merlin Games: A Randomized Proof System, and a Hierarchy of Complexity Classes , 1988, J. Comput. Syst. Sci..

[22]  H. Yuen.  Quantum detection and estimation theory , 1978, Proceedings of the IEEE.

[23]  Carsten Lund,et al.  Algebraic methods for interactive proof systems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[24]  Peter W. Shor,et al.  Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem , 2001, IEEE Trans. Inf. Theory.

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

[26]  Matthias Christandl,et al.  A quasipolynomial-time algorithm for the quantum separability problem , 2010, STOC '11.

[27]  Larry J. Stockmeyer,et al.  The Polynomial-Time Hierarchy , 1976, Theor. Comput. Sci..

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

[29]  John Watrous,et al.  Limits on the power of quantum statistical zero-knowledge , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[30]  C. Fuchs.  Distinguishability and Accessible Information in Quantum Theory , 1996, quant-ph/9601020.

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

[32]  A. Harrow,et al.  Testing Product States, Quantum Merlin-Arthur Games and Tensor Optimization , 2010, JACM.

[33]  P. Parrilo,et al.  Complete family of separability criteria , 2003, quant-ph/0308032.

[34]  Keiji Matsumoto,et al.  Quantum Certificate Verification: Single versus Multiple Quantum Certificates , 2001, ArXiv.

[35]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[36]  Runyao Duan,et al.  Bounds on the Distance Between a Unital Quantum Channel and the Convex Hull of Unitary Channels , 2012, IEEE Transactions on Information Theory.

[37]  P. Zanardi,et al.  Entangling power of quantum evolutions , 2000, quant-ph/0005031.

[38]  Sevag Gharibian,et al.  Strong NP-hardness of the quantum separability problem , 2008, Quantum Inf. Comput..

[39]  A. Winter,et al.  Distinguishability of Quantum States Under Restricted Families of Measurements with an Application to Quantum Data Hiding , 2008, 0810.2327.

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

[41]  Rahul Jain,et al.  QIP = PSPACE , 2011, JACM.

[42]  P. Parrilo,et al.  Distinguishing separable and entangled states. , 2001, Physical review letters.

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

[44]  R F Werner,et al.  Hiding classical data in multipartite quantum states. , 2002, Physical review letters.

[45]  Debbie W. Leung,et al.  Quantum data hiding , 2002, IEEE Trans. Inf. Theory.

[46]  David Deutsch,et al.  Stabilization of Quantum Computations by Symmetrization , 1997, SIAM J. Comput..

[47]  R. Cleve,et al.  Quantum fingerprinting. , 2001, Physical review letters.

[48]  David Deutsch,et al.  Stabilisation of Quantum Computations by Symmetrisation , 1996 .

[49]  David P. DiVincenzo,et al.  Hiding Quantum Data , 2002 .

[50]  I. Chuang,et al.  Quantum Computation and Quantum Information: Bibliography , 2010 .

[51]  P. Hayden,et al.  Quantum computation vs. firewalls , 2013, 1301.4504.

[52]  F. Brandão,et al.  Faithful Squashed Entanglement , 2010, 1010.1750.

[53]  John Watrous,et al.  PSPACE has constant-round quantum interactive proof systems , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[54]  Alexei Y. Kitaev,et al.  Quantum measurements and the Abelian Stabilizer Problem , 1995, Electron. Colloquium Comput. Complex..

[55]  Bill Rosgen,et al.  Testing Non-isometry Is QMA-Complete , 2009, TQC.

[56]  Gus Gutoski,et al.  Quantum Interactive Proofs with Competing Provers , 2004, STACS.

[57]  Charles H. Bennett,et al.  Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. , 1992, Physical review letters.

[58]  R. Cleve,et al.  Efficient Quantum Algorithms for Simulating Sparse Hamiltonians , 2005, quant-ph/0508139.

[59]  Xiaodi Wu,et al.  Parallel Approximation of Min-Max Problems , 2010, computational complexity.

[60]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[61]  Julia Kempe,et al.  The Complexity of the Local Hamiltonian Problem , 2004, FSTTCS.

[62]  John Watrous,et al.  Zero-knowledge against quantum attacks , 2005, STOC '06.

[63]  S. Cook,et al.  Logical Foundations of Proof Complexity: INDEX , 2010 .

[64]  R. Schumann.  Quantum Information Theory , 2000, quant-ph/0010060.

[65]  Silvio Micali,et al.  The knowledge complexity of interactive proof-systems , 1985, STOC '85.

[66]  Scott Aaronson,et al.  Quantum Computing since Democritus , 2013 .

[67]  Mark M. Wilde,et al.  From Classical to Quantum Shannon Theory , 2011, ArXiv.

[68]  Ashley Montanaro,et al.  An Efficient Test for Product States with Applications to Quantum Merlin-Arthur Games , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

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

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

[71]  Xiaodi Wu,et al.  Epsilon-net method for optimizations over separable states , 2011, Theor. Comput. Sci..

[72]  Adam D. Bookatz.  QMA-complete problems , 2012, Quantum Inf. Comput..

[73]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[74]  Alain Tapp,et al.  A Quantum Characterization Of NP , 2007, computational complexity.

[75]  M. Christandl,et al.  N-representability is QMA-complete , 2006 .

[76]  R. Werner.  An application of Bell's inequalities to a quantum state extension problem , 1989 .

[77]  Seth Lloyd,et al.  Universal Quantum Simulators , 1996, Science.

[78]  John Watrous.  PSPACE has constant-round quantum interactive proof systems , 2003, Theor. Comput. Sci..

[79]  Or Sattath,et al.  The Complexity of the Separable Hamiltonian Problem , 2011, 2012 IEEE 27th Conference on Computational Complexity.

[80]  Ashish V. Thapliyal,et al.  Entanglement-Assisted Classical Capacity of Noisy Quantum Channels , 1999, Physical Review Letters.

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

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

[83]  Keiji Matsumoto,et al.  Quantum multi-prover interactive proof systems with limited prior entanglement , 2003, J. Comput. Syst. Sci..

[84]  Andris Ambainis,et al.  Extracting quantum entanglement (general entanglement purification protocols) , 2001, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[85]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[86]  Jeroen van de Graaf,et al.  Cryptographic Distinguishability Measures for Quantum-Mechanical States , 1997, IEEE Trans. Inf. Theory.

[87]  Cécilia Lancien,et al.  Distinguishing Multi-Partite States by Local Measurements , 2012, 1206.2884.

[88]  I. Mazin,et al.  Theory , 1934 .

[89]  Silvio Micali,et al.  The Knowledge Complexity of Interactive Proof Systems , 1989, SIAM J. Comput..

[90]  Debbie W. Leung,et al.  Improving zero-error classical communication with entanglement , 2009, Physical review letters.

[91]  J. Preskill,et al.  Quantum error correction , 2001, Principles of Quantum Computation and Information.

[92]  Adam D. Smith,et al.  Authentication of quantum messages , 2001, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[93]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

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

[95]  M. Kus,et al.  Measures and dynamics of entangled states , 2005, quant-ph/0505162.

[96]  Michael A. Forbes,et al.  Improved Soundness for QMA with Multiple Provers , 2011, Electron. Colloquium Comput. Complex..

[97]  F. Brandão,et al.  Detection of multiparticle entanglement: quantifying the search for symmetric extensions. , 2011, Physical review letters.

[98]  Leonid Gurvits.  Classical deterministic complexity of Edmonds' Problem and quantum entanglement , 2003, STOC '03.

[99]  Lin Chen,et al.  Comparison of different definitions of the geometric measure of entanglement , 2013, 1308.0806.