Properties of local quantum operations with shared entanglement

Multi-party local quantum operations with shared quantum entanglement or sharedclassical randomness are studied. The following facts are established:• There is a ball of local operations with shared randomness lying within the spacespanned by the no-signaling operations and centred at the completely noisy channel.• The existence of the ball of local operations with shared randomness is employedto prove that the weak membership problem for local operations with sharedentanglement is strongly NP-hard.• Local operations with shared entanglement are characterized in terms of linearfunctionals that are "completely" positive on a certain cone K of separable Hermitianoperators, under a natural notion of complete positivity appropriate tothat cone. Local operations with shared randomness (but not entanglement) arealso characterized in terms of linear functionals that are merely positive on thatsame cone K.• Existing characterizations of no-signaling operations are generalized to the multipartysetting and recast in terms of the Choi-Jamio lkowski representation forquantum super-operators. It is noted that the standard nonlocal box is an exampleof a no-signaling operation that is separable, yet cannot be implemented by localoperations with shared entanglement.

[1]  R. Werner,et al.  Semicausal operations are semilocalizable , 2001, quant-ph/0104027.

[2]  Keiji Matsumoto,et al.  Using Entanglement in Quantum Multi-Prover Interactive Proofs , 2007, 2008 23rd Annual IEEE Conference on Computational Complexity.

[3]  S. Szarek.  The volume of separable states is super-doubly-exponentially small , 2003, quant-ph/0310061.

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

[5]  L. Gurvits.  Quantum Matching Theory (with new complexity theoretic, combinatorial and topological insights on the nature of the Quantum Entanglement) , 2002, quant-ph/0201022.

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

[7]  J. Preskill,et al.  Causal and localizable quantum operations , 2001, quant-ph/0102043.

[8]  William Slofstra,et al.  Perfect Parallel Repetition Theorem for Quantum Xor Proof Systems , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

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

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

[11]  Julia Kempe,et al.  The Unique Games Conjecture with Entangled Provers is False , 2007, Algebraic Methods in Computational Complexity.

[12]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[13]  E. Rains.  Entanglement purification via separable superoperators , 1997, quant-ph/9707002.

[14]  Howard Barnum,et al.  Better bound on the exponent of the radius of the multipartite separable ball (12 pages) , 2005 .

[15]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[16]  L. Gurvits,et al.  Largest separable balls around the maximally mixed bipartite quantum state , 2002, quant-ph/0204159.

[17]  M. Horodecki,et al.  Separability of mixed states: necessary and sufficient conditions , 1996, quant-ph/9605038.

[18]  M. Horodecki,et al.  Separability of n-particle mixed states: necessary and sufficient conditions in terms of linear maps , 2000, quant-ph/0006071.

[19]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[20]  Keiji Matsumoto,et al.  Using Entanglement in Quantum Multi-prover Interactive Proofs , 2008, Computational Complexity Conference.

[21]  Pérès.  Separability Criterion for Density Matrices. , 1996, Physical review letters.

[22]  Keiji Matsumoto,et al.  Entangled Games are Hard to Approximate , 2007, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[23]  S. Szarek.  Volume of separable states is super-doubly-exponentially small in the number of qubits , 2005 .

[24]  Julia Kempe,et al.  Unique Games with Entangled Provers are Easy , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[25]  Howard Barnum,et al.  Separable balls around the maximally mixed multipartite quantum states , 2003 .

[26]  R. Cleve,et al.  Strong Parallel Repetition Theorem for Quantum XOR Proof Systems , 2006, quant-ph/0608146.

[27]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

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

[29]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[30]  R. Cleve,et al.  Entanglement-Resistant Two-Prover Interactive Proof Systems and Non-Adaptive Private Information Retrieval Systems , 2007, 0707.1729.

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

[32]  S. Woronowicz.  Positive maps of low dimensional matrix algebras , 1976 .

[33]  Roland Hildebrand.  Entangled states close to the maximally mixed state , 2007 .

[34]  Stephanie Wehner,et al.  The Quantum Moment Problem and Bounds on Entangled Multi-prover Games , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[35]  M. Horodecki,et al.  Properties of quantum nonsignaling boxes , 2006 .

[36]  Yi-Kai Liu.  The complexity of the consistency and N-representability problems for quantum states - eScholarship , 2007, 0712.3041.

[37]  Roland Hildebrand.  Cones of ball-ball separable elements , 2005 .

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

[39]  R. Bhatia.  Positive Definite Matrices , 2007 .