SIAM Review
Journal of Computer and System Sciences - Special issue on the fourteenth annual IEE conference on computational complexity
Classical complexity and quantum entanglement
Journal of Computer and System Sciences - Special issue: STOC 2003
Computational Complexity
Optimal quantum source coding with quantum side information at the encoder and decoder
IEEE Transactions on Information Theory
An Efficient Test for Product States with Applications to Quantum Merlin-Arthur Games
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Strong NP-hardness of the quantum separability problem
Quantum Information & Computation
Quantum Information & Computation
Computational complexity of the quantum separability problem
Quantum Information & Computation
IEEE Transactions on Information Theory
Hypercontractivity, sum-of-squares proofs, and their applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Epsilon-net method for optimizations over separable states
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Quantum de finetti theorems under local measurements with applications
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Entanglement distillation by extendible maps
Quantum Information & Computation
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We present a quasipolynomial-time algorithm for solving the weak membership problem for the convex set of separable, i.e. non-entangled, bipartite density matrices. The algorithm decides whether a density matrix is separable or whether it is ε-away from the set of the separable states in time exp(O(ε-2 log|A| log|B|)), where |A| and |B| are the local dimensions, and the distance is measured with either the Euclidean norm, or with the so-called LOCC norm. The latter is an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states, each shared by two parties, using any protocol formed by quantum local operations and classical communication (LOCC) between the parties. We also obtain improved algorithms for optimizing over the set of separable states and for computing the ground-state energy of mean-field Hamiltonians. The techniques we develop are also applied to quantum Merlin-Arthur games, where we show that multiple provers are not more powerful than a single prover when the verifier is restricted to LOCC protocols, or when the verification procedure is formed by a measurement of small Euclidean norm. This answers a question posed by Aaronson et al. (Theory of Computing 5, 1, 2009) and provides two new characterizations of the complexity class QMA, a quantum analog of NP. Our algorithm uses semidefinite programming to search for a symmetric extension, as first proposed by Doherty, Parrilo and Spedialieri (Phys. Rev. A, 69, 022308, 2004). The bound on the runtime follows from an improved de Finetti-type bound quantifying the monogamy of quantum entanglement. This result, in turn, follows from a new lower bound on the quantum conditional mutual information and the entanglement measure squashed entanglement.