Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables
SIAM Journal on Optimization
A lower bound on the value of entangled binary games
Quantum Information & Computation
Properties of local quantum operations with shared entanglement
Quantum Information & Computation
Relaxed uncertainty relations and information processing
Quantum Information & Computation
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
The communication complexity of non-signaling distributions
Quantum Information & Computation
Classical and quantum partition bound and detector inefficiency
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Product-state approximations to quantum ground states
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
| Hi-index | 0.00 |
We study the quantum moment problem: Given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state rho and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on rho is specified by the conditional probability distribution, and (ii) the measurement operators satisfy the constraints. For example, the constraints might specify that some measurement operators must commute. We show that if an instance of the quantum moment problem is unsatisfiable, then there exists a certificate of a particular form proving this. Our proof is based on a recent result in algebraic geometry, the noncommutative Positivstellensatz of Helton and McCullough [Trans. Amer. Math. Soc., 356(9):3721, 2004]. A special case of the quantum moment problem is to compute the value of one-round multi-prover games with entangled provers. Under the conjecture that the provers need only share states in finite-dimensional Hilbert spaces, we prove that a hierarchy of semidefinite programs similar to the one given by Navascues, Pironioand Acin [Phys. Rev. Lett., 98:010401, 2007] converges to the entangled value of the game. Under this conjecture, it would follow that the languages recognized by a multi-prover interactive proof system where the provers share entanglement are recursive.