A quasipolynomial-time algorithm for the quantum separability problem
Proceedings of the forty-third annual ACM symposium on Theory of computing
Hypercontractivity, sum-of-squares proofs, and their applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
On QMA protocols with two short quantum proofs
Quantum Information & Computation
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
QMA variants with polynomially many provers
Quantum Information & Computation
Quantum de finetti theorems under local measurements with applications
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Adversarial hypothesis testing and a quantum stein's lemma for restricted measurements
Proceedings of the 5th conference on Innovations in theoretical computer science
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We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state psi whose maximum overlap with a product state is 1-epsilon, the test passes with probability 1-Theta(epsilon), regardless of n or the local dimensions of the individual systems. The test uses two copies of psi. We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarising channel. A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that soundness amplification is possible and that two Merlins can simulate many Merlins: QMA(k)=QMA(2) for k at least 2. Building on a previous result of Aaronson et al, this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of O(sqrt(n) polylog(n)) qubits. Among other consequences, this result implies complexity-theoretic obstructions to finding a polynomial-time algorithm to determine separability of mixed quantum states, even up to constant error, and also to proving "weak" variants of the additivity conjecture for quantum channels. Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalisation of classical linearity testing.