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Testing Product States and Managing Multiple Merlins

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Testing Product States and Managing Multiple Merlins Testing product states and managing multiple Merlins Ashley Montanaro Centre for Quantum Information and Foundations, University of Cambridge, UK Talk based on joint work with Aram Harrow arXiv:1001.0017v3, FOCS’10 Entanglement Separability Entropies Quantum channels QIP Information Computation QIP Information Computation Entanglement Separability Entropies Quantum channels QIP Information Computation Entanglement Merlin-Arthur Separability games Entropies Provable hardness Quantum Convex channels optimisation QIP Information Computation Entanglement Merlin-Arthur Separability games Entropies This talk Provable hardness Quantum Convex channels optimisation The basic problem Given a quantum state, is it entangled? Variants How are we given the input state? 01=2 0 0 1=21 B 0 0 0 0 C vs. ρ = B C @ 0 0 0 0 A 1=2 0 0 1=2 Variants Is the input state pure or mixed? ? ? i i i i j i = j 1i ... j ki vs. ρ = i pij 1ih 1j ⊗ · · · ⊗ j kih kj P Is the input state bipartite or multipartite? 1 2 vs. 1 2 3 ... k Variants What level of accuracy do we demand? SEP vs. SEP Separability testing up to accuracy : given ρ such that either ρ 2 SEP or minσ2SEP kρ - σkp > , decide which is the case. Variants Do we want to detect entanglement in all states, or just some of them? SEP SEP vs. ρ Given a bipartite mixed state ρ 2 B(Cd ⊗ Cd) as a d2-dimensional square matrix, it’s NP-hard to determine whether ρ is separable (up to accuracy1 =poly(d)). This was shown by [Gurvits ’03] for accuracy1 =exp(d) via a reduction from the NP-hard Clique problem. Later improved to1 =poly(d) by [Gharibian ’10] (using techniques of [Liu ’07]) and also (implicitly) by [Beigi ’08]. Stop press: There’s an exp(O(-2 log2 d)) algorithm for testing separability up to accuracy in the 2-norm [Brandao,˜ Christandl, Yard ’10]! Good news and bad news Given a bipartite pure state j i 2 Cd ⊗ Cd as a d2-dimensional vector, whether j i is entangled can be determined efficiently using the Schmidt decomposition. This was shown by [Gurvits ’03] for accuracy1 =exp(d) via a reduction from the NP-hard Clique problem. Later improved to1 =poly(d) by [Gharibian ’10] (using techniques of [Liu ’07]) and also (implicitly) by [Beigi ’08]. Stop press: There’s an exp(O(-2 log2 d)) algorithm for testing separability up to accuracy in the 2-norm [Brandao,˜ Christandl, Yard ’10]! Good news and bad news Given a bipartite pure state j i 2 Cd ⊗ Cd as a d2-dimensional vector, whether j i is entangled can be determined efficiently using the Schmidt decomposition. Given a bipartite mixed state ρ 2 B(Cd ⊗ Cd) as a d2-dimensional square matrix, it’s NP-hard to determine whether ρ is separable (up to accuracy1 =poly(d)). Stop press: There’s an exp(O(-2 log2 d)) algorithm for testing separability up to accuracy in the 2-norm [Brandao,˜ Christandl, Yard ’10]! Good news and bad news Given a bipartite pure state j i 2 Cd ⊗ Cd as a d2-dimensional vector, whether j i is entangled can be determined efficiently using the Schmidt decomposition. Given a bipartite mixed state ρ 2 B(Cd ⊗ Cd) as a d2-dimensional square matrix, it’s NP-hard to determine whether ρ is separable (up to accuracy1 =poly(d)). This was shown by [Gurvits ’03] for accuracy1 =exp(d) via a reduction from the NP-hard Clique problem. Later improved to1 =poly(d) by [Gharibian ’10] (using techniques of [Liu ’07]) and also (implicitly) by [Beigi ’08]. Good news and bad news Given a bipartite pure state j i 2 Cd ⊗ Cd as a d2-dimensional vector, whether j i is entangled can be determined efficiently using the Schmidt decomposition. Given a bipartite mixed state ρ 2 B(Cd ⊗ Cd) as a d2-dimensional square matrix, it’s NP-hard to determine whether ρ is separable (up to accuracy1 =poly(d)). This was shown by [Gurvits ’03] for accuracy1 =exp(d) via a reduction from the NP-hard Clique problem. Later improved to1 =poly(d) by [Gharibian ’10] (using techniques of [Liu ’07]) and also (implicitly) by [Beigi ’08]. Stop press: There’s an exp(O(-2 log2 d)) algorithm for testing separability up to accuracy in the 2-norm [Brandao,˜ Christandl, Yard ’10]! Some notes: The bounds on acceptance probability don’t depend on the local dimension d or the number of subsystems k. This is similar to classical property testing algorithms. The test can also be used to determine if a unitary operator is a tensor product. Main result Theorem ⊗ Let j i 2 (Cd) k be a pure k-partite state such that the nearest product state to j i is the state jφ1i ... jφki, where 2 jh jφ1,..., φkij = 1 - . Then there is an efficient quantum test, called the product test, that accepts with probability1 - Θ(), given two copies of j i. Main result Theorem ⊗ Let j i 2 (Cd) k be a pure k-partite state such that the nearest product state to j i is the state jφ1i ... jφki, where 2 jh jφ1,..., φkij = 1 - . Then there is an efficient quantum test, called the product test, that accepts with probability1 - Θ(), given two copies of j i. Some notes: The bounds on acceptance probability don’t depend on the local dimension d or the number of subsystems k. This is similar to classical property testing algorithms. The test can also be used to determine if a unitary operator is a tensor product. This test takes two (possibly mixed) states ρ, σ as input, returning 0 (“same”) with probability 1 1 + tr(ρ σ), 2 2 otherwise returning 1 (“different”). The swap test The product test uses the swap test as a subroutine. Swap test [Buhrman et al ’01] j0i H • H ρ SWAP σ NM The swap test The product test uses the swap test as a subroutine. Swap test [Buhrman et al ’01] j0i H • H ρ SWAP σ NM This test takes two (possibly mixed) states ρ, σ as input, returning 0 (“same”) with probability 1 1 + tr(ρ σ), 2 2 otherwise returning 1 (“different”). The product test Product test d ⊗k 1 Prepare two copies of j i 2 (C ) ; call these j 1i, j 2i. 2 Perform the swap test on each of the k pairs of corresponding subsystems of j 1i, j 2i. 3 If all of the tests returned “same”, accept. Otherwise, reject. j 1i 1 2 3 ... k j 2i 1 2 3 ... k swap swap swap swap test test test test Implemented experimentally for bipartite states by [Walborn et al ’06]. Proposed by [AM, Osborne ’08] as a means of determining whether a unitary operator is product. Our contribution: to prove correctness of the test for all k. Previous use of the product test The product test has appeared before in the literature. Originally introduced by [Mintert, Kus,´ Buchleitner ’05] as one of a family of tests for generalisations of the concurrence entanglement measure. Proposed by [AM, Osborne ’08] as a means of determining whether a unitary operator is product. Our contribution: to prove correctness of the test for all k. Previous use of the product test The product test has appeared before in the literature. Originally introduced by [Mintert, Kus,´ Buchleitner ’05] as one of a family of tests for generalisations of the concurrence entanglement measure. Implemented experimentally for bipartite states by [Walborn et al ’06]. Our contribution: to prove correctness of the test for all k. Previous use of the product test The product test has appeared before in the literature. Originally introduced by [Mintert, Kus,´ Buchleitner ’05] as one of a family of tests for generalisations of the concurrence entanglement measure. Implemented experimentally for bipartite states by [Walborn et al ’06]. Proposed by [AM, Osborne ’08] as a means of determining whether a unitary operator is product. Previous use of the product test The product test has appeared before in the literature. Originally introduced by [Mintert, Kus,´ Buchleitner ’05] as one of a family of tests for generalisations of the concurrence entanglement measure. Implemented experimentally for bipartite states by [Walborn et al ’06]. Proposed by [AM, Osborne ’08] as a means of determining whether a unitary operator is product. Our contribution: to prove correctness of the test for all k. Thus the product test measures the average purity of ρ across bipartitions. It’s immediate that Ptest(ρ) = 1 if and only if ρ is a pure product state. Our main result says: if the average entanglement across bipartitions of j i is low, j i must in fact be close to a product state. Analysing the product test Lemma Let Ptest(ρ) be the probability that the product test passes on input ρ. Then 1 P (ρ) = tr ρ2. test 2k S SX⊆[k] Analysing the product test Lemma Let Ptest(ρ) be the probability that the product test passes on input ρ. Then 1 P (ρ) = tr ρ2. test 2k S SX⊆[k] Thus the product test measures the average purity of ρ across bipartitions. It’s immediate that Ptest(ρ) = 1 if and only if ρ is a pure product state. Our main result says: if the average entanglement across bipartitions of j i is low, j i must in fact be close to a product state. Details of main result Theorem Let the nearest product state to j i be jφ1i ... jφki, and set 2 jh jφ1,..., φkij = 1 - . Then 2 3=2 2 1 - 2 + 6 Ptest(j ih j) 6 1 - + + .
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