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 - + + .