Quantum de finetti theorems under local measurements with applications

Brandão, Fernando G. S. L.; Harrow, Aram W.

doi:10.1145/2488608.2488718

Cited by 61 publications

(111 citation statements)

References 121 publications

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“…• MULTI-QSEP-CIRCUIT, a multipartite generalization of QSEP-CIRCUIT, is also decidable by a twomessage quantum interactive proof system for a wide range of parameters. The analysis of the proof system exploits a recent quantum de Finetti theorem of Brandão and Harrow [40] along with the analysis used in the proof system for QSEP-CIRCUIT. MULTI-QSEP-CIRCUIT is also NP-and QSZK-hard because QSEP-CIRCUIT trivially reduces to it, as QSEP-CIRCUIT is merely a special case of MULTI-QSEP-CIRCUIT.…”

Section: Overview Of Resultsmentioning

confidence: 99%

Two-Message Quantum Interactive Proofs and the Quantum Separability Problem

Hayden

Milner

Wilde

2013

2013 IEEE Conference on Computational Complexity

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Suppose that a polynomial-time mixed-state quantum circuit, described as a sequence of local unitary interactions followed by a partial trace, generates a quantum state shared between two parties. One might then wonder, does this quantum circuit produce a state that is separable or entangled? Here, we give evidence that it is computationally hard to decide the answer to this question, even if one has access to the power of quantum computation. We begin by exhibiting a twomessage quantum interactive proof system that can decide the answer to a promise version of the question. We then prove that the promise problem is hard for the class of promise problems with "quantum statistical zero knowledge" (QSZK) proof systems by demonstrating a polynomial-time Karp reduction from the QSZK-complete promise problem "quantum state distinguishability" to our quantum separability problem. By exploiting Knill's efficient encoding of a matrix description of a state into a description of a circuit to generate the state, we can show that our promise problem is NP-hard with respect to Cook reductions. Thus, the quantum separability problem (as phrased above) constitutes the first nontrivial promise problem decidable by a two-message quantum interactive proof system while being hard for both NP and QSZK. We also consider a variant of the problem, in which a given polynomialtime mixed-state quantum circuit accepts a quantum state as input, and the question is to decide if there is an input to this circuit which makes its output separable across some bipartite cut. We prove that this problem is a complete promise problem for the class QIP of problems decidable by quantum interactive proof systems. Finally, we show that a two-message quantum interactive proof system can also decide a multipartite generalization of the quantum separability problem.

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Section: Overview Of Resultsmentioning

confidence: 99%

Two-Message Quantum Interactive Proofs and the Quantum Separability Problem

Hayden

Milner

Wilde

2013

2013 IEEE Conference on Computational Complexity

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“…We remark that the Hamiltonians and the Bell inequalities we have studied have a finite interaction range, which in the TI case makes them particularly interesting from an experimentally-friendly perspective. Note that previous Bell inequalities for quantum many-body systems with low order correlators were specially designed for a permutationally invariant (PI) symmetry [25,26]; while being able to detect nonlocality in ground states of physical Hamiltonians such as the Lipkin-Meshkov-Glick [43] or a spin-squeezed Bose-Einstein condensate [27], the information accessible to these inequalities is bound to a de Finetti theorem [44,45], thus becoming more compatible with that produced by a separable state as the system grows [26]. This requires to increase the number of measurements with the system size in order to close the finite-statistics loophole [27].…”

Section: Discussionmentioning

confidence: 99%

Energy as a Detector of Nonlocality of Many-Body Spin Systems

et al. 2017

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We present a method to show that low-energy states of quantum many-body interacting systems in one spatial dimension are nonlocal. We assign a Bell inequality to the Hamiltonian of the system in a natural way and we efficiently find its classical bound using dynamic programming. The Bell inequality is such that its quantum value for a given state, and for appropriate observables, corresponds to the energy of the state. Thus, the presence of nonlocal correlations can be certified for states of low enough energy. The method can also be used to optimize certain Bell inequalities: in the translationally invariant (TI) case, we provide an exponentially faster computation of the classical bound and analytically closed expressions of the quantum value for appropriate observables and Hamiltonians. The power and generality of our method is illustrated through four representative examples: a tight TI inequality for 8 parties, a quasi TI uniparametric inequality for any even number of parties, ground states of spin-glass systems, and a non-integrable interacting XXZ-like Hamiltonian. Our work opens the possibility for the use of low-energy states of commonly studied Hamiltonians as multipartite resources for quantum information protocols that require nonlocality.

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“…However, they did not have a rounding algorithm, and in particular did not solve the problem of actually finding a separable state that approximately maximizes the probability of acceptance of a given one-way LOCC measurement. The techniques of the present paper were used by Brandão and Harrow [17] to solve the latter problem, and also greatly simplify the proof of [16]'s result, which originally involved relations between several measures of entanglement proved in several papers. 5 In an appendix to the full paper [11], we give a short proof of this result, specialized to the case of real vectors and polynomials of degree four (corresponding to quantum states of two systems, or two prover QMA proofs).…”

Section: Optimizing Polynomials With Nonnegative Coefficientsmentioning

confidence: 97%

“…The paper[17] was based on a previous version of this work[10] that contained only the results for nonnegative tensors.…”

mentioning

confidence: 99%

Rounding sum-of-squares relaxations

Barak

Kelner

Steurer

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-ofSquares proof system to transform a combining algorithman algorithm that maps a distribution over solutions into a (possibly weaker) solution-into a rounding algorithm that maps a solution of the relaxation to a solution of the original problem.Using this approach, we obtain algorithms that yield improved results for natural variants of several well-known problems:1. We give a quasipolynomial-time algorithm that approximates max x 2 =1 P (x) within an additive factor of ε P spectral , where ε > 0 is a constant, P is a degree d = O(1), n-variate polynomial with nonnegative coefficients, and P spectral is the spectral norm of a matrix corresponding to P 's coefficients. Beyond being of interest in its own right, obtaining such an approximation for general polynomials (with possibly negative coefficients) is a long-standing open question in quantum information theory, and our techniques have already led to improved results in this area (Brandão and Harrow, STOC '13).2. We give a polynomial-time algorithm that, given a subspace V ⊆ R n of dimension d that (almost) contains the characteristic function of a set of size n/k, finds a vector v ∈ V that satisfies Ei v 4i Ω(d −1/3 k(Ei v 2 i ) 2 ). This is a natural analytical relaxation of the problem of finding the sparsest element in a subspace, and it is also motivated by a connection to the Small-Set Expansion problem shown by Barak et al.of the previous best known algorithms for small-set expansion in a certain range of parameters.3. We use this notion of L4 vs. L2 sparsity to obtain a polynomial-time algorithm with substantially improved guarantees for recovering a planted sparse vector v in a random d-dimensional subspace of R n . If v has µn nonzero coordinates, we can recover it with high probability whenever µ O(min(1, n/d 2 )). In particular, when d √ n, this recovers a planted vector with up to Ω(n) nonzero coordinates. When d n 2/3 , our algorithm improves upon existing methods based on comparing the L1 and L∞ norms, which intrinsically require µ O 1/ √ d . We also show how this notion of L4 vs. L2 sparsity can be used to find a planted sparse vector in a random subspace, improving on a recent result of Demanet and Hand (2013).

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Quantum de finetti theorems under local measurements with applications

Cited by 61 publications

References 121 publications

Two-Message Quantum Interactive Proofs and the Quantum Separability Problem

Two-Message Quantum Interactive Proofs and the Quantum Separability Problem

Energy as a Detector of Nonlocality of Many-Body Spin Systems

Rounding sum-of-squares relaxations

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