The quantum moment problem and bounds on entangled multi-prover games

Doherty, Andrew C.; Liang, Yeong-Cherng; Toner, Ben; Wehner, Stephanie

doi:10.48550/arxiv.0803.4373

Cited by 8 publications

(14 citation statements)

References 178 publications

(548 reference statements)

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“…In a local classical model we have the maximum value of 0, while the largest violation one could get with qubits was 0.25, which could already be achieved with a maximally entangled pair of qubits (see e.g., [11,13,[16][17][18]23]). On the other hand, the best upper bounds are based on the NPA method [10] and at level three it yields the significantly higher upper bound, 0.250 875 56 [11,16]. We could even go above level three to an intermediate level in [13], and presently we have got the upper bound 0.250 875 38 at level four.…”

Section: Introductionmentioning

confidence: 80%

“…The method invented by Navascués, Pironio and Acín (NPA) [10] is based on the solution of a hierarchy of semidefinite programming (SDP) relaxations and is particularly useful since it gives better and better upper bounds on the maximum violation of an arbitrary Bell inequality by stepping to higher levels in the hierarchy. Moreover, the series of upper bounds in the hierarchy keep to the exact quantum maximum in terms of commuting measurements [11,16]. On the other hand, one can use heuristic algorithms to obtain nontrivial lower bounds in some finite dimensional Hilbert spaces on Bell inequalities, recovering the explicit form of the states and measurement operators as well.…”

Section: Introductionmentioning

confidence: 99%

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Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

2010

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The I3322 inequality is the simplest bipartite two-outcome Bell inequality beyond the ClauserHorne-Shimony-Holt (CHSH) inequality, consisting of three two-outcome measurements per party. In case of the CHSH inequality the maximal quantum violation can already be attained with local two-dimensional quantum systems, however, there is no such evidence for the I3322 inequality. In this paper a family of measurement operators and states is given which enables us to attain the largest possible quantum value in an infinite dimensional Hilbert space. Further, it is conjectured that our construction is optimal in the sense that measuring finite dimensional quantum systems is not enough to achieve the true quantum maximum. We also describe an efficient iterative algorithm for computing quantum maximum of an arbitrary two-outcome Bell inequality in any given Hilbert space dimension. This algorithm played a key role to obtain our results for the I3322 inequality, and we also applied it to improve on our previous results concerning the maximum quantum violation of several bipartite two-outcome Bell inequalities with up to five settings per party.

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Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

2010

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“…Both settings have been extensively studied in the past (see e.g. [37,93,77,42,61,60,57,56,36,62,58] for work on MIP * /QMIP and [2,20,48,49,31,21,26,14,70,32,47,75,30,80,87] for work on QMA(k)), although there are still many interesting open questions concerning them.…”

Section: Introductionmentioning

confidence: 99%

Quantum de finetti theorems under local measurements with applications

Brandão

Harrow

2013

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

108

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Quantum de Finetti theorems are a useful tool in the study of correlations in quantum multipartite states. In this paper we prove two new quantum de Finetti theorems, both showing that under tests formed by local measurements in each of the subsystems one can get a much improved error dependence on the dimension of the subsystems. We also obtain similar results for non-signaling probability distributions. We give the following applications of the results to quantum complexity theory, polynomial optimization, and quantum information theory: 7 The one-way LOCC norm is defined as X LOCC ← = max 0≤M ≤I tr(XM ), with the the maximization over all POVMs {M, I − M } that can be realized by local operations and one-way classical communication from B to A.

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“…This can be considered as a special case of a more sophisticated SDP based on the noncommutative Positivstellensatz[60,30,57], whose general performance we will not analyse.…”

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confidence: 99%

Product-state approximations to quantum ground states

Brandão

Harrow

2013

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

101

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The local Hamiltonian problem consists of estimating the ground-state energy (given by the minimum eigenvalue) of a local quantum Hamiltonian. It can be considered as a quantum generalization of constraint satisfaction problems (CSPs) and has a central role both in quantum many-body physics and quantum complexity theory. A key feature that distinguishes quantum Hamiltonians from classical CSPs is that the solutions may involve complicated entangled states. In this paper, we demonstrate several large classes of Hamiltonians for which product (i.e. unentangled) states can approximate the ground state energy to within a small extensive error.First, we show the existence of a good product-state approximation for the ground-state energy of 2-local Hamiltonians with one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state with sublinear entanglement with respect to some partition into small pieces. The approximation based on degree is a surprising difference between quantum Hamiltonians and classical CSPs, since in the classical setting, higher degree is usually associated with harder CSPs. The approximation based on low entanglement, in turn, was previously known only in the regime where the entanglement was close to zero.Estimating the energy is NP-hard by the PCP theorem, but whether it is QMA-hard is an important open question in quantum complexity theory. A positive solution would represent a quantum analogue of the PCP theorem. Since the existence of a low-energy product state can be checked in NP, the result implies that any Hamiltonian used for a quantum PCP theorem should have: (1) constant degree, (2) constant expansion, (3) a "volume law" for entanglement with respect to any partition into small parts. The result also gives a no-go to a quantum version of the PCP theorem in conjunction with parallel repetition for quantum CSPs.Second, we show that in several cases, good product-state approximations not only exist, but can be found in polynomial time: (1) 2-local Hamiltonians on any planar graph, solving an open problem of Bansal, Bravyi, and Terhal, (2) dense k-local Hamiltonians for any constant k, solving an open problem of Gharibian and Kempe, and (3) 2-local Hamiltonians on graphs with low threshold rank, via a quantum generalization of a recent result of Barak, Raghavendra and Steurer.Our work introduces two new tools which may be of independent interest. First, we prove a new quantum version of the de Finetti theorem which does not require the usual assumption of symmetry. Second, we describe a way to analyze the application of the Lasserre/Parrilo SDP hierarchy to local quantum Hamiltonians.

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The quantum moment problem and bounds on entangled multi-prover games

Abstract: The quantum moment problem and bounds on entangled multi-prover games,

Cited by 8 publications

References 178 publications

Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems

Quantum de finetti theorems under local measurements with applications

Product-state approximations to quantum ground states

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