Finite size mean-field models

Fannes, Mark; Vandenplas, Caroline

doi:10.1088/0305-4470/39/45/001

Cited by 23 publications

(40 citation statements)

References 10 publications

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“…[42], the quantum de Finetti theorem was used to show that when n d 2 , then the ground state of H is very close to a product state. In this case, finding the groundstate energy of H is equivalent to minimising tr ρK over all ρ ∈ SEP(d, d).…”

Section: Complexity-theoretic Implicationsmentioning

confidence: 99%

An Efficient Test for Product States with Applications to Quantum Merlin-Arthur Games

Harrow

Montanaro

2010

2010 IEEE 51st Annual Symposium on Foundations of Computer Science

118

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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 |ψ whose maximum overlap with a product state is 1 − , the test passes with probability 1 − Θ( ), regardless of n or the local dimensions of the individual systems. The test uses two copies of |ψ . 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 ≥ 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 e O( √ 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.

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Section: Complexity-theoretic Implicationsmentioning

confidence: 99%

An Efficient Test for Product States with Applications to Quantum Merlin-Arthur Games

Harrow

Montanaro

2010

2010 IEEE 51st Annual Symposium on Foundations of Computer Science

118

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“…These quantum de Finetti theorems are appealing not only due to their own elegance on the characterization of symmetric states, but also because of the successful applications in many-body physics [5,11,12], quantum information [9,13,14], and computational complexity theory [10,15,16].…”

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confidence: 99%

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

Smith²

2015

Phys. Rev. Lett.

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We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multifold product states. The approximation is measured by distinguishability under measurements that are implementable by fully-one-way local operations and classical communication (LOCC). Our result strengthens Brandão and Harrow's de Finetti theorem where a kind of partially-one-way LOCC measurements was used for measuring the approximation, with essentially the same error bound. As main applications, we show (i) a quasipolynomial-time algorithm which detects multipartite entanglement with an amount larger than an arbitrarily small constant (measured with a variant of the relative entropy of entanglement), and (ii) a proof that in quantum Merlin-Arthur proof systems, polynomially many provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations.

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“…Let H be an arbitrary separable Hilbert space and let Ψ N ∈ N sym H with Ψ N = 1. Assume that the sequence of k-particle density matrices γ We will also use a quantitative version of the quantum de Finetti theorem, originally proved in [8] (see [7,29,41,42,43] for variants of the proof and [15] for an earlier result in this direction). The following formulation is taken from [42,Lemma 3.4].…”

Section: Many-body Blow-upmentioning

confidence: 99%

Many-body blow-up profile of boson stars with external potentials

Nguyen

2019

Rev. Math. Phys.

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We consider a 3D quantum system of N identical bosons in a trapping potential |x| p , with p ≥ 0, interacting via a Newton potential with an attractive interaction strength a N . For a fixed large N and the coupling constant a N smaller than a critical value a * (Chandrasekhar limit mass), in an appropriate sense, the many-body system admits a ground state. We investigate the blow-up behavior of the ground state energy as well as the ground states when a N approaches a * sufficiently slowly in the limit N → ∞. The blow-up profile is given by the Gagliardo-Nirenberg solutions.2010 Mathematics Subject Classification. 81V17, 81V70.

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Finite size mean-field models

Cited by 23 publications

References 10 publications

An Efficient Test for Product States with Applications to Quantum Merlin-Arthur Games

An Efficient Test for Product States with Applications to Quantum Merlin-Arthur Games

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

Many-body blow-up profile of boson stars with external potentials

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