Numerical estimation of the relative entropy of entanglement

Zinchenko, Yuriy; Friedland, Shmuel; Gour, Gilad

doi:10.1103/physreva.82.052336

Cited by 31 publications

(27 citation statements)

References 16 publications

(27 reference statements)

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“…As for step 13, variable σ (N +1) can be given by where Z is some fixed reference point. This one-dimensional minimization can be efficiently performed using the standard derivative-based bisection scheme [38]. Using this algorithm for the Rains bound, we can easily check that it is not additive, which has been recently proved in Ref.…”

Section: Numerical Estimation Of Rains Boundmentioning

confidence: 82%

Non-Asymptotic Entanglement Distillation

Fang

Wang

Tomamichel

et al. 2019

IEEE Trans. Inform. Theory

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Non-asymptotic entanglement distillation studies the trade-off between three parameters: the distillation rate, the number of independent and identically distributed prepared states, and the fidelity of the distillation. We first study the one-shot ε-infidelity distillable entanglement under quantum operations that completely preserve positivity of the partial transpose (PPT) and characterize it as a semidefinite program (SDP). For isotropic states, it can be further simplified to a linear program. The one-shot ε-infidelity PPTassisted distillable entanglement can be transformed to a quantum hypothesis testing problem. Moreover, we show efficiently computable second-order upper and lower bounds for the non-asymptotic distillable entanglement with a given infidelity tolerance. Utilizing these bounds, we obtain the second order asymptotic expansions of the optimal distillation rates for pure states and some classes of mixed states. In particular, this result recovers the second-order expansion of LOCC distillable entanglement for pure states in [Datta/Leditzky, IEEE Trans. Inf. Theory 61:582, 2015]. Furthermore, we provide an algorithm for calculating the Rains bound and present direct numerical evidence (not involving any other entanglement measures, as in [Wang/Duan, Phys. Rev. A 95:062322, 2017]), showing that the Rains bound is not additive under tensor products. I. INDRODUCTION A. BackgroundQuantum entanglement is a striking feature of quantum mechanics and a key ingredient in many quantum information processing tasks, including teleportation [1], superdense coding [2], and quantum cryptography [3,4]. All these protocols necessarily rely on entanglement resources, especially the maximally entangled states. It is thus of great importance to develop entanglement distillation protocols to transform less useful entangled states into more suitable ones such as maximally entangled states. In general, the task of entanglement distillation aims at obtaining maximally entangled states from less-entangled bipartite states shared between two parties (Alice and Bob) and it allows them to perform local operations and classical communication (LOCC). The concept of distillable entanglement characterizes the rate at which one can asymptotically obtain maximally entangled states from a collection of identically and independently distributed (i.i.d) prepared entangled states by LOCC [5,6]. Distillation from non-i.i.d prepared states has also been considered recently [7]. Distillable entanglement is a fundamental entanglement measure which captures the resource character of entanglement. Up to now, how to calculate distillable entanglement for gen-* Electronic address: kun.fang-

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Section: Numerical Estimation Of Rains Boundmentioning

confidence: 82%

Non-Asymptotic Entanglement Distillation

Fang

Wang

Tomamichel

et al. 2019

IEEE Trans. Inform. Theory

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“…• Capacity of classical-quantum channel • Relative entropy of recovery Some tailored algorithms have previously been developed for some of these quantities [ZFG10,SSMER16] and we show that the SDP-based approximations of [FSP18] are in general much faster in addition to being more flexible. As an application, we provide a numerical counterexample for an inequality that was recently proposed in [LW14,BHOS15] concerning the relative entropy of recovery.…”

Section: Introductionmentioning

confidence: 93%

Efficient optimization of the quantum relative entropy

Fawzi

2018

J. Phys. A: Math. Theor.

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Many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states. For example, the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states. The various capacities of quantum channels can also be written in this way. We propose a unified framework to numerically compute these quantities using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in [Fawzi, Saunderson, Parrilo, Semidefinite approximations of the matrix logarithm, arXiv:1705.00812]. As a notable application, this method allows us to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery.

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“…One does not have closed expressions for the relative entropy of entanglement in general [55]. Although there are numerical methods based on semidefinite programming [56], this technique is ill-suited in terms of computational time and memory requirements for continuous variable systems; in this paper, we will simply use the Gaussian relative entropy of entanglement as an approximation. We show that the approximation is good in the regime that we care about.…”

Section: F Relative Entropy Of Entanglementmentioning

confidence: 99%

Entanglement properties of a measurement-based entanglement distillation experiment

et al. 2019

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Measures of entanglement can be employed for the analysis of numerous quantum information protocols. Due to computational convenience, logarithmic negativity is often the choice in the case of continuous variable systems. In this work, we analyse a continuous variable measurement-based entanglement distillation experiment using a collection of entanglement measures. This includes: logarithmic negativity, entanglement of formation, distillable entanglement, relative entropy of entanglement, and squashed entanglement. By considering the distilled entanglement as a function of the success probability of the distillation protocol, we show that the logarithmic negativity surpasses the bound on deterministic entanglement distribution at a relatively large probability of success. This is in contrast to the other measures which would only be able to do so at much lower probabilities, hence demonstrating that logarithmic negativity alone is inadequate for assessing the performance of the distillation protocol. In addition to this result, we also observed an increase in the distillable entanglement by making use of upper and lower bounds to estimate this quantity. We thus demonstrate the utility of these theoretical tools in an experimental setting.

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Numerical estimation of the relative entropy of entanglement

Cited by 31 publications

References 16 publications

Non-Asymptotic Entanglement Distillation

Non-Asymptotic Entanglement Distillation

Efficient optimization of the quantum relative entropy

Entanglement properties of a measurement-based entanglement distillation experiment

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