Efficient optimization of the quantum relative entropy

Fawzi, Hamza; Fawzi, Omar

doi:10.1088/1751-8121/aab285

Cited by 59 publications

(74 citation statements)

References 23 publications

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“…The relative entropy is jointly convex, which implies that it is convex in its second argument, and therefore the problem above is a convex optimization problem. Because the relative entropy can be approximated through the use of semidefinite programming [14,15], it is possible to efficiently approximate the optimization problem (116) on a computer.…”

Section: Objective Functions Based On Relative Entropymentioning

confidence: 99%

Certifying optimality for convex quantum channel optimization problems

Coutts

Girard

Watrous

2021

Quantum

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We identify necessary and sufficient conditions for a quantum channel to be optimal for any convex optimization problem in which the optimization is taken over the set of all quantum channels of a fixed size. Optimality conditions for convex optimization problems over the set of all quantum measurements of a given system having a fixed number of measurement outcomes are obtained as a special case. In the case of linear objective functions for measurement optimization problems, our conditions reduce to the well-known Holevo-Yuen-Kennedy-Lax measurement optimality conditions. We illustrate how our conditions can be applied to various state transformation problems having non-linear objective functions based on the fidelity, trace distance, and quantum relative entropy.

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Section: Objective Functions Based On Relative Entropymentioning

confidence: 99%

Certifying optimality for convex quantum channel optimization problems

Coutts

Girard

Watrous

2021

Quantum

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“…But it is not known to be efficiently computable for general channels, due to its regularization. The entanglement-assisted quantum Q E is also a strong converse for the quantum capacity [54,55] and there is a recently developed approach to efficiently compute it [56]. Quantum capacity with symmetric side channels [13], denoted as Q ss , is also an important converse bound for general channels.…”

Section: Strong Converse Ratementioning

confidence: 99%

Semidefinite Programming Converse Bounds for Quantum Communication

Wang

Fang

Duan

2019

IEEE Trans. Inform. Theory

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We derive several efficiently computable converse bounds for quantum communication over quantum channels in both the one-shot and asymptotic regime. First, we derive oneshot semidefinite programming (SDP) converse bounds on the amount of quantum information that can be transmitted over a single use of a quantum channel, which improve the previous bound from [Tomamichel/Berta/Renes, Nat. Commun. 7, 2016]. As applications, we study quantum communication over depolarizing channels and amplitude damping channels with finite resources. Second, we find an SDP strong converse bound for the quantum capacity of an arbitrary quantum channel, which means the fidelity of any sequence of codes with a rate exceeding this bound will vanish exponentially fast as the number of channel uses increases. Furthermore, we prove that the SDP strong converse bound improves the partial transposition bound introduced by Holevo and Werner. Third, we prove that this SDP strong converse bound is equal to the so-called max-Rains information, which is an analog to the Rains information introduced in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. Our SDP strong converse bound is weaker than the Rains information, but it is efficiently computable for general quantum channels. * Electronic address: xwang93@umd.edu † Electronic address: kun.fang-1@student.uts.edu.au ‡ Electronic address: runyao.duan@uts.edu.au § This work is an extended version of the previous work in [1]. arXiv:1709.00200v3 [quant-ph] 30 Sep 2018

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“…with T B denoting the partial transpose. Appendix C details a Matlab program taking advantage of recent advances in [FSP18,FF18], in order to compute the Rains relative entropy of any bipartite state. Figure 5 plots the entanglement cost of the epolarizing channel for d = 2 (qubit input), and it also plots the Rains bound on distillable entanglement in (102).…”

Section: Epolarizing Channels (Complements Of Depolarizing Channels)mentioning

confidence: 99%

“…Now consider a generic channel N A→B with Kraus operators {N i } i so that an isometric extension is given by This appendix provides a brief listing of Matlab code that can be used to compute the Rains relative entropy of a bipartite state ρ AB [Rai01, ADMVW02]. The code requires the QuantInf package in order to generate a random state [Cub], the CVX package for semi-definite programming optimization [GB14], and the CVXQuad package [Faw] for relative entropy optimization [FSP18,FF18].…”

Section: Appendix B: Relation Between Choi State Of a Complementary Cmentioning

confidence: 99%

Entanglement cost and quantum channel simulation

Wilde

2018

Phys. Rev. A

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This paper proposes a revised definition for the entanglement cost of a quantum channel N . In particular, it is defined here to be the smallest rate at which entanglement is required, in addition to free classical communication, in order to simulate n calls to N , such that the most general discriminator cannot distinguish the n calls to N from the simulation. The most general discriminator is one who tests the channels in a sequential manner, one after the other, and this discriminator is known as a quantum tester [Chiribella et al., Phys. Rev. Lett., 101, 060401 (2008)] or one who is implementing a quantum co-strategy [Gutoski et al., Symp. Th. Comp., 565 (2007)]. As such, the proposed revised definition of entanglement cost of a quantum channel leads to a rate that cannot be smaller than the previous notion of a channel's entanglement cost [Berta et al., IEEE Trans. Inf. Theory, 59, 6779 (2013)], in which the discriminator is limited to distinguishing parallel uses of the channel from the simulation. Under this revised notion, I prove that the entanglement cost of certain teleportation-simulable channels is equal to the entanglement cost of their underlying resource states. Then I find single-letter formulas for the entanglement cost of some fundamental channel models, including dephasing, erasure, three-dimensional Werner-Holevo channels, epolarizing channels (complements of depolarizing channels), as well as single-mode pure-loss and pure-amplifier bosonic Gaussian channels. These examples demonstrate that the resource theory of entanglement for quantum channels is not reversible. Finally, I discuss how to generalize the basic notions to arbitrary resource theories.

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Efficient optimization of the quantum relative entropy

Cited by 59 publications

References 23 publications

Certifying optimality for convex quantum channel optimization problems

Certifying optimality for convex quantum channel optimization problems

Semidefinite Programming Converse Bounds for Quantum Communication

Entanglement cost and quantum channel simulation

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