Quantum Markov chains, sufficiency of quantum channels, and Rényi information measures

Datta, Nilanjana; Wilde, Mark M.

doi:10.1088/1751-8113/48/50/505301

Cited by 19 publications

(27 citation statements)

References 59 publications

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“…For example, in [20] it was used to characterize the case of equality in the strong subadditivity of the von Neumann entropy [30], giving rise to the concept of a short quantum Markov chain. Moreover, sparked by a breakthrough result by Fawzi and Renner [16] relating the notion of recoverability to states with small conditional mutual information, there has been a recent surge of interest in the topic of recoverability [8,7,45,46,52,15,28]. Note that strong subadditivity is equivalent to non-negativity of the quantum conditional mutual information, and hence there is an intimate connection between recoverability and saturation of strong subadditivity.…”

Section: Introductionmentioning

confidence: 99%

Data processing for the sandwiched Rényi divergence: a condition for equality

2016

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The α-sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for α ≥ 1/2. In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the α-sandwiched Rényi divergence for all α ≥ 1/2. For the range α ∈ [1/2, 1), our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki-Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1-11, 2012) about equality in the original Araki-Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki-Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.

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Section: Introductionmentioning

confidence: 99%

Data processing for the sandwiched Rényi divergence: a condition for equality

2016

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“…used the techniques of Fawzi and Renner (2015) to establish a non-trivial lower bound on a relative entropy difference. Datta and Wilde (2015) proved that ∆ α (ρ, σ, N ) ≥ 0 for all α ∈ (1/2, 1) ∪ (1, ∞) in addition to other related statements. Berta and Tomamichel (2016) proved that the fidelity of recovery is multiplicative with respect to tensor-product states, which then simplified the proof of the bound I(A; B|C) ρ ≥ − log F (A; B|C) ρ .…”

Section: History and Further Readingmentioning

confidence: 81%

Preface to the Second Edition

Wilde¹

2016

Quantum Information Theory

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“…The fact that these limits hold for ρ, σ, and N taken as in Definition 1 and subject to supp(ρ) ⊆ supp(σ) follows from [Wil15] and the development in Appendix A. [DW15] proved that for α…”

Section: Rényi Generalizations Of the Quantum Relative Entropy Differmentioning

confidence: 99%

“…is monotone in α and never exceeds one. The recent work [DW15] addressed this open question, first by generalizing it and then proving that…”

Section: Swiveled Rényi Quantum Information Measuresmentioning

confidence: 99%

Swiveled Rényi entropies

Dupuis

Wilde

2016

Quantum Inf Process

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This paper introduces "swiveled Rényi entropies" as an alternative to the Rényi entropic quantities put forward in [Berta et al., Phys. Rev. A 91, 022333 (2015)]. What distinguishes the swiveled Rényi entropies from the prior proposal of Berta et al. is that there is an extra degree of freedom: an optimization over unitary rotations with respect to particular fixed bases (swivels). A consequence of this extra degree of freedom is that the swiveled Rényi entropies are ordered, which is an important property of the Rényi family of entropies. The swiveled Rényi entropies are however generally discontinuous at α = 1 and do not converge to the von Neumann entropy-based measures in the limit as α → 1, instead bounding them from above and below. Particular variants reduce to known Rényi entropies, such as the Rényi relative entropy or the sandwiched Rényi relative entropy, but also lead to ordered Rényi conditional mutual informations and ordered Rényi generalizations of a relative entropy difference. Refinements of entropy inequalities such as monotonicity of quantum relative entropy and strong subadditivity follow as a consequence of the aforementioned properties of the swiveled Rényi entropies. Due to the lack of convergence at α = 1, it is unclear whether the swiveled Rényi entropies would be useful in one-shot information theory, so that the present contribution represents partial progress toward this goal.

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Quantum Markov chains, sufficiency of quantum channels, and Rényi information measures

Cited by 19 publications

References 59 publications

Data processing for the sandwiched Rényi divergence: a condition for equality

Data processing for the sandwiched Rényi divergence: a condition for equality

Preface to the Second Edition

Swiveled Rényi entropies

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