Quantum entropies, Schur concavity and dynamical semigroups

Aniello, Paolo

doi:10.1088/1742-6596/804/1/012003

Cited by 7 publications

(7 citation statements)

References 24 publications

(51 reference statements)

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“…They are formed by those density operators of a (finite) fixed rank whose eigenvalues form a uniform (i.e., constant) probability distribution. Note, moreover, that the set S k (H) u has a precise meaning in relation with the natural majorization preorder in S (H) (see [21,22], and references therein). It consists of those elements that are minimal, with respect to majorization, among all elements of S (H) of rank not larger than k.…”

Section: Final Remarks and Conclusionmentioning

confidence: 99%

Symmetry witnesses

Aniello¹,

Chruściński²

2017

J. Phys. A: Math. Theor.

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A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with respect to an injective densely defined linear operator in the Banach space of selfadjoint trace class operators (if and) only if this operator is a symmetry transformation. According to a linear version of Wigner's theorem, the set of pure states -the rank-one projections -is a symmetry witness. We show that an analogous result holds for the set of projections with a fixed rank (with some mild constraint on this rank, in the finite-dimensional case). It turns out that this result provides a complete classification of the sets of projections with a fixed rank that are symmetry witnesses. These particular symmetry witnesses are projectable; i.e., reasoning in terms of quantum states, the sets of 'uniform' density operators of corresponding fixed rank are symmetry witnesses too.

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Section: Final Remarks and Conclusionmentioning

confidence: 99%

Symmetry witnesses

Aniello¹,

Chruściński²

2017

J. Phys. A: Math. Theor.

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“…Notice that S α (ρ) is of the form f α (ρ), where f α is a Schur-concave function on the convex set of states, which attains a strict global maximum at ρ = ρ ⋆ ; see sect. 2 of [56] (also see remarks 9 and 10 ibidem), or propositions 2 and 3 of [57]. Therefore, by lemma 1 of [56], and taking into account remark 8 ibidem, S α (ρ) is not decreased by a positive trace-preserving map if and only if this is unital.…”

Section: Entropic Criteria Of Non-markovianitymentioning

confidence: 83%

“…where S α (ρ) = 1 1−α log Trρ α stands for the Rényi entropy of order α, for α ∈ (0, 1) ∪ (1, +∞), and S 1 (ρ) = lim α→1 S α (ρ). Therefore, using results from the recent papers [56,57], one can easily derive the following Proposition 6. Let t → Λ t be P-divisible and α ∈ (0, +∞).…”

Section: Entropic Criteria Of Non-markovianitymentioning

confidence: 99%

Quantum entropy and non-Markovian evolution

Aniello,

Bae,

Chruscinski

2018

Preprint

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Entropy, and its temporal evolution, play a central role in the foundations of quantum theory and in modern quantum technologies. Here we study, in particular, the relations between thein general, non-Markovian -evolution of an open quantum system, the notions of divisibility of a dynamical map and of distinguishability of quantum states, and the temporal behaviour of various entropy-related quantities such as the Rényi (and sandwiched Rényi) divergences, and the so-called min-and max-conditional entropies. This, in turn, gives rise to an operational meaning of (non-)Markovianity.

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“…• The behaviour of a quantum entropy [57][58][59] -not necessarily the von Neumann entropy -wrt to a stochastic product is another interesting issue. In particular, it is natural to wonder whether the twirled stochastic product is, say, entropy-nondecreasing; i.e., whether the entropy of the product of two states is not smaller than the entropy of each of these states.…”

Section: Final Remarks Conclusion and Perspectivesmentioning

confidence: 99%

A class of stochastic products on the convex set of quantum states

Aniello¹

2019

J. Phys. A: Math. Theor.

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We introduce the notion of stochastic product as a binary operation on the convex set of quantum states (the density operators) that preserves the convex structure, and we investigate its main consequences. We consider, in particular, stochastic products that are covariant wrt a symmetry action of a locally compact group. We then construct an interesting class of group-covariant, associative stochastic products, the so-called twirled stochastic products. Every binary operation in this class is generated by a triple formed by a square integrable projective representation of a locally compact group, by a probability measure on that group and by a fiducial density operator acting in the carrier Hilbert space of the representation. The salient properties of such a product are studied. It is argued, in particular, that, extending this binary operation from the density operators to the whole Banach space of trace class operators, this space becomes a Banach algebra, a so-called twirled stochastic algebra. This algebra is shown to be commutative in the case where the relevant group is abelian. In particular, the commutative stochastic products generated by the Weyl system are treated in detail. Finally, the physical interpretation of twirled stochastic products and various interesting connections with the literature are discussed.

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Quantum entropies, Schur concavity and dynamical semigroups

Cited by 7 publications

References 24 publications

Symmetry witnesses

Symmetry witnesses

Quantum entropy and non-Markovian evolution

A class of stochastic products on the convex set of quantum states

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