Maps on positive definite matrices preserving Bregman and Jensen divergences

Molnár, Lajos; Pitrik, József; Virosztek, Dániel

doi:10.1016/j.laa.2016.01.010

Cited by 16 publications

(18 citation statements)

References 11 publications

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“…As for Bregman divergences (see, e.g., Section 1 in [12]), we do something similar. We write A = t I , t > 0 and B = s I , s > 0 into…”

Section: Proofssupporting

confidence: 55%

“…However, let us begin with the following remark. We have already mentioned that in our previous works [12] and [18] we presented structural results concerning maps on the positive definite or semidefinite cones preserving Bregman divergences, or Jensen divergences, or f -divergences. Therefore, it is necessary to make it clear that what we obtain in the present paper, our main result Theorem 3 is a really new result, it is independent from the previous ones.…”

Section: Proofsmentioning

confidence: 99%

“…In [12] we described the structure of all bijective maps on the positive definite cone L ++ (H ) which preserve the Bregman divergence corresponding to any differentiable convex function on the positive reals with derivative bounded from below and unbounded from above. In addition, we considered the cases of the particular functions x → x log x, x > 0 (von Neumann divergence, in other words, Umegaki's relative entropy) and that of x → − log x (Stein's loss).…”

Section: Introductionmentioning

confidence: 99%

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Maps on positive definite operators preserving the quantum $$\chi _\alpha ^2$$ χ α 2 -divergence

et al. 2017

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ABSTRACT. We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least twodimensional complex Hilbert space which preserve the quantum χ 2 α -divergence for some α ∈ [0, 1]. We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.

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“…As for Bregman divergences (see, e.g., Section 1 in [12]), we do something similar. We write A = t I , t > 0 and B = s I , s > 0 into…”

Section: Proofssupporting

confidence: 55%

Section: Proofsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Maps on positive definite operators preserving the quantum $$\chi _\alpha ^2$$ χ α 2 -divergence

et al. 2017

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“…We denote its integral by G(x) =´x x0 g(x )dx where x 0 will be chosen later. The Bregman matrix divergence [43,44] is [45] …”

Section: Quantum Generalized CI and Thermodynamic Coherence Measuresmentioning

confidence: 99%

Additional energy-information relations in thermodynamics of small systems

Uzdin

2017

Phys. Rev. E

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The Clausius inequality (CI) form of the second law of thermodynamics relates information changes (entropy) to changes in the first moment of the energy (heat and indirectly also work). Are there similar relations between other moments of the energy distribution, and other information measures, or is the Clausius inequality a one of a kind instance of the energy-information paradigm? If there are additional relations, can they be used to make predictions on measurable quantities? Changes in the energy distribution beyond the first moment (average heat or work) are especially important in small systems which are often very far from thermal equilibrium. The generalized Clausius inequalities (GCI's), here derived, provide positive answers to the two questions above and add another layer to the fundamental connection between energy and information. To illustrate the utility of the new GCI's, we find scenarios where the GCI's yield tighter constraints on performance (e.g. in thermal machines) compared to the second law. To obtain the GCI's we use the Bregman divergence -a mathematical tool found to be highly suitable for energy-information studies. The quantum version of the GCI's provides a thermodynamic meaning to various quantum coherence measures. It is intriguing to fully map the regime of validity of the GCI's and extend the present results to more general scenarios including continuous systems and particles exchange with the baths.

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“…As recent literature on investigations in this direction we mention our paper [10] where bijective maps on the cone of positive definite matrices preserving rather general Bregman divergences or Jensen divergences were described and also the paper [17] by Virosztek who managed to determine the structure of all bijective maps on the cone of all positive semidefinite matrices which preserve a general quantum f -divergence.…”

Section: Introductionmentioning

confidence: 99%