11 research outputs found
Investigating Properties of a Family of Quantum Renyi Divergences
Audenaert and Datta recently introduced a two-parameter family of relative
R\'{e}nyi entropies, known as the --relative R\'{e}nyi entropies.
The definition of the --relative R\'{e}nyi entropy unifies all
previously proposed definitions of the quantum R\'{e}nyi divergence of order
under a common framework. Here we will prove that the
--relative R\'{e}nyi entropies are a proper generalization of the
quantum relative entropy by computing the limit of the - divergence
as approaches one and is an arbitrary function of . We
also show that certain operationally relevant families of R\'enyi divergences
are differentiable at . Finally, our analysis reveals that the
derivative at evaluates to half the relative entropy variance, a
quantity that has attained operational significance in second-order quantum
hypothesis testing.Comment: 15 pages, v2: journal versio
Monotonicity of a relative Rényi entropy
We show that a recent definition of relative Rényi entropy is monotone
under completely positive, trace preserving maps. This proves a recent
conjecture of Müller-Lennert et al. [“On quantum Rényi entropies: A
new definition, some properties,” J. Math. Phys. 54, 122203 (2013); e-print
arXiv:1306.3142v1; see also e-print arXiv:1306.3142]
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Convergence Rates for Quantum Evolution and Entropic Continuity Bounds in Infinite Dimensions
By extending the concept of energy-constrained diamond norms, we obtain
continuity bounds on the dynamics of both closed and open quantum systems in
infinite-dimensions, which are stronger than previously known bounds. We
extensively discuss applications of our theory to quantum speed limits,
attenuator and amplifier channels, the quantum Boltzmann equation, and quantum
Brownian motion. Next, we obtain explicit log-Lipschitz continuity bounds for
entropies of infinite-dimensional quantum systems, and classical capacities of
infinite-dimensional quantum channels under energy-constraints. These bounds
are determined by the high energy spectrum of the underlying Hamiltonian and
can be evaluated using Weyl's law
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Convergence Rates for Quantum Evolution and Entropic Continuity Bounds in Infinite Dimensions
Abstract: By extending the concept of energy-constrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite dimensions, which are stronger than previously known bounds. We extensively discuss applications of our theory to quantum speed limits, attenuator and amplifier channels, the quantum Boltzmann equation, and quantum Brownian motion. Next, we obtain explicit log-Lipschitz continuity bounds for entropies of infinite-dimensional quantum systems, and classical capacities of infinite-dimensional quantum channels under energy-constraints. These bounds are determined by the high energy spectrum of the underlying Hamiltonian and can be evaluated using Weyl’s law