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Register a new userWeak quasi-factorization for the Belavkin-Staszewski relative entropy
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Added by
Antonio P\\'erez Hern\\'andez
Publication date
2021
fields
Physics
and research's language is
English
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Quasi-factorization-type inequalities for the relative entropy have recently proven to be fundamental in modern proofs of modified logarithmic Sobolev inequalities for quantum spin systems. In this paper, we show some results of weak quasi-factorization for the Belavkin-Staszewski relative entropy, i.e. upper bounds for the BS-entropy between two bipartite states in terms of the sum of two conditional BS-entropies, up to some multiplicative and additive factors.
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In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard $f$-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal $f$-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.
The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-$sigma$-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.
In this paper a general definition of quantum conditional entropy for infinite-dimensional systems is given based on recent work of Holevo and Shirokov arXiv:1004.2495 devoted to quantum mutual and coherent informations in the infinite-dimensional case. The properties of the conditional entropy such as monotonicity, concavity and subadditivity are also generalized to the infinite-dimensional case.
Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Slomczynski and Zyczkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Luders-von Neumann instruments and partitions.
The quantum theory of indirect measurements in physical systems is studied. The example of an indirect measurement of an observable represented by a self-adjoint operator $mathcal{N}$ with finite spectrum is analysed in detail. The Hamiltonian generating the time evolution of the system in the absence of direct measurements is assumed to be given by the sum of a term commuting with $mathcal{N}$ and a small perturbation not commuting with $mathcal{N}$. The system is subject to repeated direct (projective) measurements using a single instrument whose action on the state of the system commutes with $mathcal{N}$. If the Hamiltonian commutes with the observable $mathcal{N}$ (i.e., if the perturbation vanishes) the state of the system approaches an eigenstate of $mathcal{N}$, as the number of direct measurements tends to $infty$. If the perturbation term in the Hamiltonian does textit{not} commute with $mathcal{N}$ the system exhibits jumps between different eigenstates of $mathcal{N}$. We determine the rate of these jumps to leading order in the strength of the perturbation and show that if time is re-scaled appropriately a maximum likelihood estimate of $mathcal{N}$ approaches a Markovian jump process on the spectrum of $mathcal{N}$, as the strength of the perturbation tends to $0$.
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