arXiv subjects

Milan Mosonyi

Publications and source records attributed to Milan Mosonyi.

13 recordsLinked to original sources

arXiv2018

Strong converse exponent for classical-quantum channel coding

We determine the exact strong converse exponent of classical-quantum channel coding, for every rate above the Holevo capacity. Our form of the exponent is an exact analogue of Arimoto's, given as a transform of the Renyi capacities with parameters alpha>1. It is important to note that, unlike in the classical case, there are many inequivalent ways to define the Renyi divergence of states, and hence the Rényi capacities of channels. Our exponent is in terms of the Renyi capacities corresponding to a version of the Renyi divergences that has been introduced recently in [Müller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013)], and [Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014)]. Our result adds to the growing body of evidence that this new version is the natural definition for the purposes of strong converse problems.

quant-ph↗

arXiv2017

Different quantum f-divergences and the reversibility of quantum operations

The concept of classical $f$-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the Rényi divergences. Various quantum versions of this concept, and more narrowly, the concept of Rényi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard $f$-divergences), Matsumoto's maximal $f$-divergences, measured $f$-divergences, and sandwiched and $α$-$z$-Rényi divergences. In this paper we give a systematic overview of the various concepts of quantum $f$-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum $f$-divergence by a quantum operation. In particular, we compare the standard and the maximal $f$-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum $f$-divergences are strictly different for non-commuting operators unless $f$ is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard $f$-divergences. We also study the monotonicity of the $α$-$z$-Rényi divergences under the special class of bistochastic maps that leave one of the arguments of the Rényi divergence invariant, and determine domains of the parameters $α,z$ where monotonicity holds, and where the preservation of the $α$-$z$-Rényi divergence implies the reversibility of the quantum operation.

math-ph↗

arXiv2014

Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies

We show that the new quantum extension of Renyi's α-relative entropies, introduced recently by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Renyi relative entropies depends on the parameter α: for α<1, the right choice seems to be the traditional definition, whereas for α>1 the right choice is the newly introduced version. As a sideresult, we show that the new Renyi α-relative entropies are asymptotically attainable by measurements for α>1, and give a new simple proof for their monotonicity under completely positive trace-preserving maps.

quant-ph↗

arXiv2013

A smooth entropy approach to quantum hypothesis testing and the classical capacity of quantum channels

We use the smooth entropy approach to treat the problems of binary quantum hypothesis testing and the transmission of classical information through a quantum channel. We provide lower and upper bounds on the optimal type II error of quantum hypothesis testing in terms of the smooth max-relative entropy of the two states representing the two hypotheses. Using then a relative entropy version of the Quantum Asymptotic Equipartition Property (QAEP), we can recover the strong converse rate of the i.i.d. hypothesis testing problem in the asymptotics. On the other hand, combining Stein's lemma with our bounds, we obtain a stronger ($\ep$-independent) version of the relative entropy-QAEP. Similarly, we provide bounds on the one-shot $\ep$-error classical capacity of a quantum channel in terms of a smooth max-relative entropy variant of its Holevo capacity. Using these bounds and the $\ep$-independent version of the relative entropy-QAEP, we can recover both the Holevo-Schumacher-Westmoreland theorem about the optimal direct rate of a memoryless quantum channel with product state encoding, as well as its strong converse counterpart.

quant-ph↗

arXiv2012

Quantum state discrimination bounds for finite sample size

In the problem of quantum state discrimination, one has to determine by measurements the state of a quantum system, based on the a priori side information that the true state is one of two given and completely known states, rho or sigma. In general, it is not possible to decide the identity of the true state with certainty, and the optimal measurement strategy depends on whether the two possible errors (mistaking rho for sigma, or the other way around) are treated as of equal importance or not. Results on the quantum Chernoff and Hoeffding bounds and the quantum Stein's lemma show that, if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copies, and the decay rate is given by a certain statistical distance between rho and sigma (the Chernoff distance, the Hoeffding distances, and the relative entropy, respectively). While these results provide a complete solution to the asymptotic problem, they are not completely satisfying from a practical point of view. Indeed, in realistic scenarios one has access only to finitely many copies of a system, and therefore it is desirable to have bounds on the error probabilities for finite sample size. In this paper we provide finite-size bounds on the so-called Stein errors, the Chernoff errors, the Hoeffding errors and the mixed error probabilities related to the Chernoff and the Hoeffding errors.

quant-ph↗

arXiv2009

Relation between the Dynamics of the Reduced Purity and Correlations

A general property of the relation between the dynamics of the reduced purity and correlations is investigated in quantum mechanical systems. We show that a non-zero time-derivative of the reduced purity of a system implies the existence of non-zero correlations with its environment under any unbounded Hamiltonians with finite variance. This shows the role of local dynamical information on the correlations, as well as the role of correlations in the mechanism of purity change.

quant-ph↗

arXiv2009

Generalized relative entropies and the capacity of classical-quantum channels

We provide lower and upper bounds on the information transmission capacity of one single use of a classical-quantum channel. The lower bound is expressed in terms of the Hoeffding capacity, that we define similarly to the Holevo capacity, but replacing the relative entropy with the Hoeffding distance. Similarly, our upper bound is in terms of a quantity obtained by replacing the relative entropy with the recently introduced max-relative entropy in the definition of the divergence radius of a channel.

quant-ph↗

arXiv2009

Quantum hypothesis testing with group symmetry

The asymptotic discrimination problem of two quantum states is studied in the setting where measurements are required to be invariant under some symmetry group of the system. We consider various asymptotic error exponents in connection with the problems of the Chernoff bound, the Hoeffding bound and Stein's lemma, and derive bounds on these quantities in terms of their corresponding statistical distance measures. A special emphasis is put on the comparison of the performances of group-invariant and unrestricted measurements.

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arXiv2008

Hypothesis testing for Gaussian states on bosonic lattices

The asymptotic state discrimination problem with simple hypotheses is considered for a cubic lattice of bosons. A complete solution is provided for the problems of the Chernoff and the Hoeffding bounds and Stein's lemma in the case when both hypotheses are gauge-invariant Gaussian states with translation-invariant quasi-free parts.

quant-ph↗

arXiv2008

Large deviations and Chernoff bound for certain correlated states on a spin chain

In this paper we extend the results of Lenci and Rey-Bellet on the large deviation upper bound of the distribution measures of local Hamiltonians with respect to a Gibbs state, in the setting of translation-invariant finite-range interactions. We show that a certain factorization property of the reference state is sufficient for a large deviation upper bound to hold and that this factorization property is satisfied by Gibbs states of the above kind as well as finitely correlated states. As an application of the methods the Chernoff bound for correlated states with factorization property is studied. In the specific case of the distributions of the ergodic averages of a one-site observable with respect to an ergodic finitely correlated state the spectral theory of positive maps is applied to prove the full large deviation principle.

math-ph↗

arXiv2008

Free energy density for mean field perturbation of states of a one-dimensional spin chain

Motivated by recent developments on large deviations in states of the spin chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expression of free energy density in the presence of a mean field type perturbation. We extend their results from the product state case to the Gibbs state case in the setting of translation-invariant interactions of finite range. In the special case of a locally faithful quantum Markov state, we clarify the relation between two different kinds of free energy densities (or pressure functions).

math-ph↗

arXiv2007

Error exponents in hypothesis testing for correlated states on a spin chain

We study various error exponents in a binary hypothesis testing problem and extend recent results on the quantum Chernoff and Hoeffding bounds for product states to a setting when both the null-hypothesis and the counter-hypothesis can be correlated states on a spin chain. Our results apply to states satisfying a certain factorization property; typical examples are the global Gibbs states of translation-invariant finite-range interactions as well as certain finitely correlated states.

quant-ph↗

arXiv2003

Structure of sufficient quantum coarse-grainings

Let H and K be Hilbert spaces and T be a coarse-graining from B(H) to B(K). Assume that density matrices D_1 and D_2 acting on H are given. In the paper the consequences of the existence of a coarse-graining S from B(K) to B(H) satisfying ST(D_1)=D_1 and ST(D_2)=D_2 are given. (This condition means the sufficiency of T for D_1 and D_2.) Sufficiency implies a particular decomposition of the density matrices. This decomposition allows to deduce the exact condition for equality in the strong subadditivity of the von Neumann entropy.

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