There are different inequivalent ways to define the Renyi capacity of a channel for a fixed input distribution $P$. In a 1995 paper Csiszar has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an o
perational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of Renyi capacity, defined in terms of the sandwiched quantum Renyi divergences, has the same operational interpretation in the strong converse problem of classical-quantum channel coding. Denoting the constant composition strong converse exponent for a memoryless classical-quantum channel $W$ with composition $P$ and rate $R$ as $sc(W,R,P)$, our main result is that [ sc(W,R,P)=sup_{alpha>1}frac{alpha-1}{alpha}left[R-chi_{alpha}^*(W,P)right], ] where $chi_{alpha}^*(W,P)$ is the $P$-weighted sandwiched Renyi divergence radius of the image of the channel.
We present two general approaches to obtain the strong converse rate of quantum hypothesis testing for correlated quantum states. One approach requires that the states satisfy a certain factorization property; typical examples of such states are the
temperature states of translation-invariant finite-range interactions on a spin chain. The other approach requires the differentiability of a regularized Renyi $alpha$-divergence in the parameter $alpha$; typical examples of such states include temperature states of non-interacting fermionic lattice systems, and classical irreducible Markov chains. In all cases, we get that the strong converse exponent is equal to the Hoeffding anti-divergence, which in turn is obtained from the regularized Renyi divergences of the two states.
We show that the new quantum extension of Renyis alpha-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 alpha: for alpha<1, the right choice seems to be the traditional definition, whereas for alpha>1 the right choice is the newly introduced version. As a sideresult, we show that the new Renyi alpha-relative entropies are asymptotically attainable by measurements for alpha>1, and give a new simple proof for their monotonicity under completely positive trace-preserving maps.
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 replaci
ng 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.
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 sho
w 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.
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.
