Arleta Szkoła

Asymptotic Error Rates in Quantum Hypothesis Testing

We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing.The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support.The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein’s Lemma and quantum Sanov’s theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.

THE CHERNOFF LOWER BOUND FOR SYMMETRIC QUANTUM HYPOTHESIS TESTING

We consider symmetric hypothesis testing in quantum statistics, where the hypotheses are density operators on a finite-dimensional complex Hilbert space, representing states of a finite quantum system. We prove a lower bound on the asymptotic rate exponents of Bayesian error probabilities. The bound represents a quantum extension of the Chernoff bound, which gives the best asymptotically achievable error exponent in classical discrimination between two probability measures on a finite set. In our framework, the classical result is reproduced if the two hypothetic density operators commute. Recently, it has been shown elsewhere [Phys. Rev. Lett. 98 (2007) 160504] that the lower bound is achievable also in the generic quantum (noncommutative) case. This implies that our result is one part of the definitive quantum Chernoff bound.

A Quantum Version of Sanov's Theorem

We present a quantum version of Sanovs theorem focussing on a hypothesis testing aspect of the theorem: There exists a sequence of typical subspaces for a given set Ψ of stationary quantum product states asymptotically separating them from another fixed stationary product state. Analogously to the classical case, the separating rate on a logarithmic scale is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set Ψ. While in the classical case the separating subsets can be chosen universally, in the sense that they depend only on the chosen set of i.i.d. processes, in the quantum case the choice of the separating subspaces depends additionally on the reference state.

Effective Complexity and Its Relation to Logical Depth

Effective complexity measures the information content of the regularities of an object. It has been introduced by Gell-Mann and Lloyd to avoid some of the disadvantages of Kolmogorov complexity. In this paper, we derive a precise definition of effective complexity in terms of algorithmic information theory. We analyze rigorously its basic properties such as effective simplicity of incompressible binary strings and existence of strings that have effective complexity close to their lengths. Since some of the results have appeared independently in the context of algorithmic statistics by Gács , we discuss the relation of effective complexity to the corresponding complexity measures, in particular to Kolmogorov minimal sufficient statistics. As our main new result we show a remarkable relation between effective complexity and Bennetts logical depth: If the effective complexity of a string x exceeds a certain explicit threshold then that string must have astronomically large depth; otherwise, the depth can be arbitrarily small.

Typical Support and Sanov Large Deviations of Correlated States

Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov’s theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states.

Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno’s Theorem

In classical information theory, entropy rate and algorithmic complexity per symbol are related by a theorem of Brudno. In this paper, we prove a quantum version of this theorem, connecting the von Neumann entropy rate and two notions of quantum Kolmogorov complexity, both based on the shortest qubit descriptions of qubit strings that, run by a universal quantum Turing machine, reproduce them as outputs.

The Data Compression Theorem for Ergodic Quantum Information Sources

We extend the data compression theorem to the case of ergodic quantum information sources. Moreover, we provide an asymptotically optimal compression scheme which is based on the concept of high probability subspaces. The rate of this compression scheme is equal to the von Neumann entropy rate.

Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach

We study the continuity of an abstract generalization of the maximum-entropy inference - a maximizer. It is defined as a right-inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body. Using convex geometry we prove, amongst others, the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations. Further, we use numerical range methods in the case of quantum inference which refers to two observables. One result is a complete characterization of points of discontinuity for

An asymptotic error bound for testing multiple quantum hypotheses

3\times 3

Exponential error rates in multiple state discrimination on a quantum spin chain

matrices.