Dénes Petz

Quantum entropy and its use

I Entropies for Finite Quantum Systems.- 1 Fundamental Concepts.- 2 Postulates for Entropy and Relative Entropy.- 3 Convex Trace Functions.- II Entropies for General Quantum Systems.- 4 Modular Theory and Auxiliaries.- 5 Relative Entropy of States of Operator Algebras.- 6 From Relative Entropy to Entropy.- 7 Functionals of Entropy Type.- III Channeling Transformation and Coarse Graining.- 8 Channels and Their Transpose.- 9 Sufficient Channels and Measurements.- 10 Dynamical Entropy.- 11 Stationary Processes.- IV Perturbation Theory.- 12 Perturbation of States.- 13 Variational Expression of Perturbational Limits.- V Miscellanea.- 14 Central Limit and Quasi-free Reduction.- 15 Thermodynamics of Quantum Spin Systems.- 16 Entropic Uncertainty Relations.- 17 Temperley-Lieb Algebras and Index.- 18 Optical Communication Processes.

Quantum information theory and quantum statistics

Prerequisites from Quantum Mechanics.- Information and its Measures.- Entanglement.- More About Information Quantities.- Quantum Compression.- Channels and Their Capacity.- Hypothesis Testing.- Coarse-grainings.- State Estimation.- Appendix: Auxiliary Linear and Convex Analysis.

Monotone metrics on matrix spaces

Abstract The study of monotone inner products under stochastic mappings on the space of matrices was initiated by Morozova and Chentsov, motivated by information geometry. They did not show a monotone metric, but proposed several candidates. The main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them. It turns out that there is a correspondence between monotone metrics and operator means in the sense of Kubo and Ando. It follows that all proposals of Morozova and Chentsov are indeed monotone metrics.

The proper formula for relative entropy and its asymptotics in quantum probability

Umegakis relative entropyS(ω,ϕ)=TrDω(logDω−logDϕ) (of states ω and ϕ with density operatorsDω andDϕ, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegakis one. As a byproduct, the inequality TrA logAB ≧TrA(logA+logB) is obtained for positive definite matricesA andB.

Quasi-entropies for finite quantum systems

Abstract Convexity properties of entropy-like functionals on states of a finite dimensional algebra are discussed. The treatment covers both the quantum mechanical and the classical cases. The purpose is to generalize Liebs convexity theorem and the monotonicity of the relative entropy using the Jensen inequality of operator convex functions. From the quasi-entropies defined here the quantum version of Renyis α- entropies can be deduced.

Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality

We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so–called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically meaningful sense. This characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the Holevo bound on accessible information.

Geometries of quantum states

The quantum analog of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cramer–Rao‐type inequalities or the generalization of the Berry phase to mixed states. They are shown to be stochastically monotone here. All stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them. A class of metrics can be extended to pure states and a constant multiple of the Fubini–Study metric appears in the extension.

Sufficient subalgebras and the relative entropy of states of a von Neumann algebra

A subalgebraM0 of a von Neumann algebraM is called weakly sufficient with respect to a pair (φ,ω) of states if the relative entropy of φ and ω coincides with the relative entropy of their restrictions toM0. The main result says thatM0 is weakly sufficient for (φ,ω) if and only ifM0 contains the Radon-Nikodym cocycle [Dφ,Dω]t. Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.

Geometry of canonical correlation on the state space of a quantum system

A Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo–Mori/Bogoliubov scalar product). This metric is infinitesimally induced by the (nonsymmetric) relative entropy functional or the von Neumann entropy of density matrices. Hence its geometry expresses maximal uncertainty. It is proven that the metric is monotone under stochastic mappings, however, an example shows that it is not the only such Riemannian metric. This fact is remarkable because in the probabilistic case, the Markovian monotonicity property characterizes the Fisher information metric. The essential difference appears in the curvatures of a classical state space and a quantum one. A conjecture is made that the scalar curvature is monotone with respect to the ‘‘more mixed’’ (statistical) partial order of density matrices. Furthermore, an information inequality resembling the Cramer–Rao inequality of classical statistics is established. The inequality provides a lower bound for the c...

The Golden-Thompson trace inequality is complemented

We prove a class of trace inequalities which complements the Golden-Thompson inequality. For example, Tr(epA#epB)2/p⩽ Tr eA+B holds for all p > 0 when A and B are Hermitian matrices and # denotes the geometric mean. We also prove related trace inequalities involving the logarithmic function; namely p−1Tr X log Yp/2XpYp/2⩽ Tr X(log X+log Y) ⩽ p−1Tr X log Xp/2YpXp/2 for all p > 0 when X and Y are nonnegative matrices. These inequalities supply lower and upper bounds on the relative entropy.