On entropy for general quantum systems

Banach Center Publications

We define a new quantum dynamical entropy for a C *-algebra automorphism with an invariant state (and for an appropriate 'approximating' subalgebra), which entropy is a 'hybrid' of the two alternative definitions by Connes, Narnhofer and Thirring resp. by Alicki and Fannes (and earlier, Lindblad). We report on this entropy's properties and on three examples.

Entropy

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.

The British Journal for the Philosophy of Science, 2003

Shenker has claimed that Von Neumann's argument for identifying the quantum mechanical entropy with the Von Neumann entropy, SðrÞ ¼ Àktrðr log rÞ, is invalid. Her claim rests on a misunderstanding of the idea of a quantum mechanical pure state. I demonstrate this, and provide a further explanation of Von Neumann's argument.

2021

Logical entropy gives a measure, in the sense of measure theory, of the distinctions of a given partition of a set, an idea that can be naturally generalized to classical probability distributions. Here, we analyze how fundamental concepts of this entropy and other related definitions can be applied to the study of quantum systems, leading to the introduction of the quantum logical entropy. Moreover, we prove several properties of this entropy for generic density matrices that may be relevant to various areas of quantum mechanics and quantum information. Furthermore, we extend the notion of quantum logical entropy to post-selected systems.

Advances in Mathematics of Communications, 2019

We propose a generalization of the quantum relative entropy by considering the geodesic on a manifold formed by all the invertible density matrices P. This geodesic is defined from a deformed exponential function ϕ which allows to work with a wider class of families of probability distributions. Such choice allows important flexibility in the statistical model. We show and discuss some properties of this proposed generalized quantum relative entropy.

Foundations of Physics, 2013

The classical concept of entropy was successfully extended to quantum mechanics by the introduction of the density operator formalism. However, further extensions to quantum decaying states have been hampered by conceptual difficulties associated to the particular nature of these states. In this work we address this problem, by (i) pointing out the difficulties that appear when one tries a consistent definition for this entropy, and (ii) building up a plausible formalism for it, which is based on the use of coherent complex states in the context of a path integration.

The novel concept of quantum logical entropy is presented. We prove several basic properties of this entropy with regard to density matrices. The properties are similar to those of the familiar von Neumann entropy, but it turns out that some of the proofs are easier using logical entropy. We thereby motivate a different approach for the assignment of quantum entropy to density matrices.

Communications in Mathematical Physics, 1987

The definition of the dynamical entropy is extended for automorphism groups of C* algebras. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.

The von Neumann entropy is a key quantity in quantum information theory and, roughly speaking, quantifies the amount of quantum information contained in a state when many identical and independent (i.i.d.) copies of the state are available, in a regime that is often referred to as being asymptotic. In this work, we provide a new operational characterization of the von Neumann entropy which neither requires an i.i.d. limit nor any explicit randomness. We do so by showing that the von Neumann entropy fully characterizes single-shot state transitions in unitary quantum mechanics, as long as one has access to a catalyst — an ancillary system that can be re-used after the transition — and an environment which has the effect of dephasing in a preferred basis. Building upon these insights, we formulate and provide evidence for the catalytic entropy conjecture, which states that the above result holds true even in the absence of decoherence. If true, this would prove an intimate connection between single-shot state transitions in unitary quantum mechanics and the von Neumann entropy. Our results add significant support to recent insights that, contrary to common wisdom, the standard von Neumann entropy also characterizes single-shot situations and opens up the possibility for operational single-shot interpretations of other standard entropic quantities. We discuss implications of these insights to readings of the third law of quantum thermodynamics and hint at potentially profound implications to holography.

Physical Review A, 2002

The theory of noncommutative dynamical entropy and quantum symbolic dynamics for quantum dynamical systems is analised from the point of view of quantum information theory. Using a general quantum dynamical system as a communication channel one can define different classical capacities depending on the character of resources applied for encoding and decoding procedures and on the type of information sources. It is shown that for Bernoulli sources the entanglement-assisted classical capacity, which is the largest one, is bounded from above by the quantum dynamical entropy defined in terms of operational partitions of unity. Stronger results are proved for the particular class of quantum dynamical systems-quantum Bernoulli shifts. Different classical capacities are exactly computed and the entanglementassisted one is equal to the dynamical entropy in this case.