Bounds on general entropy measures (original) (raw)

An Upper Bound on Quantum Entropy

Following ref [1], a classical upper bound for quantum entropy is identified and illustrated, 0 ≤ Sq ≤ ln(eσ2/ 2􏰁), involving the variance σ2 in phase space of the classical limit distribution of a given system. A fortiori, this further bounds the corresponding information-theoretical generalizations of the quantum entropy proposed by Re ́nyi.

A classical bound on quantum entropy A classical bound on quantum entropy

A classical upper bound for quantum entropy is identified and illustrated, 0 S q ln(eσ 2 /2h), involving the variance σ 2 in phase space of the classical limit distribution of a given system. A fortiori, this further bounds the corresponding information-theoretical generalizations of the quantum entropy proposed by Rényi.

Quantum entanglement and entropy

Physical Review A, 2001

Entanglement is the fundamental quantum property behind the now popular field of quantum transport of information. This quantum property is incompatible with the separation of a single system into two uncorrelated subsystems. Consequently, it does not require the use of an additive form of entropy. We discuss the problem of the choice of the most convenient entropy indicator, focusing our attention on a system of 2 qubits, and on a special set, denoted by ℑ. This set contains both the maximally and the partially entangled states that are described by density matrices diagonal in the Bell basis set. We select this set for the main purpose of making more straightforward our work of analysis. As a matter of fact, we find that in general the conventional von Neumann entropy is not a monotonic function of the entanglement strength. This means that the von Neumann entropy is not a reliable indicator of the departure from the condition of maximum entanglement. We study the behavior of a form of non-additive entropy, made popular by the 1988 work by Tsallis. We show that in the set ℑ, implying the key condition of non-vanishing entanglement, this non-additive entropy indicator turns out

Relative Entropy of Entanglement and Restricted Measurements

Physical Review Letters, 2009

We introduce variants of relative entropy of entanglement based on the optimal distinguishability from unentangled states by means of restricted measurements. In this way, we are able to prove that the standard regularized entropy of entanglement is strictly positive for all multipartite entangled states. In particular, this implies that the asymptotic creation of a multipartite entangled state by means of local operations and classical communication always requires the consumption of a non-local resource at a strictly positive rate.

On upper bound for the quantum entropy

Linear Algebra and its Applications, 2001

A new upper bound for the von Neumann entropy of a state of a compound quantum system is given. This leads to a log-Sobolev inequality on the matrix algebra.

Connections between relative entropy of entanglement and geometric measure of entanglement

Quantum Information and Computation

As two of the most important entanglement measures---the entanglement of formation and the entanglement of distillation---have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures---the relative entropy of entanglement and the geometric measure of entanglement---are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.

Maximal entanglement versus entropy for mixed quantum states

Physical Review A, 2003

Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of (entanglement and mixedness) measures chosen. Moreover, for certain combinations, the forms of the maximally entangled mixed states can change discontinuously at a specific value of the entropy.

Comparison of the relative entropy of entanglement and negativity

Physical Review A, 2008

It is well known that for two qubits the upper bounds of the relative entropy of entanglement (REE) for a given concurrence as well as the negativity for a given concurrence are reached by pure states. We show that, by contrast, there are two-qubit mixed states for which the REE for some range of a fixed negativity is higher than that for pure states. Moreover, we demonstrate that a mixture of a pure entangled state and pure separable state orthogonal to it is likely to give the maximal REE. By noting that the negativity is a measure of entanglement cost under operations preserving positivity of partial transpose, our results provide an explicit example of operations such that, even though the entanglement cost for an exact preparation is the same, the entanglement of distillation of a mixed state can exceed that of pure states. This means that the entanglement manipulation via a pure state can result in a larger entanglement loss than that via a mixed state.

Bounds on relative entropy of entanglement for multi-party systems

Journal of Physics A: Mathematical and General, 2001

We present upper and lower bounds to the relative entropy of entanglement of multi-party systems in terms of the bi-partite entanglements of formation and distillation and entropies of various subsystems. We point out implications of our results to the local reversible convertibility of multi-party pure states and discuss their physical basis in terms of deleting of information.