Information, Relative Entropy of Entanglement, and Irreversibility

Abstract

Previously proposed measures of entanglement, such as entanglement of formation and assistance, are shown to be special cases of the relative entropy of entanglement. The difference between these measures for an ensemble of mixed states is shown to depend on the availability of classical information about particular members of the ensemble. Based on this, relations between relative entropy of entanglement and mutual information are derived.In quantifying entanglement, a number of measures have been proposed. For bipartite pure states, ρ AB , the Von Neumann entropy of the reduced density matrix of either subsystem, ρ A or ρ B , has been found to be a good and unique measure, [1], [2]. Relative entropy of entanglement has been proposed as a measure which extends to mixed states, [3], [4]. Loosely speaking, it quantifies how 'far' an entangled state is from the set of disentangled states. Entanglement of mixed states has also been characterised by the 'entanglement of formation', [5], [6], and by the 'entanglement of distillation', [5]. Rather surprisingly, use of entanglement in mixed states is not reversible in the sense that all the entanglement required to construct a particular mixed state cannot be distilled out again, so the entanglement of formation is greater than the entanglement of distillation, [4]. In this paper, we clarify the role of classical information about the identity of particular members of an ensemble of mixed states, and show that the loss of such information is responsible for the difference between the entanglement of formation and the entanglement of distillation. We provide a unifying frame-work for entanglement measures by showing how previously proposed measures are special cases of the relative entropy of entanglement. This gives a strong 1

The role of relative entropy in quantum information theory

“…There is a corresponding reduction in the relative entropy of entanglement, from the entanglement of the original pure state, E RE ( ψ (MA):B ψ (MA):B |), to the entanglement of the mixed state E RE (ρ ε A:(BM) ) for all decompositions ε arising as the result of an orthogonal measurement on the memory. It is possible to prove, using the non-increase of relative entropy under local operations, that when the mutual information loss is added to the relative entropy of entanglement of the mixed state E RE (ρ ε A:(BM) ), the result is greater than the relative entropy of entanglement of the original pure state, E RE ( ψ (MA):B ψ (MA):B |), (Henderson and Vedral, 2000).…”

Describing mixed spin-space entanglement of pure states of indistinguishable particles using an occupation-number basis

“…It is already known that for such states the von Neumann entropy provides the correct measure of entanglement between two distinguishable subsystems [3]. We do not address the case of an overall mixed state, for which the definition of an entanglement measure is more subtle [4].…”

Quantum jumps and entropy production

“…On the ground of the weak coupling master equation for the open system's density matrix the corresponding expressions can again be shown to have useful convexity properties and to satisfy a minimal entropy production principle [7], as well as a kind of quantum Onsager relations [8]. The concept of relative entropy is also useful in quantum information theory [9], e. g. in the quantification of entanglement [10,11] and of the success of quantum teleportation [12]. A general account of the theory of quantum entropy may be found in [13].…”