Entanglement-induced state ordering under local operations

Abstract

We analyze how entanglement between two components of a bipartite system behaves under the action of local channels of the form E ⊗ I. We show that a set of maximally entangled states is by the action of E ⊗ I transformed into the set of states that exhibit the same degree of entanglement. Moreover, this degree represents an upper bound on entanglement that is available at the output of the channel irrespective what is the input state of the composite system. We show that within this bound the the entanglement-induced state ordering is "relative" and can be changed by the action of local channels. That is, two states ̺ The success of quantum information theory [1,2] is intimately related to the phenomenon of quantum entanglement. The better we will understand properties of this purely quantum phenomenon the deeper will be our insight into the quantum realm. Even though the importance of quantum entanglement has been clearly acknowledged by founding fathers of quantum mechanics [3], the true potential of this phenomenon has been appreciated just recently with the development of quantum information science. Over last ten years many results of fundamental importance illuminating properties of quantum entanglement have been reported. In spite of all the progress, there are still many questions that are to be answered. In particular, criteria of non-separability of arbitrarily-dimensional bi-partite systems, the study of intrinsic multi-partite entanglement in composite quantum systems [4,5], or the role of quantum entanglement in macroscopic systems is presently under investigation [6]. One of the problems that has attracted interest of researchers for quite some time is the issue of "proper" measures of entanglement [7].In general, we can identify two conceptually different approaches in various attempt to define measures of entanglement. These can be named as i) the operational approach, and ii) the formal (abstract) approach. The operational approach is based on an assumption that there exists a process, or an information protocol, in which the quantum entanglement plays the role of a new resource that provides some improvement in the performance of the protocol compared to its "classical analogue". The second approach [7] is based on postulation of the desired properties that an entanglement measure has to satisfy and defines a functional with these properties (see below).In this paper we will adopt this second approach. We will start our discussion by addressing a specific question concerning the entanglement measures. First, we will answer the question: Do local operations preserve the entanglement-induced ordering? Then we will analyze how entangled states that exhibit the same degree of entanglement are transformed under the action of local channels.Let us start with a trivial observation: States of bipartite quantum systems can be either entangled or separable (this is almost a tautological statement since the presence of entanglement is defined as an absence of the separability, and vice versa). On the oth...

Ultimate entanglement robustness of two-qubit states against general local noises

“…In this paper, we extend these results to the case of general local unital channels Υ A 1 (t) ⊗ Υ B 2 (t) and prove that the maximally entangled state + is optimal for the transmission of entanglement through such channels. It is tempting to conclude that the maximally entangled state + exhibits ultimate robustness to general local two-qubit noises Φ A 1 (t) ⊗ Φ B 2 (t); however, this is not true [38][39][40] and we show that explic- arXiv:1708.08208v2 [quant-ph] 21 Jan 2018 itly in this paper. Moreover, we analytically find the initial two-qubit state , which is the most robust to a given nonunital local two-qubit dynamical map Φ 1 (t) ⊗ Φ 2 (t).…”

Entanglement sharing via qudit channels: Nonmaximally entangled states may be necessary for one-shot optimal singlet fraction and negativity

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