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Register a new userQuantifying entanglement in terms of an operational way
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Quantifying entanglement is one of the most important tasks in the entanglement theory. In this paper, we establish entanglement monotones in terms of an operational approach, which is closely connected with the state conversion from pure states to the objective state by the local operations and classical communications (LOCC). It is shown that any good entanglement quantifier defined on pure states can induce an entanglement monotone for all density matrices. We especially show that our entanglement monotone is the maximal one among all that have the same form for pure states. In some particular cases, our proposed entanglement monotones turned to be equivalent to the convex roof construction, which hence gains an operational meaning. Some examples are given to demonstrate the different cases.
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Based on the monogamy of entanglement, we develop the technique of quantum conditioning to build an {it additive} entanglement measure: the conditional entanglement of mutual information. Its {it operational} meaning is elaborated to be the minimal net flow of qubits in the process of partial state merging. The result and conclusion can also be generalized to multipartite entanglement cases.
We introduce two operational entanglement measures which are applicable for arbitrary multipartite (pure or mixed) states. One of them characterizes the potentiality of a state to generate other states via local operations assisted by classical communication (LOCC) and the other the simplicity of generating the state at hand. We show how these measures can be generalized to two classes of entanglement measures. Moreover, we compute the new measures for pure few-partite systems and use them to characterize the entanglement contained in a three-qubit state. We identify the GHZ- and the W-state as the most powerful pure three-qubit states regarding state manipulation.
Quantifying quantum coherence is a key task in the resource theory of coherence. Here we establish a good coherence monotone in terms of a state conversion process, which automatically endows the coherence monotone with an operational meaning. We show that any state can be produced from some input pure states via the corresponding incoherent channels. It is especially found that the coherence of a given state can be well characterized by the least coherence of the input pure states, so a coherence monotone is established by only effectively quantifying the input pure states. In particular, we show that our proposed coherence monotone is the supremum of all the coherence monotones that give the same coherence for any given pure state. Considering the convexity, we prove that our proposed coherence measure is a subset of the coherence measure based on the convex roof construction. As an application, we give a concrete expression of our coherence measure by employing the geometric coherence of a pure state. We also give a thorough analysis on the states of qubit and finally obtain series of analytic coherence measures.
The unextendibility or monogamy of entangled states is a key property of quantum entanglement. Unlike conventional ways of expressing entanglement monogamy via entanglement measure inequalities, we develop a state-dependent resource theory to quantify the unextendibility of bipartite entangled states. First, we introduce a family of entanglement measures called unextendible entanglement. Given a bipartite state $rho_{AB}$, the key idea behind these measures is to minimize a divergence between $rho_{AB}$ and any possibly reduced state $rho_{AB}$ of an extension $rho_{ABB}$ of $rho_{AB}$. These measures are intuitively motivated by the fact that the more a bipartite state is entangled, the less that each of its individual systems can be entangled with a third party. Second, we show that the unextendible entanglement is an entanglement monotone under two-extendible operations, which include local operations and one-way classical communication as a special case. Unextendible entanglement has several other desirable properties, including normalization and faithfulness. As applications, we show that the unextendible entanglement provides efficiently computable benchmarks for the rate of perfect secret key distillation or entanglement distillation, as well as for the overhead of probabilistic secret key or entanglement distillation.
A new theory-independent noncontextuality inequality is presented [Phys. Rev. Lett. 115, 110403 (2015)] based on Kochen-Specker (KS) set without imposing the assumption of determinism. By proposing novel noncontextuality inequalities, we show that such result can be generalized from KS set to the noncontextuality inequalities not only for state-independent but also for state-dependent scenario. The YO-13 ray and $n$ cycle ray are considered as examples.
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