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arXiv:0811.2047 · Physical Review A

Entanglement monogamy of multipartite higher-dimensional quantum systems using convex-roof extended negativity

Jeong San Kim, Anirban Das, Barry C. Sanders

Published Jan 30, 2009• 208 citations
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Abstract

We propose replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MOE). We show that all proven MOE relations using concurrence can be rephrased in terms of CREN. Furthermore, we show that higher-dimensional (qudit) extensions of MOE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MOE in terms of concurrence. We further test the CREN version of MOE for qudits by considering fully or partially coherent mixtures of a qudit $W$-class state with the vacuum and show that the CREN version of MOE for qudits is satisfied in this case as well. The CREN version of MOE for qudits is thus a strong conjecture with no obvious counterexamples.

DOI
10.1103/physreva.79.012329

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We propose replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MOE). We show that all proven MOE relations using concurrence can be rephrased in terms of CREN. Furthermore, we show that higher-dimensional (qudit) extensions of MOE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MOE in terms of concurrence. We further test the CREN version of MOE for qudits by considering fully or partially coherent mixtures of a qudit WWW-class state with the vacuum and show that the CREN version of MOE for qudits is satisfied in this case as well. The CREN version of MOE for qudits is thus a strong conjecture with no obvious counterexamples.
In this paper, we investigate the generalised monogamy inequalities of convex-roof extended negativity (CREN) in multi-level systems. The generalised monogamy inequalities provide the upper and lower bounds of bipartite entanglement, which are obtained by using CREN and the CREN of assistance (CRENOA). Furthermore, we show that the CREN of multi-qubit pure states satisfies some monogamy relations. Additionally, we test the generalised monogamy inequalities for qudits by considering the partially coherent superposition of a generalised W-class state in a vacuum and show that the generalised monogamy inequalities are satisfied in this case as well.
We propose the square of convex-roof extended negativity (SCREN) as a powerful candidate to characterize strong monogamy of multiparty quantum entanglement. We first provide a strong monogamy inequality of multiparty entanglement using SCREN and show that the tangle-based multiqubit strong-monogamy inequality can be rephrased by SCREN. We further show that the SCREN strong-monogamy inequality is still true for the counterexamples that violate tangle-based the strong-monogamy inequality in higher-dimensional quantum systems other than qubits. We also analytically show that SCREN strong-monogamy inequality is true for a large class of multiqudit states, a superposition of multiqudit generalized WWW-class states and vacuums. Thus SCREN is a good alternative to characterize the strong monogamy of entanglement even in multiqudit systems.
In this thesis I study multipartite entanglement sharing, and use entanglement to examine quantum phase transitions (QPT) and equilibrium in closed quantum systems. ? In the first part, I discuss on replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MOE). I show that all proven MOE relations using concurrence can be rephrased in terms of CREN. Furthermore, I show that higher-dimensional (qudit) extensions of MOE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MOE in terms of concurrence. I further test the CREN version of MOE for qudits by considering fully or partially coherent mixtures of a qudit W-class state with the vacuum and show that the CREN version of MOE for qudits is satisfied in this case as well. Hence I prove that the CREN version of MOE for qudits is a strong conjecture with no obvious counterexamples. ? In the second part, I show how entanglement entropy can be used to characterize phases and study equilibrium dynamics in closed quantum systems. As an example I study the Ising spin chain with a Dzyaloshinskii-Moriya interaction, focusing on the static and dynamic properties of the entanglement entropy. I show that the effects of the additional anisotropic interaction on the phase diagram and on the dynamics of the system are captured by the properties of the entanglement entropy. In particular, the model provides a way to study the quench dynamics in a system with an energy current. I consider quenches starting from an initial excited state of the Ising spin chain, and I analyze the effects of different initial conditions.
We provide evidence for the validity of strong monogamy inequality of multiparty quantum entanglement using the square of convex-roof extended negativity (SCREN). We first consider a large class of multiqudit mixed states that are in a partially coherent superposition of a generalized WWW-class state and the vacuum, and provide some useful properties about this class of states. We show that monogamy inequality of multiqudit entanglement in terms of SCREN holds for this class of states. We further show that SCREN strong monogamy inequality of multiqudit entanglement also holds for this class of states. Thus SCREN is a good alternative for characterizing the monogamous and strongly monogamous properties of multiqudit entanglement.
One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the αth power of logarithmic negativity, and a newly defined good measure of entanglement, LCREN, obey a class of general monogamy inequalities in multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems for α≥4ln2. We provide a class of general polygamy inequalities of multiqubit systems in terms of logarithmic convex-roof extended negativity of assistance (LCRENoA) for 0≤β≤2. Given that the logarithmic negativity and LCREN are not convex, these results are surprising. Using the power of the logarithmic negativity and LCREN, we further establish a class of tight monogamy inequalities of multiqubit systems, 2⊗2⊗3 systems and 2⊗2⊗2n systems in terms of the αth power of logarithmic negativity and LCREN for α≥4ln2. We also show that the βth power of LCRENoA obeys a class of tight polygamy inequalities of multiqubit systems for 0≤β≤2.