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Communications in Theoretical Physics

Monogamy Relations for Squared Entanglement Negativity*

Feng Liu

Published Oct 1, 2016• 9 citations

Abstract

Abstract This paper contains two main contents. In the first part, we provide two counterexamples of monogamy inequalities for the squared entanglement negativity: one three-qutrit pure state which violates of the He—Vidal monogamy conjecture, and one four-qubit pure state which disproves the squared-negativity-based Regula—Martino—Lee—Adesso-class strong monogamy conjecture. In the second part, we investigate the sharing of the entanglement negativity in a composite cavity-reservoir system using the corresponding multipartite entanglement scores, and then we find that there is no simple dominating relation between multipartite entanglement scores and the entanglement negativity in composite cavity-reservoir systems. As a by-product, we further validate that the entanglement of two cavity photons is a decreasing function of the evolution time, and the entanglement will suddenly disappear interacting with independent reservoirs.

DOI: 10.1088/0253-6102/66/4/407

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Abstract This paper contains two main contents. In the first part, we provide two counterexamples of monogamy inequalities for the squared entanglement negativity: one three-qutrit pure state which violates of the He—Vidal monogamy conjecture, and one four-qubit pure state which disproves the squared-negativity-based Regula—Martino—Lee—Adesso-class strong monogamy conjecture. In the second part, we investigate the sharing of the entanglement negativity in a composite cavity-reservoir system using the corresponding multipartite entanglement scores, and then we find that there is no simple dominating relation between multipartite entanglement scores and the entanglement negativity in composite cavity-reservoir systems. As a by-product, we further validate that the entanglement of two cavity photons is a decreasing function of the evolution time, and the entanglement will suddenly disappear interacting with independent reservoirs.

Monogamy is a defining feature of entanglement, having far-reaching applications. Recently, Regula et al. [Phys. Rev. Lett. 113, 110501 (2014)] proposed a stronger version of monogamy relation for concurrence. We have extended the strong monogamy inequality for another entanglement measure, viz., negativity. In particular, we have concentrated on the four-qubit system and provided a detailed study on the status of strong monogamy on pure states. Further, we have analytically provided some classes of states for which negativity and squared negativity satisfy strong monogamy. Numerical evidences have also been shown in proper places. Our analysis also provides cases where strong monogamy is violated.

We analyze the entanglement distribution and the two-qubit residual entanglement in multipartite systems. For a composite system consisting of two cavities interacting with independent reservoirs, it is revealed that the entanglement evolution is restricted by an entanglement monogamy relation derived here. Moreover, it is found that the initial cavity-cavity entanglement evolves completely to the genuine four-partite cavities-reservoirs entanglement in the time interval between the sudden death of cavity-cavity entanglement and the birth of reservoir-reservoir entanglement. In addition, we also address the relationship between the genuine block-block entanglement form and qubit-block form in the interval.

The entanglement in multipartite quantum system is hard to characterize and quantify, although it has been intensively studied in bipartite systems. The monogamy of entanglement, as a special property of multipartite systems, shows the distribution of entanglement in the system. In this paper, we investigate the monogamy relations for multi-qubit systems. By using two entangled measures, namely the concurrence C and the negativity Nc, we establish tighter monogamy inequalities for their α-th power than those in all the existing ones. We also illustrate the tightness of our results for some classes of quantum states.

Entanglement negativity is a measure of mixed-state entanglement increasingly used to investigate and characterize emerging quantum many-body phenomena, including quantum criticality and topological order. We present two results for the entanglement negativity: a disentangling theorem, which allows the use of this entanglement measure as a means to detect whether a wave function of three subsystems

A,\phantom{\rule{0.28em}{0ex}}B

, and

C

factorizes into a product state for parts

A{B}_{1}

and

{B}_{2}C

; and a monogamy relation conjecture based on entanglement negativity, which states that if

A

is very entangled with

B

, then

A

cannot be simultaneously very entangled also with

C

.