Local basis-dependent noise-induced Bell-nonlocality sudden death in tripartite systems

doi:10.1016/j.physleta.2007.11.036

Physics Letters A

Volume 372, Issue 13, 24 March 2008, Pages 2212-2216

https://doi.org/10.1016/j.physleta.2007.11.036 Get rights and content

Abstract

We demonstrate that multipartite Bell-inequality violations can be fully destroyed in finite time in three-qubit systems subject only to the mechanism of local external asymptotic dephasing noise. This broadens the study of local-noise-induced sudden death of nonlocal behavior, extending it beyond the realm of bipartite systems, to which it had previously been restricted.

Introduction

It has been increasingly recognized that the rates of loss of joint-state coherence and of entanglement, two fundamental characteristics of quantum states, may differ within a composite system subject to local external noise [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Moreover, it has been shown that for some classes of states entanglement sudden death (ESD) [8], the disentanglement of bipartite systems in finite time subject only to the mechanism of basis-dependent local phase noise, occurs. Thus, qualitative as well as quantitative differences in coherence and nonlocality have been demonstrated [4], [5], [6], [7], [8], [9], [10]. Here, this investigation is further advanced.

Previously, such differences of noise-induced behavior have been carefully explored only in bipartite systems. The study of ESD under local dephasing noise has not been demonstrated in any multipartite system of more than two components because properly quantifying multipartite entanglement, particularly for mixed states which it involves by necessity, is problematic for systems of more than two qudits [11], [12]. Nonetheless, as we demonstrate here, one may still demonstrate the existence of local-noise induced death of nonlocal behavior with tools currently at hand. In particular, one can find classes of states in which generalized Bell-nonlocality in multipartite systems can go to zero in finite time while state coherence continues to be maintained for all finite times, an effect which we term Bell-nonlocality sudden death (BNSD). Here, BNSD is demonstrated in the tripartite context: the destruction of nonlocality as measured by the extent of violation of tripartite Bell inequalities in finite time under basis-dependent multi-local asymptotic dephasing noise is demonstrated in a class of initially Bell-nonlocal pure states of three-qubit systems, namely, the W class of states.

The extension of the consideration of the sudden death of nonlocal properties due to local dephasing noise beyond the bipartite case to the tripartite case is important because tripartite systems can exhibit fundamental characteristics impossible in bipartite systems, even at the three-qubit level, for example, see [13], [14]. The exhibition of BNSD illuminates the quantum–classical transition, quantum measurement, and quantum information processing where joint-state coherence and nonlocality are typically considered crucial. Our demonstration of BNSD shows that quantum information processing may be even more challenging to carry out in a noisy environment than previously thought.

Section snippets

Model: Initial state and noise model

There are two distinct classes of entangled pure states for three-qubit systems, the GHZ class and the W class, each represented by a characteristic state related to all others of their class by stochastic local operations and classical communication [15].¹ Here, we restrict our attention to systems initially prepared in pure states of the W class in order to show that tripartite

Bell-nonlocality sudden death

In the multi-local noise environment described in the previous section, when a three-qubit system is prepared at the initial time $t = 0$ in the generic pure W state $ρ (0) = | W_{g} 〉〈 W_{g} |$ , the time-evolved state $ρ (t)$ , that is, the solution of Eq. (2) is $ρ (t) = (\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {| {\bar{a}}_{1} |}^{2} & {\bar{a}}_{1} {\bar{a}}_{2}^{*} γ_{B} γ_{C} & 0 & {\bar{a}}_{1} {\bar{a}}_{4}^{*} γ_{A} γ_{C} & 0 & 0 & 0 \\ 0 & {\bar{a}}_{2} {\bar{a}}_{1}^{*} γ_{B} γ_{C} & {| {\bar{a}}_{2} |}^{2} & 0 & {\bar{a}}_{2} {\bar{a}}_{4}^{*} γ_{A} γ_{B} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {\bar{a}}_{4} {\bar{a}}_{1}^{*} γ_{A} γ_{C} & {\bar{a}}_{4} {\bar{a}}_{2}^{*} γ_{A} γ_{B} & 0 & {| {\bar{a}}_{4} |}^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}) .$ The off-diagonal elements of this matrix undergo a simple exponential decay, the three-qubit state fully

Conclusions

We have demonstrated the destruction of Bell-nonlocal behavior under basis-dependent local asymptotic dephasing noise, as measured by the extent of violation of a tripartite Bell inequality, in finite time while state coherence remains for all finite times in a class of initial states of three-qubit systems. This illuminates the quantum–classical transition, and quantum information processing in particular, because it shows in the multipartite context that nonlocal behavior can be lost simply

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Local basis-dependent noise-induced Bell-nonlocality sudden death in tripartite systems

Abstract

Introduction

Section snippets

Model: Initial state and noise model

Bell-nonlocality sudden death

Conclusions

Opt. Commun.

Phys. Lett. A

Phys. Rev. B

Phys. Rev. B

Phys. Rev. A

Phys. Rev. Lett.

Phys. Rev. Lett.

Science

Phys. Rev. A

Phys. Rev. A