Disorder-Robust Entanglement Transport

Clemens Gneiting, Daniel Leykam, and Franco Nori

Phys. Rev. Lett. 122, 066601 – Published 12 February 2019

Abstract

We study the disorder-perturbed transport of two noninteracting entangled particles in the absence of backscattering. This situation is, for instance, realized along edges of topological insulators. We find profoundly different responses to disorder-induced dephasing for the center-of-mass and relative coordinates: While a mirror symmetry protects even highly delocalized relative states when resonant with the symmetry condition, delocalizations in the center of mass [e.g., two-particle ( $N = 2$ ) $N 00 N$ states] remain fully sensitive to disorder. We demonstrate the relevance of these differences to the example of interferometric entanglement detection. Our platform-independent analysis is based on the treatment of disorder-averaged quantum systems with quantum master equations.

Received 23 July 2018
Revised 7 November 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.066601

Physics Subject Headings (PhySH)

Edge states Quantum entanglement Topological insulators Transport phenomena

Methods in transport Quantum master equation

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Clemens Gneiting^1,*, Daniel Leykam², and Franco Nori^1,3

¹Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
²Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea
³Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*clemens.gneiting@riken.jp

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Vol. 122, Iss. 6 — 15 February 2019

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Figure 1
(a) Disorder-perturbed transport of two-particle entangled edge modes. While topologically protected against backscattering, perturbations (schematic, blue) along the paths of the particles still cause disorder-induced dephasing, deteriorating the possibility to detect and/or harness their entanglement. (b) Disorder-induced dephasing degrades, e.g., two-particle coherence effects, such as (anti-)bunching at beam splitters. If cascaded, the disorder impact accumulates.
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Figure 2
Disorder influence [Eq. (4b)] for a pair of propagating edge modes [Gaussian disorder correlations, $t = 10 ℓ / v$ , values increase from 0 (blue)]. (a) While the center-of-mass coordinate shows a dephasing behavior similar to a single particle, with coherences between coordinates $x_{cm}$ and $x_{cm}^{'}$ homogeneously degrading with their increasing separation, (b) coherences between mirror points $x_{rel}$ and $- x_{rel}$ of the relative coordinate remain unaffected by disorder-induced dephasing, regardless of their spatial separation. This is because both particles acquire the same disorder phases which thus cancel out.
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Figure 3
Disorder-perturbed evolution of the relative-state superposition (5). (a) While the center-of-mass state $⟨ x_{cm}, x_{L} | {\bar{ρ}}_{RI} (t) | x_{cm}^{'}, x_{L} ⟩$ undergoes a decay of coherences, (b) the correlations in $⟨ (x_{1}, x_{2} | {\bar{ρ}}_{RI} (t) | x_{1}, x_{2} ⟩$ between the particle coordinates $x_{1}$ and $x_{2}$ remain unaffected. Depending on how well the mirror condition $x_{L} = - x_{R}$ is met, the visibility loss in the interference pattern displayed by the relative momentum is (c) controllable or (d) exceedingly detrimental.
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Figure 4
Disorder-perturbed transport of the two-particle ( $N = 2$ ) $N 00 N$ state (7). (a) The mirror-point symmetry protects the coherences in the relative coordinate. (b) Its absence in the delocalized center-of-mass coordinate, however, causes rapid and substantial visibility loss in the center-of-mass momentum interference, highlighting the difference in the disorder sensitivity between different choices of entangled states.
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Figure 5
Disorder-perturbed evolution of relative-state superpositions Eq. (5) in the Haldane model. (a) Correlations in real space coordinates $x_{1}$ and $x_{2}$ exhibit broadening due to the edge state dispersion. (b) Interference in the relative momentum is only robust when the mirror condition is satisfied (solid red line). The mirror-broken state $x_{L} = - 32$ , $x_{R} = 8$ (blue dashed line) has significantly lower visibility.
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