Scale-invariant freezing of entanglement

Titas Chanda, Tamoghna Das, Debasis Sadhukhan, Amit Kumar Pal, Aditi Sen(De), and Ujjwal Sen

Phys. Rev. A 97, 062324 – Published 15 June 2018

Abstract

We show that bipartite entanglement in an one-dimensional quantum spin model undergoing time evolution due to local Markovian environments can be frozen over time. We demonstrate this by using a number of paradigmatic quantum spin models including the anisotropic $X Y$ model in the presence of a uniform and an alternating transverse magnetic field (ATXY), the $X X Z$ model, the $X Y Z$ model, and the $J_{1} - J_{2}$ model involving the next-nearest-neighbor interactions. We show that the length of the freezing interval, for a chosen pair of nearest-neighbor spins, may remain independent of the length of the spin chain, for example, in paramagnetic phases of the ATXY model, indicating a scale invariance. Freezing of entanglement is found to be robust against a change in the environment temperature, the presence of disorder in the system, and whether the noise is dissipative or not dissipative. Moreover, we connect the freezing of entanglement with the propagation of information through a quantum many-body system, as considered in the Lieb-Robinson theorem. We demonstrate that the variation of the freezing duration exhibits a quadratic behavior against the distance of the nearest-neighbor spin pair from the noise source, obtained from exact numerical simulations, in contrast to the linear one as predicted by the Lieb-Robinson theorem.

Received 4 January 2017
Revised 13 February 2018

DOI:https://doi.org/10.1103/PhysRevA.97.062324

Physics Subject Headings (PhySH)

Open quantum systems & decoherence Phase diagrams Quantum entanglement Quantum information processing Quantum memories Quantum phase transitions XY model

1-dimensional spin chains

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

Titas Chanda¹, Tamoghna Das^1,2, Debasis Sadhukhan¹, Amit Kumar Pal^1,3, Aditi Sen(De)¹, and Ujjwal Sen¹

¹Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211019, India
²Institute of Informatics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-308 Gdańsk, Poland
³Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom

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Issue

Vol. 97, Iss. 6 — June 2018

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Images

Figure 1
Schematic representation of a 1D system of $L$ spins, of which $N_{d}$ spins, labeled as $d_{i}, i = 1, 2, ..., N_{d}$ , act as the doors and interact with independent environments, denoted by $E_{d_{i}}$ . The enlarged portion describes the local repetitive interaction between the environment and a door in the system. The spin $d_{1}$ in the 1D QSM acts as the door and interacts with a copy of the environment for a short interval of time $δ t$ . In the $n th$ interval of duration $δ t$ , the interacting copy of the environment is $E_{n}^{d_{1}}$ . Note here that during the same $n th$ interval of duration $δ t$ , along with the door $d_{1}$ , the door $d_{i}$ in the system $(i \neq 1)$ is also interacting with the copy $E_{n}^{d_{i}}$ of the environment.
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Figure 2
Freezing dynamics of NN entanglement. The NN entanglement freezes in the PM-II phase of the ATXY model for (a) dissipative LRQI and (b) local phase-damping noise, where the time axis is in logarithmic scale; the system parameters used in this figure are given in Table 1. (c) Similar dynamics is observed in the case of the AFM phase in the TXXZ model, where we choose $Δ = 1.5$ and $h_{1} / J = 0.1$ . All the axes in all the figures are dimensionless.
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Figure 3
Time dynamics of entanglement of the spin pair $(1, 2)$ (i) for the PM-II phase of the ATXY model under LRQI, (ii) under local dephasing (with model parameters given in Table 1), and (iii) for the AFM phase of the TXXZ model under LRQI, where we choose $Δ = 1.5$ and $h_{1} / J = 0.1$ . All the axes in the figure are dimensionless.
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Figure 4
Scale invariance. The behavior of $τ_{F}^{i, i + 1}$ against $i$ , for $6 \leq L \leq 11$ , in the (a) PM-I, (b) PM-II, and (c) AFM phases of the ATXY model, with the chosen system parameters given in Table 1. Different point types correspond to different values of $L$ . In (a) and (b) the points corresponding to $L = 11$ are joined by a solid line, which clearly exhibits the monotonicity, while such monotonic behavior is not present in the case in (c). The green dashed curves, in all the figures, show the variation of freezing terminal $τ_{F}^{LR}$ , as predicted by the Lieb-Robinson theorem, with $i$ (see the discussion in Sec. 3d). All quantities plotted are dimensionless.
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Figure 5
Variation of NN quenched entanglement with respect to $t$ under LRQI in the case of the disordered ATXY model, with $h_{2}^{i} / J$ as the disordered system parameter. The values of $h_{2}^{i} / J$ are chosen from a Gaussian distribution of mean $〈 h_{2} / J 〉 = 1.2$ with standard deviation 0.3 and we set $γ = 0.8$ and $h_{1}^{i} / J = 0$ . The time axis is in logarithmic scale and all the axes are dimensionless.
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Figure 6
Variation of $τ_{F}^{10, 11}$ with $N_{d}$ in the PM-I phase of the ATXY model with $L = 11$ , where doors are added one by one in the system, starting from spin 1. All quantities plotted are dimensionless.
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