Emergence of entanglement with temperature and time in factorization-surface states

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

Phys. Rev. A 97, 012316 – Published 16 January 2018

Abstract

There exist zero-temperature states in quantum many-body systems that are fully factorized, thereby possessing vanishing entanglement, and hence being of no use as resource in quantum information processing tasks. Such states can become useful for quantum protocols when the temperature of the system is increased, and when the system is allowed to evolve under either the influence of an external environment, or a closed unitary evolution driven by its own Hamiltonian due to a sudden change in the system parameters. Using the one-dimensional anisotropic XY model in a uniform and an alternating transverse magnetic field, we show that entanglement of the thermal states, corresponding to the factorization points in the space of the system parameters, revives once or twice with increasing temperature. We also study the closed unitary evolution of the quantum spin chain driven out of equilibrium when the external magnetic fields are turned off, and show that considerable entanglement is generated during the dynamics, when the initial state has vanishing entanglement. Interestingly, we find that creation of entanglement for a pair of spins is possible when the system is made open to an external heat bath, interacting with the system through that spin-pair via a repetitive quantum interaction.

Received 6 July 2017

DOI:https://doi.org/10.1103/PhysRevA.97.012316

Physics Subject Headings (PhySH)

Magnetism Phase diagrams Quantum correlations in quantum information Quantum information processing

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

Authors & Affiliations

Titas Chanda¹, Tamoghna Das¹, Debasis Sadhukhan¹, Amit Kumar Pal^1,2, Aditi Sen(De)¹, and Ujjwal Sen¹

¹Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211019, India
²Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom

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Vol. 97, Iss. 1 — January 2018

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Images

Figure 1
Phase boundaries and factorization lines on the phase plane of the ATXY model. (a) Phase boundaries corresponding to PM-I $\leftrightarrow$ AFM [Eq. (6)] and PM-II $\leftrightarrow$ AFM [Eq. (7)], for $γ = 0.8$ , are represented by dashed and dotted lines on the $(λ_{1}, λ_{2})$ plane. The factorization line [Eq. (8)] is represented by the continuous line on the $(λ_{1}, λ_{2})$ plane. (b) Factorization lines corresponding to different values of $λ_{2}$ are marked on $(λ_{1}, γ)$ plane. The dashed and the short-dashed lines represent the ATXY model, while the continuous line corresponds to the UXY model ( $λ_{2} = 0$ ). All the axes in both figures are dimensionless.
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Figure 2
Emergence of entanglement in thermal state corresponding to Hamiltonian parameters on the factorization surface. (a) Variation of LN as a function of $β_{S}$ for different values of $γ$ , with $λ_{2} = 1$ and $λ_{1}$ being fixed by the condition of the factorization line given in Eq. (8). The variation shows two successive revivals of entanglement, separated by a complete collapse, on the $β_{S}$ axes. The second revival of entanglement at $β_{S} = β_{S}^{R_{2}}$ is separated from the complete collapse of LN at $β_{S} = β_{S}^{C_{1}}$ by a finite difference, which increases with the values of $γ$ in the range $γ \leq 0.45$ . For $γ = 0.35$ , $β_{S}^{R_{1, 2}}$ and $β_{S}^{C_{1, 2}}$ are marked with vertical lines. Moreover, for $γ = 0.25$ , we find $L_{m}^{(2)} > L_{m}^{(1)}$ . (b) Variation of $L$ as a function of $β_{S}$ and $γ$ , for $λ_{2} = 1$ , with $λ_{1}$ being fixed by Eq. (8). Different shades in the figure represent different values of LN. (c) Map of the $L = 0$ region (shaded region) on the $(λ_{1}, β_{S})$ plane, with $γ = 0.6$ , and $λ_{2} = 1.0$ . (Inset) Variation of LN as a function of $λ_{1}$ for specimen values of $β_{S}$ . Note that for $β_{S} = 100$ , i.e., for sufficiently low temperature, the zero-entanglement region on the $λ_{1}$ axes is effectively a point, corresponding to the factorization point for fixed values of $γ$ and $λ_{2}$ , satisfying Eq. (8). All quantities plotted are dimensionless.
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Figure 3
Propagation of thermal entanglement after starting off from the factorization line under closed unitary evolution. The variation of LN as a function of $t$ and $λ_{1}$ with (a) $γ = 0.6$ , and (b) $γ = 0.8$ , where $λ_{2}$ is fixed by Eq. (8). (Insets) Variation of LN as a function of $t$ for different values of $λ_{1}$ . The axes in all the figures are dimensionless.
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Figure 4
Frozen-time snapshots of the $L \neq 0$ regions on the $(β_{S}, γ)$ plane. The shaded regions in the figures represent the regions on the $(β_{S}, γ)$ plane where $L = 0$ while the white regions represent $L \neq 0$ . The left column of figures correspond to the UXY model ( $λ_{2} = 0$ ), while the right column is for the ATXY model ( $λ_{2} = 1$ ). The snapshots are taken at $t = 0, 2, 10$ and 40. The value of $λ_{1}$ is fixed by Eq. (8) for all the points on the $(β_{S}, γ)$ plane. All quantities plotted are dimensionless.
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Figure 5
Snapshots of the $L \neq 0$ regions on the $(β_{S}, γ)$ plane at $t = 0$ for a finite-size system, specifically for $N = 10$ . The shaded regions in the figures represent the regions on the $(β_{S}, γ)$ plane where $L = 0$ . The left figure corresponds to the UXY model ( $λ_{2} = 0$ ), while the right one is for the ATXY model ( $λ_{2} = 1$ ). The value of $λ_{1}$ is fixed by Eq. (8) on the $(β_{S}, γ)$ plane. The horizontal lines in the figures represent the model with $γ = 0.5$ , where a double revival of LN takes place with varying $β_{S}$ [compare with Fig. 2], mimicking the behavior of entanglement of the model in the thermodynamic limit. All quantities plotted are dimensionless.
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Figure 6
Open system dynamics of entanglement under repetitive quantum interaction after starting off from the factorization line. The variation of LN as a function of $t$ and $λ_{1}$ with (a) $γ = 0.6$ , and (b) $γ = 0.8$ . The variations of LN as a function of $t$ for different values of $λ_{1}$ are given in (c) for $γ = 0.6$ and (d) for $γ = 0.8$ . Entanglement generation under closed vs open dynamics can be made by comparing insets of Figs. 3 and 3 and Figs. 6 and 6 in above figures. Although in a closed unitary evolution, entanglement can be preserved for a long time while it is not possible in an open dynamics considered in this paper. All quantities plotted are dimensionless.
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