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Coherence, entanglement, and quantumness in closed and open systems with conserved charge, with an application to many-body localization

Katarzyna Macieszczak, Emanuele Levi, Tommaso Macrì, Igor Lesanovsky, and Juan P. Garrahan
Phys. Rev. A 99, 052354 – Published 31 May 2019
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  • INTRODUCTION
  • COHERENCE AND ENTANGLEMENT
  • VANISHING SEPARABILITY THRESHOLDS FROM…
  • BOUNDS ON ENTANGLEMENT AND DISCORD…
  • NEGATIVITY IN THE MANY-BODY LOCALIZED…
  • APPLICATION TO A MANY-BODY LOCALIZED…
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    While the scaling of entanglement in a quantum system can be used to distinguish many-body quantum phases, it is usually hard to quantify the amount of entanglement in mixed states of open quantum systems, while measuring entanglement experimentally, even for the closed systems, requires in general quantum state tomography. In this work we show how to remedy this situation in system with a fixed or conserved charge, e.g., density or magnetization, due to an emerging relation between quantum correlations and coherence. First, we show how, in these cases, the presence of multipartite entanglement or quantumness can be faithfully witnessed simply by detecting coherence in the quantum system, while bipartite entanglement or bipartite quantum discord are implied by asymmetry (block coherence) in the system. Second, we prove that the relation between quantum correlations and coherence is also quantitative. Namely, we establish upper and lower bounds on the amount of multipartite and bipartite entanglement in a many-body system with a fixed local charge, in terms of the amount of coherence and asymmetry present in the system. Importantly, both for pure and mixed quantum states, these bounds are expressed as closed formulas, and furthermore, for bipartite entanglement, are experimentally accessible by means of the multiple quantum coherence spectra. In particular, in one-dimensional systems, our bounds may detect breaking of the area law of entanglement entropy. We illustrate our results on the example of a many-body localized system, also in the presence of dephasing.

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    • Received 8 October 2018

    DOI:https://doi.org/10.1103/PhysRevA.99.052354

    ©2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Many-body localizationQuantum coherence & coherence measuresQuantum discordQuantum entanglementQuantum metrology
    Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsNuclear Physics

    Authors & Affiliations

    Katarzyna Macieszczak1,2, Emanuele Levi1,2, Tommaso Macrì3,4, Igor Lesanovsky1,2, and Juan P. Garrahan1,2

    • 1School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
    • 2Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom
    • 3International Institute of Physics, 59078-400 Natal, Rio Grande do Norte, Brazil
    • 4Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, Rio Grande do Norte, Brazil

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    Vol. 99, Iss. 5 — May 2019

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    • Figure 1
      Figure 1

      Coherence and asymmetry implies multipartite and bipartite quantum correlations in the presence of fixed or conserved charge. A state ρ of the conserved charge Q, [ρ,Q]=0, is block diagonal [light blue (light gray) squares] with respect to the charge eigenspaces (with values of Q denoted by q), while when a charge value is fixed [(middle) light blue square shaded into red], Qρ=qρ, it is supported within only a single block corresponding to q eigenspace. When the charge is local, Q=∑k=1NQ(k), and Q(k) is nondegenerate for each subsystem, it uniquely defines the separable basis without coherence [dark blue (dark gray) squares]. As we show in Secs. 3b and 3d, when the charge Q is fixed or conserved, any coherence in this basis (faithfully) implies MPE or MPD, respectively. Furthermore, when the system is divided into two parts A and B, the local charges Q(A)≡∑k∈AQ(k) are in general degenerate. Nevertheless, the block coherence (asymmetry) with respect to Q(A) eigenspaces [blue (gray)] still implies BPE or BPD when the charge Q is fixed or conserved, respectively, as we show in Sec. 3c.

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    • Figure 2
      Figure 2

      QFI in an MBL system. Throughout the paper we illustrate our results with the example of a many-body localized system, an XXZ chain with strong disordered longitudinal field (see details in Sec. 6). The figure shows the evolution of the QFI per number of subsystems N. We consider four different phase encodings: x magnetization Mx≡∑k=1NSkx [green (top) line], the difference of z magnetization between two halves of the chain δMz≡∑k=1N/2Skz−∑k=N/2+1NSkz [black (bottom) line], and x and z imbalance Ix,z≡∑k=1N(−1)kSkx,z [blue and gray (at Jt=10−1 lower and upper middle) lines, respectively]. The QFI witnesses entanglement only if the corresponding separability threshold QFIsep/N=1 [red dashed (horizontal) line] is crossed [cf. Eq. (9)]. Due to the conservation of Mz by the dynamics, for an initial state with a fixed Mz, the spin axes x and y are equivalent (and the QFI for My and Iy is equal to the QFI for Mx and Ix, respectively). For observables commuting with Mz (here δMz and Iz) the separability threshold is reduced to zero (see Sec. 3). The parameters of the dynamics [cf. Eqs. (52) and (56)] are N=14 spins, V/J=2, h/J=5, and γ/J=0.

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    • Figure 3
      Figure 3

      Witnessing bipartite entanglement in an MBL system with conserved charge. We show for the XXZ chain of Sec. 6 that not only MPE can be witnessed, but also BPE, provided the phase encoding observable commutes with subsystem charges Q(A) and Q(B) in the bipartition. (a) For closed dynamics (γ/J=0) the QFI of the z-magnetization difference between two halves of the chain (solid lines) witnesses BPE at all times t>0. Similarly, the QFI of the z imbalance (dashed lines) witnesses BPE in the staggered partition (ABAB...AB instead of AA⋯AB...BB) at all times t. The inset shows that the asymptotic values (taken from Jt=104) of the QFI for the z-magnetization difference (circles) and the z imbalance (triangles), as a function of size N, grow with system size (cf. Ref. [43]). For the z imbalance the asymptotic value grows with N even after rescaling by the system size. (b) In the presence of local dephasing (γ/J=2×10−4) the QFI of the magnetization difference (solid lines) decays in comparison with the closed case (dotted lines), while the variance (dashed line) increases, thus overestimating the QFI [cf. Eq. (A7)]. The inset shows that both the QFI (solid lines) and its lower bound in terms of the curvature (dotted lines) (defined in Appendix pp1) witness BPE for the observable chosen as the magnetization difference, although the curvature decays at a faster rate dependent on the system size [40, 63]. The parameters of the dynamics [cf. Eqs. (52) and (56)] are N=6,8,10,12,14 [yellow, green, red, blue, and black (grayscale: light gray to black), respectively, open case only up to N=12], V/J=2, and h/J=5; gray (bottom) curves correspond to the noninteracting case V/J=0 with N=12 (closed dynamics) and N=8 (open dynamics). Here the QFI is independent of system size as entanglement obeys the area law.

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    • Figure 4
      Figure 4

      Coherence as an upper bound on the MPE in an MBL system. We show coherence (10), which is a faithful upper bound on MPE [cf. Eq. (38)], for the XXZ chain of Sec. 6. The curves are rescaled by the effective system size Neff=log2(NN/2) of the zero z-magnetization subspace. In the presence of dephasing, coherence decays at a rate proportional to the dephasing strength γ and is weakly dependent on the interaction strength, but not on the system size. The inset shows that the coherence in the closed dynamics (γ=0) at t=1/J (triangles) and t=103/J (circles) follows the same scaling with system size. The parameters of the dynamics [cf. Eqs. (52) and (56)] are N=6,8,10,12,14 spins [yellow, green, blue, red, and black (grayscale: light gray to black), respectively], V/J=2, h/J=5, and γ/J=2×10−4; gray (bottom) curves are the noninteracting case V/J=0 with N=8 (here results are independent of system size as entanglement obeys the area law).

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    • Figure 5
      Figure 5

      Bipartite entanglement and asymmetry in an MBL system. (a) Entanglement entropy S(ρA) (solid lines) and asymmetry Sblock of half-chain magnetization (dashed lines) [cf. (46)] for the XXZ chain of Sec. 6, with closed dynamics. The asymmetry initially follows the area law (Jt<1), which is broken at later times, in analogy with the entanglement entropy. The inset shows that the asymptotic value of the asymmetry (taken from Jt=104) scales as log2(N/2+1) with system size (cf. Fig. 10). (That of the entanglement entropy, not shown, scales as N, as expected.) (b) Similar to (a), but with dephasing, so for −S(ρ)+Sblock (solid lines) and −S(ρ)+S(ρA) (dashed lines). Here the decay rate of the asymmetry is independent of the system size, but depends on the interaction strength (cf. Fig. 10). (c) Entanglement entropy (solid lines) for the staggered bipartition (ABAB...AB instead of AA...AB⋯BB) and corresponding asymmetry of the staggered magnetization. The curves are scaled by N/2 and log2(N/2), respectively. The inset shows that the asymptotic value of the entanglement entropy per site (taken from Jt=104) shows an additional weak dependence on N. (d) Same as (c) but for the dissipative case (subtracting the von Neumann entropy S). In the presence of dephasing, the decay of both sets of curves is independent of the system size, but depends on the interaction strength. The parameters of the dynamics [cf. Eqs. (52) and (56)] are N=6,8,10,12,14 [yellow, green, red, blue, and black (grayscale: light gray to black), respectively, open case only up to N=12], V/J=2, h/J=5, and with γ/J=2×10−4 for the dissipative case [(b) and (d)]; gray (bottom) curves correspond to the noninteracting case V/J=0 and N=12 (closed dynamics) and N=8 (open dynamics), but in both cases the results follow area laws.

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    • Figure 6
      Figure 6

      Negativity and experimentally accessible lower bound in a closed MBL system. (a) We show the negativity of entanglement [Eq. (50)] for the XXZ chain of Sec. 6. The negativity (solid lines) initially follows an area law for times Jt<2 (with coefficient dependent on the interaction strength for Jt>1), but at later times becomes dependent on the system size. The lower bound by l1block [Eq. (51)] is also shown (dashed lines) (cf. Fig. 7). The inset shows that the asymptotic value of N in the closed dynamics (at Jt=104) scales exponentially with the system size N (note the logarithmic scale of the vertical axis). (b) We show the lower bounds l1block≥l1,Mz(A)block on the convex roof of the negativity of entanglement [cf. Eq. (51)]. Both l1block (solid lines) and the experimentally accessible l1,Mz(A)block (dashed lines) initially follow the area law for times Jt<2, and at later times become dependent on the system size. In particular, the asymptotic value of l1block for the noninteracting system (gray) is crossed by l1,Mz(A)block for sizes above N=8. The inset shows that the asymptotic values of l1block (circles) and l1,Mz(A)block (triangles) in the closed dynamics (at Jt=104) increase with the system size. The parameters of the dynamics [cf. Eqs. (52) and (56)] are N=6,8,10,12,14 spins [yellow, green, red, blue, and black (grayscale: light gray to black), respectively], V/J=2, and h/J=5; gray (bottom) curves correspond to the noninteracting case V/J=0 with (a) N=12 and (b) N=8 (b). Light blue stars in (a) correspond to a more strongly interacting system with V/J=5 and N=12.

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    • Figure 7
      Figure 7

      Lower bound on BPE in an open MBL system. We show the lower bound on the convex roof on the negativity of entanglement by l1block [cf. Eq. (51)] for the XXZ chain of Sec. 6. The solid curve is for the closed case and the dashed curves for the dissipative case. Decay in the presence of dephasing depends on both the system size and interaction strength. The inset shows that the asymptotic value of l1block(ρ) in the closed dynamics at t=103/J (circles) grows approximately linearly with N (for N=8,10,12,14). The parameters of the dynamics [cf. Eqs. (52) and (56)] are N=6,8,10,12,14 spins [yellow, green, red, blue, and black (grayscale: light gray to black), respectively, open case only up to N=12], V/J=2, h/J=5, and γ/J=2×10−4; gray curves correspond to the noninteracting case V/J=0 with N=12 (closed dynamics) and with N=8 (open dynamics) (here results are independent of system size as the system follows an area law). Light blue stars in (a) correspond to a more strongly interacting system with V/J=5 and N=12.

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    • Figure 8
      Figure 8

      Observable averages in MBL and thermal phases: imbalance (solid lines) and average half-chain magnetizations difference (dashed lines) for the XXZ chain of Sec. 6. They both show a quick approach to stationarity for and in contrast to the behavior of quantum correlations (cf. Fig. 5). In particular, there is no appreciable difference in timescales between the interacting and noninteracting cases. The parameters of the dynamics [cf. Eqs. (52) and (56)] are N=6,8,10,12,14 [yellow, green, red, blue, and black (grayscale: light gray to black), respectively, open case only up to N=12], V/J=2, and h/J=5; gray (top solid and bottom dashed) curves are for V/J=0 and N=8. The bottom inset shows that in the thermal phase (V/J=2 and h/J=1) the asymptotic values decrease with the system size [green (gray) N=8 and blue (darker gray) N=12]. The top inset shows that the average imbalance in the presence of dephasing γ/J=2×10−4 (dashed curves) and 10−3 (dotted curves) decays to zero.

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    • Figure 9
      Figure 9

      Correlations in the thermal phase of an MBL system: relative entropy coherence [black (top) curve] (10), a lower bound on the relative entropy of entanglement [108] [green (lower middle) curve] −S(ρ)+S(ρA), half-chain magnetization asymmetry [red (bottom) curve] (15), and mutual information (total classical and quantum correlations) I=(A,B)≡S(ρA)+S(ρB)−S(ρ) [gray (upper middle) curve] in thermal phase of MBL system (V/J=2, h/J=1, and N=8). Solid lines corresponds to the closed case, while the open case with dephasing is illustrated by dashed (γ/J=2×10−4) and dotted (γ/J=10−3) lines. The inset shows the closed case for N=8 (solid lines) and N=12 (dashed lines) spins.

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    • Figure 10
      Figure 10

      Growth of asymmetry in an open MBL system: asymmetry of half magnetization [cf. Eq. (46)] for the XXZ chain of Sec. 6, as also shown in Fig. 5. Decay in the presence of dephasing (γ/J=2×10−4) depends on the interaction strength, but not the system size. The parameters of the dynamics [cf. Eqs. (52) and (56)] are N=6,8,10,12,14 [yellow, green, red, blue, and black (grayscale: light gray to black), respectively, open case only up to N=12], V/J=2, and h/J=5; light blue stars are for interaction strength V/J=5 and N=12 and gray curves correspond to the noninteracting case V/J=0 with N=8. The inset shows that although the asymmetry saturates one order of magnitude earlier that the entanglement entropy [cf. Fig. 5], its growth lasts for one order of magnitude longer than in the thermal phase, where asymptotic asymmetry is proportional to log2(N/2) (cf. Appendix pp1). The parameters in the inset are V/J=2, h/J=5, and N=8,12 [green (gray) solid line and blue (darker gray) solid line], V/J=2, h/J=1, and N=8,12 [green (gray) dashed line and blue (darker gray) dashed line], V/J=0, h/J=5, and N=8,12 [light gray dotted line and dark gray dotted line].

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    • Figure 11
      Figure 11

      Protocol for the curvature and the MQC spectrum. For a system state ρ being a result of the dynamics from |ψ0〉 (steps 1 and 2), its coherence with respect to the eigenbasis of an observable M can be accessed by unitary perturbation encoding a phase ϕ (step 3), followed by a measurement (step 4 and 5) of the overlap between ρ(ϕ) and the unperturbed state ρ. This measurement scheme can also be used to estimate the encoded phase value ϕ, and in the case of (noninteracting) closed dynamics it corresponds to Ramsey spectroscopy [135].

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    • Figure 12
      Figure 12

      Experimentally accessible lower bounds on BPE in an MBL system. We show the dynamics of l2block(ρ) (solid lines) and C(Pz(AB),ρ), where the parity Pz(AB)≡(−1)δM/2 for the XXZ discorded chain of N=6,8,10,12,14 spins [yellow, green, red, blue, and black (grayscale: light gray to black), respectively]. The inset shows the asymptotic values of l2block(ρ) at Jt=104. The parameters of the dynamics (52) and (56) are V/J=2, h/J=5, and γ/J=0, while the gray (bottom) curve corresponds to the noninteracting case V/J=0 and N=12.

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