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Witnessing non-Markovian dynamics through correlations

Dario De Santis, Markus Johansson, Bogna Bylicka, Nadja K. Bernardes, and Antonio Acín
Phys. Rev. A 102, 012214 – Published 13 July 2020
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  • INTRODUCTION
  • NON-MARKOVIAN DYNAMICS
  • CORRELATION MEASURES AS WITNESSES OF…
  • ENTANGLEMENT MEASURES
  • MUTUAL INFORMATION
  • CORRELATION MEASURES DETECTING ALMOST…
  • QUASICORRELATION MEASURES
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Non-Markovian effects in an open-system dynamics are usually associated to information backflows from the environment to the system. However, the way these backflows manifest and how to detect them is unclear. A natural approach is to study the backflow in terms of the correlations the evolving system displays with another unperturbed system during the dynamics. In this work, we study the power of this approach to witness non-Markovian dynamics using different correlation measures. We identify simple dynamics where the failure of completely positive divisibility is in one-to-one correspondence with a correlation backflow. We then focus on specific correlation measures, such as those based on entanglement and the mutual information, and identify their strengths and limitations. We conclude with a study of a recently introduced correlation measures based on state distinguishability and see how, for these measures, adding an extra auxiliary system enlarges the set of detectable non-Markovian dynamics.

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    • Received 17 June 2019
    • Revised 13 May 2020
    • Accepted 21 May 2020

    DOI:https://doi.org/10.1103/PhysRevA.102.012214

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Open quantum systems & decoherenceQuantum correlations in quantum information
    1. Techniques
    Non-Markovian processes
    Quantum Information, Science & Technology

    Authors & Affiliations

    Dario De Santis1, Markus Johansson1, Bogna Bylicka1, Nadja K. Bernardes2, and Antonio Acín1,3

    • 1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
    • 2Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil
    • 3ICREA–Institucio Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain

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    Vol. 102, Iss. 1 — July 2020

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    Images

    • Figure 1
      Figure 1

      The Stinespring-Kraus representation theorem allows to simulate any Markovian evolution with a sequence of unitary interactions between S and different environments that are discarded during the evolution. Top: Given a Markovian evolution {Λt}t, the intermediate map Vs,t between any two times t and s≥t is CPTP. The CPTP map Λt induces the evolution ρS(0)→ρS(t) and the CPTP map Vs,t induces the intermediate evolution ρS(t)→ρS(s). Bottom: The same evolution can be simulated with subsequent interactions between S and two different initially uncorrelated environments. First, E1, initialized in σE1, interacts with S up to time t, where the interaction is realized by a unitary USE1t,0. E1 is discarded at time t and S is coupled with a a different environment E2, initialized in σE2. The evolution to time s is implemented by a unitary USE2s,t. Since E1 is not in contact with S after time t, no information that S loses during the time interval [0,t] can be recovered when S interacts with E2, i.e., in [t,s]. This scenario can be considered for any t and s≥t and therefore no information that S loses in one or more time intervals can be recovered later in time.

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    • Figure 2
      Figure 2

      Left: in the first setting, an initial state between system S and ancilla A is used. An increase of correlations between these two parts witnesses the presence of non-Markovian effects. Right: in our second extended setting, the whole setup consists of three parts, the systems S and A as before, plus an extra ancilla A′. An increase of the correlations over the bipartition A versus SA′ is used to witness non-Markovian evolutions.

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    • Figure 3
      Figure 3

      Depiction of the trajectory of the evolution of a maximally entangled state ϕSA+, where the system S evolves under an entanglement breaking ΛtEB. Therefore, if t>tEB, any initial state ρSA(0)∈QSA=S(HSA) is evolved into a separable state ρSA(t)∈SSA. Suppose that ΛtEB is non-Markovian but CP-divisible in [0,tEB] and with a non-CP intermediate map Vs,tEB for s>t>tEB. In this case, it is not possible to witness backflows of any entanglement measure. Indeed, entanglement is zero in the set of separable states SSA.

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    • Figure 4
      Figure 4

      The mutual information relative to its value at t=0.1, I(ρ−(t,ε))/I(ρ−(0.1,ε)) as a function of the dimensionless parameter t for values of ε between 10−3 and 10−500 (black curves). With successively smaller ε the mutual information increases in a larger part of the interval 0.13437≲t≲0.31416 where the dynamics is non-CP-divisible (light blue area), and the t where the mutual information begins to increase approaches the beginning of the interval (gray vertical lines). For ε=10−500 the increase in mutual information begins at t≃0.1352.

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    • Figure 5
      Figure 5

      The dimensionless coefficients α(δτ) (dark blue curve) and β(δτ) (black curve) of the leading order of the series expansion of ddtI(ρ−(t,ε)) in ε as functions of the dimensionless parameter δτ for −10−6≤δτ≤10−6. The dynamics is non-Markovian for δτ>0 (light blue area). For δτ≤0 the coefficient β(δτ) is non-negative while α(δτ) is negative and therefore the leading-order term of the expansion is negative for any ε. For δτ>0 both β(δτ) and α(δτ) are negative and, therefore, for sufficiently small ε the leading-order term of the expansion is positive. For δτ=0 the value of β(δτ) is zero to within numerical precision.

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    • Figure 6
      Figure 6

      Plots of the dimensionless rates γz(t1,2NM)(t)=−tanh(t−t1,2NM)/5 of the quasieternal non-Markovian model for t1NM=1 (blue) and t2NM=4 (orange) as functions of the dimensionless parameter t. The differences between Λt(1) and Λt(4) are given by the different values of the integral of γz(1)(t) and γz(4)(t) [see Eqs. (14) and (51)]. Let V¯ be the P intermediate map of Λt(1) in the time interval (t1′,t1′′)=(2,3) (left pink region). V¯ is also the intermediate map of Λt(4) that occurs in a time interval shifted by ΔtNM=t2NM−t1NM=3, i.e., in (t2′,t2′′)=(5,6) (right pink region). The difference between the images of the two maps before the action of V¯ is given by the contractive action of ΛΔtNM(4) (orange region). This result follows from γz(1)(t)=γz(4)(t+ΔtNM) [Eqs. (14) and (51)].

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    • Figure 7
      Figure 7

      Plots of Im(Λt(tNM)) for different values of the dimensionless parameter tNM and time t as subsets of the Bloch sphere, i.e., the qubit state space. First, we set the value (purple) tNM=t1NM=1 and we consider times (a) t=t1′=2 and (b) t=t1′′=3. The map that transforms Im(Λ2(1)) into Im(Λ3(1)) is the P intermediate map V3,2(1)=V¯. By considering (pink) tNM=t2NM=4 and times (c) t=t2′=5 and (d) t=t2′′=6, the map that transforms Im(Λ5(4)) into Im(Λ6(4)) is again V¯, namely V6,5(4)=V3,2(1)=V¯ [see Eq. (55)]. Therefore, while the map that transforms (a) into (b) and (c) into (d) is V¯ in both cases, in the first case (a) the starting set where V¯ acts is larger than in the second case (c). In general, the more we increase the value of tNM, the smaller is Im(Λt′(tNM)) at the time t′ when V¯ starts to act [see Eq. (62)].

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    • Figure 8
      Figure 8

      The measurement scenario where Alice, measuring her side of ρAB with an n-ouput ME-POVM {PA,i}i=1n, produces on Bob's side the EES given by Eq. (76).

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