Measuring non-Markovianity via incoherent mixing with Markovian dynamics

Dario De Santis and Vittorio Giovannetti

Phys. Rev. A 103, 012218 – Published 21 January 2021

Abstract

We introduce a measure of non-Markovianity based on the minimal amount of extra Markovian noise we have to add to the process via incoherent mixing, in order to make the resulting transformation Markovian too at all times. We show how to evaluate this measure by considering the set of depolarizing evolutions in arbitrary dimension and the set of dephasing evolutions for qubits.

Received 11 May 2020
Accepted 24 November 2020

DOI:https://doi.org/10.1103/PhysRevA.103.012218

Physics Subject Headings (PhySH)

Open quantum systems Quantum channels

Markovian processes Non-Markovian processes

Quantum Information, Science & Technology

Authors & Affiliations

Dario De Santis ¹ and Vittorio Giovannetti²

¹ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
²NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy

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Vol. 103, Iss. 1 — January 2021

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Figure 1
Example of a noncontinuous Markovian characteristic function (above) $f^{M} (t) \in F^{M}$ and a noncontinuous non-Markovian characteristic function (below) $f^{N M} (t) \in F^{N M}$ for $d = 2$ . Given Eq. (8), any characteristic function has to assume values in $I_{D} = [- 1 / 3, 1]$ . Discontinuities are underlined by dotted dashed lines. $f^{M} (t)$ , when continuous, satisfies ${CM}_{1} (τ)$ , i.e., it does not increase its distance from zero. When $f^{M} (t)$ is not continuous it satisfies ${CM}_{2} (τ)$ : for the times $τ = 1, 2, 5$ , and 8, we have $ξ (f^{M} (1)) = 0.83, ξ (f^{M} (2)) = - 0.33, ξ (f^{M} (5)) = - 0.27$ and $ξ (f^{M} (8)) = 0$ . Since $f^{M} (8^{+}) = 0, f^{M} (t)$ has to be equal to 0 for any $t > 8$ . The times when $f_{C}^{N M} (t)$ has a non-Markovian behavior are colored in purple. This characteristic function shows both time intervals and times of discontinuity when, respectively, ${CM}_{1} (τ)$ and ${CM}_{2} (τ)$ are violated. Indeed, for $τ = 7$ and 9 we have non-Markovian discontinuities $ξ (f^{N M} (7)) = 2.39$ and $ξ (f^{N M} (t)) = - \infty$ , while at $τ = 8$ we have $ξ (f^{N M} (8)) = 0$ , i.e., a Markovian discontinuity. The temporal parameter $t$ in the plots is expressed in arbitrary unit.
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Figure 2
Example of a continuous Markovian characteristic function (above) $f_{C}^{M} (t) \in F_{C}^{M}$ and a continuous non-Markovian characteristic function (below) $f_{C}^{N M} (t) \in F_{C}^{N M}$ for $d = 2$ . Given Eq. (8), any characteristic function has to assume values in $I_{D} = [- 1 / 3, 1]$ . $f_{C}^{M} (t)$ is nonincreasing and assumes values in [0,1]. $f_{C}^{N M} (t)$ assumes values in $I_{D} = [- 1 / 3, 1]$ (horizontal lines) and violates the Markovian condition ${CM}_{1} (τ)$ in the time intervals colored in purple, i.e., when it increases its distance from zero. Dashed lines underline the times when the respective time derivatives are noncontinuous. The temporal parameter $t$ in the plots is expressed in arbitrary unit.
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Figure 3
Plots of $f_{C}^{N M} (t) = e^{- 2 t / 5} cos (t)$ (yellow), the corresponding optimal Markovian characteristic function $h_{C}^{M} (t)$ (blue) and $f^{(p)} (t)$ for different values of $p$ (dashed lines) in the time interval $t \in [0, 7 π / 2]$ . The inset shows their behavior for $t \geq 5.90$ . In this example $T_{1}^{-} ≃ (π / 2, 2.76), T_{1}^{+} ≃ (3 π / 2, 5.90)$ and $T_{2}^{-} ≃ (5 π / 2, 9.04)$ are the time intervals of non-Markovianity of $f^{N M} (t)$ and $Θ_{1}^{N M} ≃ - 0.31, Δ_{1}^{N M} ≃ 0.09$ and $Θ_{2}^{N M} ≃ - 0.02$ are the corresponding non-Markovian gaps. The value of the measure of non-Markovianity is $p (D_{C}^{N M} | D^{M}) ≃ 0.30$ . If $p = 0.5 > p (D_{C}^{N M} | D_{C}^{M}), f^{(p)} (t) \in F_{C}^{M}$ is monotonically decreasing (green dashed line). If $p = p (D_{C}^{N M} | D_{C}^{M}) ≃ 0.30, f^{(p)} (t) \in F_{C}^{M}$ is monotonically decreasing and constant when ${\dot{f}}^{N M} (t) > 0$ (red dashed line). If $p = 0.15 < p (D_{C}^{N M} | D_{C}^{M}), f^{(p)} (t) \in F_{C}^{N M}$ is not monotonic nor positive in more than one time interval (purple dashed line).
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Figure 4
Plots of $f_{1}^{M} (t), f_{Θ}^{N M} (t)$ , and $f^{(p)} (t)$ for a non-Markovian gap $Θ = - Θ^{N M} = 0.1$ and $p = p (D_{Θ}^{N M} | D) ≃ 0.77$ . The time interval of non-Markovianity $T^{-} = (2, 3)$ of $f_{0.1}^{N M} (t)$ is colored in purple. Since $Θ < 1 / 6$ , the optimal Markovian characteristic function is $f_{1}^{M} (t)$ .
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Figure 5
Plots of $f_{2}^{M} (t), f_{Θ}^{N M} (t)$ , and $f^{(p)} (t)$ for a non-Markovian gap $Θ = - Θ^{N M} = 0.2$ and $p = p (D_{Θ}^{N M} | D) ≃ 0.65$ . The time interval of non-Markovianity $T^{-} = (2, 3)$ of $f_{0.2}^{N M} (t)$ is colored in purple. Since $Θ > 1 / 6$ , the optimal Markovian characteristic function is $f_{2}^{M} (t)$ .
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Figure 6
Plots of $f_{1}^{M} (t) = 1$ (orange), $f_{2}^{M} (t)$ (purple), and $f^{(p)} (t) = p f_{1}^{M} (t) + (1 - p) f_{2}^{M} (t)$ for $p = 0.75$ (red), where $t_{N C, 1} = 1$ and $t_{N C, 2} = 2$ . While $f_{1, 2}^{M} (t)$ satisfy the conditions (15) at any time, this is not the case for $f^{(0.75)} (t)$ : it shows a non-Markovian discontinuity at $t_{N C, 2} = 2$ when ${CM}_{2} (2)$ is violated. Indeed, $f^{(0.75)} (2^{-}) = 0, f^{(0.75)} (2^{+}) > 0$ and $ξ (f^{(0.75)} (2)) = + \infty$ .
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Physical Review A

covering atomic, molecular, and optical physics and quantum information

Measuring non-Markovianity via incoherent mixing with Markovian dynamics

Dario De Santis and Vittorio Giovannetti

Phys. Rev. A 103, 012218 – Published 21 January 2021

Abstract

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Physical Review A

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Measuring non-Markovianity via incoherent mixing with Markovian dynamics

Dario De Santis and Vittorio Giovannetti

Phys. Rev. A 103, 012218 – Published 21 January 2021

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