Characterizing non-Markovianity via quantum interferometric power

Himadri Shekhar Dhar, Manabendra Nath Bera, and Gerardo Adesso

Phys. Rev. A 91, 032115 – Published 18 March 2015

Abstract

Non-Markovian evolution in open quantum systems is often characterized in terms of the backflow of information from environment to system and is thus an important facet in investigating the performance and robustness of quantum information protocols. In this work, we explore non-Markovianity through the breakdown of monotonicity of a metrological figure of merit, called the quantum interferometric power, which is based on the minimal quantum Fisher information obtained by local unitary evolution of one part of the system, and can be interpreted as a quantifier of quantum correlations beyond entanglement. We investigate our proposed non-Markovianity indicator in two relevant examples. First, we consider the action of a single-party dephasing channel on a maximally entangled two-qubit state, by applying the Jamiołkowski-Choi isomorphism. We observe that the proposed measure is consistent with established non-Markovianity quantifiers defined using other approaches based on dynamical divisibility, distinguishability, and breakdown of monotonicity for the quantum mutual information. Further, we consider the dynamics of two-qubit Werner states, under the action of a local, single-party amplitude damping channel, and observe that the nonmonotonic evolution of the quantum interferometric power is more robust than the corresponding one for entanglement in capturing the backflow of quantum information associated with the non-Markovian process. Implications for the role of non-Markovianity in quantum metrology and possible extensions to continuous variable systems are discussed.

Received 19 January 2015

DOI:https://doi.org/10.1103/PhysRevA.91.032115

Authors & Affiliations

Himadri Shekhar Dhar^1,2, Manabendra Nath Bera¹, and Gerardo Adesso³

¹Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
²School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India
³School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 91, Iss. 3 — March 2015

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
A scheme showing the protocol under consideration. The system $a$ interacts with the environment, while the quantum interferometric power is computed by applying local unitaries on the ancilla part $b$ , which serves as the measuring apparatus.
Reuse & Permissions
Figure 2
In this figure, we compare the measure $N_{Q}$ (= $N_{Q}^{0}$ ), with other quantifiers of non-Markovianity, for a single-qubit dephasing channel with an Ohmic spectral density, $J (ω) = α ω_{c} {(ω / ω_{c})}^{S} exp (ω / ω_{c})$ . The blue dashed region, corresponding to $γ (t) >$ 0, is the Markovian regime. This corresponds to $N = 0$ , for all known measures of non-Markovianity. The region $γ (t) <$ 0 denotes the non-Markovian regime and we plot the behavior of the measures based on QIP ( $N_{Q}$ , green bar), quantum mutual information ( $N_{I}$ , red bar), and divisibility criteria ( $N_{R}$ , purple bar). We observe that the non-Markovian regime lies in the super-Ohmic region, $S > 2$ . The behavior of the measures, $N_{Q}$ and $N_{I}$ , is similar: Both indicators initially increase, followed by a decrease, with increasing $S$ , with vanishing values for $S >$ 5. However, in contrast the measure $N_{R}$ monotonically increases with $S$ in the non-Markovian region. This implicitly shows that the measures based on QIP and mutual information are independent of the divisibility criteria. The inset figure shows the measures of non-Markovianity around the critical transition parameter value, $S = 2$ . We note that, for this model, the non-Markovianity measure based on distinguishability is identical to the one derived in this work.
Reuse & Permissions
Figure 3
The flow of QIP $(Q)$ in a two-qubit system with maximal entanglement at $t = 0$ under a single-qubit amplitude damping channel. The initial entanglement is set by setting $r = 1$ in the Werner state given by Eq. (30). $Q (ρ_{a b})$ is equal to $| J_{t} |$ , as shown by Eq. (32). We observe the flow of $Q$ in both Markovian and non-Markovian regimes of the dynamical evolution. The flow under the Markovian regime (red dashed line) corresponds to the ratio of the reservoir correlation to system relaxation, $λ / γ_{0} = 10$ . Under the non-Markovian regime, the flow is shown for $λ / γ_{0} = 0.5$ (blue dotted line) and $λ / γ_{0} = 0.1$ (green solid line). The system-reservoir frequency detuning is set at $δ = 0.01 γ_{0}$ . The backflow of quantum correlation, in terms of QIP, is observed by the nonmonotonic increase of $Q$ during the evolution.
Reuse & Permissions
Figure 4
The flow of QIP $(Q)$ for an initially mixed two-qubit system under a single-qubit amplitude damping channel. The initial mixed state is obtained for $r = 0.45$ , in the Werner state given by Eq. (30).The parameters pertaining to the Lorentzian reservoir spectral distribution are set at $λ = 0.01 γ_{0}$ and $δ = 0.001 γ_{0}$ . The figure shows the non-Markovian evolution of $Q (ρ_{a b})$ (blue solid line), concurrence $C (ρ_{a b})$ (black dotted line), quantum mutual information $I (ρ_{a b})$ (red broad-dashed line), and the scaled function $| J_{t} | / 2$ (brown dashed line).
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information