Interference between lossy quantum evolutions activates information backflow

TL;DR

This work shows that IB, a hallmark of non-Markovianity, can be activated by coherently interfering two lossy quantum evolutions that individually lack IB, without requiring indefinite causal order. It compares two coherent-control schemes—coherent path interference and quantum-switch—and demonstrates that interference over trajectories can yield broader IB activation and greater robustness to control noise. Using a specific eternal-non-Markovian model, the authors derive explicit Bloch-vector transfer relations and analytical IB conditions, revealing a nontrivial ordering where the interferometric scheme surpasses the switch after finite times. These results illuminate how coherent control of trajectories can enhance information recovery from the environment, with potential experimental advantages and open questions about general applicability to other lossy non-Markovian dynamics.

Abstract

Quantum evolutions are often non-unitary and in such cases, they are frequently regarded as lossy. Such lossiness, however, does not necessarily persist throughout the evolution, and there can often be intermediate time-spans during which information ebbs in the environment to re-flood the system -- an event known as information backflow. This phenomenon serves as a well-established and sufficient indicator of non-Markovian behavior of open quantum dynamics. Nevertheless, not all non-Markovian dynamics exhibit such backflow. We find that when interference is allowed between two quantum evolutions that individually generate non-Markovianity and yet do not exhibit information backflow, it becomes possible to retrieve information from the environment. Furthermore, we show that this setup involving coherently-controlled quantum operation trajectories provides enhanced performance and is more robust compared to an alternate coherently-controlled arrangement of the quantum switch.

Figures (5)

• Figure 1: Depiction of the two paradigmatic coherent control configurations. We depict here the two paradigmatic coherent control configurations of quantum channels, viz. (a) the coherent control over causal orders and (b) the coherent control over quantum operation trajectories.
• Figure 2: IB or not: Time-dynamics of the distance between two states for superposing trajectories and for quantum-switch. The dynamical natures of the trace distance between the two states, $\cos\frac{\theta_1}{2}\ket{0}+\sin\frac{\theta_1}{2}\ket{1}$ and $\cos\frac{\theta_2}{2}\ket{0}+\sin\frac{\theta_2}{2}\ket{1}$, with $\theta_1+\theta_2=\frac{5\pi}{9}$, are plotted in the two panels. Panel (a) assumes coherent control over the lossy dynamics and we obtain IB at intermediate times, while panel (b) assumes the switch configuration for the same, where we obtain no IB. The vertical axes are dimensionless. The horizontal axes are also dimensionless, if we rescale time to time $\times J/\hbar$, where $J$ is a multiplicative constant, of the $\gamma_i$ in the master equations, with the unit of energy.
• Figure 3: Comparison of IB-activation regimes for initial states restricted to a common azimuthal plane. The vertical axis represents evolution time, while the horizontal axis denotes $\Theta=\theta_1+\theta_2$, the sum of Bloch-sphere polar angles for the two input states. Since, the IB admissible range is independent of $\Phi$ in the quantum-switch configuration (see Corollary \ref{['cor_max_co_qs']}), the blue region marks the $(\Theta$, time$)$ pairs that activate IB for all azimuthal planes. The combined red region indicates the $(\Theta$, time$)$ pairs enabling IB in the coherently controlled path configuration on the $\Phi=0$ plane. Note that $\Phi=0$ yields the smallest IB-admitting region in this setup, and therefore serves as a lower-bound case for comparison. The solid green line corresponds to $t=\frac{1}{2}\ln 5$ while the dotted purple one represents $t_0=\frac{1}{2}\ln 8.85$. The plots show that although quantum switching allows a wider IB-admissible initial state space when $t\gtrsim\frac{1}{2}\ln 5$, the ordering reverses at later finite times and remains so asymptotically.
• Figure 4: Regions of the initial Bloch state space that allow IB activation are displayed at two finite time points, with $\Phi=\frac{\pi}{6}$ held fixed. (a) At $t=0.874719$, the quantum-switch setup supports IB for a wider subset of initial states than the interference setup. (b) At $t=1.20472$, this relation is exactly reversed, as the interference setup enables IB for a strictly larger initial-state region. This reversal emphasizes that earlier activation in time for some states does not ensure a persistent advantage across the full state space at later finite times.
• Figure 5: Comparison of IB-supporting regions at different finite time instances, with both initial states restricted to the $xz$-plane of the Bloch sphere under noisy control where $p$ denotes the noise parameter. (a) At $t=\frac{1}{2}\ln\left(1+\frac{4}{p}\right)+0.01$, the quantum switch enables IB over a larger initial-state region and with enhanced robustness in control than is possible in the interference-based setup. (b) At $t=\frac{1}{2}\ln\left(1+\frac{4}{p}\right)+0.6$, the previous ordering is reversed, as the interference setup admits IB for a strictly larger set of initial states and under more robust control conditions.

Theorems & Definitions (14)

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• Proposition 1
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• Corollary 1
• Proposition 2
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• Corollary 2
• Theorem 1
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• Proposition 3