Entropic witness for quantum memory in open system dynamics

TL;DR

The paper addresses distinguishing quantum from classical memory in open quantum dynamics using only reduced-system information. It derives a dimension-agnostic entropic witness based on von Neumann entropies that bounds the original process-tensor criterion, enabling practical detection in arbitrary dimensions. The approach is demonstrated on non-Markovian qudit damping and extended to continuous-variable Gaussian dynamics, with explicit expressions for lossy channels and a non-Markovian damped oscillator. This entropic criterion provides a computationally efficient tool for certifying quantum memory in a broad class of open-system dynamics, with potential experimental applicability.

Abstract

The dynamics of open quantum system are often modeled by non-Markovian processes that account for memory effects arising from interactions with the environment. It is well-known that the memory provided by the environment can be classical or quantum in nature. Remarkably, the quantumness of the memory can be witnessed locally by measurements on the open system alone, without requiring access to the environment. However, existing witnesses are computationally challenging for systems beyond qubits. In this work, we present a tractable criterion for quantum memory based on the von Neumann entropy, which is easily computable for systems of any dimension. Using this witness, we investigate the nature of memory in a class of physically motivated finite-dimensional qudit dynamics. Moreover, we demonstrate that this criterion is also suitable for detecting quantum memory in continuous-variable systems. As an illustrative example, we analyze non-Markovian Gaussian dynamics of a damped harmonic oscillator.

Figures (4)

• Figure 1: Time evolution of the entropic quantities used in Eq. \ref{['eq:entropic-criterion']} for a system of dimension $d=4$ and $\gamma/\omega = 0.05$. The initial state $\rho^{\mathcal{S}\!\mathcal{A}}_0$ of system and ancilla in Eq. \ref{['eq:choi-state']} is chosen to be the maximally entangled state $\vert{\Phi_+}\rangle = \sum_{l=0}^{d-1} \vert{ll}\rangle/\sqrt{d}$. The minimum of $S_\mathcal{S}$ at time $t_1$ is smaller than the subsequent maximum of $-S_{S|A}$ and $-S_{A|S}$ at time $t_2$. Thus, quantum memory is demonstrated by the entropic criterion in Eq. \ref{['eq:entropic-criterion']}. The Markovian damping of the memory qubit [see Eq. \ref{['eq:GKSL']}] prevents us from revealing the quantum memory at later times. The second revival of the conditional entropies is already too small to satisfy Eq. \ref{['eq:entropic-criterion']}.
• Figure 2: We plot $\Delta S$ from Eq. \ref{['eq:deltaS']} as a function of $\gamma/\omega$ for different dimensions $d$ of the qudit system $\mathcal{S}$. The times $t_1$ and $t_2$ in Eq. \ref{['eq:deltaS']} are chosen to minimize $\Delta S$ for the respective parameters (see main text and Fig. \ref{['fig:time-evolution']}). For $d>2$ there is a critical $\gamma/\omega$ above which the verification of the quantum memory by means of the entropic witness becomes impossible for the given dynamics.
• Figure 3: Minimal value of $\Delta S$ obtained by varying $r>0$ for every combination of $\eta_1$ and $\eta_2$. The diagonal dashed line describes the case that $\eta_1 = \eta_2$ and every combination of $\eta_1$ and $\eta_2$ lying beneath this line correspond to dynamical maps $\mathcal{E}_{t_1}$ and $\mathcal{E}_{t_2}$ which can only be realized using quantum memory since there is an arbitrary small $r>0$ such that $\Delta S<0$. In general, the detection of quantum memory gets easier for larger differences of $\eta_1$ and $\eta_2$. We also plot boundaries $\Delta S = 0$ for three different fixed values of the squeezing parameter $r$ (dashed lines). Stronger squeezing shrinks the parameter region for which quantum memory can be detected. This is interesting since it tells us that greater entanglement in the initial state $\rho^{\mathcal{S}\!\mathcal{A}}_0$ diminishes the ability of the entropic criterion to witness quantum memory.
• Figure 4: The loss parameter $\eta_t = 1 - \left|c_t\right|^2$ for $|g|^2=1$, $\kappa=1/4$, $\omega=1$, and $\Omega=1$. The non-monotonous behavior cannot be explained by classical memory effects, as we can always find two times $t_2 > t_1$ for which $\eta_{t_2} < \eta_{t_1}$, a signature of quantum memory as shown in Sec. \ref{['sec:lossy']}.

Theorems & Definitions (3)

• Definition 1
• Theorem 1
• Theorem 2