On the emergence of quantum memory in non-Markovian dynamics

TL;DR

This work investigates the emergence of quantum memory in non-Markovian dynamics using a quantum homogenizer model. By analyzing how reservoir initialization and inter-ancilla interactions influence memory, it shows that genuinely quantum memory is required only under certain structured environmental correlations, with a Bell marginal maximizing leading-order entanglement transfer and establishing a memory-onset threshold at $\eta = \pi/3$. GHZ-type reservoirs can realized classical memory unless perturbed to introduce local entanglement, in which case quantum memory can be activated. The findings have practical implications for solid-state quantum computing, suggesting memory-aware error mitigation and decoders, and offer physical criteria for engineering or suppressing quantum memory through reservoir design and coupling strength.

Abstract

The emergence of memory is a hallmark feature of non-Markovian dynamics. However, the type of memory -- classical or quantum -- required to realize certain dynamics remains unknown. We study the quantum homogenizer as a minimal model of non-Markovian evolution and identify the physical conditions under which genuinely quantum memory becomes necessary. Using entanglement measures and relying only on the local dynamics as a witness, we prove both analytically and numerically the type of memory depends not merely on the dynamics itself, but also on the reservoir's initial entanglement structure, and in particular the propagation of non-classical correlations within it. For different bi- or multi-partite reservoir initializations, we establish a correspondence between interaction strength and entanglement generation. We provide physical criteria and an activation lower bound for the onset of quantum memory. The results may inform us how environmental correlations govern the transition from classical to quantum memory in open quantum systems.

Figures (3)

• Figure 1: The Markovian dynamics setup. Schematic of the quantum homogenizer, where a system qubit $S$ in state $\rho_{S}$ (green square) interacts sequentially ($k=1,...,N$) with the identically prepared $R$ qubits $\xi^{(k)}$ (orange circle). The partial-SWAP operator $\mathcal{U}_{{S},{k}}$, as in Eq. (\ref{['eq:pswap']}), mediates the unitary two-body operations. Similar to a noisy quantum channel alexander, it is essentially a convex combination of the identity and SWAP gates.
• Figure 2: Numerical simulation of the composite dynamics. (a) Depiction of the non-Markovian evolution induced by the action of Eq. (\ref{['eq:composite']}) on a bath of four qubits initialized in a product state. The coupling strength $\eta \in [0, \pi/2]$ is drawn every $k$ from a Gaussian distribution and $N$ is the number of interactions. For performance comparison, we also show (in green) the same evolution but mediated by the traditional homogenizer instead. (b)(c) Corresponding heatmap representations of the quantum state evolution via Eq. (\ref{['eq:pswap']}) and Eq. (\ref{['eq:composite']}), respectively. Notably, we observe identical behavior as reported in Ref. vlatko2cswap2: while the difference in convergence accuracy (i.e., effectiveness) between partial-SWAP (Markovian) and non-Markovian machines vanishes with increasing bath size, the trajectories (transient dynamics) of the evolving states in both scenarios differ significantly.
• Figure 3: The entanglement differences $C^{\#}(\tilde{L}^{(k)}) - C(\tilde{M}^{(k)})$ and $C^{\#}(\tilde{L}^{(k)}) - C(\tilde{K}^{(k)})$ as a function of the coupling strength $\eta$ for $R$ initialized in (a) the Bell state, (b) the GHZ state $\ket{111}+\ket{000}$, (c) asymmetric GHZ state $\ket{000}+\ket{101}$, and (d) perturbed GHZ state of the form $\sqrt{\alpha} \ket{100} + \sqrt{1-\alpha}$GHZ, with perturbation strength $\alpha=\{0.1, 0.3, 0.5\}$. In all cases, $\varphi_{S+R}^{(0)}$ evolves for $k=2$ steps under Eq. (\ref{['eq:composite']}) for the two schemes, respectively. Numerically, for Bell, asymmetric, and perturbed GHZ state initializations (with $\alpha=0.5$), the concurrence witness Eq. (\ref{['eq:ineq']}) becomes negative above a threshold of $\eta$. In those regimes, quantum memory is needed. The lower bound is $\eta = \pi/3$ saturated in the setting of (a). For GHZ-type $R$, the concurrence remains positive and thus only classical memory suffices. However, for any non-zero value of the non-trivial perturbation $\alpha$, Eq. (\ref{['eq:ineq']}) is satisfied, where higher values of $\alpha$ evoke quantum memory earlier.

Theorems & Definitions (10)

• Theorem 1: informal
• Theorem 2: informal
• Proposition 1
• Theorem 2: Linear transfer of entanglement between the reservoir and the system
• proof
• Proposition 1: Optimal reservoir initialization
• proof
• Theorem 1: Multipartite entanglement transfer
• proof
• Corollary 1