Hamiltonian characterisation of multi-time processes with classical memory

TL;DR

The paper addresses how non-Markovian memory in open quantum systems can be connected to explicit system–environment dynamics within the process-matrix framework. It proves a sufficient condition: a time-dependent Hamiltonian of the form $H(t)=\sum_j S_j(t)\otimes \mathcal{E}_j(t)$ with commuting environmental parts yields a classical-memory, mixed-unitary process $W=\sum_\nu p(\nu)\widetilde{W}_\nu$, with a time-independent environment basis $\{|\nu\rangle\}$. Furthermore, any mixed-unitary process admits a dilation to a quantum circuit using controlled unitaries with the environment acting as a fixed control in the same basis, thereby unifying Hamiltonian/open-system and process-matrix descriptions. The work also clarifies how simple Hamiltonians with product eigenstates can generate CCC processes and discusses examples and limitations, outlining future work toward continuous-time and continuous-variable extensions. Overall, the results provide a concrete link between traditional Hamiltonian models and the process-matrix approach to non-Markovianity, with implications for memory detection and noise mitigation in quantum devices.

Abstract

A central problem in open quantum systems is the characterization of non-Markovian processes, where an environment retains the memory of its interaction with the system. A key distinction is whether or not this memory can be simulated classically, as this can lead to efficient modelling and noise mitigation. Powerful tools have been developed recently within the process matrix formalism, a framework that conveniently characterizes all multi-time correlations through a sequence of measurements. This leads to a detailed classification of classical and quantum-memory processes and provides operational procedures to distinguish between them. However, these results leave open the question of what type of system-environment interactions lead to classical memory. More generally, process-matrix methods lack a direct connection to joint system-environment evolution, a cornerstone of open-system modelling. In this work, we characterize Hamiltonian and circuit-based models of system-environment interactions leading to classical memory. We show that general time-dependent Hamiltonians with product eigenstates, and where the environment's eigenstates form a time-independent, orthonormal basis, always produce a particular type of classical memory: probabilistic mixtures of unitary processes. Equivalently, these Hamiltonians are characterized as commuting with a complete set of observables on the environment. Additionally, we show that the most general type of classical memory processes can be generated by a quantum circuit in which the system and environment interact through a specific class of controlled unitaries. Our results establish the first strong link between process-matrix methods and traditional Hamiltonian-based approaches to open quantum systems.

Figures (7)

• Figure 1: Non-Markovian processes with different memories. (a) The general non-Markovian process with initial joint system-environment state $\rho^{A_I^1E^1}$ and subsequent system-environment unitary interaction $U^n:A_O^nE^n{\to} A_I^{n{+}1}E^{n+1}$ for $n{\in}\{1,2,{\cdots},{N-1}\}$.(b) A Markovian process, A Markovian process, where the environment is reset to a fresh state $\tau$ at each timestep via trace-and-replace channels (orange dashed box), preventing memory retention. (c) A classical-memory process, where an entanglement-breaking channel acts on the environment at each step (orange dashed box), preserving only classical correlations across time. The "gaps" in the circuits represent the possibility of performing arbitrary operations on the system at the given times, under the assumption that their duration is negligible compared to the interaction time.
• Figure 2: Depiction of Theorem \ref{['Thm:direct_cause_extension']} for a three-time mixed unitary process $W{=}\sum_{\nu} p(\nu) W_{\nu}$ with $W_{\nu} {=} \rho_{\nu}^{A_I^1}{\otimes}\left|\widetilde{U}_{\nu}^1 \left \rangle\! \left \rangle\!\right \langle \! \right \langle \widetilde{U}_{\nu}^1\right |^{A_O^1A_I^1}{\otimes}\left|\widetilde{U}_{\nu}^2 \left \rangle\! \left \rangle\!\right \langle \! \right \langle \widetilde{U}_{\nu}^2\right |^{A_O^2A_I^3}$ being a Markovian process. Here $\rho_{\nu}$ is an initial state at $A_I^1$ the system, and $\left|\widetilde{U}_{\nu}^k \left \rangle\! \left \rangle\!\right \langle \! \right \langle \widetilde{U}_{\nu}^k\right |$ is the CJ representation of the unitary $\widetilde{U}_{\nu}^k$, with $k {\in} \{1,2\}$. The right-hand side of the picture shows that any such mixed unitary process can be dilated in terms of controlled unitaries $\widetilde{U}^k=\sum_{\nu}\widetilde{U}_{\nu}^k\otimes \left|\nu\middle\rangle\!\middle\langle\nu\right|^E$, $k {\in} \{1,2\}$, and an initial quantum-classical system-environment state $\rho^{A^1_IE}=\sum_{\nu}p(\nu)\rho_{\nu}^{A_I^1}{\otimes}\left|\nu\middle\rangle\!\middle\langle\nu\right|$. See Appendix \ref{['App:proof_direct_cause_ext']} for the proof.
• Figure 3: Circuits representing different classes of classical memory, with time oriented from left to right. (a) Full classical memory with dependence on past (including stochastic), and (b) classical common cause memory. The dashed circles represent the sites.
• Figure 4: Tracing over controlled unitaries in a fixed basis is equivalent to a sum over classically controlled unitaries. Time is from left to right. Note that controls on the same environment become conditioned on the same classical index.
• Figure 5: Implementing a stochastically controlled unitary. The operator in the dashed box is also a unitary. Time is from left to right.

Theorems & Definitions (4)

• Theorem 1
• Corollary 1
• Conjecture 1
• Theorem 2