Dissipation alters modes of information encoding in small quantum reservoirs near criticality

TL;DR

This work investigates how dissipation and dynamical instability near a critical point shape information encoding in a minimal driven-dissipative quantum reservoir composed of two coupled Kerr-nonlinear oscillators. Using Partial Information Decomposition, the authors quantify when the reservoir encodes inputs redundantly versus synergistically, revealing a transition to synergy at $J$ close to $|\Delta|$, driven by the competition between soft and fast collective modes and their overdamped dynamics due to dissipation. They show that synergy enhances short-term memory but may hinder long-term retention, whereas large dissipation yields robust redundant encoding that supports fading memory; memory performance is thus governed by a trade-off between sensitivity to recent inputs and stability. The findings offer a nuanced information-theoretic perspective for designing QRC systems, highlighting how tuning dissipation and coupling near dynamical bifurcations enables control over encoding modes and memory capabilities in small quantum reservoirs.

Abstract

Quantum reservoir computing (QRC) has emerged as a promising paradigm for harnessing near-term quantum devices to tackle temporal machine learning tasks. Yet identifying the mechanisms that underlie enhanced performance remains challenging, particularly in many-body open systems where nonlinear interactions and dissipation intertwine in complex ways. Here, we investigate a minimal model of a driven-dissipative quantum reservoir described by two coupled Kerr-nonlinear oscillators, an experimentally realizable platform that features controllable coupling, intrinsic nonlinearity, and tunable photon loss. Using Partial Information Decomposition (PID), we examine how different dynamical regimes encode input drive signals in terms of redundancy (information shared by each oscillator) and synergy (information accessible only through their joint observation). Our key results show that, near a critical point marking a dynamical bifurcation, the system transitions from predominantly redundant to synergistic encoding. We further demonstrate that synergy amplifies short-term responsiveness, thereby enhancing immediate memory retention, whereas strong dissipation leads to more redundant encoding that supports long-term memory retention. These findings elucidate how the interplay of instability and dissipation shapes information processing in small quantum systems, providing a fine-grained, information-theoretic perspective for analyzing and designing QRC platforms.

Figures (12)

• Figure 1: Schematic of two coupled Kerr-nonlinear oscillators. Each cavity $i$ features a Kerr nonlinearity $U_i$ and a photon-loss rate $\gamma_i$. A time-dependent drive $F(t)$ (green arrows) injects identical signals into both cavities, while coherent tunneling of strength $J$ (violet arrow) couples the two modes. We measure the mean fields $\alpha_i = \langle \hat{a}_i(t)\rangle$ to probe the system’s response. The Hamiltonian is specified by Eq. \ref{['eqn:Hamiltonian']}, and the Lindblad equation \ref{['eqn:Lindblad']} governs this driven-dissipative dynamics. We assume both cavities have the same detuning $\Delta$ from the drive frequency. This work investigates how the readouts $\alpha_i(t)$ encode the time-dependent drive $F(t)$ across different dynamical regimes of the coupled Kerr oscillators.
• Figure 2: Classical mutual information $I\bigl(s:(X_1,X_2)\bigr)$, compared to $I(s:X_1)$ and $I(s:X_2)$ in (left) a mean-field regime and (right) a quantum regime. In the mean field regime, $I\bigl(s:(X_1,X_2)\bigr)$ exceeds $I(s:X_1)$ or $I(s:X_2)$ alone near $J = |\Delta|$, hinting at synergy. On the other hand, in the quantum regime, $I\bigl(s:(X_1,X_2)\bigr)$ is comparable to, but not always exceeding, the sum of $I(s:X_1)$ and $I(s:X_2)$.
• Figure 3: Normalized synergy (left) and normalized redundancy (right) vs. the coupling $J$. A pronounced peak near $J \approx |\Delta|$ marks the crossover from predominantly redundant to more synergistic encoding. In the fully quantum description, enhanced quantum correlations can favor redundancy even at the transition, whereas second-order cumulants interpolate between mean-field and quantum descriptions.
• Figure 4: Effective potential around the steady state $\alpha_{1,c}=\alpha_{2,c}=0$ (red dot) in the $\Delta < 0, J > 0$ regime, projected onto $\mathrm{Im}(\alpha_1)=\mathrm{Im}(\alpha_2)=0$. (Left) When $J < |\Delta|$, the steady state is weakly unstable in all directions, with no soft modes present, and the system predominantly encodes inputs redundantly. (Center) At the critical point $J = |\Delta|$, flat directions appear, marking the onset of soft modes. In this near-critical regime, collective oscillations enhance synergistic encoding. (Right) For $J > |\Delta|$, the potential deforms into a saddle, with two stable and two unstable directions. Here, the soft modes again disappear, and the system encodes inputs redundantly.
• Figure 5: Retarded Green’s function poles \ref{['eqn:Greenfunction']} in the complex-frequency plane as $J$ increases, illustrating the evolution of slow modes $\omega_s$ (orange dots) and fast modes $\omega_f$ (blue dots). For $J < |\Delta|$, both slow and fast modes coexist, and the system tends to encode inputs more redundantly. Near the critical point $J = |\Delta|$, the real part of $\omega_s$ approaches zero, indicating a disappearance of slow collective oscillations and an overdamped decay. In this near-critical regime, collective oscillations due to underdamped fast modes dominate and enhance synergistic encoding. For $J > |\Delta|$, the slow modes shift away from zero frequency, reducing synergy and transitioning the system back toward redundant encoding. This interplay between slow and fast modes governs how the system transitions from predominantly redundant to synergistic processing and back again as $J$ crosses the critical point.