Quantum fluctuation dynamics of open quantum systems with collective operator-valued rates, and applications to Hopfield-like networks

TL;DR

It is shown that, asymptotically and at the description level of quantum fluctuations, only a very weak amount of quantum correlations, in the form of quantum discord, emerges beyond classical correlations.

Abstract

We consider a class of open quantum many-body systems that evolves in a Markovian fashion, the dynamical generator being in GKS-Lindblad form. Here, the Hamiltonian contribution is characterized by an all-to-all coupling, and the dissipation features local transitions that depend on collective, operator-valued rates, encoding average properties of the system. These types of generators can be formally obtained by generalizing, to the quantum realm, classical (mean-field) stochastic Markov dynamics, with state-dependent transitions. Focusing on the dynamics emerging in the limit of infinitely large systems, we build on the exactness of the mean-field equations for the dynamics of average operators. In this framework, we derive the dynamics of quantum fluctuation operators, that can be used in turn to understand the fate of quantum correlations in the system. We apply our results to quantum generalized Hopfield associative memories, showing that, asymptotically and at the mesoscopic scale only a very weak amount of quantum correlations, in the form of quantum discord, emerges beyond classical correlations.

Figures (4)

• Figure 1: Classical, collective state-dependent rates. Representation of transitions in stochastic Markovian processes for systems made of classical Ising spins. Each one of the particle can be found in the configuration $\bullet / \circ$, corresponding to, e.g., excited state and ground state, respectively. In panel (a) a classical stochastic non-interacting dynamics is sketched. It consists of independent spin-flips, $\bullet {{\, }}^{\to} /_{ \leftarrow} \circ$, occurring at rates $\gamma_{\circ /\bullet}$, respectively. Here, transitions rates are independent of the state of neighboring particles. Panel (b) represents an instance of kinetically-constrained model, in which the central particle can change its state only if the neighboring ones are both in the excited state $\bullet \,$. Finally, in panel (c), a collective all-to-all model is displayed, as the dynamics sketched in panel (b) features here transition rates that depend on the square of the density of excited particles, $n_\bullet$.
• Figure 2: Phase diagram of the quantum Hopfield-like model. At high temperature the mean-field equations for the overlaps admits a unique, vanishing solution, this corresponding to the paramagnetic phase. At low temperature the stationary solutions are characterized by finite overlap values (colorbar). This phase is referred to as retrieval phase. The (white) region between the paramagnetic and the retrieval one is referred to as a limit-cycle phase, and it displays self-sustained oscillations at stationarity.
• Figure 3: Quantum and classical correlations. (a) One way quantum discord, and (b) one-way classical correlations between the two large-spin operators of the one-memory case, $p=1$. In the displayed parameter regime, the system-average operators identify a retrieval region (low temperature) and a paramagnetic one (high temperature).
• Figure 4: Limit-cycle phase. (a) For the case $p=1$, we display the standard deviation $o_{\mathrm{sd}}$ of the overlap $o_z(t)$ with respect to the stationary solutions, at long times. We set initial conditions such that $o_{\alpha}(0) \approx 0$, for $\alpha = y, z$. In the blue region $o_{\mathrm{sd}}$ is finite, signalling a limit-cycle phase. (b) With $T=0.6$, the two panels display the parametric plot and the flux diagram of the vector field $(o_y(t),o_z(t))$, for $\Omega=0.35$ (bottom panel), and $\Omega=0.6$ (upper panel). The latter shows a closed orbit, that in general characterizes limit-cycle phases. Colors for the flux diagram identify the norm of the corresponding vector field, which increases from purple to yellow.

Theorems & Definitions (26)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Lemma 4
• Lemma 5