Verifying Quantum Memory in the Dynamics of Spin Boson Models

TL;DR

The paper addresses how to verify quantum memory in non-Markovian spin-boson dynamics by comparing two locally defined criteria: one based on single-intervention process tensors and one based on dynamical maps. It employs numerically exact matrix product operator influence-functionals (MPO-IF) via uniTEMPO to construct process tensors for spin-boson and two-spin-boson models across Lorentzian and Ohmic baths. The results show that the process-tensor criterion reliably detects quantum memory at low temperatures and in stationary regimes, while the map-based criterion often misses memory except in highly coherent, resonant settings. Overall, process tensors offer a more complete and robust diagnostic of environment-induced quantum memory, with practical implications for experimental verification and control of non-Markovian quantum dynamics.

Abstract

We investigate the nature of memory effects in the non-Markovian dynamics of spin boson models. Local quantum memory criteria can be used to indicate that the reduced dynamics of an open system necessarily requires a quantum memory in its environment. We apply two such criteria, derived from different definitions put forward in the literature, to spin boson and two-spin boson models. For the computation of dynamical maps and process tensors, we employ a numerically exact method for non-Markovian open system dynamics based on matrix product operator influence functionals, that can be applied across broad parameter regimes. We find that, with access to single-intervention process tensors, one can generally predict quantum memory in the dynamics at low temperatures. Given instead only the dynamical map, we are still able to detect quantum memory in the case of resonant environments at short evolution times. Moreover, we confirm quantum memory in the stationary dynamical regime using process tensors with the correlated steady state of system and environment as initial condition.

Figures (9)

• Figure 1: Upper: Influence matrix (time step $\delta t$) expressed as a temporal uniform matrix product operator. Open legs correspond to free indices that can be contracted with local quantum operations. Lower: Process tensor for a single intervention $\mathcal{A}_1$ after $t_1$ time steps, written as a matrix product operator according to Eq. \ref{['eq:PT_MPO']}.
• Figure 2: Upper: A single-intervention process tensor has classical memory if and only if it can be written as a composition of conditional instruments, as in Eq. \ref{['eq:classical_memory_pt']}. Lower: For all process tensors with classical memory, the Choi state is a separable state according to Eq. \ref{['eq:classical_memory_pt_sep']}.
• Figure 3: Extraction of the two dynamical maps $\mathcal{E}_{t_1}$ and $\mathcal{E}_{t_2}$ from the single-intervention process tensor $\mathcal{T}_{t_2,t_1}$ (see Fig. \ref{['fig:im']}, $t_2=t_1+t$). For the map at $t_2$ one simply connects the outgoing and ingoing legs of the intervention at $t_1$. To recover the intermediate map $\mathcal{E}_{t_1}$ one leaves the outgoing leg open and contracts the ingoing leg with an arbitrary state $\sigma_0$, traced out at the end.
• Figure 4: Deformation of the Bloch sphere under the dynamics of the one-spin-boson model with Lorentzian bath with parameters $\gamma=0.075\Omega$, $\omega_0=1.5\Omega$, $g=\Omega$. The specific maps depicted in the inserts correspond to the two time points for which quantum memory is maximal according to the map criterion in the upper left panel in Fig. \ref{['fig:1']}. The coherent dynamics leads to strong revivals in the Bloch sphere volume which violate all common notions of Markovianity.
• Figure 5: Quantum memory criteria for the spin-boson model with Lorentzian bath (left: single spin, $\gamma=0.075\Omega$, $\omega_0=1.5\Omega$, $g=\Omega$; right: two spins, $\gamma=0.05\Omega$, $\omega_0=1.5\Omega$, $g=\Omega$). The figures show the quantities $\Delta(t_1,t_2)$ from Eq. \ref{['eq:map_crit_witness']} (top) and $C_<[\chi]$ from Eq \ref{['eq:conc_main_text']} (bottom) in color-code (color label below each figure). Positive values (blue color) imply quantum memory for the two times $(t_1,t_2)$, respectively. For the chosen fine-tuned parameters, both criteria find quantum memory.

Theorems & Definitions (1)

• Theorem 1