Computable measures of non-Markovianity for Gaussian free fermion systems

Abstract

We investigate measures of non-Markovianity in open quantum systems governed by Gaussian free fermionic dynamics. Standard indicators of non-Markovian behavior, such as the BLP and LFS measures, are revisited in this context. We show that for Gaussian states, trace-based distances -- specifically the Hilbert-Schmidt norm -- and second-order Rényi mutual information can be efficiently expressed in terms of two-point correlation functions, enabling practical computation even in systems where the full density matrix is intractable. Crucially, this framework remains valid even when the density matrix of the system is an average over stochastic Gaussian trajectories, yielding a non-Gaussian state. We present efficient numerical protocols based on this structure and demonstrate their feasibility through a small-scale simulation. Our approach opens a scalable path to quantifying non-Markovianity in interacting or measured fermionic systems, with applications in quantum information and non-equilibrium quantum dynamics.

Figures (2)

• Figure 1: (a) Sketch of the system under consideration. The system $S$ is composed by fermionic sites along which the electrons can hop with strength $t_1$. The ancillary degrees of freedom $B$ are also given by fermions that hop with strength $t_2$. $S$ and $B$ are coupled by fermionic hopping $t_{12}$ and they undergo a total Markovian dynamics. When the $LFS$ non-Markovianity measure is considered, an ancilla $A$ is coupled to the system, and $S+A$ is initialized in a completely entangled state $\ket{\Psi}$. (b) The BLP non-Markovianity measure studies the time evolution of two different initial conditions $\rho_{S,1}$ and $\rho_{S,2}$, which converge to the same steady state $\rho_{S,\infty}$. When the dynamics is Markovian, the distance between the density matrices decay in a monotonic way and $\mathcal{N}_{BLP}=0$ (left). On the other hand when the dynamics is non-Markovian the two density matrices do not converge monotonically to the steady state and $\mathcal{N}_{BLP}>0$ (right). (c) The LFS non-Markovianity measure studies the time evolution of the mutual information $I_2$ between the system and the ancilla $A$. When $I_2$ decays monotonicly (green curve) the system is Markovian and $\mathcal{N}_{LFS}=0$. When $I_2$ exhibits times where it increases, then the system is non-Markovian and $\mathcal{N}_{LFS}>0$ (red curve).
• Figure 2: Plots of the computational time required to calculate $d_2$ for a system with $L$ fermionic sites (plus $L$ sites for the ancillary degrees of freedom). a) Computational time required for a simulation of the full quantum state using QuTiP for $L=2,3,4,5,6$; the blue points indicate the data obtained from the simulations, while the red dashed line indicates an exponential fit to the data, from which we estimate $T_{\mathrm{comp}}\sim2^{4.1L}$, in line with the theoretical prediction $\sim2^{2(2L)}\sim2^{4L}$. For $L=2,3$ the computational time is dominated by the overhead of the algorithm, while for larger $L$ the exponential increase is very evident. b) Computational time required for a simulation using quantum trajectories for Gaussian states for $L$ ranging between $L=2$ and $L=256$; the blue points indicate the data obtained from the simulations, while the red dashed line indicates a power-law fit to the data, from which we estimate $T_{\mathrm{comp}}\sim L^{2.38}$. Also in this case, $T_{\mathrm{comp}}$ is dominated by the computational overhead at small system sizes, and then follows a power-law behavior. c) Comparison of $T_{\mathrm{comp}}$ for the two methods. The Gaussian method has a larger overhead then the full state simulation method, which makes computationally more costly at low $L$. However, already for $L\approx6$, the Gaussian method requires less resources; for $L\gtrsim10$ the Gaussian method is the only one feasible.