Non-Markovianity in collision models with initial intra-environment correlations

TL;DR

This work analyzes non-Markovian open quantum dynamics in collision models by introducing intra-environment correlations generated operationally on overlapping ancilla groups. It shows that the resulting dynamics can be mapped to a Markovian composite collision model, with memory length $L$ encoding the memory depth, enabling standard Markovian analyses while capturing non-Markovian features. In an all-qubit setup with $L=2$, non-Markovianity arises only when ancilla-ancilla entanglement is present in the interacting memory block, and crucially depends on the order of system-ancilla and ancilla-ancilla collisions; reversing the order yields entirely Markovian dynamics. The results establish a direct link between intra-environment correlations and memory effects, provide explicit decoherence-function expressions, and suggest experimental implementations and extensions to quantum thermodynamics and information processing.

Abstract

Collision models (CMs) describe an open system interacting in sequence with elements of an environment, termed ancillas. They have been established as a useful tool for analyzing non-Markovian open quantum dynamics based on the ability to control the environmental memory through simple feedback mechanisms. In this work, we investigate how ancilla-ancilla entanglement can serve as a mechanism for controlling the non-Markovianity of an open system, focusing on an operational approach to generating correlations within the environment. To this end, we first demonstrate that the open dynamics of CMs with sequentially generated correlations between groups of ancillas can be mapped onto a composite CM, where the memory part of the environment is incorporated into an enlarged Markovian system. We then apply this framework to an all-qubit CM, and show that non-Markovian behavior emerges only when the next incoming pair of ancillas are entangled prior to colliding with the system. On the other hand, when system-ancilla collisions precede ancilla-ancilla entanglement, we find the open dynamics to always be Markovian. Our findings highlight how certain qualitative features of inter-ancilla correlations can strongly influence the onset of system non-Markovianity.

Figures (7)

• Figure 1: The model. An environment $E$ of identical ancillas $A_1,...,A_n$, each in the state $\rho_A$, travels to the left; they reach a mechanism whereby groups of size $L$ become sequentially correlated through an operation ${\mathcal{W}}$ ($L=2$ in the schematic). After exiting the 'correlation tube', overlapping groups of $L$ ancillas end up correlated. The system $S$ then interacts with each ancilla, one by one, via a unitary operation ${\mathcal{U}}$.
• Figure 2: The map $\mathcal{M}$ (acting on $\rho_S\otimes\rho^{\otimes L-1}_A$) viewed in two different ways according to \ref{['20m']} and \ref{['anfin']}, with the chronological order of operations labeled by (i)-(iii).(a) The map represented by \ref{['20m']}. In the first step (i), the ancillas ${A}_1,...,{A}_L$ are correlated via ${\mathcal{W}}_{[L,1]}$ (the case where $\mathcal{W}$ performs pairwise collisions is shown). This is followed by (ii)${S}$ colliding with ${A}_1$, and (iii)${A}_1$ being traced out. The ancillas are then shifted: ${A}_2$ takes the role of the previous ${A}_1$ and so on, and a new ancilla ${A}_{L+1}$ is introduced in place of ${A}_L$. (b) The map represented by \ref{['anfin']}. A different sequence of operations (i)-(iii) is now applied, with ${\mathcal{W}}_{[L,1]}$ and ${\mathcal{U}}_{S1}$ replaced by ${\mathcal{W}}'_{[L,1]}$ and ${\mathcal{U}}_{SL}$, and where the partial trace is taken over the incoming ancilla $A_L$. As such, $S$ interacts with the same group of ancillas $M$ with repeated applications of \ref{['anfin']}.
• Figure 3: The maps $\mathcal{M}$ and $\mathcal{M}'$ schematically represented in the case of $L=2$.(a) The system $S$ interacts with the memory ancilla $M$ ($A_1$) and non-memory ancilla $A_2$ as per Fig. \ref{['fig2']}(b). (b) The map $\mathcal{M}'$ introduced in Ciccarello2013Ciccarello2013aMcCloskey2014Campbell2018 to describe the dynamics of the analogous CCM in which there are no initial correlations, but where AA collisions occur via $\mathcal{W}'_{[2,1]}$after$S$ collides with $M$. In contrast to ${\mathcal{M}}$, the order of the operations (i) and (ii) is reversed, in addition to $S$ colliding directly with the memory ancilla.
• Figure 4: Non-Markovianity measure $\mathcal{N}_{BLP}$, \ref{["27'"]}, evaluated to $n=100$ collisions, for which the non-Markovianity reaches its asymptotic value (increasing $n$ further does not change the results). The ancilla state is $\rho_{A}=|+\rangle\langle +|_A$ (see \ref{['rhoa']}). The bright regions correspond to a higher degree of non-Markovianity, while the dark regions correspond to parameters $(\varepsilon, \tau)$ for which the behavior is fully Markovian. The right panel shows that $\mathcal{N}_{BLP}$ is symmetric under reflections about $\varepsilon=\frac{\pi}{2}$ and $\tau=\frac{\pi}{2}$. The left panel is an enlargement of the region enclosed by the inner red square in the right panel.
• Figure 5: Snapshots of the non-Markovianity measure$\mathcal{N}_{BLP}$ shown in Fig. \ref{['fig4']} for (a)$\tau=0.15\pi$, and (b)$\varepsilon=0.195\pi$. In both panels, the red dashed lines indicate the values $\varepsilon=\frac{\pi}{4}$ and $\tau=\frac{\pi}{4}$ where the measure vanishes (see \ref{["31'"]} and \ref{["32'"]}). (c), (d) Corresponding dynamics of$|D(n)|$as a function of the collision number$n$. The gray lines represent instances where $|D(n)|$ decreases monotonically, indicating Markovian behavior.

Theorems & Definitions (4)

• Theorem 1
• Proposition 1
• Proposition 2
• Lemma 1