Memory effects in repeated uses of quantum channels

TL;DR

The paper addresses memory effects that arise when quantum channels are reused without resetting, impacting quantum state transfer and entanglement distribution. It develops an analytic framework for $U(1)$-symmetric channels, showing that the $n$-th use fidelity can be expressed in terms of single-particle amplitudes plus a history memory factor $A_{n-1}$. For free-fermion (quadratic) Hamiltonians, memory contributions simplify to Motzkin-path–type sums over past amplitudes, enabling explicit forms in PST chains. The results reveal that even small readout timing errors compound with repeated uses, often driving fidelity below LOCC limits and reducing quantum capacity, with broad implications for scalable and energy-efficient quantum networks.

Abstract

Quantum Information Processing (QIP) tasks can be efficiently formulated in terms of quantum dynamical maps, whose formalism is able to provide the appropriate mathematical representation of the evolution of open quantum systems. A key QIP task is quantum state transfer (QST) aimed at sharing quantum information between distant nodes of a quantum network, enabling, e.g. quantum key distribution and distributed quantum computing. QST has primarily been addressed insofar by resetting the quantum channel after each use, thus giving rise to memoryless channels. Here we consider the case where the quantum channel is continuously used, without implementing time- and resource- consuming resetting operations. We derive a general, analytical expression for the $n^{\mathrm{th}}$-use average QST fidelity for $U(1)$-symmetric channels and apply our formalism to a perfect QST channel in the presence of imperfect readout timing. We show that even relatively small readout timing errors give rise to memory effects which have a highly detrimental impact on subsequent QST tasks.

Figures (5)

• Figure 1: A quantum channel (green spheres) for the transfer of quantum information between two quantum registers: a sender register (red spheres) and a receiver register (blue spheres).
• Figure 2: Mixed-degree rooted tree showing the transition amplitudes entering the term $A_{n-1}\left(t_{n-1};t_1\right)$ in \ref{['eq_fid_gen']}. The relevant transition amplitudes are along the Motzkin paths marked in red. Each vertex (blue dot) shows the number of excitations inside the channel after the readout procedure. The edges represent transition amplitudes between the states with the number of excitations between connected vertices.
• Figure 3: Average $n\nthscript{th}$-use fidelity for a chain of fixed length $N=6$ in \ref{['eq_Chris']} for different readout timing errors $\delta$. The red dashed line reports the LOCC limit $\frac{2}{3}$. Apart from the ideal scenario of $\delta=0\%$, each subsequent use of the quantum channel lowers the fidelity.
• Figure 4: Average fidelity for a fixed readout timing error of $\delta=1\%$ for different chain lengths in \ref{['eq_Chris']}. Each curve represents a different number of uses $n=1,2,\dots,5$. Already after a few uses, chains in the order of $10^3$ sites fall below the LOCC limit (red dashed line). The inset shows the average $n\nthscript{th}$-use fidelity along vertical lines of the main plot for selected lengths $N$.
• Figure 5: Entanglement distribution for the 1- (blue curve) and 2-use (red curve) across the quantum channel modeled by \ref{['eq_Chris']} with $N=10$ as a function of time. Whereas the 1-use admits a non-zero concurrence between $s'$ and $r$ at any $t\neq 0$ or $\pi$, the 2-use admits only a reduced time-window for entanglement distribution.