Entanglement and Optimal Timing in Discriminating Quantum Dynamical Processes

TL;DR

The paper addresses the problem of distinguishing two open quantum dynamical processes in a single-shot setting by jointly optimizing the input state (potentially entangled with an ancilla) and the interrogation time. It develops a framework based on time-dependent channels ${\cal E}_t^{(i)}$ and the Helstrom/distance formalism, identifying when entanglement yields a true advantage via the diamond norm. Through detailed analysis of two Pauli dynamical maps, it shows that entangled inputs can enable finite-time, lower-error discrimination in several cases (e.g., coplanar decays and depolarising vs dephasing), while in other cases entanglement offers no benefit. The results illuminate how timing and quantum correlations jointly enhance discrimination of dynamical processes, with implications for quantum metrology, sensing, and quantum illumination.

Abstract

I investigate the problem of optimally discriminating between two open quantum dynamical processes in a single-shot scenario, with the goal of minimizing the error probability of identification. This task involves optimising both the input state -- potentially entangled with an ancillary system that remains isolated from the dynamics -- and the time at which the resulting time-dependent quantum channels, induced by the two distinct dynamical maps, becomes most distinguishable. To illustrate the complexity and richness of this problem, I focus on Pauli dynamical maps and their associated families of time-dependent Pauli channels. I identify a regime in which separable strategies require waiting indefinitely for the dynamics to reach the stationary state, whereas entangled input states enable optimal discrimination at a finite time, with a strict reduction in error probability. These results highlight the crucial interplay between entanglement and timing in enhancing the distinguishability of quantum dynamical processes.

Figures (5)

• Figure 1: Minimum error probability for discriminating at time $t$ a dephasing process ${\bm\gamma }^{(1)} =(0,0,1)$ from those with ${\bm\gamma }^{(2)} =(0,0,\gamma ^{(2)})$, where $\gamma ^{(2)}=0.25$, $0.5$, and $4.0$ (solid, dashed, and dot-dashed lines, respectively). Entanglement provides no advantage at any time.
• Figure 2: Minimum error probabilities for discriminating between two coplanar decaying processes with ${\bm\gamma }^{(1)} =(1,1,0)$ and ${\bm\gamma }^{(2)} =(.2,.2,0)$ at time $t$, with and without entanglement (solid and dashed lines, respectively). Side entanglement strictly improves discrimination at any time.
• Figure 3: Minimum error probability for discriminating two depolarising processes at time $t$, with ${\bm\gamma }^{(1)} =(1,1,1)$ and ${\bm\gamma }^{(2)}=(0.2,0.2,0.2)$, with and without entanglement assistance (solid and dashed lines, respectively).
• Figure 4: Ultimate minimum error probabilities in discriminating between two depolarising processes as a function of the ratio of the decay rates, with and without entanglement assistance (solid and dashed lines, respectively). In both cases, the optimal discrimination time is given by Eq. (\ref{['tss']}).
• Figure 5: Minimum error probability for discriminating the depolarising process ${\bm\gamma }^{(1)} =(1,1,1)$ from a dephasing process ${\bm\gamma }^{(2)} =(0,0,\gamma ^{(2)})$ with and without side entanglement (solid and dashed lines, respectively), for different values of $\gamma ^{(2)}$: $10$ (top left), $0.5$ (top right), $0.3785$ (bottom left), and $0.2$ (bottom right).