Single shot distinguishability of noisy quantum channels
Satyaki Manna
TL;DR
This work analyzes single-shot discrimination of noisy quantum channels using either product or entangled probes, and develops a unified framework based on CPTP maps and Helstrom-type distinguishability. Across depolarizing, dephasing, amplitude-damping, noisy unitary, and erasure channels, it derives exact or regime-dependent optimal probing strategies, including when maximally or non-maximally entangled states provide advantages. Key findings include: maximally entangled probes are optimal for distinguishing two qubit depolarizing channels; single-system probes suffice for two dephasing channels (and generalizations) with entanglement offering no improvement; amplitude-damping discrimination exhibits noise-parameter regimes where either product or entangled probes dominate, and non-maximally entangled probes can be optimal; certain mixed-unitary channels allow perfect discrimination with non-maximally entangled probes, while others favor product probes; erasure channels show that entanglement yields no advantage. The results illuminate when entanglement is a resource for channel discrimination and outline open questions for more complex channel sets and cross-type discrimination, with implications for quantum communication and metrology.
Abstract
Among the intriguing features of quantum theory, the problem of distinguishing quantum channels is of fundamental interest. In this paper, we focus on the single-shot discrimination of two noisy quantum channels using two distinct classes of probes: single-system (product) probes and entangled probes. Our aim is to identify optimal probing state for specific discrimination tasks and to analyze the necessity and role of entanglement in enhancing channel distinguishability. We show that maximally entangled probes are optimal for discriminating two qubit depolarizing channels, with any nonzero entanglement providing an advantage over single-system probes. In contrast, for dephasing channels in arbitrary dimensions, we prove that single-system probe can be optimal and that entanglement offers no improvement, even when the dephasing unitary is generalized. For qubit amplitude-damping channels, we identify distinct noise-dependent regimes in which either single-system probe outperforms maximally entangled probes and vice-versa. Moreover, we demonstrate that non-maximally entangled probes can act as the optimum probe if the noise parameters restricted to certain values in this task. We also present examples of noisy unitary channels for which discrimination is possible using non-maximally entangled probe, while both single-systems and maximally entangled probes fail. We introduce another class of noisy unitary channels for which perfect discrimination is achievable with a single system, while maximally entangled probes are insufficient. Finally, we show that two erasure channels can be optimally discriminated using any pure single-system probe, with no advantage gained from entanglement.