Bayesian and Markovian classical feedforward for discriminating qubit channels

TL;DR

This work analyzes multi-shot discrimination between two qubit channels using separable inputs and adaptive Helstrom measurements with classical feedforward. It compares Bayesian and Markovian strategies against a global nonlocal benchmark, finding that Bayesian gives only a modest advantage in a restricted parameter region, while the Markovian strategy frequently approaches global performance, notably in amplitude-damping scenarios. By deriving explicit single-shot and multi-shot discriminability formulas for depolarizing, bit-flip, and amplitude-damping channels and numerically optimizing input parameters, the study clarifies when local adaptive schemes can approximate optimal global strategies and highlights practical implications for adaptive quantum channel discrimination. The results suggest that, despite limitations, local adaptive strategies—especially Markovian—are promising for efficient discrimination with minimal quantum memory and entanglement, and point to future work on backward optimization and extension to other channel families.

Abstract

We address the issue of multishot discrimination between two qubit channels by invoking a simple adaptive protocol that employs Helstrom measurement at each step and classical information feedforward, beside separable inputs. We contrast the performance of Bayesian and Markovian strategies. We show that the former is only slightly advantageous and for a limited parameters' region.

Figures (7)

• Figure 1: Left: Schematic representation of channel discrimination through a local adaptive strategy. $\psi$ (resp. $\rho$) denotes the input (resp. output) state at each step. The symbol "?" stands for the quantum channel picked up from a binary ensemble. Dashed (resp. solid) lines refer to the flow of classical (resp. quantum) information. Right: Schematic representation of channel discrimination with global measurement.
• Figure 2: Difference between $P_{succ}^{(3)}$ for Bayesian strategy and $P_{succ}^{(3)}$ for Markovian strategy vs $\eta_0,\eta_1$ for depolarizing channels.
• Figure 3: Success probability $P_{succ}^{(n)}$ vs $n$ computed in different points (a), (b) of the $\eta_0,\eta_1$ plane for depolarizing channels. Red squares, Orange triangles, Blue dots correspond to global, Bayesian and Markovian strategies respectively.
• Figure 4: Difference between $P_{succ}^{(3)}$ for Bayesian strategy and $P_{succ}^{(3)}$ for Markovian strategy vs $\eta_0,\eta_1$ for bit-flip channels.
• Figure 5: Success probability $P_{succ}^{(n)}$ vs $n$ computed in different points (a), (b) of the $\eta_0,\eta_1$ plane for bit-flip channels. Red squares, Orange triangles, Blue dots correspond to global, Bayesian and Markovian strategies respectively.