Initial archival release (v0.0.1) of the working repository for an ongoing theoretical project on memory-restricted adaptive quantum channel discrimination. The project investigates the boundary between regimes in which adaptive strategies genuinely outperform parallel probing and regimes in which they do not, focusing on the small-N, noisy, memory-limited setting that has received comparatively little systematic treatment. The eventual paper: provisionally titled Where Adaptivity Actually Helps: Memory-Constrained Phase Diagrams for Few-Use Quantum Channel Discrimination, studies a family of binary qubit-channel discrimination problems including rotated amplitude damping, dephasing-with-drift, and generalised Pauli and non-Pauli mixtures. Recent literature has developed constrained-separability formulations for memory-limited strategies and tensor-network methods for larger adaptive protocols, but a compact atlas of the underlying transition surfaces in the few-use regime appears to be missing. This project aims to fill that gap. This release contains the initial project scaffolding. Keywords: quantum channel discrimination; adaptive strategies; quantum memory; semidefinite programming; Pauli channels; amplitude damping; phase diagrams; quantum information theory Related identifiers: companion to prior work on time-bin photonic qudit calibration (https://arxiv.org/abs/2604.27009); independent project, distinct subject area within quant-ph. License: MIT for code, CC-BY 4.0 for the manuscript. Version: v0.0.1 (initial scaffolding release)
Initial archival release (v0.0.1) of the working repository for an ongoing theoretical project on memory-restricted adaptive quantum channel discrimination. The project investigates the boundary between regimes in which adaptive strategies genuinely outperform parallel probing and regimes in which they do not, focusing on the small-N, noisy, memory-limited setting that has received comparatively little systematic treatment. The eventual paper: provisionally titled Where Adaptivity Actually Helps: Memory-Constrained Phase Diagrams for Few-Use Quantum Channel Discrimination, studies a family of binary qubit-channel discrimination problems including rotated amplitude damping, dephasing-with-drift, and generalised Pauli and non-Pauli mixtures. Recent literature has developed constrained-separability formulations for memory-limited strategies and tensor-network methods for larger adaptive protocols, but a compact atlas of the underlying transition surfaces in the few-use regime appears to be missing. This project aims to fill that gap. This release contains the initial project scaffolding. Keywords: quantum channel discrimination; adaptive strategies; quantum memory; semidefinite programming; Pauli channels; amplitude damping; phase diagrams; quantum information theory Related identifiers: companion to prior work on time-bin photonic qudit calibration (https://arxiv.org/abs/2604.27009); independent project, distinct subject area within quant-ph. License: MIT for code, CC-BY 4.0 for the manuscript. Version: v0.0.1 (initial scaffolding release)
Quantum memories are a crucial precondition in many protocols for processing quantum information. A fundamental problem that illustrates this statement is given by the task of channel discrimination, in which an unknown channel drawn from a known random ensemble should be determined by applying it for a single time. In this paper, we characterise the quality of channel discrimination protocols when the quantum memory, quantified by the auxiliary dimension, is limited. This is achieved by formulating the problem in terms of separable quantum states with additional affine constraints that all of their factors in each separable decomposition obey. We discuss the computation of upper and lower bounds to the solutions of such problems which allow for new insights into the role of memory in channel discrimination. In addition to the single-copy scenario, this methodological insight allows to systematically characterise quantum and classical memories in adaptive channel discrimination protocols. Especially, our methods enabled us to identify channel discrimination scenarios where classical or quantum memory is required, and to identify the hierarchical and non-hierarchical relationships within adaptive channel discrimination protocols.
Discriminating between noisy quantum processes is a central primitive for quantum communication, metrology, and computing. While discrimination limits for finite-dimensional channels are well understood, the continuous-variable setting, particularly under experimentally relevant energy constraints, remains significantly less developed. In this work, we establish an energy-constrained chain rule for the Belavkin-Staszewski channel divergence, which yields a fundamental upper bound on the error exponents achievable by fully adaptive, energy-constrained quantum channel discrimination protocols. We then derive efficiently computable bounds on asymmetric error exponents for energy-constrained discrimination of bosonic dephasing and loss-dephasing channels. Specifically, we show that three operationally relevant quantities -- the measured relative entropy, the Umegaki relative entropy, and the geometric Renyi divergence -- admit semidefinite program (SDP) formulations when the input energy is bounded and the Hilbert space is suitably truncated. Applying these tools, we demonstrate that optimal probes for these channels under energy constraints are Fock-diagonal, and we also enable numerically precise evaluation of bounds on achievable error exponents across discrimination strategies ranging from separable to fully adaptive. The resulting SDPs provide practical benchmarks for quantum-limited sensing in low-energy bosonic platforms.
Adaptiveness is a key principle in information processing including statistics and machine learning. We investigate the usefulness adaptive methods in the framework of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel and observing outputs. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and nonadaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) We prove that for classical-quantum channels, adaptive and nonadaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding, Stein) settings. (2) The first separation between adaptive and nonadaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for nonadaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. (3) We prove, in some sense generalizing the previous statement, that for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior in the asymptotic limit to nonadaptive product state strategies. (4) As an application of our findings, we address the discrimination power of an arbitrary quantum channel and show that adaptive strategies with classical feedback and no quantum memory at the input do not increase the discrimination power of the channel beyond nonadaptive tensor product input strategies.
We present a broad investigation of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel multiple times and observing its output at the end. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and non-adaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. Our main result is the first separation between adaptive and non-adaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for non-adaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. Full details in [1].
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.
We present a broad investigation of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel and observing its output. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and non-adaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) The first separation between adaptive and non-adaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for non-adaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. (2) We prove that for classical-quantum channels, adaptive and non-adaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding, Stein) settings. (3) We prove, in some sense generalizing the previous statement, that for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior in the asymptotic limit to non-adaptive product state strategies. (4) As an application of our findings, we address the discrimination power of quantum channels and show that adaptive strategies with classical feedback and no quantum memory at the input do not increase the discrimination power of entanglement-breaking channel beyond non-adaptive tensor product input strategies.