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IEEE Transactions on Quantum Engineering

Quantum Detection Over Quantum Channels With Uncertainty

Jeremy Johnston, Xiaodong Wang

Published Oct 27, 2025• 0 citations
Publisher

Abstract

In quantum state discrimination, typically the design of measurement operators or probe state is formulated assuming the set of possible states is perfectly known, but this may yield designs which are sensitive to deviations in the realized set of states. For example, the channel through which a transmitted state is sent may not be deterministic, but instead characterized by a classical distribution over quantum channels. In this paper, we consider the design of measurement schemes and probe states for quantum detection over an uncertain quantum channel. We present stochastic gradient-based algorithms to maximize the expected performance over the channel distribution under two design scenarios, joint design and two-stage design. We consider various design objectives, including the detection probability and mutual information, with the latter leading to a hybrid scheme consisting of von Neumann measurement and classical hypothesis test. Furthermore, we introduce a channel discrimination scheme which leverages the isometric extension of a quantum channel which increases the channel distinguishability, while simultaneously reducing the effective dimension and thereby reducing the optimization complexity. In addition, we apply amortized optimization to train a recurrent neural network in order to improve the convergence speed of the proposed algorithms. Finally, we apply the proposed algorithms to multi-copy channel discrimination as well as a novel joint channel-state discrimination scenario.

DOI
10.1109/tqe.2025.3625774

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In quantum state discrimination, typically the design of measurement operators or probe state is formulated assuming the set of possible states is perfectly known, but this may yield designs which are sensitive to deviations in the realized set of states. For example, the channel through which a transmitted state is sent may not be deterministic, but instead characterized by a classical distribution over quantum channels. In this paper, we consider the design of measurement schemes and probe states for quantum detection over an uncertain quantum channel. We present stochastic gradient-based algorithms to maximize the expected performance over the channel distribution under two design scenarios, joint design and two-stage design. We consider various design objectives, including the detection probability and mutual information, with the latter leading to a hybrid scheme consisting of von Neumann measurement and classical hypothesis test. Furthermore, we introduce a channel discrimination scheme which leverages the isometric extension of a quantum channel which increases the channel distinguishability, while simultaneously reducing the effective dimension and thereby reducing the optimization complexity. In addition, we apply amortized optimization to train a recurrent neural network in order to improve the convergence speed of the proposed algorithms. Finally, we apply the proposed algorithms to multi-copy channel discrimination as well as a novel joint channel-state discrimination scenario.
In quantum state discrimination, typically the design of measurement operators or probe state is formulated assuming the set of possible states is perfectly known, but this may yield designs which are sensitive to deviations in the realized set of states. For example, the channel through which a transmitted state is sent may not be deterministic, but instead characterized by a classical distribution over quantum channels. In this paper, we consider the design of measurement schemes and probe states for quantum detection over an uncertain quantum channel. We present stochastic gradient-based algorithms to maximize the expected performance over the channel distribution for various design objectives, including the detection probability and mutual information. Furthermore, we introduce a scheme that leverages the isometric extension of a quantum channel to measure the channel output in an enlarged Hilbert space such that the channels are more distinguishable, while simultaneously reducing the effective dimension and thereby reducing the optimization complexity. Finally, we apply the proposed algorithms to multicopy channel discrimination.
Distinguishing one quantum channel from another becomes a next natural question following the advent of quantum state discrimination. Here, we address the problem of quantum channel discrimination with the separable states and the entangled states input. Two schemes, based on the simulator for quantum networks and channels framework (SQUANCH) and the quantum machine learning toolbox (QMLT), are proposed to perform the quantum channel discrimination task. In the first scheme, we give an example of several qubit channels, verify that the performance metrics of channel discrimination can be improved by the entangled states. In the second scheme, we train a quantum circuit learning model to classify quantum channels using the Adam stochastic optimization algorithm. Although we consider the distinguishability in the case of only two possible channels, this model can be extended to any number of channels. This method can provide a good solution for distinguishing some complex quantum channels.
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.
Here, we study the problem of decoding information transmitted through unknown quantum states. We assume that Alice encodes an alphabet into a set of orthogonal quantum states, which are then transmitted to Bob. However, the quantum channel that mediates the transmission maps the orthogonal states into non-orthogonal states, possibly mixed. If an accurate model of the channel is unavailable, then the states received by Bob are unknown. In order to decode the transmitted information we propose to train a measurement device to achieve the smallest possible error in the discrimination process. This is achieved by supplementing the quantum channel with a classical one, which allows the transmission of information required for the training, and resorting to a noise-tolerant optimization algorithm. We demonstrate the training method in the case of minimum-error discrimination strategy and show that it achieves error probabilities very close to the optimal one. In particular, in the case of two unknown pure states, our proposal approaches the Helstrom bound. A similar result holds for a larger number of states in higher dimensions. We also show that a reduction of the search space, which is used in the training process, leads to a considerable reduction in the required resources. Finally, we apply our proposal to the case of the phase flip channel reaching an accurate value of the optimal error probability.
In this work, we consider optimal discrimination among quantum states that are sent through a quantum channel. We show the conditions on a quantum channel and an ensemble of states to preserve a measurement for optimal state discrimination over the channel. In particular, we show that for an ensemble of states where the states are given with equal a priori probabilities, an optimal measurement can be preserved over any quantum channel by applying local operations and classical communication, that is, by manipulating the quantum states before and after the channel application. Examples are provided for illustration in the case of quantum communication protocols. Our results can be readily applied to optimal state discrimination over various types of noise.
Abstract In this paper, we mainly study the indistinguishability of classically simulable measurements (measurements that can be effectively simulated on classical computers) in the tasks of quantum channel discrimination. First, we investigate the limitations of classically simulable measurements under unambiguous and perfect discrimination strategies. For unambiguous discrimination, we establish a conditional limitation result for channel-induced output-state discrimination scenarios, while for perfect discrimination we derive a sufficient condition under which classically simulable measurements fail to perfectly distinguish quantum channels. When considering the minimum-error discrimination strategy, we formulate the corresponding bilinear optimization problem and discuss its formal Lagrange dual formulation. Subsequently, we give the lower bound of the minimum-error probability obtained by using classically simulable measurements in the minimum-error discrimination task under the adaptive protocol. Furthermore, we give two measures of the nonstabilizerness (magic) of measurements. We prove that one measure we proposed can be used as an exact quantifier for the advantage in some tasks of quantum state discrimination. Finally, we investigate the relationship between classically simulable measurements and the data hiding ratio.