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Scientific Reports

Single-shot comparison of random quantum channels and measurements

Marcin Markiewicz, Łukasz Pawela, Zbigniew Puchała

Published Sep 14, 2026• 0 citations
Publisher

Abstract

We analyze the task of deciding whether two unknown quantum devices are identical or independent, for arbitrary quantum channels and generalized measurements (POVMs) drawn at random from Haar-Stinespring ensembles, in the single-shot regime in which each device is used exactly once. Our contribution is twofold. First, in the symmetric setting, in which both types of error are treated on an equal footing, we derive a closed-form expression for the optimal success probability and show that it is achieved by a universal, parameter-free strategy: an antisymmetric two-copy input state followed by a projection onto the symmetric and antisymmetric subspaces of the outputs, with no additional ancillary systems required. Second, in the asymmetric (hypothesis-testing) setting we reduce the problem to a simple linear program and derive explicit type-I/type-II error trade-off curves for deterministic fixed-input protocols, for which the optimal input depends on the allowed type-I error; remarkably, classical randomization—equivalently, an untouched side register correlated with the input choice—can strictly improve the asymmetric performance by convexifying the deterministic trade-off, in sharp contrast to the symmetric setting. Finally, we show that the type-I error can be made to vanish without trivializing the test only in the environment-free case, corresponding to isometric channels, and for von Neumann measurements; for any genuinely noisy dynamics both types of error remain strictly positive in every nontrivial comparison scheme.

DOI
10.1038/s41598-026-70695-z

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Quantum theory of unambiguous measurements

Michal Sedlák

Quantum theory of unambiguous measurements In the present paper I formulate a framework that accommodates many unambiguous discrimination problems. I show that the prior information about any type of constituent (state, channel, or observable) allows us to reformulate the discrimination among finite number of alternatives as the discrimination among finite number of average constituents. Using this framework I solve several unambiguous tasks. I present a solution to optimal unambiguous comparison of two ensembles of unknown quantum states. I consider two cases: 1) The two unknown states are arbitrary pure states of qudits. 2) Alternatively, they are coherent states of single-mode optical fields. For this case I propose simple and optimal experimental setup composed of beam-splitters and a photodetector. As a second tasks I consider an unambiguous identification (UI) of coherent states. In this task identical quantum systems are prepared in coherent states and labeled as unknown and reference states, respectively. The promise is that one reference state is the same as the unknown state and the task is to find out unambiguously which one it is. The particular choice of the reference states is unknown to us, and only the probability distribution describing this choice is known. In a general case when multiple copies of unknown and reference states are available I propose a scheme consisting of beamsplitters and photodetectors that is optimal within linear optics. UI can be considered as a search in a quantum database, whose elements are the reference states and the query is represented by the unknown state. This perspective motivated me to show that reference states can be recovered after the measurement and might be used (with reduced success rate) in subsequent UI. Moreover, I analyze the influence of noise in preparation of coherent states on the performance of the proposed setup. Another problem I address is the unambiguous comparison of a pair of unknown qudit unitary channels. I characterize all solutions and identify the optimal ones. I prove that in optimal experiments for comparison of unitary channels the entanglement is necessary. The last task I studied is the unambiguous comparison of unknown non-degenerate projective measurements. I distinguish between measurement devices with apriori labeled and unlabeled outcomes. In both cases only the difference of the measurements can be concluded unambiguously. For the labeled case I derive the optimal strategy if each unknown measurement is used only once. However, if the apparatuses are not labeled, then each measurement device must be used (at least) twice. In particular, for qubit measurement apparatuses with unlabeled outcomes I derive the optimal test state in the two-shots scenario.

arXiv:1003.2448•Acta Physica Slovaca Reviews and Tutorials•Dec 1, 2009•18 citations
We analyze the task of deciding whether two unknown quantum devices are identical or independent, for arbitrary quantum channels and generalized measurements (POVMs) drawn at random from Haar-Stinespring ensembles, in the single-shot regime in which each device is used exactly once. Our contribution is twofold. First, in the symmetric setting, in which both types of error are treated on an equal footing, we derive a closed-form expression for the optimal success probability and show that it is achieved by a universal, parameter-free strategy: an antisymmetric two-copy input state followed by a projection onto the symmetric and antisymmetric subspaces of the outputs, with no additional ancillary systems required. Second, in the asymmetric (hypothesis-testing) setting we reduce the problem to a simple linear program and derive explicit type-I/type-II error trade-off curves for deterministic fixed-input protocols, for which the optimal input depends on the allowed type-I error; remarkably, classical randomization—equivalently, an untouched side register correlated with the input choice—can strictly improve the asymmetric performance by convexifying the deterministic trade-off, in sharp contrast to the symmetric setting. Finally, we show that the type-I error can be made to vanish without trivializing the test only in the environment-free case, corresponding to isometric channels, and for von Neumann measurements; for any genuinely noisy dynamics both types of error remain strictly positive in every nontrivial comparison scheme.
Comparison of quantum objects is the task of determining relational properties, such as whether two unknown objects are the same or different, without identifying the objects themselves. Multiple copies of unknown states or multiple queries to unknown channels are natural resources for improving comparison, and the optimal strategy for pure-state comparison with multiple copies is known. For unitary-channel comparison with multiple queries, however, the optimal strategy has been unclear because different queries can be arranged in parallel, sequentially, adaptively, or in more general causal structures. We study comparison of two unknown ddd-dimensional unitary channels using N1N_1N1​ and N2N_2N2​ queries to the two channels, respectively, under the promise that they are either identical Haar-random unitaries or independent Haar-random unitaries. We optimize over valid general testers, a class containing ordinary quantum testers, classically controlled causal orders, and indefinite-causal-order processes. We characterize the optimal minimum-error and one-sided unambiguous comparison probabilities for arbitrary finite query numbers. The upper bound in fact holds for a larger positive relaxation, while the optimum is attained by a valid parallel tester. The optimum is determined by a finite representation-theoretic parameter built from Young diagrams and Littlewood--Richardson coefficients. For fixed N1N_1N1​, the performance is saturated for every N2≥(d−1)N1N_2\geq(d-1)N_1N2​≥(d−1)N1​. By contrast, the corresponding pure-state comparison parameter has no finite-N2N_2N2​ saturation and reaches its known-reference limit only asymptotically, yielding continued success-probability improvement in the nontrivial decision regime. This highlights a sharp difference between comparison tasks for states and channels, analogous to distinctions known in quantum discrimination.
In this work, we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage of entanglement. We quantify the distance between two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance between the measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product, the program finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.
We study a variant of quantum hypothesis testing wherein an additional ‘inconclusive’ measurement outcome is added, allowing one to abstain from attempting to discriminate the hypotheses. The error probabilities are then conditioned on a successful attempt, with inconclusive trials disregarded. We completely characterise this task in both the single-shot and asymptotic regimes, providing exact formulas for the optimal error probabilities. In particular, we prove that the asymptotic error exponent of discriminating any two quantum states$\rho andandand\sigma isgivenbytheHilbertprojectivemetricis given by the Hilbert projective metricisgivenbytheHilbertprojectivemetricD_{\max }(\rho \|\sigma ) + D_{\max }(\sigma \| \rho )inasymmetrichypothesistesting,andbytheThompsonmetricin asymmetric hypothesis testing, and by the Thompson metricinasymmetrichypothesistesting,andbytheThompsonmetric\max \! \big \{ D_{\max }(\rho \|\sigma ),\, D_{\max }(\sigma \| \rho ) \big \}$in symmetric hypothesis testing. This endows these two quantities with fundamental operational interpretations in quantum state discrimination. Our findings extend to composite hypothesis testing, where we show that the asymmetric error exponent with respect to any convex set of density matrices is given by a regularisation of the Hilbert projective metric. We apply our results also to quantum channels, showing that no advantage is gained by employing adaptive or even more general discrimination schemes over parallel ones, in both the asymmetric and symmetric settings. Our state discrimination results make use of no properties specific to quantum mechanics and are also valid in general probabilistic theories.
This paper concerns the problem of quantum measurement compression with side information in the one-shot setting with shared-randomness. In this problem, Alice shares a pure quantum state with Bob and the reference system. She performs a measurement on her registers and wishes to communicate the outcome to Bob using shared-randomness and classical communication. The outcome that Bob receives must be correctly correlated with the reference system and his own registers. Our goal is to concurrently minimize the classical communication and shared-randomness cost. The suggested protocol presented in this paper is based on convex-split and position based decoding. The communication is upper bounded in terms of smooth max and hypothesis testing relative entropies. A second protocol addresses the task of strong randomness extraction in the presence of quantum side information. The protocol provides an error guarantee in terms of relative entropy (as opposed to trace distance) and extracts close to the optimal number of uniform bits. As an application, we provide a new achievability result for the task of quantum measurement compression without feedback, in which Alice does not need to know the outcome of the measurement. The result achieves the optimal number of bits communicated and the required number of bits of shared-randomness, for the same task in the asymptotic and i.i.d. setting.
We develop a device-independent framework for testing quantum channels. That is, we falsify a hypothesis about a quantum channel based only on an observed set of input-output correlations. Formally, the problem consists of characterizing the set of input-output correlations compatible with any arbitrary given quantum channel. For binary (i.e. two input symbols, two output symbols) correlations, we show that extremal correlations are always achieved by orthogonal encodings and measurements, irrespective of whether or not the channel preserves commutativity. We further provide a full, closed-form characterization of the sets of binary correlations in the case of: (i) any dihedrally covariant qubit channel (such as any Pauli and amplitude-damping channels) and (ii) any universally-covariant commutativity-preserving channel in an arbitrary dimension (such as any erasure, depolarizing, universal cloning and universal transposition channels).
Quantum theory of unambiguous measurements In the present paper I formulate a framework that accommodates many unambiguous discrimination problems. I show that the prior information about any type of constituent (state, channel, or observable) allows us to reformulate the discrimination among finite number of alternatives as the discrimination among finite number of average constituents. Using this framework I solve several unambiguous tasks. I present a solution to optimal unambiguous comparison of two ensembles of unknown quantum states. I consider two cases: 1) The two unknown states are arbitrary pure states of qudits. 2) Alternatively, they are coherent states of single-mode optical fields. For this case I propose simple and optimal experimental setup composed of beam-splitters and a photodetector. As a second tasks I consider an unambiguous identification (UI) of coherent states. In this task identical quantum systems are prepared in coherent states and labeled as unknown and reference states, respectively. The promise is that one reference state is the same as the unknown state and the task is to find out unambiguously which one it is. The particular choice of the reference states is unknown to us, and only the probability distribution describing this choice is known. In a general case when multiple copies of unknown and reference states are available I propose a scheme consisting of beamsplitters and photodetectors that is optimal within linear optics. UI can be considered as a search in a quantum database, whose elements are the reference states and the query is represented by the unknown state. This perspective motivated me to show that reference states can be recovered after the measurement and might be used (with reduced success rate) in subsequent UI. Moreover, I analyze the influence of noise in preparation of coherent states on the performance of the proposed setup. Another problem I address is the unambiguous comparison of a pair of unknown qudit unitary channels. I characterize all solutions and identify the optimal ones. I prove that in optimal experiments for comparison of unitary channels the entanglement is necessary. The last task I studied is the unambiguous comparison of unknown non-degenerate projective measurements. I distinguish between measurement devices with apriori labeled and unlabeled outcomes. In both cases only the difference of the measurements can be concluded unambiguously. For the labeled case I derive the optimal strategy if each unknown measurement is used only once. However, if the apparatuses are not labeled, then each measurement device must be used (at least) twice. In particular, for qubit measurement apparatuses with unlabeled outcomes I derive the optimal test state in the two-shots scenario.