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Aperture-Limited Quantum State Discrimination

Joshua Ross Gottlieb

Published Mar 11, 2026• 0 citations
Publisher

Abstract

Quantum hypothesis testing studies the distinguishability of quantum states using measurements.Standard formulations assume access to arbitrary POVMs. In realistic experimentalsettings, however, observers are restricted to specific measurement interfaces. In this work westudy quantum state discrimination under such measurement constraints.Given a measurement family M representing an observer aperture, we analyze the optimaldiscrimination of two quantum states using only measurements from M. We show that theasymptotic error exponent is governed by the aperture-restricted relative entropy introduced inour previous work. This provides an operational interpretation of observer aperture as a limiton achievable discrimination rates.

DOI
10.5281/zenodo.18960686

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Quantum hypothesis testing studies the distinguishability of quantum states using measurements.Standard formulations assume access to arbitrary POVMs. In realistic experimentalsettings, however, observers are restricted to specific measurement interfaces. In this work westudy quantum state discrimination under such measurement constraints.Given a measurement family M representing an observer aperture, we analyze the optimaldiscrimination of two quantum states using only measurements from M. We show that theasymptotic error exponent is governed by the aperture-restricted relative entropy introduced inour previous work. This provides an operational interpretation of observer aperture as a limiton achievable discrimination rates.
We introduce a formal notion of observer aperture, representing the measurement interfaceavailable to an observer interacting with a quantum system. Given a restricted family of POVMsM, we define the aperture-restricted distinguishability between two quantum states as thesupremum of classical relative entropy between the induced outcome distributions. This quantitycan be viewed as a measurement-restricted variant of measured quantum relative entropy. Weprove monotonicity under aperture expansion and establish a data-processing inequality underquantum channels via Heisenberg pullback of measurement families. We further show thatthe quantity is subadditive across independent systems. The resulting framework provides anoperational interpretation of observer bandwidth connecting measurement constraints, coarse-graining, and accessible information within standard quantum information theory.
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.
We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relative entropies between the two states. Moreover, if we allow joint measurements on multiple samples, the rates are increased to the respective quantum relative entropies. We also fully characterize the achievable error exponents for non-adaptive strategies and provide numerical evidence showing that adaptive measurements are necessary to achieve our bounds.
The minimum error probability for distinguishing between two quantum states is bounded by the Helstrom limit, derived under the assumption that measurement strategies are restricted to positive operator-valued measurements. We explore scenarios in which the error probability for discriminating two quantum states can be reduced below the Helstrom bound under some constrained access of resources, indicating the use of measurement operations that go beyond the standard positive operator-valued measurements framework. We refer to such measurements as non-positive operator-valued measurements. While existing literature often associates these measurements with initial entanglement between the system and an auxiliary, followed by joint projective measurement and discarding the auxiliary, we demonstrate that initial entanglement between system and auxiliary is not necessary for the emergence of such measurements in the context of state discrimination. Interestingly, even initial product states can give rise to effective non-positive measurements on the subsystem, and achieve sub-Helstrom discrimination error when discriminating quantum states of the subsystem.
The ultimate limits of quantum state discrimination are often thought to be captured by asymptotic bounds that restrict the achievable error probabilities, notably the quantum Chernoff and Hoeffding bounds. Here we study hypothesis testing protocols that are permitted a probability of producing an inconclusive discrimination outcome, and investigate their performance when this probability is suitably constrained. We show that even by allowing an arbitrarily small probability of inconclusiveness, the limits imposed by the quantum Hoeffding and Chernoff bounds can be significantly exceeded. This completely circumvents the conventional trade-offs between error exponents in hypothesis testing while incurring only a vanishingly small overhead over conventional approaches. Such improvements over standard state discrimination are robust and can be obtained even when an exponentially vanishing probability of inconclusive outcomes is demanded. Relaxing the constraints on the inconclusive probability can enable even larger advantages, but this comes at a price. We show a 'strong converse' property of this setting: targeting error exponents beyond those achievable with vanishing inconclusiveness necessarily forces the probability of inconclusive outcomes to converge to one. By exactly quantifying the rate of this convergence, we give a complete characterisation of the trade-offs between error exponents and rates of conclusive outcome probabilities. Overall, our results provide a comprehensive asymptotic picture of how the allowance for inconclusive measurement outcomes reshapes optimal quantum hypothesis testing.
This expository article gives an overview of the theory of hypothesis testing of quantum states in finite dimensional Hilbert spaces. Optimal measurement strategy for testing binary quantum hypotheses, which result in minimum error probability, is discussed. Collective and individual adaptive measurement strategies in testing hypotheses in the multiple copy scenario, with various upper and lower bounds on error probability, are outlined. A brief account on quantum channel discrimination and the role of entangled states in achieving enhanced precision in the task of channel discrimination is given.