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arXiv:1504.03740 · Physical Review A

Cost of postselection in decision theory

Joshua Combes, Christopher Ferrie

Published Aug 19, 2015• 9 citations
arXivHTMLPDF

Abstract

Postselection is the process of discarding outcomes from statistical trials that are not the event one desires. Postselection can be useful in many applications where the cost of getting the wrong event is implicitly high. However, unless this cost is specified exactly, one might conclude that discarding all data is optimal. Here we analyze the optimal decision rules and quantum measurements in a decision theoretic setting where a prespecified cost is assigned to discarding data. Our scheme interpolates between unambiguous state discrimination (when the cost of postselection is zero) and a minimum error measurement (when the cost of postselection is maximal). We also relate our formulation to previous approaches which focus on minimizing the probability of indecision.

DOI
10.1103/physreva.92.022117

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Postselection is the process of discarding outcomes from statistical trials that are not the event one desires. Postselection can be useful in many applications where the cost of getting the wrong event is implicitly high. However, unless this cost is specified exactly, one might conclude that discarding all data is optimal. Here we analyze the optimal decision rules and quantum measurements in a decision theoretic setting where a prespecified cost is assigned to discarding data. Our scheme interpolates between unambiguous state discrimination (when the cost of postselection is zero) and a minimum error measurement (when the cost of postselection is maximal). We also relate our formulation to previous approaches which focus on minimizing the probability of indecision.
Quantum state discrimination involves finding the best measurement to distinguish between quantum states in a given set. Typically, one wants the POVM which produces the minimal amount of error, or the maximal amount of confidence. We show that this task can easily be cast as an example of Bayesian experimental design, which makes explicit the decision process involved when discriminating between states. This highlights the common implicit assumption that the agent simply decides on the state indicated by the measurement outcome. This naive decision leads to anomalous results, which are resolved when using the optimal decision strategy. Further we generalise state discrimination to POVMs with an arbitrary number of elements, and generalise some well-known results.
In hypothesis testing with quantum states, given a black box containing one of the two possible states, measurement is performed to detect in favor of one of the hypotheses. In postselected hypothesis testing, a third outcome is added, corresponding to not selecting any of the hypotheses. In postselected scenario, minimum error one-shot symmetric hypothesis testing is characterized in literature conditioned on the fact that one of the selected outcomes occur. We proceed further in this direction to give the set of all possible measurements that lead to the minimum error. We have given an arbitrary error-minimizing measurement in a parametric form. Note that not selecting any of the hypotheses decimates the quality of testing. We further give an example to show that these measurements vary in quality. There is a need to discuss the quality of postselected hypothesis testing. We then characterize the quality of postselected hypothesis testing by defining a new metric acceptance and give expression of acceptance for an arbitrary error-minimizing measurement in terms of some parameters of the measurement. On the set of measurements that achieve minimum error, we have maximized the acceptance, and given an example which achieves that, thus giving an example of the best possible measurement in terms of acceptance.
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 derive the optimal measurement for quantum-state discrimination without a priori probabilities, i.e., in a minimax strategy instead of the usually considered Bayesian one. We consider both minimal-error and unambiguous discrimination problems, and provide the relation between the optimal measurements according to the two schemes. We show that there are instances in which the minimum risk cannot be achieved by an orthogonal measurement, and this is a common feature of the minimax estimation strategy.
We study an optimum measurement for quantum state discrimination, which maximizes the probability of correct results when the probability of inconclusive results is fixed at a given value. The measurement describes minimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimized maximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixed value reaches a definite minimum. Using operator conditions that determine the optimum measurement, we derive analytical solutions for the discrimination of two mixed qubit states, including the case of two pure states occurring with arbitrary prior probabilities, and for the discrimination of NNN symmetric states, both pure and mixed. We also consider a case where the given density operators resolve the identity operator, and we specify the optimality conditions for partially symmetric states. Moreover, we show that from the complete solution for arbitrary values of the fixed rate of inconclusive results one can always obtain the optimum measurement in another strategy where the error rate is fixed, and vice versa.