General Probabilistic Theories provide the most general mathematical framework for the theory of probability in an operationally natural manner, and generalize classical and quantum theories. In this article, we study state-discrimination problems in general probabilistic theories using a Bayesian strategy. After re-formulation of the theories with mathematical rigor, we first prove that an optimal observable to discriminate any (finite) number of states always exists in the most general setting. Next, we revisit our recently proposed geometric approach for the problem and show that, for two-state discrimination, this approach is indeed effective in arbitrary dimensional cases. Moreover, our method reveals an operational meaning of Gudder’s “intrinsic metric ” by means of the optimal success probability, which turns out to be a generalization of the trace distance for quantum systems. As its by-product, an information-disturbance theorem in general probabilistic theories is derived, generalizing its well known quantum version.
General Probabilistic Theories provide the most general mathematical framework for the theory of probability in an operationally natural manner, and generalize classical and quantum theories. In this article, we study state-discrimination problems in general probabilistic theories using a Bayesian strategy. After re-formulation of the theories with mathematical rigor, we first prove that an optimal observable to discriminate any (finite) number of states always exists in the most general setting. Next, we revisit our recently proposed geometric approach for the problem and show that, for two-state discrimination, this approach is indeed effective in arbitrary dimensional cases. Moreover, our method reveals an operational meaning of Gudder’s “intrinsic metric ” by means of the optimal success probability, which turns out to be a generalization of the trace distance for quantum systems. As its by-product, an information-disturbance theorem in general probabilistic theories is derived, generalizing its well known quantum version.
General probabilistic theories provide the most general mathematical framework for the theory of probability in an operationally natural manner and generalize classical and quantum theories. In this article, we study state discrimination problems in general probabilistic theories using a Bayesian strategy. After reformulation of the theories with mathematical rigor, we first prove that an optimal observable to discriminate any (finite) number of states always exists in the most general setting. Next, we revisit our recently proposed geometric approach for the problem and show that for two-state discrimination, this approach is indeed effective in arbitrary dimensional cases. Moreover, our method reveals an operational meaning of Gudder’s “intrinsic metric” by means of the optimal success probability, which turns out to be a generalization of the trace distance for quantum systems. As its by-product, an information disturbance theorem in general probabilistic theories is derived, generalizing its well known quantum version.
We investigate a state discrimination problem in operationally the most general framework to use a probability, including both classical, quantum theories, and more. In this wide framework, introducing closely related family of ensembles (which we call a Helstrom family of ensembles) with the problem, we provide a geometrical method to find an optimal measurement for state discrimination by means of Bayesian strategy. We illustrate our method in two-level quantum systems and in a probabilistic model with square-state space to reproduce, e.g., the optimal success probabilities for binary state discrimination and N numbers of symmetric quantum states. The existences of families of ensembles in binary cases are shown both in classical and quantum theories in any generic cases.
Abstract It is a key issue to characterize the model of standard quantum theory out of general models by an operational condition. The framework of general probabilistic theories (GPTs) is a new information theoretical approach to single out standard quantum theory. It is known that traditional properties, for example, Bell-CHSH inequality, are not sufficient to single out standard quantum theory among possible models in GPTs. As a more precise property, we focus on the bound of the performance for an information task called state discrimination in general models. We give an equivalent condition for outperforming the minimum discrimination error probability under the standard quantum theory given by the trace norm. Besides, by applying the equivalent condition, we characterize standard quantum theory out of general models in GPTs by the bound of the performance for state discrimination.
For a collection of states indexed by [n]=1,…,n, let F(H) denote the optimal unnormalized minimum discrimination error for each nonempty subfamily H⊆[n]. We call F the discrimination profile and characterize exactly which profiles can arise in a general probabilistic theory (GPT). For fixed n, the GPT region Gn is a bounded rational polytope, giving a hierarchy Cn⊆Qn⊆Gn of classical, quantum, and GPT profiles, analogous to the local--quantum--no-signalling hierarchy in Bell theory.
For n=3 we determine the classical and GPT polytopes explicitly. Within G3, the classical region is characterized by β:=F(123)−F(12)−F(13)−F(23)≤0, while the GPT maximum is 1. For qubit triples, the maximal value is attained by the trine, βtr=33/2−2, and we prove the dimension-independent quantum bound βQ≤2/3, with a slight further improvement. Hence 0=βC<33/2−2≤βQ≤2/3<1=βGPT. We conjecture βQ=βtr and prove this for arbitrary pure-state triples, together with further supporting evidence.
For arbitrary n, every classical facet admitting a GPT violation already admits a quantum violation in dimension at most 3. A four-state qubit example shows that quantum success profiles need not be submodular, unlike for n=3. Finally, we exhibit a four-dimensional quantum triple whose discrete profile is classical for all subfamilies but becomes nonclassical when unequal priors are allowed.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
Distinguishing different quantum states is a fundamental task having practical applications in information processing. Despite the effort devoted so far, however, strategies for optimal discrimination are known only for specific examples. In this paper we consider the problem of minimum-error quantum state discrimination where one attempts to minimize the average error. We show the general structure of minimum-error state discrimination as well as useful properties to derive analytic solutions. Based on the general structure, we present a geometric formulation of the problem, which can be applied to cases where quantum state geometry is clear. We also introduce equivalent classes of sets of quantum states in terms of minimum-error discrimination: sets of quantum states in an equivalent class that share the same guessing probability. In particular, for qubit states where the state geometry is found with the Bloch sphere, we illustrate that for an arbitrary set of qubit states, the minimum-error state discrimination with equal prior probabilities can be analytically solved, that is, optimal measurement and the guessing probability are explicitly obtained.