We consider optimal state discrimination in a general convex operational framework, so-called generalized probabilistic theories (GPTs), and present a general method of optimal discrimination by applying the complementarity problem from convex optimization. The method exploits the convex geometry of states but not other detailed conditions or relations of states and effects. We also show that properties in optimal quantum state discrimination are shared in GPTs in general: (i) no measurement sometimes gives optimal discrimination, and (ii) optimal measurement is not unique.
We consider optimal state discrimination in a general convex operational framework, so-called generalized probabilistic theories (GPTs), and present a general method of optimal discrimination by applying the complementarity problem from convex optimization. The method exploits the convex geometry of states but not other detailed conditions or relations of states and effects. We also show that properties in optimal quantum state discrimination are shared in GPTs in general: (i) no measurement sometimes gives optimal discrimination, and (ii) optimal measurement is not unique.
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.
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.
A state discrimination problem in an operational probabilistic theory (OPT) is investigated in diagrammatic terms. It is well-known that, in the case of quantum theory, if a state set has a certain symmetry, then there exists a minimum-error measurement having the same type of symmetry. However, to our knowledge, it is not yet clear whether this property also holds in a more general OPT. We show that it also holds in OPTs, i.e., for a symmetric state set, there exists a minimum-error measurement that has the same type of symmetry. It is also shown that this result can be utilized to optimize over a restricted class of measurements, such as sequential or separable measurements.
This thesis deals with General Probabilistic Theories (GPTs) and Entanglement Structures (ESs). An ES is a possible structure of a quantum composite system in GPTs, which is not uniquely determined as the Standard Entanglement Structure (SES). It is an important problem to find reasonable postulates that determine various ESs as the SES. In order to solve this problem, this thesis explores the diversity of ESs. The topics of this thesis are roughly divided into two parts. First, this thesis considers state discrimination in ESs. As a main result, this thesis gives equivalent conditions for a given measurement in ESs to have a performance superior to standard quantum theory. Second, this thesis focuses on symmetry and self-duality. As a main result, this thesis gives derivations of the SES by symmetric conditions. On the other hand, this thesis clarifies that there are infinitely many self-dual ESs, even if they cannot be distinguished from the SES by a certain physical experiment with small errors.
Generalized probabilistic theories (GPTs) provide a framework in which a range of possible theories can be examined, including classical theory, quantum theory, and those beyond. In general, enlarging the state space of a GPT leads to fewer possible measurements because the additional states give stronger constraints on the set of effects, the constituents of measurements. This can have implications for information processing. In box world, for example, a GPT in which any no-signaling distribution can be realized, there is no analog of a measurement in the Bell basis and hence the analog of entanglement swapping is impossible. A comprehensive study of measurements on multiple systems in box world has been lacking. Here we consider such measurements in detail, distinguishing those that can be performed by interacting with individual systems sequentially (termed wirings), and the more interesting set of those that cannot. We compute all the possible box-world effects for cases with small numbers of inputs, outputs, and parties, identifying those that are wirings. The large state space of box world leads to a small effect space and hence the effects of box world are widely applicable in GPTs. We also show some possible uses of nonwirings for information processing by studying state discrimination, nonlocality distillation, and the box-world analog of nonlocality without entanglement. Finally, we connect our results to the study of logically consistent classical processes and to the composition of contextuality scenarios. By enhancing understanding of measurements in box world, our results could be useful in studies of possible underlying principles on which quantum theory can be based.
We consider a fundamental operational task, distinguishing systems in different states, in the framework of generalized probabilistic theories and provide a general formalism of minimum-error discrimination of states in convex optimization. With the formalism established, we show that the distinguishability is generally a global property assigned to the ensemble of given states rather than other details of a given state space or pairwise relations of given states. Then, we consider bipartite systems where ensemble steering is possible, and show that show that with two operational tasks, ensemble steering and the no-signaling condition, the distinguishability is tightly determined. The result is independent to the structure of the state space. This concludes that the distinguishability is generally determined by the compatibility between two tasks, ensemble steering on states and the non-signaling principle on probability distributions of outcomes.