As a modern approach for the foundation of quantum theory, existing studies of general probabilistic theories gave various models of states and measurements that are quite different from quantum theory. In this Letter, to seek a more realistic situation, we investigate models approximately close to quantum theory. We define larger measurement classes that are smoothly connected with the class of POVMs via a parameter, and investigate the performance of perfect discrimination. As a result, we give a sufficient condition of perfect discrimination, which shows a significant improvement beyond the class of POVMs.
As a modern approach for the foundation of quantum theory, existing studies of general probabilistic theories gave various models of states and measurements that are quite different from quantum theory. In this Letter, to seek a more realistic situation, we investigate models approximately close to quantum theory. We define larger measurement classes that are smoothly connected with the class of POVMs via a parameter, and investigate the performance of perfect discrimination. As a result, we give a sufficient condition of perfect discrimination, which shows a significant improvement beyond the class of POVMs.
Abstract We address perfect discrimination of two separable states. When available states are restricted to separable states, we can theoretically consider a larger class of measurements than the class of measurements allowed in quantum theory. The framework composed of the class of separable states and the above extended class of measurements is a typical example of general probabilistic theories. In this framework, we give a necessary and sufficient condition to discriminate two separable pure states perfectly. In particular, we derive measurements explicitly to discriminate two separable pure states perfectly, and find that some non-orthogonal states are perfectly distinguishable. However, the above framework does not improve the capacity , namely, the maximum number of states that are simultaneously and perfectly distinguishable.
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.
We identify an operational principle that singles out Projection-Valued Measures (PVMs) among general Positive Operator-Valued Measures (POVMs), bridging the modern quantum measurement theory and the traditional formulation based on projective measurements of physical observables. We reformulate Ozawa's intersubjectivity condition, which requires inter-observer agreement of the measurement outcomes, in a quantitative manner within the framework of generalized probabilistic theories. We prove that (i) a POVM is a PVM if and only if its every coarse-graining is intersubjective, and (ii) a system is classical if and only if intersubjectivity is preserved under any coarse-graining, establishing a complete characterization of the physical observables and the classical theory. Furthermore, measurements with intersubjectivity are sufficiently rich for the informational tasks of state tomography and state discrimination, testifying to its operational significance in quantum and beyond information processing.
Quantum mechanics has enjoyed a multitude of successes since its formulation in the early twentieth century. At the same time, it has generated puzzles that persist to this day. These puzzles have inspired a large literature in physics and philosophy. There are two distinct approaches. One is to assume that quantum theory is exact, but that the interpretive postulates need modification, to eliminate apparent contradictions. The second approach is to assume that quantum mechanics is not exact, but instead is a very accurate approximation to a deeper level theory, which reconciles the deterministic and probabilistic aspects. We will focus on this second approach.
This book is devoted to aspects of the foundations of quantum mechanics in which probabilistic and statistical concepts play an essential role. The main part of the book concerns the quantitative statistical theory of quantum measurement, based on the notion of positive operator-valued measures. During the past years there has been substantial progress in this direction, stimulated to a great extent by new applications such as Quantum Optics, Quantum Communication and high-precision experiments. The questions of statistical interpretation, quantum symmetries, theory of canonical commutation re
General probabilistic theories (GPTs) are theoretical models having states, measurements, and probabilities of obtaining measurement outcomes.For example, classical probability theory and quantum theory are GPTs with probability vectors and density matrices as states, respectively.In the study of GPTs, one often describes a finite number of GPTs A 1 , . . ., A n together, as one GPT A = A 1 • • • A n .In this case, we say that A is a whole system of subsystems A 1 , . . ., A n .If A 1 , . . ., A n are quantum systems, then A is called a locally quantum system.A quantum system is locally quantum, but there are many other locally quantum systems.In this thesis, we study three topics associated with quantum systems and locally quantum systems.The first one is capacity.For a GPT, the maximum number of simultaneously and perfectly distinguishable states is called the capacity.It is known that the capacity does not really depend on the GPT.In this thesis, we propose a statement S to determine the capacities of special locally quantum systems, and prove a weaker statement WS.Statement WS is a statement associated with tensor product spaces over C. Our proof works even if replacing the scalar field C with an arbitrary infinite field F. We also show a partial result for finite fields.The second one is differential privacy (DP).DP was born in the study to utilize private data while protecting the data (privacy-preserving data mining, PPDM).For this reason, DP has been studied by using classical probability theory, and has almost never been studied in quantum information theory.In this thesis, we define a quantum version of DP (called classical-quantum DP) and investigate its mathematical aspects.We define the set CQ n of all classical-quantum differentially private n-tuples of quantum states and the subset EC n of CQ n that is essentially classical.We prove that EC n = CQ n if n = 2 and EC n = CQ n if n ≥ 3.The third one is perfect discrimination of two states.In quantum theory, orthogonality is a necessary and sufficient condition to distinguish two states perfectly (denote this equivalence by E).However, equivalence E does not hold for a general GPT.In this thesis, we construct continuous one-parameter families of locally quantum systems in a certain natural manner, and investigate perfect discrimination of two states in the framework of those locally quantum systems.These one-parameter families contain a locally quantum system sufficiently close to a quantum system.Nevertheless, we show that they violate equivalence E.