Mathematical structure of positive operator valued measures and applications

R Robbert Beukema

Hayato Arai, Masahito Hayashi

It is important problem to clarify the class of implementable quantum measurements from both fundamental and applicable viewpoints. Positive-Operator-Valued Measure (POVM) measurements are implementable by the indirect measurement methods, and the class is the largest class determined by the mathematical structure of Hilbert space. However, if we assume probabilistic consistency in our operations instead of the structure of Hilbert space, we can deal with Non-Positive-Operator-Valued Measure (N-POVM) measurements in the framework of General Probabilistic Theories (GPTs). N-POVM measurements are not considered as implementable, but this paper gives a constructive way to implement N-POVM measurements by POVM measurements and post-selection in quantum theory when we restrict the domain of target states. Besides, we show that a post-selected POVM measurement is regarded as an N-POVM measurement in a restricted domain. These results provide a new relationship between N-POVM measurements in GPTs and post-selection.

Sachin .

We present a comprehensive technical review of quantum measurement theory, emphasizing the mathematical structures underlying the measurement problem and their role in modern quantum information science. Beginning with spectral theory and the Born rule, we develop the theory of positive operator-valued measures (POVMs), Kraus operators, and completely positive channels as essential generalizations of projective measurements. We discuss how decoherence theory addresses the preferred-basis and interference problems while leaving the outcome problem formally unresolved, motivating interpretational responses ranging from collapse models to many-worlds scenarios. Throughout, we emphasize the applicability of POVM theory to quantum state discrimination, quantum cryptography, and quantum error correction. The review prioritizes physical motivation and mathematical hints over detailed proofs, suitable for researchers transitioning into quantum foundations or quantum information theory.

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau

Andrei Yu. Khrennikov, Irina Basieva

Recently foundational issues of applicability of the formalism of quantum mechanics (QM) to cognitive psychology, decision making, and psychophysics attracted a lot of interest. In particular, in \cite{DKBB} the possibility to use of the projection postulate and representation of "mental observables" by Hermitian operators was discussed in very detail. The main conclusion of the recent discussions on the foundations of "quantum(-like) cognitive psychology" is that one has to be careful in determination of conditions of applicability of the projection postulate as a mathematical tool for description of measurements of observables represented by Hermitian operators. To represent some statistical experimental data (both physical and mental) in the quantum(-like) way, one has to use generalized quantum observables given by positive operator-valued measures (POVMs). This paper contains a brief review on POVMs which can be useful for newcomers to the field of quantum(-like) studies. Especially interesting for cognitive psychology is a variant of the formula of total probability (FTP) with the interference term derived for incompatible observables given by POVMs. We present an interpretation of the interference term from the psychological viewpoint. As was shown before, the appearance of such a term (perturbing classical FTP) plays the important role in cognitive psychology, e.g., recognition of ambiguous figures and the disjunction effect. The interference term for observables given by POVMs has much more complicated structure than the corresponding term for observables given by Hermitian operators. We elaborate cognitive interpretations of different components of the POVMs-interference term and apply our analysis to a quantum(-like) model of decision making.