arXiv subjects

Masanao Ozawa

Publications and source records attributed to Masanao Ozawa.

At least 19 recordsLinked to original sources

arXiv2026

Limitations of Quantum Measurements and Operations of Scattering Type under the Energy Conservation Law

It is important to improve the accuracy of quantum measurements and operations both in engineering and fundamental physics. It is known, however, that the achievable accuracy of measurements and unitary operations are generally limited by conservation laws according to the Wigner-Araki-Yanase theorem (WAY theorem) and its generalizations. Although many researches have extended the WAY theorem quantitatively, most of them, as well as the original WAY theorem, concern only additive conservation laws like the angular momentum conservation law. In this paper, we explore the limitation incurred by the energy conservation law, which is universal but is one of the non-additive conservation laws. We present a lower bound for the error of a quantum measurement using a scattering process satisfying the energy conservation law. We obtain conditions that a control system Hamiltonian must fulfill in order to implement a controlled unitary gate with zero error when a scattering process is considered. We also show the quantitative relationship between the upper bound of the gate fidelity of a controlled unitary gate and the energy fluctuation of systems when a target system and a control system are both one qubit.

quant-ph↗

arXiv2026

Quantum-like Cognition in Process Theories: An Analysis

Various effects in human cognition, often considered `non-classical', have been argued to be most naturally modelled by quantum-like models of decision making. We extend this approach to describe models of cognition and decision-making in general probabilistic process theories, which include both classical probabilistic models and quantum instrument models as special cases. We show how many aspects of quantum-like cognition can be described diagrammatically in process theories, before using our approach to assess the arguments for quantum-like models. While standard Bayesian classical models are insufficient, we prove that any sequential decision data can in fact be given a more general form of classical instrument model, and see that even simple deterministic models can exhibit all cognitive effects. Restricting attention to instruments induced by measurements, such as classical Bayesian and quantum POVM models, rules out such a result, but is challenged by the fact that such instruments cannot account for certain effects. Finally, we argue that to strictly rule out classical instrument models one should make use of parallel composition in the modelling of joint decisions, and find real world cognitive data violating Bell inequalities.

q-bio.NC↗

arXiv2025

Quantum set theory: quantum conditionals and order of observable

A difficulty in quantum logic is the well-known arbitrariness in choosing a binary operation for conditional among three principal candidates called the Sasaki, the contrapositive Sasaki, and the relevance conditional, mainly chosen from syntactical grounds. A fundamental problem remains to clarify their semantical differences manifest in operational concepts in quantum theory. Here, we attempt such an analysis through quantum set theory, developing models of quantum set theory built upon quantum logics with those three conditionals, each of which defines different quantum logical truth-value assignment for set theoretical statements. We show that each of them satisfies the transfer principle to determine the truth values of theorems of the ZFC set theory and defines the internal reals bijectively corresponding to the observables of the quantum system under consideration. Then, the truth values of their equality relations are identical irrespective of the chosen conditionals. Interestingly, however, their order relations exhibit a strong dependence on the specific conditional employed, while the order relation attains full truth value if and only if Olson's spectral order relation holds. We further characterize the order relation in terms of experimentally accessible relations for outcomes of successive projective measurements of the corresponding observables, showing that each choice has its own operational meaning with symmetry between the Sasaki and the contrapositive Sasaki conditionals, in contrast to the majority view that favors the Sasaki conditional. Our findings reveal that quantum set theory yields empirically testable predictions concerning state-dependent binary relations between quantum observables, thereby extending Born's probabilistic interpretation from propositions to relations.

quant-ph↗

arXiv2025

Intersubjectivity and value reproducibility of outcomes of quantum measurements

Every measurement determines a single value as its outcome, and yet quantum mechanics predicts it only probabilistically. The Kochen-Specker theorem and Bell's inequality are often considered to reject a realist view but favor a skeptical view that measuring an observable does not mean ascertaining the value that it has, but producing the outcome, having only a personal meaning. However, precise analysis supporting this view is unknown. Here, we show that a quantum mechanical analysis turns down this view. Supposing that two observers simultaneously measure the same observable, we can well pose the question as to whether they always obtain the same outcome, or whether the probability distributions are the same, but the outcomes are uncorrelated. Contrary to the widespread view in favor of the second, we shall show that quantum mechanics predicts that only the first case occurs. We further show that any measurement establishes a time-like entanglement between the observable to be measured and the meter after the measurement, which causes the space-like entanglement between the meters of different observers. We also show that our conclusion cannot be extended to measurements of so-called 'generalized' or 'unsharp' observables, suggesting a demand for reconsidering the notion of observables in foundations of quantum mechanics.

physics.gen-ph↗

arXiv2025

Incorporating episodic memory into quantum models of judgment and decision

An important challenge for quantum theories of cognition and decision concerns the incorporation of memory for recently made judgments and their effects on later judgments. First, we review a general approach to measurement based on system plus environment representations of states and measurement instruments. These more general measurement models provide ways to incorporate effects of recent judgments on later judgments. Then we compare three different measurement models that are based on these more general measurement operations to a puzzling collection of question order effect findings.

physics.soc-ph↗

arXiv2025

Quantum-like cognition and decision making in the light of quantum measurement theory

We characterize the class of quantum measurements that matches the applications of quantum theory to cognition (and decision making) - quantum-like modeling. Projective measurements describe the canonical measurements of the basic observables of quantum physics. However, the combinations of the basic cognitive effects, such as the question order and response replicability effects, cannot be described by projective measurements. We motivate the use of the special class of quantum measurements, namely {\it sharp repeatable non-projective measurements} - ${\cal SR\bar{P}}. $ This class is practically unused in quantum physics. Thus, physics and cognition explore different parts of quantum measurement theory. Quantum-like modeling isn't automatic borrowing of the quantum formalism. Exploring the class ${\cal SR\bar{P}}$ highlights the role of {\it noncommutativity of the state update maps generated by measurement back action.} Thus, ``non-classicality'' in quantum physics as well as quantum-like modeling for cognition is based on two different types of noncommutativity, of operators (observables) and instruments (state update maps): {\it observable-noncommutativity} vs. {\it state update-noncommutativity}. We speculate that distinguishing quantum-like properties of the cognitive effects are the expressions of the latter, or possibly both.

cs.AI↗

arXiv2025

Coupling quantum-like cognition with the neuronal networks within generalized probability theory

The past few years have seen a surge in the application of quantum theory methodologies and quantum-like modeling in fields such as cognition, psychology, and decision-making. Despite the success of this approach in explaining various psychological phenomena such as order, conjunction, disjunction, and response replicability effects there remains a potential dissatisfaction due to its lack of clear connection to neurophysiological processes in the brain. Currently, it remains a phenomenological approach. In this paper, we develop a quantum-like representation of networks of communicating neurons. This representation is not based on standard quantum theory but on generalized probability theory (GPT), with a focus on the operational measurement framework. Specifically, we use a version of GPT that relies on ordered linear state spaces rather than the traditional complex Hilbert spaces. A network of communicating neurons is modeled as a weighted directed graph, which is encoded by its weight matrix. The state space of these weight matrices is embedded within the GPT framework, incorporating effect observables and state updates within the theory of measurement instruments a critical aspect of this model. This GPT based approach successfully reproduces key quantum-like effects, such as order, non-repeatability, and disjunction effects (commonly associated with decision interference). Moreover, this framework supports quantum-like modeling in medical diagnostics for neurological conditions such as depression and epilepsy. While this paper focuses primarily on cognition and neuronal networks, the proposed formalism and methodology can be directly applied to a wide range of biological and social networks.

physics.soc-ph↗

arXiv2024

Disturbance Evaluation Circuit in Quantum Measurement

According to the uncertainty principle, every quantum measurement accompanies disturbance. In particular, accurate sequential measurements need the accurate control of disturbance. However, the correct role of disturbance in the uncertainty principle has been known only recently. Understanding the disturbance is crucial for understanding the fundamentals of physics, and accurately evaluating the disturbance is important for quantum technologies such as quantum information processing and quantum metrology. Therefore, the experimental evaluation of the disturbance is a significant challenge in those fields. In this study, we propose a novel evaluation method for the quantum root-mean-square (QRMS) disturbance and compare its performance with the existing approaches, known as the three-state method (TSM) and the weak measurement method (WMM). Our method establishes a correspondence between the QRMS disturbance of the measurement and the second-order derivative of the decoherence induced in a newly introduced weak probe system with respect to the coupling strength of the weak interaction at its zero-limit. Furthermore, we demonstrate the effectiveness of our method in comparison with the other two through a simulation and experiment using a quantum computer. The results capture the key features of the TSM, WMM, and our method, providing insights into the strengths and limitations of these methods.

quant-ph↗

arXiv2023

Logical Characterization of Contextual Hidden-Variable Theories based on Quantum Set Theory

While non-contextual hidden-variable theories are proved to be impossible, contextual ones are possible. In a contextual hidden-variable theory, an observable is called a beable if the hidden-variable assigns its value in a given measurement context specified by a state and a preferred observable. Halvorson and Clifton characterized the algebraic structure of beables as a von Neumann subalgebra, called a beable subalgebra, of the full observable algebra such that the probability distribution of every observable affiliated therewith admits the ignorance interpretation. On the other hand, we have shown that for every von Neumann algebra there is a unique set theoretical universe such that the internal "real numbers" bijectively correspond to the observables affiliated with the given von Neumann algebra. Here, we show that a set theoretical universe is associated with a beable subalgebra if and only if it is ZFC-satisfiable, namely, every theorem of ZFC set theory holds with probability equal to unity. Moreover, we show that there is a unique maximal ZFC-satisfiable subuniverse "implicitly definable", in the sense of Malament and others, by the given measurement context. The set theoretical language for the ZFC-satisfiable universe, characterized by the present work, rigorously reconstructs Bohr's notion of the "classical language" to describe the beables in a given measurement context.

quant-ph↗

arXiv2023

Quantum Measurement Theory for Systems with Finite Dimensional State Spaces

In this paper, we present a general theory of finite quantum measurements, for which we assume that the state space of the measured system is a finite dimensional Hilbert space and that the possible outcomes of a measurement is a finite set of real numbers. We develop the theory in a deductive manner from the basic postulates for quantum mechanics and a few plausible axioms for general quantum measurements. We derive an axiomatic characterization of all the physically realizable finite quantum measurements. Mathematical tools necessary to describe measurement statistics, such as POVMs and quantum instruments, are not assumed at the outset, but we introduce them as natural consequences of our axioms. Our objective is to show that those mathematical tools can be naturally derived from obvious theoretical requirements.

quant-ph↗

arXiv2022

Nondistributivity of human logic and violation of response replicability effect in cognitive psychology

The aim of this paper is to promote quantum logic as one of the basic tools for analyzing human reasoning. We compare it with classical (Boolean) logic and highlight the role of violation of the distributive law for conjunction and disjunction. It is well known that nondistributivity is equivalent to incompatibility of logical variables -- the impossibility to assign jointly the two-valued truth values to these variables. A natural question arises as to whether quantum logical nondistributivity in human logic can be tested experimentally. We show that testing the response replicability effect (RRE) in cognitive psychology is equivalent to testing nondistributivity -- under the prevailing conjecture that the mental state update generated by observation is described as orthogonal projection of the mental state vector (the projective update conjecture of Wang and Busemeyer). A simple test of RRE is suggested. In contrast to the previous works in quantum-like modeling, we proceed in the state-dependent framework; in particular, distributivity, compatibility, and RRE are considered in a fixed mental state. In this framework, we improve the previous result on the impossibility to combine question order and response replicability effects by using (von Neumann-Lüders) projective measurements.

q-bio.NC↗

arXiv2022

Quantum Disturbance without State Change: Soundness and Locality of Disturbance Measures

It is often supposed that a quantum system is not disturbed without state change. In a recent debate, this assumption is used to claim that the operator-based disturbance measure, a broadly used disturbance measure, has an unphysical property. Here, we show that a quantum system possibly incurs an operationally detectable disturbance without state change to rebut the claim. Moreover, we establish the reliability, formulated as soundness and locality, of the operator-based disturbance measure, which, we show, quantifies the disturbance on an observable that manifests in the time-like correlation even in the case where its probability distribution does not change.

quant-ph↗

arXiv2021

Quantum Set Theory: Transfer Principle and De Morgan's Laws

In quantum logic, introduced by Birkhoff and von Neumann, De Morgan's Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeuti's quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value assignment for bounded quantifiers that satisfies De Morgan's Laws. To justify the new assignment, we prove the Transfer Principle, showing that this assignment of a truth value to every bounded ZFC theorem has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula. We study the most general class of truth value assignments and obtain necessary and sufficient conditions for them to satisfy the Transfer Principle, to satisfy De Morgan's Laws, and to satisfy both. For the class of assignments with polynomially definable logical operations, we determine exactly 36 assignments that satisfy the Transfer Principle and exactly 6 assignments that satisfy both the Transfer Principle and De Morgan's Laws.

quant-ph↗

arXiv2021

From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti

Gaisi Takeuti introduced Boolean valued analysis around 1974 to provide systematic applications of Boolean valued models of set theory to analysis. Later, his methods were further developed by his followers, leading to solving several open problems in analysis and algebra. Using the methods of Boolean valued analysis, he further stepped forward to construct set theory based on quantum logic, as the first step to construct "quantum mathematics", a mathematics based on quantum logic. While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. As quantum logic is intrinsic and empirical, the results of the quantum set theory can be experimentally verified by quantum mechanics. In this paper, we analyze Takeuti's mathematical world view underlying his program from two perspectives: set theoretical foundations of modern mathematics and extending the notion of sets to multi-valued logic. We outlook the present status of his program, and envisage the further development of the program, by which we would be able to take a huge step forward toward unraveling the mysteries of quantum mechanics that have persisted for many years.

quant-ph↗

arXiv2021

Error-Disturbance Relation in Stern-Gerlach Measurements

Although Heisenberg's uncertainty principle is represented by a rigorously proven relation about intrinsic uncertainties in quantum states, Heisenberg's error-disturbance relation (EDR) has been commonly believed to be another aspect of the principle. Based on the recent development of universally valid reformulations of Heisenberg's EDR, we study the error and disturbance of Stern-Gerlach measurements of a spin-1/2 particle. We determine the range of the possible values of the error and disturbance for arbitrary Stern-Gerlach apparatuses with the orbital degree prepared in an arbitrary Gaussian state. We show that their error-disturbance region is close to the theoretical optimal and actually violates Heisenberg's EDR in a broad range of experimental parameters. We also show the existence of orbital states in which the error is minimized by the screen at a finite distance from the magnet, in contrast to the standard assumption.

quant-ph↗

arXiv2020

Violation of Heisenberg's error-disturbance relation by Stern-Gerlach measurements

Although Heisenberg's uncertainty principle is represented by a rigorously proven relation about intrinsic indeterminacy in quantum states, Heisenberg's error-disturbance relation (EDR) has been commonly believed as another aspect of the principle. However, recent developments of quantum measurement theory made Heisenberg's EDR testable to observe its violations. Here, we study the EDR for Stern-Gerlach measurements. In a previous report [arXiv:1910.07929], it has been pointed out that their EDR is close to the theoretical optimal. The present note reports that even the original Stern-Gerlach experiment in 1922, the available experimental data show, violates Heisenberg's EDR. The results suggest that Heisenberg's EDR is more ubiquitously violated than it has long been supposed.

quant-ph↗

arXiv2020

Reforming Takeuti's Quantum Set Theory to Satisfy De Morgan's Laws

In 1981, Takeuti introduced set theory based on quantum logic by constructing a model analogous to Boolean-valued models for Boolean logic. He defined the quantum logical truth value for every sentence of set theory. He showed that equality axioms do not hold, while axioms of ZFC set theory hold if appropriately modified with the notion of commutators. Here, we consider the problem in Takeuti's quantum set theory that De Morgan's laws do not hold for bounded quantifiers. We construct a counter-example to De Morgan's laws for bounded quantifiers in Takeuti's quantum set theory. We redefine the truth value for the membership relation and bounded existential quantification to ensure that De Morgan's laws hold. Then, we show that the truth value of every theorem of ZFC set theory is lower bounded by the commutator of constants therein as quantum transfer principle.

quant-ph↗

arXiv2020

Modeling combination of question order effect, response replicability effect, and QQ-equality with quantum instruments

We continue to analyze basic constraints on the human decision making from the viewpoint of quantum measurement theory (QMT). As it has been found, the conventional QMT based on the projection postulate cannot account for the combination of the question order effect (QOE) and the response replicability effect (RRE). This was alarming finding for quantum-like modeling of decision making. Recently, it was shown that this difficulty can be resolved by using of the general QMT based on quantum instruments. In the present paper we analyse the problem of the combination of QOE, RRE, and the well-known QQ-equality (QQE). This equality was derived by Busemeyer and Wang and it was shown (in a joint paper with Solloway and Shiffrin) that statistical data from many social opinion polls satisfy it. Here we construct quantum instruments satisfying QOE, RRE and QQE. The general features of our approach are formalized with postulates that generalize (the Wang-Busemeyer) postulates for quantum-like modeling of decision making. Moreover, we show that our model closely reproduces the statistics of the well-known Clinton-Gore Poll data with a prior belief state independent of the question order. This model successfully corrects for the order effect in the data to determine the "genuine" distribution of the opinions in the Poll. The paper also provides an accessible introduction to the theory of quantum instruments - the most general mathematical framework for quantum measurements.

q-bio.NC↗