arXiv subjects

Masanao Ozawa

Publications and source records attributed to Masanao Ozawa.

At least 37 records · Page 2Linked to original sources

arXiv2020

Quantum-like modeling in biology with open quantum systems and instruments

We present the novel approach to mathematical modeling of information processes in biosystems. It explores the mathematical formalism and methodology of quantum theory, especially quantum measurement theory. This approach is known as {\it quantum-like} and it should be distinguished from study of genuine quantum physical processes in biosystems (quantum biophysics, quantum cognition). It is based on quantum information representation of biosystem's state and modeling its dynamics in the framework of theory of open quantum systems. This paper starts with the non-physicist friendly presentation of quantum measurement theory, from the original von Neumann formulation to modern theory of quantum instruments. Then, latter is applied to model combinations of cognitive effects and gene regulation of glucose/lactose metabolism in Escherichia coli bacterium. The most general construction of quantum instruments is based on the scheme of indirect measurement, in that measurement apparatus plays the role of the environment for a biosystem. The biological essence of this scheme is illustrated by quantum formalization of Helmholtz sensation-perception theory. Then we move to open systems dynamics and consider quantum master equation, with concentrating on quantum Markov processes. In this framework, we model functioning of biological functions such as psychological functions and epigenetic mutation.

physics.bio-ph↗

arXiv2020

Neutron optical test of completeness of quantum root-mean-square errors

One of the major problems in quantum physics has been to generalize the classical root-mean-square error to quantum measurements to obtain an error measure satisfying both soundness (to vanish for any accurate measurements) and completeness (to vanish only for accurate measurements). A noise-operator based error measure has been commonly used for this purpose, but it has turned out incomplete. Recently, Ozawa proposed a new definition for a noise-operator based error measure to be both sound and complete. Here, we present a neutron optical demonstration for the completeness of the new error measure for both projective (or sharp) as well as generalized (or unsharp) measurements.

quant-ph↗

arXiv2020

Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.

quant-ph↗

arXiv2020

A Bridge Between Q-Worlds

Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding quantum mechanics by reformulating parts of the theory inside of non-classical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, `Q-worlds'. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.

quant-ph↗

arXiv2020

Heisenberg's original derivation of the uncertainty principle and its universally valid reformulations

Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schrödinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.

quant-ph↗

arXiv2019

Soundness and completeness of quantum root-mean-square errors

Defining and measuring the error of a measurement is one of the most fundamental activities in experimental science. However, quantum theory shows a peculiar difficulty in extending the classical notion of root-mean-square (rms) error to quantum measurements. A straightforward generalization based on the noise-operator was used to reformulate Heisenberg's uncertainty relation for the accuracy of simultaneous measurements to be universally valid and made the conventional formulation testable to observe its violation. Recently, its reliability was examined based on an anomaly that the error vanishes for some imprecise measurements, in which the meter does not commute with the measured observable. Here, we propose an improved definition for a quantum generalization of the classical rms error, which is state-dependent, operationally definable, and perfectly characterizes precise measurements. Moreover, it is shown that the new notion maintains the previously obtained universally valid uncertainty relations and their experimental confirmations without changing their forms and interpretations, in contrast to a prevailing view that a state-dependent formulation for measurement uncertainty relation is not tenable.

quant-ph↗

arXiv2018

Universally valid uncertainty relations in general quantum systems

We study universally valid uncertainty relations in general quantum systems described by general $σ$-finite von Neumann algebras to foster developing quantitative analysis in quantum systems with infinite degrees of freedom such as quantum fields. We obtain the most stringent measurement-disturbance relation ever, applicable to systems with infinite degrees of freedom, by refining the proofs given by Branciard and one of the authors (MO) for systems with finite degrees of freedom. In our proof the theory of the standard form of von Neumann algebras plays a crucial role, incorporating with measurement theory for local quantum systems recently developed by the authors.

quant-ph↗

arXiv2018

Measuring processes and repeatability hypothesis

Srinivas [Commun. Math. Phys. 71 (1980), 131-158] proposed a postulate in quantum mechanics that extends the von Neumann-Lueders collapse postulate to observables with continuous spectrum. His collapse postulate does not determine a unique state change, but depends on a particular choice of an invariant mean. To clear the physical significance of employing different invariant means, we construct different measuring processes of the same observable satisfying the Srinivas collapse postulate corresponding to any given invariant means. Our construction extends the von Neumann type measuring process with the meter being the position observable to the one with the apparatus prepared in a non-normal state. It is shown that the given invariant mean corresponds to the momentum distribution of the apparatus in the initial state, which is determined as a non-normal state, called a Dirac state, such that the momentum distribution is the given invariant mean and that the position distribution is the Dirac measure.

quant-ph↗

arXiv2017

Orthomodular-Valued Models for Quantum Set Theory

In 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo-Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti's model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even non-polynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti's model and leads to a much more flexible approach to quantum set theory.

quant-ph↗

arXiv2017

Operational Meanings of Orders of Observables Defined through Quantum Set Theories with Different Conditionals

In quantum logic there is well-known arbitrariness in choosing a binary operation for conditional. Currently, we have at least three candidates, called the Sasaki conditional, the contrapositive Sasaki conditional, and the relevance conditional. A fundamental problem is to show how the form of the conditional follows from an analysis of operational concepts in quantum theory. Here, we attempt such an analysis through quantum set theory (QST). In this paper, we develop quantum set theory based on quantum logics with those three conditionals, each of which defines different quantum logical truth value assignment. We show that those three models satisfy the transfer principle of the same form to determine the quantum logical truth values of theorems of the ZFC set theory. We also show that the reals in the model and the truth values of their equality are the same for those models. Interestingly, however, the order relation between quantum reals significantly depends on the underlying conditionals. We characterize the operational meanings of those order relations in terms of joint probability obtained by the successive projective measurements of arbitrary two observables. Those characterizations clearly show their individual features and will play a fundamental role in future applications to quantum physics.

quant-ph↗

arXiv2015

Residual error-disturbance uncertainties in successive spin-1/2 measurements tested in matter-wave optics

The indeterminacy inherent in quantum measurement is an outstanding character of quantum theory, which manifests itself typically in Heisenberg's error-disturbance uncertainty relation. In the last decade, Heisenberg's relation has been generalized to hold for completely general quantum measurements. Nevertheless, the strength of those relations has not been clarified yet for mixed quantum states. Recently, a new error-disturbance uncertainty relation (EDUR), stringent for generalized input states, has been introduced by one of the present authors. A neutron-optical experiment is carried out to investigate this new relation: it is tested whether error and disturbance of quantum measurements disappear or persist in mixing up the measured ensemble. Our results exhibit that measurement error and disturbance remain constant independent of the degree of mixture. The tightness of the new EDUR is confirmed, thereby validating the theoretical prediction.

quant-ph↗

arXiv2015

Measurement theory in local quantum physics

In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated by CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional (AFD) von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts (DHR-DR) theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.

math-ph↗

arXiv2015

Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements

Information-theoretic definitions for noise and disturbance in quantum measurements were given in Phys. Rev. Lett. 112, 050401 (2014) and a state-independent noise-disturbance uncertainty relation was obtained. Here, we derive a tight noise-disturbance uncertainty relation for complementary qubit observables and carry out an experimental test. Successive projective measurements on the neutron's spin-1/2system, together with a correction procedure which reduces the disturbance, are performed. Our experimental results saturate the tight noise-disturbance uncertainty relation for qubits when an optimal correction procedure is applied.

quant-ph↗

arXiv2015

Comment on "Proof of Heisenberg's error-disturbance principle"

Recently, Kosugi [arXiv:1504.03779v2 [quant-ph]] argued that Heisenberg's error-disturbance relation (EDR) must be interpreted as being between the resolution, the preparational error for the post-measurement observable, and the disturbance. He further claimed that Heisenberg's EDR can be proven to hold true in general, when the meter observable is modified as one of its functions. Here, some comments are given to suggest that the above claims are not supported.

quant-ph↗

arXiv2015

Quantum Limits of Measurements and Uncertainty Principle

In this paper, we show how the Robertson uncertainty relation gives certain intrinsic quantum limits of measurements in the most general and rigorous mathematical treatment. A general lower bound for the product of the root-mean-square measurement errors arising in joint measurements of noncommuting observables is established. We give a rigorous condition for holding of the standard quantum limit (SQL) for repeated measurements, and prove that if a measuring instrument has no larger root-mean-square preparational error than the root-mean-square measurement errors then it obeys the SQL. As shown previously, we can even construct many linear models of position measurement which circumvent this condition for the SQL.

quant-ph↗

arXiv2015

Realization of Measurement and the Standard Quantum Limit

This paper, following [M. Ozawa, Phys. Rev. Lett. 60, 385 (1988)], reports a refutation of the claim that for monitoring the position of a free mass such as gravitational-wave interferometers the sensitivity is limited by the so called standard quantum limit (SQL) due to the uncertainty principle. The latest proof of the SQL is analyzed to revleal an unsupported assumption on quantum measurements. Quantum measurement theory is introduced to give a general criterion for physically realizable measurements in quantum mechanics. A theory of approximate position measurements is developed to obtain a rigorous condition for the SQL and also to show that a precise position measurement can leave the object in an arbitrary family of states independent of the input state. This concludes that Yuen's proposal of breaking the SQL by a contractive state measurement, a measurement of the position leaving the free mass in a state with the position uncertainty decreasing in time, is physically realizable in principle. To enforce this conclusion, a model for error-free position measurement that leaves the object in a contractive state is constructed with a solvable Hamiltonian for measuring interaction. Finally, this model is shown to break the SQL with arbitrary accuracy.

quant-ph↗

arXiv2014

Quantum Set Theory Extending the Standard Probabilistic Interpretation of Quantum Theory (Extended Abstract)

The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality relation for a pair of arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the probabilistic interpretation of quantum theory to define the probability of equality relation for a pair of arbitrary observables. Applications of this new interpretation to measurement theory are discussed briefly.

quant-ph↗

arXiv2014

Mathematical foundations of quantum information: Measurement and foundations

The purpose of this paper is to survey some topics on mathematical foundations of quantum information developed mainly by the present author and co-workers for the last three decades. The topics include an axiomatic construction of quantum measurement theory based on completely positive map-valued measures, a universally valid new formulation of the uncertainty principle for error and disturbance in quantum measurements, the Wigner-Araki-Yanase limit of quantum measurements, the accuracy limit of quantum computing based on conservation laws, and a quantum interpretation based on quantum set theory.

quant-ph↗