Measurement theory for physics

A highly abstracted theory of measurement is synthesized from classical measurement theory, fuzzy set theory, generalized information theory, and predicate calculus. The theory does not require specific truth value concepts, nor does it specify what subsets of the reals can be observed, thus avoiding the usual fundamental difficulties. Problems such as the definition of systems, the significance of observations, numerical scales and observables, etc. are examined. The general logico-algebraic approach to quantum/classical physics is justified as a special case of measurement theory.     

Stochastic interpretations of nonrelativistic quantum theory

A critique of the causla and classical stochastic interpretations of nonrelativistic quantum mechanics is presented. The only way that the classical stochastic formulation can be made compatible with the theory of quantum measurement is to extend the probability measure density for fluctuating paths to the complex domain. In doing so, we obtain the generalized stochiastic formulation in which the methods of classical probability theory can be used to describe the quantum mechanical phenomenon of interfering alternatives. Illustrative examples from quantum theory are used to show the complete compatibility between the traditional and generalized stochastic interpretations of quantum mechanics. Work supported in part by a contribution from the CNR.     

Quantum Generalized Measurement and Deterministic Generation of Maximum Entangled Pure State

We propose the concept of the quantum generalized projector measurement （QGPM） for finite-dimensional quantum systems by studying the quantum generalized measurement. This research reveals a distinguished property of this quantum generalized measurement： no matter what the system state is prior to the measurement and what the result of the measurement occurs, the state of the system after the measurement can be collapsed into any specified pure state, i.e., the state of quantum system can be deterministically reduced to any specified pure state just by a single QGPM. Subsequently. QGPM can be used to deterministically generate the maximum entangled pure state for quantum systems. We give three concrete theoretic schemes of generating the maximum quantum entangled pure states for two 2-Jevel particles, three 2-level particles and two 3-Jevel particles, respectively.     

Quantum Generalized Measurement and Deterministic Generation of Maximum Entangled Pure State

We propose the concept of the quantum generalized projector measurcment (QGPM) for finite-dimensional quantum systems by studying the quantum generalized measurement. This research reveals a distinguished property of this quantum generalized measurement: no matter what the system state is prior to the measurement and what the result of the measurement occurs, the state of the system after the measurement can be collapsed into any specified pure state, i.e., the state of quantum system can be deterministically reduced to any specified pure state just by a single QGPM. Subsequently, QGPM can be used to deterministically generate the maximum entangled pure state for quantum systems. We give three concrete theoretic schemes of generating the maximum quantum entangled pure stazes for two 2-level particles, three 2-level particles and two 3-level particles, respectively.     

Pairwise correlations in a three-partite quantum system from a prequantum random field

Prequantum classical statistical field theory (PCSFT) is a model that provides the possibility to represent the averages of quantum observables (including correlations of observables on subsystems of a composite system) as averages with respect to fluctuations of classical random fields. In view of the PCSFT terminology, quantum states are classical random fields. The aim of our approach is to represent all quantum probabilistic quantities by means of classical random fields. We obtain the classical-random-field representation for pairwise correlations in three-partite quantum systems. The three-partite case (surprisingly) differs substantially from the bipartite case. As an important first step, we generalized the theory developed for pure quantum states of bipartite systems to the states given by density operators.     

Quantum theory of continuous measurements and its applications in quantum optics

We present an overview of the quantum theory of continuous measurements and discuss some of its important applications in quantum optics. Quantum theory of continuous measurements is the appropriate generalization of the conventional formulation of quantum theory, which is adequate to deal with counting experiments where a detector monitors a system continuously over an interval of time and records the times of occurrence of a given type of event, such as the emission or arrival of a particle. We first discuss the classical theory of counting processes and indicate how one arrives at the celebrated photon counting formula of Mandel for classical optical fields. We then discuss the inadequacies of the so called quantum Mandel formula. We explain how the unphysical results that arise from the quantum Mandel formula are due to the fact that the formula is obtained on the basis of an erroneous identification of the coincidence probability densities associated with a continuous measurement situation. We then summarize the basic framework of the quantum theory of continuous measurements as developed by Davies. We explain how a complete characterization of the counting process can be achieved by specifying merely the measurement transformation associated with the change in the state of the system when a single event is observed in an infinitesimal interval of time. In order to illustrate the applications of the quantum theory of continuoius measurements in quantum optics, we first derive the photon counting probabilities of a single-mode free field and also of a single-mode field in interaction with an external source. We then discuss the general quantum counting formula of Chmara for a multi-mode electromagnetic field coupled to an external source. We explain how the Chmara counting formula is indeed the appropriate quantum generalization of the classical Mandel formula. To illustrate the fact that the quantum theory of continuous measurements has other diverse applications in quantum optics, besides the theory of photodetection, we summarize the theory of ‘quantum jumps’ developed by Zoller, Marte and Walls and Barchielli, where the continuous measurements framework is employed to evaluate the statistics of photon emission events in the resonance fluorescence of an atomic system.     

Generalized measure theory

It is argued that a reformulation of classical measure theory is necessary if the theory is to accurately describe measurements of physical phenomena. The postulates of a generalized measure theory are given and the fundamentals of this theory are developed, and the reader is introduced to some open questions and possible applications. Specifically, generalized measure spaces and integration theory are considered, the partial order structure is studied, and applications to hidden variables and the logic of quantum mechanics are given.     

Unification theory of classical statistical uncertainty relation and quantum uncertainty relation and its applications

General classical statistical uncertainty relation is deduced and generalized to quantum uncertainty relation. We give a general unification theory of the classical statistical and quantum uncertainty relations, and prove that the classical limit of quantum mechanics is just classical statistical mechanics. It is shown that the classical limit of the general quantum uncertainty relation is the general classical uncertainty relation. Also, some specific applications show that the obtained theory is self-consistent and coincides with those from physical experiments.     

Generalized quantum measurement

We overcome one of Bell's objections to ‘quantum measurement’ by generalizing the definition to include systems outside the laboratory. According to this definition, a generalized quantum measurement takes place when the value of a classical variable is influenced significantly by an earlier state of a quantum system. A generalized quantum measurement can then take place in equilibrium systems, provided the classical motion is chaotic. This Letter deals with this classical aspect of quantum measurement, assuming that the Heisenberg cut between the quantum dynamics and the classical dynamics is made at a very small scale. For simplicity, a gas with collisions is modelled by an “Arnold gas”.     

Geometry of the Parameter Space of a Quantum System: Classical Point of View

The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin-half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.     

The bond-algebraic approach to dualities

An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the ?₂ Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan–Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long-standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.     

Convex Structures and Effect Algebras

Effect algebras have important applications inthe foundations of quantum mechanics and in fuzzyprobability theory. An effect algebra that possesses aconvex structure is called a convex effect algebra. Our main result shows that any convex effectalgebra admits a representation as a generating initialinterval of an ordered linear space. This result isanalogous to a classical representation theorem for convex structures due to M. H. Stone. We alsogive a relationship between a convex effect algebra anda statistical model called a convex effect-statespace.     

A derivation of local commutativity from macrocausality using a quantum mechanical theory of measurement

A theory of the joint measurement of quantum mechanical observables is generalized in order to make it applicable to the measurement of the local observables of field theory. Subsequently, the property of local commutativity, which is usually introduced as a postulate, is derived by means of the theory of measurement from a requirement of mutual nondisturbance, which, for local observables performed at a spacelike distance from each other, is interpreted as a requirement of macrocausality. Alternative attempts at establishing a deductive relationship between relativistic causality and local commutativity are reviewed, but found wanting, either because of the assumption of an unwarranted objectivity of the object system (algebraic approach) or because of the use of a projection postulate (operational approach). Finally, the quantum mechanical nonobjectivity is related to certain features of nonlocality which are present in the formalism of quantum mechanics.     

Quantum filtering for multiple input multiple output systems driven by arbitrary zero-mean jointly Gaussian input fields

In this paper, we treat the quantum filtering problem for multiple input multiple output (MIMO) Markovian open quantum systems coupled to multiple boson fields in an arbitrary zero-mean jointly Gaussian state, using the reference probability approach formulated by Bouten and van Handel as a quantum version of a well-known method of the same name from classical nonlinear filtering theory, and exploiting the generalized Araki-Woods representation of Gough. This includes Gaussian field states such as vacuum, squeezed vacuum, thermal, and squeezed thermal states as special cases. The contribution is a derivation of the general quantum filtering equation (or stochastic master equation as they are known in the quantum optics community) in the full MIMO setup for any zero-mean jointly Gaussian input field states, up to some mild rank assumptions on certain matrices relating to the measurement vector.     

Information flow due to controlled interference in entangled systems

We point out that controlled quantum interference corresponds to measurement in an incomplete basis and implies a nonlocal transfer of classical information. A test of whether such a generalized measurement is permissible in quantum theory is presented. An erratum to this article is available at .     

Quantum probability and unified approach to quantization and dynamics

A simplified derivation of the Gudder-Hemion quantum probability formula is proposed. Defining configurations as the classical (q, p) deterministic states and generalized action as the (quantum) generating function of a canonical transformation, we obtain the usual quantization rules (for arbitrary polynomial quantities) and derive the Schrödinger wave equation on the same grounds. This approach suggests a statistical interpretation of the wave function in terms of the classical canonical transformations.     

Topological Chern—Simons vortices in the O (3) σ -model

Based on the phase-space path integral (functional integral) for a system with a regular or singular Lagrangian, the generalized Ward identities for phase space generating functional under the global transformation in phase space are derived respectively. The canonical Noether theorem at the quantum level is also established. It is pointed out that the connection between the symmetries and conservation laws in classical theories, in general,is no longer preserved in quantum theories. The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta as usually performed. Applying the present formulation to Yang-Mills theory, the quantal BRS conserved quantity and Ward-Takahashi identity for BRS tranformation are derived; the Ward identities for gaugeghost proper vertices and new quantal conserved quantity are also found. In comparison of quantal conservation laws with those one deriving from configuration-space path integral using the Faddeev-Popov(F-P) trick is discussed. A precise study of path-integral quantisation for a nonlinear sigma model with Hopf and Chern-Simons (CS) terms is reexamined. It has been shown that the angular momentum at the quantum level is equal to classical (Noether ) one. Applying our formulation to non-Abelian CS theory, the quantal conserved angular momentum of this system is obtained which differs from classical one in that one needs to take into account the contribution of angular momenta of ghost fields.     

From Wave-Particle to Features-Event Complementarity

The terms wave and particle are of classical origin and are inadequate in dealing with the novelties of quantum mechanics with respect to classical physics. In this paper we propose to substitute the wave-particle terminology with that of features-event complementarity. This approach aims at solving some of the problems affecting quantum-mechanics since its birth. In our terminology, features are what is responsible for one of the most characterizing aspects of quantum mechanics: quantum correlations. We suggest that an (uninterpreted) basic ontology for quantum mechanics should be thought of as constituted by events, features and their dynamical interplay, and that its (interpreted) theoretical ontology (made up by three classes of theoretical entities: states, observables and properties) does not isomorphically correspond to the uninterpreted ontology. Operations, i.e. concrete interventions within the physical world, like preparation, premeasurement and measurement, together with reliable inferences, assure the bridge between interpreted and uninterpreted ontology.