On the quantum theory of sequential measurements

The quantum theory of sequential measurements is worked out and is employed to provide an operational analysis of basic measurement theoretical notions such as coexistence, correlations, repeatability, and ideality. The problem of the operational definition of continuous observables is briefly revisited, with a special emphasis on the localization observable. Finally, a brief overview is given of possible applications of the theory to various fields and problems in quantum physics.     

Indeterminacy relations and simultaneous measurements in quantum theory

This paper concerns derivations and interpretations of the uncertainty relations. The exclusive validity of the statistical interpretation is called into question. An individualistic interpretation, formulated by means of the concept of unsharp observables, is justified through a model of a joint measurement of position and momentum.     

Repeatable measurements in quantum theory: Their role and feasibility

Recent advantages in experimental quantum physics call for a careful reconsideration of the measurement process in quantum mechanics. In this paper we describe the structure of the ideal measurements and their status among the repeatable measurements. Then we provide an exhaustive account of the interrelations between repeatability and the apparently weaker notions of value reproducible or first- kind measurements. We demonstrate the close link between repeatable measurements and discrete observables and show how the ensuing measurement limitations for continuous observables can be lifted in a way that is in full accordance with actual experimental practice. We present examples of almost repeatable measurements of continuous observables and some realistic models of weakly disturbing measurements.Dedicated to Peter Mittelstaedt on the occasion of his 65th birthday.Leaving the Institute for Theoretical Physics, University of Cologne, D-50937 Cologne, Germany.     

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.     

Optimal entropic uncertainty relation for successive measurements in quantum information theory

We derive an optimal bound on the sum of entropic uncertainties of two or more observables when they are sequentially measured on the same ensemble of systems. This optimal bound is shown to be greater than or equal to the bounds derived in the literature on the sum of entropie uncertainties of two observables which are measured on distinct but identically prepared ensembles of systems. In the case of a two-dimensional Hilbert space, the optimum bound for successive measurements of two-spin components, is seen to be strictly greater than the optimal bound for the case when they are measured on distinct ensembles, except when the spin components are mutually parallel or perpendicular     

Concerning measurements in quantum theory: A critique of a recent proposal

Conclusion Jauch's theory of measurement, if satisfactory, would have provided an elegant and very simple solution to all of the standard difficulties found in other versions of the measurement process in quantum theory. We have examined this theory and found it to contain an important element of obscurity (that concerned with just exactly under what conditions the reductions may be performed) and ambiguity (namely concerning which reductions are to be performed), but most importantly we have concluded that it was both inconsistent and lacking in physical plausibility.I wish to acknowledge Dr. J. Bub and members of my graduate seminar, stimulating conversations with whom served to clarify many of the issues in this paper.     

On quantum measurements

Is the reduction of the wavefunction a real physical process? A mental experiment providing an answer to this question is considered.     

Nonideal quantum measurements

A partial ordering in the class of observables ( positive operator-valued measures, introduced by Davies and by Ludwig) is explored. The ordering is interpreted as a form of nonideality, and it allows one to compare ideal and nonideal versions of the same observable. Optimality is defined as maximality in the sense of the ordering. The framework gives a generalization of the usual (implicit) definition of self-adjoint operators as optimal observables (von Neumann), but it can, in contrast to this latter definition, be justified operationally. The nonideality notion is compared to other quantum estimation theoretic methods. Measures for the amount of nonideality are derived from information theory.     

Repeated quantum non-demolition measurements

 We describe, in detail, theoretical and experimental aspects related to our recently reported repeated back-action evading measurements performed using two travelling-wave optical parametric amplifiers in series. The state of the observable being measured is almost perfectly preserved after two successive measurements. The final signal, and the intermediate measurement of the two successive setups are quantum correlated with conditional variances that lie in the quantum regime. Moreover, we show that the two independent measurements are quantum correlated up to 30%. Received: 11 April 1996/Revised version: 18 July 1996     

Mesoscopic quadratic quantum measurements

We develop a theory of quadratic quantum measurements by a mesoscopic detector. It is shown that the quadratic measurements should have nontrivial quantum information properties, providing, for instance, a simple way of entangling two noninteracting qubits. We also calculate the output spectrum of a detector with both linear and quadratic response, continuously monitoring two qubits.     

Maximum confidence quantum measurements

We consider the problem of discriminating between states of a specified set with maximum confidence. For a set of linearly independent states unambiguous discrimination is possible if we allow for the possibility of an inconclusive result. For linearly dependent sets an analogous measurement is one which allows us to be as confident as possible that when a given state is identified on the basis of the measurement result, it is indeed the correct state.     

Pruned quantum theory

Comments are given on controversial problems of interpretation of quantum mechanics and quantal measurements.     

Discrete quantum theory

This paper is concerned with tracing the implications of two ideas as they affect quantum theory. One, which descends from Leibniz and Mach, is that there is no space-time continuum, but that which are involved are spacial and temporal relations involving the distant matter of the universe. The other is that our universe is finite. The picture of the world to which we are led is that of an enormous space-time Feynman diagram whose vertices are events. A consequence of finiteness is that between each pair of events, along a world line, there can be only finitely many intermediate events. A further change is that we are no longer required to believe that particles need be anywhere between events. The paper takes up nonrelativistic quantum theory in a way that is consistent with these ideas. By considering analogies between the Wiener and the Feynman integrals, and between the Wiener process and related discrete processes, there is obtained a straightforward theory for the Feynman integral. Propagators are worked out for many of the cases relevant to the nonrelativistic theory.The paper shows that, even when there are, along each world-line, no more than one event per Compton wavelength, agreement is good with the usual Schrödinger theory.Research supported in part by the NSF.     

Quantum strategies of quantum measurements

In the classical Monty Hall problem, one player can always win with probability 2/3. We generalize the problem to the quantum domain and show that a fair two-party zero-sum game can be carried out if the other player is permitted to adopt quantum measurement strategy.     

Generalization of the von Neumann theorems to irreproducible measurements within quantum field theory. IV

It is shown that the theorem of the first part of this cycle of papers necessarily, by virtue of only the fundamental principles of quantum theory, implies the existence of two fundamentally different types of evolution of isolated quantum macrosystems, i.e., the S and PS evolutions of pure states. As for isolated quantum microsystems, they can only S-evolve. The paper considers the fundamental specific singularities of PS evolution, which follow strictly from the theorem itself, as well as its corollary, proved in the first part [1] of this cycle of papers, also only on the basis of the fundamental principles of quantum theory. These singularities consist in the fact that, regardless of the commutativity of any observable of a system described by a vector in physical Hilbert space with the S-operator of this system, the probability distribution of this observable during measurement (with an appropriate instrument ), providing complete information about the characteristic of the system, is not conserved during the process of PS evolution. Nor is even the expectation of the result of measurement of the observable conserved with time.Paper presented at a session of the Department of Nuclear Physics, Academy of Sciences of the USSR (Moscow, February, 1978).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 63–72, June, 1979.The author is indebted to Prof. S. N. Sokolov (Institute of High-Energy Physics, Serpukhov) for his invaluable discussion of the Everett concept.