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Operational Characterization of Simultaneous Measurements in Quantum Mechanics

Ozawa, M

Published Jan 1, 2000• 5 citations
OpenAlex

Abstract

Quantum mechanics predicts the joint probability distribution of the outcomes of simultaneous measurements of commuting observables, but, in the state of the art, has lacked the operational definition of simultaneous measurements. The question is answered as to when the consecutive applications of measuring apparatuses give a simultaneous measurement of their observables. For this purpose, all the possible state reductions caused by measurements of an observable is also characterized by their operations.

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Quantum mechanics predicts the joint probability distributions of the outcomes of simultaneous measurements of commuting observables, but the current formulation lacks the operational definition of simultaneous measurements. In order to provide foundations of joint statistics of local general measurements on entangled systems in a general theoretical framework, the question is answered as to under what condition the outputs of two measuring apparatuses satisfy the joint probability formula for simultaneous measurements of their observables. For this purpose, all the possible state changes caused by measurements of an observable are characterized, and the notion of disturbance in measurement is formalized in terms of operations derived by the measuring interaction.

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Quantum mechanics is based on Schrodinger’s wave function with linear superposition of vectors in a Hilbert space. Due to superposition, multiple physical states are considered simultaneously for the same time point, but when the corresponding quantum mechanical experiments are realized, there is only one outcome and not a superposition of pointer positions. This observation characterizes the quantum mechanical measurement problem, which for about a century found multiple interpretations, which can be classified as potentiality, as proposed by Heisenberg, or as real physical entities, for instance fields. Potentialities are yet inexistent in reality and unobservable in the present, since they represent a possible future. Thus, they can only be an intra-mental representation of a future process and not a physical extra-mental process ongoing in the present. There is coexistence of a physical object observable in reality and a mental prediction by an observer concerning the future behavior of this object. The characterization of quantum mechanical formalism by some physicists corresponds to a prediction and therefore to a mental representation of the future behavior of elementary particles and not to a description of an ongoing extra-mental process. Prediction in life and science has some characteristics, which resemble the measurement problem, since it undergoes a “multiple to one” reduction of mental potentialities to only one observable real outcome. Besides quantum mechanics, such reductions can also be found with many kinds of predictions, in classical physics for regular and irregular dynamical processes and in humans for programming future actions. Superposition in classical physics only concerns variable effects, but in quantum mechanics includes variable effects and non-localizable causes. Quantum mechanics, interpreted as prediction of future outcomes with multiple potentialities but only one realization, would resemble the general prediction problem in life and science and thereby loos its weird aspects.

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In classical mechanics, as we know, all observables commute with each other and therefore belong to the same “commuting set.” This means that – in principle – we can measure all observables simultaneously. Then, the state manifests itself as a collection of properties (of values of observables) (see Sec. 1.1), and as such it is observable. On the other hand, quantum-mechanical observables may not commute (see Subsec. 2.1.5). As a consequence, the state is not just a collection of observable values (see also Sec. 2.3) so that we may ask ourselves whether it is observable or not, and therefore raise the following question: What is the nature of the wave function? Is it simply a mathematical tool, i.e. does it only represent the sum of all that we can know about a system? Or does it have an ontological status? Since the early days of quantum mechanics, this question has been very popular. As a first attempt, de Broglie tried to interpret the wave function as describing, in classical terms, a wave field that envelopes a classical particle (see also Subsec. 16.3.2). This theory should predict the existence of empty waves, but several experiments have shown negative results (see also the end of Sec. 9.6). More recently, many physicists have tried to propose some methods in order to measure the wave function. Here we shortly report on some of these proposals and discuss to what extent the quantum state can be measured.

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arXiv:1303.0078•AIP conference proceedings•Jan 1, 2014•4 citations
Quantum mechanics predicts the joint probability distribution of the outcomes of simultaneous measurements of commuting observables, but, in the state of the art, has lacked the operational definition of simultaneous measurements. The question is answered as to when the consecutive applications of measuring apparatuses give a simultaneous measurement of their observables. For this purpose, all the possible state reductions caused by measurements of an observable is also characterized by their operations.
Quantum mechanics predicts the joint probability distributions of the outcomes of simultaneous measurements of commuting observables, but the current formulation lacks the operational definition of simultaneous measurements. In order to provide foundations of joint statistics of local general measurements on entangled systems in a general theoretical framework, the question is answered as to under what condition the outputs of two measuring apparatuses satisfy the joint probability formula for simultaneous measurements of their observables. For this purpose, all the possible state changes caused by measurements of an observable are characterized, and the notion of disturbance in measurement is formalized in terms of operations derived by the measuring interaction.
The state of a quantum-mechanical system after a simultaneous measurement of conjugate variables such as coordinate and momentum is derived. In the case of the minimum-uncertainty, or ideal, simultaneous measurement, the state of the system after the measurement is a coherent state corresponding to that representing a minimum-uncertainty wave packet. An operator formalism to describe simultaneous measurements of noncommuting observables is introduced which parallels that familiar for a single measurement, and an expression for the joint probability distribution predicting the results of an ideal simultaneous measurement is derived and applied to the example of a harmonic oscillator. It is found that the minimum uncertainty in a simultaneous measurement of noncommuting variables has two causes: (1) the unavoidable perturbations introduced by the measuring process, and (2) the unavoidable lack of precision in the state of the system itself. For position and momentum measurements, these two independent uncertainties contribute to give a net minimum uncertainty of $\ensuremath{\Delta}{q}^{\ensuremath{'}}\ensuremath{\Delta}{p}^{\ensuremath{'}}=\ensuremath{\hbar}$. It is also shown that the formalism of simultaneous measurement leads to a well-defined quantum-mechanical phase space.
The possibility of performing simultaneous measurements in quantum mechanics is investigated in the context of the Curie-Weiss model for a projective measurement. Concretely, we consider a spin--12\frac{1}{2}21​ system simultaneously interacting with two magnets, which act as measuring apparatuses of two different spin components. We work out the dynamics of this process and determine the final state of the measuring apparatuses, from which we can find the probabilities of the four possible outcomes of the measurements. The measurement is found to be nonideal, as (i) the joint statistics do not coincide with the one obtained by separately measuring each spin component, and (ii) the density matrix of the spin does not collapse in either of the measured observables. However, we give an operational interpretation of the process as a generalized quantum measurement, and show that it is fully informative: The expected value of the measured spin components can be found with arbitrary precision for sufficiently many runs of the experiment.
Quantum mechanics is based on Schrodinger’s wave function with linear superposition of vectors in a Hilbert space. Due to superposition, multiple physical states are considered simultaneously for the same time point, but when the corresponding quantum mechanical experiments are realized, there is only one outcome and not a superposition of pointer positions. This observation characterizes the quantum mechanical measurement problem, which for about a century found multiple interpretations, which can be classified as potentiality, as proposed by Heisenberg, or as real physical entities, for instance fields. Potentialities are yet inexistent in reality and unobservable in the present, since they represent a possible future. Thus, they can only be an intra-mental representation of a future process and not a physical extra-mental process ongoing in the present. There is coexistence of a physical object observable in reality and a mental prediction by an observer concerning the future behavior of this object. The characterization of quantum mechanical formalism by some physicists corresponds to a prediction and therefore to a mental representation of the future behavior of elementary particles and not to a description of an ongoing extra-mental process. Prediction in life and science has some characteristics, which resemble the measurement problem, since it undergoes a “multiple to one” reduction of mental potentialities to only one observable real outcome. Besides quantum mechanics, such reductions can also be found with many kinds of predictions, in classical physics for regular and irregular dynamical processes and in humans for programming future actions. Superposition in classical physics only concerns variable effects, but in quantum mechanics includes variable effects and non-localizable causes. Quantum mechanics, interpreted as prediction of future outcomes with multiple potentialities but only one realization, would resemble the general prediction problem in life and science and thereby loos its weird aspects.
In classical mechanics, as we know, all observables commute with each other and therefore belong to the same “commuting set.” This means that – in principle – we can measure all observables simultaneously. Then, the state manifests itself as a collection of properties (of values of observables) (see Sec. 1.1), and as such it is observable. On the other hand, quantum-mechanical observables may not commute (see Subsec. 2.1.5). As a consequence, the state is not just a collection of observable values (see also Sec. 2.3) so that we may ask ourselves whether it is observable or not, and therefore raise the following question: What is the nature of the wave function? Is it simply a mathematical tool, i.e. does it only represent the sum of all that we can know about a system? Or does it have an ontological status? Since the early days of quantum mechanics, this question has been very popular. As a first attempt, de Broglie tried to interpret the wave function as describing, in classical terms, a wave field that envelopes a classical particle (see also Subsec. 16.3.2). This theory should predict the existence of empty waves, but several experiments have shown negative results (see also the end of Sec. 9.6). More recently, many physicists have tried to propose some methods in order to measure the wave function. Here we shortly report on some of these proposals and discuss to what extent the quantum state can be measured.
Quantum mechanics does not permit joint measurements of non-commuting observables.However, it is possible to measure the weak value of a projection operator, followed by the precise measurement of a different property.The results can be interpreted as complex joint probabilities of the two non-commuting measurement outcomes.Significantly, it is possible to predict the outcome of completely different measurements by combining the joint probabilities of the initial state with complex conditional probabilities relating the new measurement to the possible combinations of measurement outcomes used in the characterization of the quantum state.We can therefore conclude that the complex conditional probabilities observed in weak measurements describe fundamental state-independent relations between non-commuting properties that represent the most fundamental form of universal laws in quantum physics.