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Physical Review Letters

Curious New Statistical Prediction of Quantum Mechanics

David Z. Albert, Yakir Aharonov, Susan S. D’Amato

Published Jan 7, 1985• 69 citations
Publisher

Abstract

Whether quantum mechanical systems, within the interval between two measurements, can be assigned definite, simultaneous values of noncommuting observables has been the subject of a very old dispute. A certain widely held assumption about such systems has been crucial to that dispute. That assumption turns out to be wrong: A previously unknown prediction of quantum mechanics, which fails to satisfy that assumption, is described here, and the consequences of that failure are considered.

DOI
10.1103/physrevlett.54.5

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Whether quantum mechanical systems, within the interval between two measurements, can be assigned definite, simultaneous values of noncommuting observables has been the subject of a very old dispute. A certain widely held assumption about such systems has been crucial to that dispute. That assumption turns out to be wrong: A previously unknown prediction of quantum mechanics, which fails to satisfy that assumption, is described here, and the consequences of that failure are considered.
The outcomes of quantum measurements are generally considered to be random, but despite the fact that this randomness is an important element in quantum information theory, its nature is not well understood. In this thesis, we study several issues relating to the origin and certification of quantum randomness and unpredictability. One of the key results in forming our understanding of quantum mechanics as an intrinsically indeterministic theory is the Kochen-Specker theorem, which shows the impossibility to consistently assign simultaneous noncontextual definite values to all quantum mechanical observables prior to measurement. However, the theorem, under the assumption that any definite values must be noncontextual, only strictly shows that some observables must be value indefinite. We strengthen this result, proving a stronger variant of the Kochen-Specker theorem showing that, under the same assumption, if a system is prepared in an arbitrary state j i, then every observable A is value indefinite unless j i is an eigenstate of A. The indeterministic nature of quantum measurements does little to explain how the quality of quantum randomness differs from classical randomness. We show that, subject to certain physical assumptions, a sequence of bits generated by the measurement of value indefinite observables is guaranteed, in the infinite limit, to be strongly incomputable. We further discuss how this can be used to build a quantum random number generator certified by value indefiniteness. Next, we study the notion of unpredictability, which is central to the concept of (quantum) randomness. In doing so, we propose a formal model of prediction that can be used to asses the predictability of arbitrary physical experiments. We investigate how the quantum features of value indefiniteness and complementarity can be used to certify different levels of unpredictability, and show that the outcome of a single measurement of a value indefinite quantum observable is formally unpredictable. Finally, we study the relation between this notion of unpredictability and the computability-theoretic certification of quantum randomness.
One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes. It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle). It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations).
525. IN preceding chapters we saw how the classical system of mechanics failed to account for certain phenomena of physics, and it became clear that these could only be explained in terms of a new system of dynamics which must be supposed to supersede the old classical system when we have to deal with phenomena conditioned by the fine-scale structure of matter. The first instance of the failure of the classical mechanics was provided by the phenomenon of black-body radiation. On the assumption that black-body radiation represents a state of thermodynamical equilibrium, we found that the observed distribution of radiant energy in the spectrum of an ideal black body could be explained on the supposition that the energy of each part of the radiation-producing mechanism fell into “quanta.” Corresponding to a simple harmonic vibration of frequency v , there was supposed to be an energyquantum of amount hv , where h is Planck's universal constant of which the value has already been given. Calling this energy ∈ , it was found that the energy of vibration could be 0, ∈, 2∈, 3∈,… or any integral multiple of ∈ , but could not contain fractional parts of ∈ . The energy, after retaining one of these values for a certain time, had to be supposed to jump abruptly to some other value, which in turn was retained until another jump occurred.
Should we doubt the exactness of the predictive quantum rules of calculation? Although this question is sometimes raised in connection with the one on how to physically understand quantum mechanics, these two questions should not be mixed up. It is recalled here that even the first one is stil an object of controversy, and it is shown (a) that in one specific case the arguments put forward in support of such doubts are hardly cogent but (b) that, nevertheless, at least in one specific other context, the question is worth attention. This is the problem of repeated imperfect measurements. Relative to it, a theoretical possibility is shown of discriminating between the thesis that the quantum rules are exact and a powerful theory of which it is proved that it cannot be reconciled with the assumption.
A recent result concerning the assignment of well-defined values to certain noncommuting observables in quantum ensembles which have been both preselected and post-selected appears to show that quantum mechanics is "contextual." Extending the argument to the case of measurements on two separated spin-\textonehalf{} systems suggests that quantum mechanics is nonlocal. The significance of these phenomena is evaluated, and it is shown that they have no bearing on the question of contextuality or nonlocality.