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Foundations of Physics

Present, predicted, and hidden probabilities

Louis de Broglie, Georges Lochak, Juan Alberto Beswick, Jos� Vassalo Pereira

Published Feb 1, 1976• 11 citations
Publisher
DOI
10.1007/bf00708659

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No-Signalling is a fundamental constraint on the probabilistic predictions made by physical theories. It is usually justified in terms of the constraints imposed by special relativity. However, this justification is not as clear-cut as is usually supposed. We shall give a different perspective on this condition by showing an equivalence between No-Signalling and Lambda Independence, or "free choice of measurements", a condition on hidden-variable theories which is needed to make no-go theorems such as Bell's theorem non-trivial. More precisely, we shall show that a probability table describing measurement outcomes is No-Signalling if and only if it can be realized by a Lambda-Independent hidden-variable theory of a particular canonical form, in which the hidden variables correspond to non-contextual deterministic predictions of measurement outcomes. The key proviso which avoids contradiction with Bell's theorem is that we consider hidden-variable theories with signed probability measures over the hidden variables - i.e. negative probabilities. Negative probabilities have often been discussed in the literature on quantum mechanics. We use a result proved previously in "The Sheaf-theoretic Structure of Locality and Contextuality" by Abramsky and Brandenburger, which shows that they give rise to, and indeed characterize, the entire class of No-Signalling behaviours. In the present paper, we put this result in a broader context, which reveals the surprising consequence that the No-Signalling condition is equivalent to the apparently completely different notion of free choice of measurements.

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The Question of the Completeness of Quantum Mechanics When a classical system is in a pure state, that is, when its preparation represents maximal information, the value of every physical quantity is uniquely predicted. In other words, classical pure states are dispersion-free, and classical mechanics is deterministic. Only in case the state is nonpure, thus representing nonmaximal information about the system, does probability enter into the theory. Things are different with quantum systems. Even in case the state of the system is pure, it is impossible to predict a definite value of every physical quantity: only the probability distributions of the physical quantities are given. Pure quantum states have dispersion (though, of course, they can be dispersion-free for particular physical quantities), and probability enters into quantum theory at a much more fundamental level. The basic idea of so-called hidden-variable theories is to reject the fundamental role of probability in quantum theory and to argue that probability arises because quantum states, even the pure ones, do not represent the ultimate information about the system. Accordingly, it is conjectured that there are certain hidden variables, not yet subject to experimental detection, which would complete the information carried by the quantum states, thus giving rise to so precise a knowledge of the system that probabilities would disappear and precise values of all physical quantities would be uniquely determined.

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Transition Probabilities

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Undergraduate lecture notes in physics•Jan 1, 2013•1 citation
No-Signalling is a fundamental constraint on the probabilistic predictions made by physical theories. It is usually justified in terms of the constraints imposed by special relativity. However, this justification is not as clear-cut as is usually supposed. We shall give a different perspective on this condition by showing an equivalence between No-Signalling and Lambda Independence, or "free choice of measurements", a condition on hidden-variable theories which is needed to make no-go theorems such as Bell's theorem non-trivial. More precisely, we shall show that a probability table describing measurement outcomes is No-Signalling if and only if it can be realized by a Lambda-Independent hidden-variable theory of a particular canonical form, in which the hidden variables correspond to non-contextual deterministic predictions of measurement outcomes. The key proviso which avoids contradiction with Bell's theorem is that we consider hidden-variable theories with signed probability measures over the hidden variables - i.e. negative probabilities. Negative probabilities have often been discussed in the literature on quantum mechanics. We use a result proved previously in "The Sheaf-theoretic Structure of Locality and Contextuality" by Abramsky and Brandenburger, which shows that they give rise to, and indeed characterize, the entire class of No-Signalling behaviours. In the present paper, we put this result in a broader context, which reveals the surprising consequence that the No-Signalling condition is equivalent to the apparently completely different notion of free choice of measurements.
The Question of the Completeness of Quantum Mechanics When a classical system is in a pure state, that is, when its preparation represents maximal information, the value of every physical quantity is uniquely predicted. In other words, classical pure states are dispersion-free, and classical mechanics is deterministic. Only in case the state is nonpure, thus representing nonmaximal information about the system, does probability enter into the theory. Things are different with quantum systems. Even in case the state of the system is pure, it is impossible to predict a definite value of every physical quantity: only the probability distributions of the physical quantities are given. Pure quantum states have dispersion (though, of course, they can be dispersion-free for particular physical quantities), and probability enters into quantum theory at a much more fundamental level. The basic idea of so-called hidden-variable theories is to reject the fundamental role of probability in quantum theory and to argue that probability arises because quantum states, even the pure ones, do not represent the ultimate information about the system. Accordingly, it is conjectured that there are certain hidden variables, not yet subject to experimental detection, which would complete the information carried by the quantum states, thus giving rise to so precise a knowledge of the system that probabilities would disappear and precise values of all physical quantities would be uniquely determined.