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Completing Quantum Mechanics with Quantized Hidden Variables

S. J. van Enk

Published Jun 10, 2015• 0 citations
OpenAlex

Abstract

I explore the possibility that a quantum system S may be described completely by the combination of its standard quantum state $|\psi\rangle$ and a (hidden) quantum state $|\phi\rangle$ (that lives in the same Hilbert space), such that the outcome of any standard projective measurement on the system S is determined once the two quantum states are specified. I construct an algorithm that retrieves the standard quantum-mechanical probabilities, which depend only on $|\psi\rangle$, by assuming that the (hidden) quantum state $|\phi\rangle$ is drawn at random from some fixed probability distribution Pr(.) and by averaging over Pr(.). Contextuality and Bell nonlocality turn out to emerge automatically from this algorithm as soon as the dimension of the Hilbert space of S is larger than 2. If $|\phi\rangle$ is not completely random, subtle testable deviations from standard quantum mechanics may arise in sequential measurements on single systems.

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Completing Quantum Mechanics with Quantized Hidden Variables

S. J. van Enk

I explore the possibility that a quantum system S may be described completely by the combination of its standard quantum state $|ψ\rangle$ and a (hidden) quantum state $|ϕ\rangle$ (that lives in the same Hilbert space), such that the outcome of any standard projective measurement on the system S is determined once the two quantum states are specified. I construct an algorithm that retrieves the standard quantum-mechanical probabilities, which depend only on $|ψ\rangle$, by assuming that the (hidden) quantum state $|ϕ\rangle$ is drawn at random from some fixed probability distribution Pr(.) and by averaging over Pr(.). Contextuality and Bell nonlocality turn out to emerge automatically from this algorithm as soon as the dimension of the Hilbert space of S is larger than 2. If $|ϕ\rangle$ is not completely random, subtle testable deviations from standard quantum mechanics may arise in sequential measurements on single systems.

arXiv:1506.03485•Jun 10, 2015•0 citations

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I explore the possibility that a quantum system S may be described completely by the combination of its standard quantum state ∣ψ⟩|\psi\rangle∣ψ⟩ and a (hidden) quantum state ∣ϕ⟩|\phi\rangle∣ϕ⟩ (that lives in the same Hilbert space), such that the outcome of any standard projective measurement on the system S is determined once the two quantum states are specified. I construct an algorithm that retrieves the standard quantum-mechanical probabilities, which depend only on ∣ψ⟩|\psi\rangle∣ψ⟩, by assuming that the (hidden) quantum state ∣ϕ⟩|\phi\rangle∣ϕ⟩ is drawn at random from some fixed probability distribution Pr(.) and by averaging over Pr(.). Contextuality and Bell nonlocality turn out to emerge automatically from this algorithm as soon as the dimension of the Hilbert space of S is larger than 2. If ∣ϕ⟩|\phi\rangle∣ϕ⟩ is not completely random, subtle testable deviations from standard quantum mechanics may arise in sequential measurements on single systems.
I explore the possibility that a quantum system S may be described completely by the combination of its standard quantum state ∣ψ⟩|ψ\rangle∣ψ⟩ and a (hidden) quantum state ∣ϕ⟩|ϕ\rangle∣ϕ⟩ (that lives in the same Hilbert space), such that the outcome of any standard projective measurement on the system S is determined once the two quantum states are specified. I construct an algorithm that retrieves the standard quantum-mechanical probabilities, which depend only on ∣ψ⟩|ψ\rangle∣ψ⟩, by assuming that the (hidden) quantum state ∣ϕ⟩|ϕ\rangle∣ϕ⟩ is drawn at random from some fixed probability distribution Pr(.) and by averaging over Pr(.). Contextuality and Bell nonlocality turn out to emerge automatically from this algorithm as soon as the dimension of the Hilbert space of S is larger than 2. If ∣ϕ⟩|ϕ\rangle∣ϕ⟩ is not completely random, subtle testable deviations from standard quantum mechanics may arise in sequential measurements on single systems.
The question of a hidden variable interpretation of quantum contextuality in the Mermin-Peres square is considered. The Kochen-Specker theorem implies that quantum mechanics may be interpreted as a contextual hidden-variable theory. It is shown that such a hidden-variable description can be viewed as either contextual in the random variables mapping hidden states to observable outcomes or in the probability measure on the hidden state space. The latter view suggests that this apparent contextuality may be interpreted as a simple consequence of measurement disturbance, wherein the initial hidden state is altered through interaction with the measuring device, thereby giving rise to a possibly different final hidden-variable state from which the measurement outcome is obtained. In light of this observation, a less restrictive and, arguably, more reasonable definition of noncontextuality is suggested. To prove that such a description is possible, an explicit and, in this sense, noncontextual hidden-variable model is constructed, which reproduces all quantum theoretic predictions for the Mermin-Peres square. A critical analysis of some recent and proposed experimental tests of contextuality is also provided. Although the discussion is restricted to a four-dimensional Hilbert space, the approach and conclusions are expected to generalize to any Hilbert space.
Bell's theorem rests on the following fundamental condition for a local system: P(a,b|alpha, beta, lambda)= P(a|alpha, lambda)P(b|beta, lambda). Here a and b are the outcomes respectively for measurements alpha on one side, and beta on the other, of an experiment involving two entangled particles traveling in opposite directions from a source. The parameter lambda (the set of "hidden variables") represents a more complete description of the joint state of the two particles. Because of lambda, the joint probability of detection is now dependent only on lambda and the local measurement setting of alpha; similarly for the other side and the setting beta. From this equation John Bell derived a simple inequality that is violated by the predictions of quantum mechanics, which is generally taken to imply that quantum mechanics is a nonlocal theory. But, by combining Richard Feynman's formulation of quantum mechanics with a model of particle interaction described by David Deutsch, we develop a system (the "space of all paths," SP) that (1) is immediately seen to replicate the predictions of quantum mechanics, (2) has a single outcome for each quantum event (unlike MWI on which it is partly based), and (3) contains the set lambda of hidden variables consisting of all possible paths from the source to the detectors on each side of the two-particle experiment. However, the set lambda is nonmeasurable, and therefore the above equation is meaningless in SP.
This paper discusses some tests of Bell-like inequalities not requiring entangled states. The proposed tests are based on consecutive measurements on a single qubit. Available hidden-variable models for a single qubit [see, e.g., J. S. Bell, Rev. Mod. Phys. 38, 447 (1966)] reproduce the predictions of quantum mechanics and hence violate the Bell-like inequalities addressed in this paper. It is shown how this fact is connected with the state ``collapse'' and with its random nature. Thus, it becomes possible to test truly realistic and deterministic hidden-variable models. In this way, it can be shown that a hidden-variable model should entail at least one of the following features: (i) nonlocality, (ii) contextuality, or (iii) discontinuous measurement-dependent probability functions. The last two features are put to the test with the experiments proposed in this paper. A hidden-variable model that is noncontextual and deterministic would be at variance with some predictions of quantum mechanics. Furthermore, the proposed tests are more likely to be loophole-free, as compared to former ones.
No physical measurement can be performed with infinite precision. This leaves a loophole in the standard no–go arguments against non–contextual hidden variables. All such arguments rely on choosing special sets of quantum–mechanical observables with measurement outcomes that cannot be simulated non–contextually. As a consequence, these arguments do not exclude the hypothesis that the class of physical measurements in fact corresponds to a dense subset of all theoretically possible measurements with outcomes and quantum probabilities that can be recovered from a non–contextual hidden–variable model. We show here by explicit construction that there are indeed such non–contextual hidden–variable models, both for projection–valued and positive–operator–valued measurements.
A new interpretation of quantum mechanics is presented in which there is one hidden variable. This could be described as “partial realism” where sometimes quantum variables have real values before we measure and sometimes they don’t. While one hidden variable is insufficient to explain Bell nonlocality, it is allowed by Bell’s inequality. This can help solve the measurement problem and is in principle testable. In addition, a theory of nonlocality emerges directly from this assumption. From the perspective of an outside observer, not correlated with an observed system, we treat all interactions within the system as unitary. However, from the perspective of an “observer” inside the system, we treat all new entanglements as “micromeasurements”, as a projection onto some basis of measurement. One particle can act as a measurement device on another and give the other particle a defined value on one possible measurement axis from an inside perspective. This means that the wave function, from the outside view, will at times describe an objective, fundamentally stochastic uncertainty, at other times a subjective uncertainty that reflects only our lack of knowledge, and at yet other times a combination of both. The standard wave function describes the total uncertainty, the minimal uncertainty that is present for any observer outside of the system, whereas for an “observer” inside the system, only the objective, fundamentally stochastic part of the uncertainty remains, and this can be described by an inner wave function. The hidden variable is invisible to the outside observer.