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Completely Real? A Critical Note on the Theorems by Colbeck and Renner

Ronnie Hermens

Published Jul 4, 2019• 0 citations
OpenAlex

Abstract

In a series of papers Colbeck and Renner (2011, 2015a,b) claim to have shown that the quantum state provides a complete description for the prediction of future measurement outcomes. In this paper I argue that thus far no solid satisfactory proof has been presented to support this claim. Building on the earlier work of Landsman (2015) and Leegwater (2016) I highlight the implicit use of an assumption concerning the way unitary evolution is to be represented in any possible completion of quantum mechanics. I show that the assumption is quite crucial to the proof of the claim and argue that it is unwarranted. I further discuss a possible validation for a restricted version of this assumption that is based on considerations of measurement processes. I argue that also this restricted version is unsatisfactory.

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Completely real? A critical note on the claims by Colbeck and Renner

Ronnie Hermens

In a series of papers Colbeck and Renner (2011, 2015a, 2015b) claim to have shown that the quantum state provides a complete description for the prediction of future measurement outcomes. In this paper I argue that thus far no solid satisfactory proof has been presented to support this claim. Building on the earlier work of Leifer (2014), Landsman (2015) and Leegwater (2016), I present and prove two results that only partially support this claim. I then discuss the arguments by Colbeck, Renner and Leegwater concerning how these results are to generalize to the full claim. This argument turns out to hinge on the implicit use of an assumption concerning the way unitary evolution is to be represented in any possible completion of quantum mechanics. I argue that this assumption is unsatisfactory and that possible attempts to validate it based on measurement theory also do not succeed.

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In a series of papers Colbeck and Renner claim to have shown that the quantum state provides a complete description for the prediction of future measurement outcomes. In this paper I argue that thus far no solid satisfactory proof has been presented to support this claim. Building on the earlier work of Leifer, Landsman and Leegwater, I present and prove two results that only partially support this claim. I then discuss the arguments by Colbeck, Renner and Leegwater concerning how these results are to generalize to the full claim. This argument turns out to hinge on the implicit use of an assumption concerning the way unitary evolution is to be represented in any possible completion of quantum mechanics. I argue that this assumption is unsatisfactory and that possible attempts to validate it based on measurement theory also do not succeed.

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In this thesis a recent claim by Roger Colbeck and Renato Renner is reviewed [18]. This claim concerns the completeness of quantum mechanics. Colbeck and Renner state that no theory can be constructed that is compatible with quantum mechanics, and gives more detailed predictions through supplemental variables added to the description of the state of a system. Their only assumption is that measurement settings can be choosen freely. We will argue that their result is for the most part valid and interesting, but the freedom assumption is not so innocent as it is presented, and parts of the result need to be clarified. Therefore, we will present a reformulation of the result. In our formulation the result is a no-go theorem for hidden variable theories satisfying Parameter Independence. This is a generalization of Bell’s theorem. Performed in the period 01-11-2012 14-08-2012

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Quantum theory (QT) provides statistical predictions for various physical phenomena. To verify these predictions a considerable amount of data has been accumulated in the 'measurements' performed on the ensembles of identically prepared physical systems or in the repeated 'measurements' on some trapped 'individual physical systems'. The outcomes of these measurements are, in general, some numerical time series registered by some macroscopic instruments. The various empirical probability distributions extracted from these time series were shown to be consistent with the probabilistic predictions of QT. More than 70 years ago the claim was made that QT provided the most complete description of 'individual' physical systems and outcomes of the measurements performed on 'individual' physical systems were obtained in an intrinsically random way. Spin polarization correlation experiments (SPCEs), performed to test the validity of Bell inequalities, clearly demonstrated the existence of strong long-range correlations and confirmed that the beams hitting far away detectors somehow preserve the memory of their common source which would be destroyed if the individual counts of far away detectors were purely random. Since the probabilities describe the random experiments and are not the attributes of the 'individual' physical systems, the claim that QT provides a complete description of 'individual' physical systems seems not only unjustified but also misleading and counter productive. In this paper, we point out that we even do not know whether QT is predictably complete because it has not been tested carefully enough. Namely, it was not proven that the time series of existing experimental data did not contain some stochastic fine structures that could have been averaged out by describing them in terms of the empirical probability distributions. In this paper, we advocate various statistical tests that could be used to search for such fine structures in the data and to answer the title question of this paper. In our opinion a proper understanding of the statistical character of QT and of its limitations is crucial in the domains such as quantum optics and quantum information.

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In a series of papers Colbeck and Renner (2011, 2015a,b) claim to have shown that the quantum state provides a complete description for the prediction of future measurement outcomes. In this paper I argue that thus far no solid satisfactory proof has been presented to support this claim. Building on the earlier work of Landsman (2015) and Leegwater (2016) I highlight the implicit use of an assumption concerning the way unitary evolution is to be represented in any possible completion of quantum mechanics. I show that the assumption is quite crucial to the proof of the claim and argue that it is unwarranted. I further discuss a possible validation for a restricted version of this assumption that is based on considerations of measurement processes. I argue that also this restricted version is unsatisfactory.
In a series of papers Colbeck and Renner (2011, 2015a, 2015b) claim to have shown that the quantum state provides a complete description for the prediction of future measurement outcomes. In this paper I argue that thus far no solid satisfactory proof has been presented to support this claim. Building on the earlier work of Leifer (2014), Landsman (2015) and Leegwater (2016), I present and prove two results that only partially support this claim. I then discuss the arguments by Colbeck, Renner and Leegwater concerning how these results are to generalize to the full claim. This argument turns out to hinge on the implicit use of an assumption concerning the way unitary evolution is to be represented in any possible completion of quantum mechanics. I argue that this assumption is unsatisfactory and that possible attempts to validate it based on measurement theory also do not succeed.
In a series of papers Colbeck and Renner claim to have shown that the quantum state provides a complete description for the prediction of future measurement outcomes. In this paper I argue that thus far no solid satisfactory proof has been presented to support this claim. Building on the earlier work of Leifer, Landsman and Leegwater, I present and prove two results that only partially support this claim. I then discuss the arguments by Colbeck, Renner and Leegwater concerning how these results are to generalize to the full claim. This argument turns out to hinge on the implicit use of an assumption concerning the way unitary evolution is to be represented in any possible completion of quantum mechanics. I argue that this assumption is unsatisfactory and that possible attempts to validate it based on measurement theory also do not succeed.
In this thesis a recent claim by Roger Colbeck and Renato Renner is reviewed [18]. This claim concerns the completeness of quantum mechanics. Colbeck and Renner state that no theory can be constructed that is compatible with quantum mechanics, and gives more detailed predictions through supplemental variables added to the description of the state of a system. Their only assumption is that measurement settings can be choosen freely. We will argue that their result is for the most part valid and interesting, but the freedom assumption is not so innocent as it is presented, and parts of the result need to be clarified. Therefore, we will present a reformulation of the result. In our formulation the result is a no-go theorem for hidden variable theories satisfying Parameter Independence. This is a generalization of Bell’s theorem. Performed in the period 01-11-2012 14-08-2012
It was argued [1] that there can be no extension of quantum mechanics with improved predictive power on a measurement freely chosen, independently of any event that is not in its future light cone. The assumption of measurement choice was criticized [2] to be too strong to be physically necessary and extensions of quantum mechanics were shown [3] to be possible under a more relaxed measurement assumption. Here I point out an error in the criticism and observe that the actual mistake of the no-go theorem lies in an unwarranted assumption implicitly made in the proof of [1]. Hence, quantum mechanics is guaranteed to have the maximal predictive power only in situations of complete certainty and complete uncertainty about measurement outcomes. I then show that the measurement assumption can be further relaxed without affecting the conclusion on the predictive power of quantum mechanics versus alternative theories. I further study the optimal predicative improvement over quantum mechanics of local spin measurements on a pair of entangled qubits by any alternative theory and conjecture a strict upper bound.
In three papers, Colbeck and Renner [Nat. Commun. 2, 411 (2011); Phys. Rev. Lett. 108, 150402 (2012); e-print arXiv:1208.4123] argued that “no alternative theory compatible with quantum theory and satisfying the freedom of choice assumption can give improved predictions.” We give a more precise version of the formulation and proof of this remarkable claim. Our proof broadly follows theirs, which relies on physically well motivated axioms, but to fill in some crucial details, certain technical assumptions have had to be added, whose physical status seems somewhat obscure.
Quantum theory (QT) provides statistical predictions for various physical phenomena. To verify these predictions a considerable amount of data has been accumulated in the 'measurements' performed on the ensembles of identically prepared physical systems or in the repeated 'measurements' on some trapped 'individual physical systems'. The outcomes of these measurements are, in general, some numerical time series registered by some macroscopic instruments. The various empirical probability distributions extracted from these time series were shown to be consistent with the probabilistic predictions of QT. More than 70 years ago the claim was made that QT provided the most complete description of 'individual' physical systems and outcomes of the measurements performed on 'individual' physical systems were obtained in an intrinsically random way. Spin polarization correlation experiments (SPCEs), performed to test the validity of Bell inequalities, clearly demonstrated the existence of strong long-range correlations and confirmed that the beams hitting far away detectors somehow preserve the memory of their common source which would be destroyed if the individual counts of far away detectors were purely random. Since the probabilities describe the random experiments and are not the attributes of the 'individual' physical systems, the claim that QT provides a complete description of 'individual' physical systems seems not only unjustified but also misleading and counter productive. In this paper, we point out that we even do not know whether QT is predictably complete because it has not been tested carefully enough. Namely, it was not proven that the time series of existing experimental data did not contain some stochastic fine structures that could have been averaged out by describing them in terms of the empirical probability distributions. In this paper, we advocate various statistical tests that could be used to search for such fine structures in the data and to answer the title question of this paper. In our opinion a proper understanding of the statistical character of QT and of its limitations is crucial in the domains such as quantum optics and quantum information.