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arXiv:2405.17733

Contradictions or Curiosities? On Kent's Critique of the Masanes--Galley--Müller Derivation of the Quantum Measurement Postulates

Blake C. Stacey

Published May 28, 2024• 0 citations
arXivHTMLPDF

Abstract

Adrian Kent has recently criticized Masanes, Galley and Müller's work on postulates for quantum mechanics. MGM claim to find two contradictions in Kent's criticism. I argue that neither is a true contradiction unless some other premise is added.
DOI
10.48550/arxiv.2405.17733

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Quantum physics has a measurement problem, and it’s not the one the field usually means: it’s that physics has built a precise theory of measurement outcomes without a theory of what measurement structurally is. The Galley-Kent exchange makes this gap unusually visible. Galley, Masanes, and Müller argued that quantum measurement postulates follow necessarily from structural postulates plus the possibility of state estimation. Kent refuted this, constructing alternative measurement postulates consistent with those assumptions. We argue that both positions operate one level above the question that would resolve their debate: what is the minimum structure that makes any measurement coherent in the first place? We show that this floor — two faces, one non-directed comparator, and the invariant sameness that makes their difference determinate — forces complex Hilbert space as the unique minimal field consistent with its constraints, and forces the Born rule as the unique probability assignment that closes both faces simultaneously. Galley et al. are correct that measurement postulates are necessary, but the proof does not live inside quantum mechanics. Kent's alternatives are formally consistent but do not instantiate coherent measurement: they either abandon ratio structure or treat measurement as directed rather than comparative. The proof of necessity lives below quantum mechanics, not inside it.
Adrian Kent has recently presented a critique \cite{Kent} of our paper \cite{MGM} in which he claims to refute our main result: the measurement postulates of quantum mechanics can be derived from the rest of postulates, once we assume that the set of mixed states of a finite-dimensional Hilbert space is finite-dimensional. To construct his argument, Kent considers theories resulting from supplementing quantum mechanics with hypothetical ``post-quantum'' measurement devices. We prove that each of these theories contains pure states (i.e. states of maximal knowledge) which are not rays of the Hilbert space, in contradiction with the ``pure state postulate'' of quantum mechanics. We also prove that these alternatives violate the finite-dimensionality of mixed states. Each of these two facts separately invalidates the refutation. In this note we also clarify the assumptions used in \cite{MGM} and discuss the notions of pure state, physical system, and the sensitivity of the structure of the state space under modifications of the measurements or the dynamics.
Adrian Kent has recently presented a critique [arXiv:2307.06191] of our paper [Nat. Comms. 10, 1361 (2019)] in which he claims to refute our main result: the measurement postulates of quantum mechanics can be derived from the rest of postulates, once we assume that the set of mixed states of a finite-dimensional Hilbert space is finite-dimensional. To construct his argument, Kent considers theories resulting from supplementing quantum mechanics with hypothetical "post-quantum" measurement devices. We prove that each of these theories contains pure states (i.e. states of maximal knowledge) which are not rays of the Hilbert space, in contradiction with the "pure state postulate" of quantum mechanics. We also prove that these alternatives violate the finite-dimensionality of mixed states. Each of these two facts separately invalidates the refutation. In this note we also clarify the assumptions used in [Nat. Comms. 10, 1361 (2019)] and discuss the notions of pure state, physical system, and the sensitivity of the structure of the state space under modifications of the measurements or the dynamics.
Masanes, Galley and Müller \cite{masanes2019measurement} argue that the measurement postulates of non-relativistic quantum mechanics follow from the structural postulates together with an assumption they call the "possibility of state estimation". Their argument also relies on what they term a "theory-independent characterization of measurements for single and multipartite systems". We refute their conclusion, giving explicit examples of non-quantum measurement and state update rules that satisfy all their assumptions. We also show that their "possibility of state estimation" assumption is neither necessary nor sufficient to ensure a sensible notion of state estimation within a theory whose states are described by the quantum formalism. We further show their purportedly "theory-independent" characterization assumes several properties of quantum measurements that exclude plausible alternative types of measurement. We illustrate all these points with specific alternative measurement postulates and post-measurement state update rules. We conclude that, contrary to some folklore, quantum mechanics is by no means an island in theory-space. It can consistently be extended by rules for obtaining information about quantum states other than via POVMs. Whether such rules are realised in nature, for example in linking quantum theory and gravity, is an empirical question that cannot be resolved by theoretical analysis alone.