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Physics Letters A

Distance between quantum states, statistical inference and the projection postulate

Dennis Dieks, P.J. Veltkamp

Published Aug 1, 1983• 13 citations
Publisher
DOI
10.1016/0375-9601(83)90092-0

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This chapter is a step towards understanding why quantum nonlocalityis a misleading concept. Metaphorically speaking, quantum nonlocality isJanus faced. One face is an apparent nonlocality of the state update basedon the Luders projection postulate. It can be referred as intrinsic quantumnonlocality. And the other face is subquantumnonlocality: by introducing a special model with hidden variables onederives the Bell inequality and claims that its violation implies the existenceof mysterious instantaneous influences between distant physical systems(Bell nonlocality). According to the Luders projection postulate, aquantum measurement performed on one of the two distant entangled physicalsystems, say on S1, modifies instantaneously the state of S2. Therefore, ifthe quantum state is considered to be an attribute of the individual physicalsystem (Copenhagen interpretation) and if one assumes thatexperimental outcomes are produced in a random way one arrives at the contradiction. It is a primary source of speculation about aspooky action at the distance. But Einstein had already pointed out that the quantum paradoxes disappear, ifone adopts the statistical interpretation.

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Albert and Loewer’s Mentaculus is an ambitious unification project that aims to account for the reversibility problem, the thermodynamic arrow of time, the epistemic arrow, the influence and causal/counterfactual arrows, the applicability of statistical reasoning to macroscopic regularities, and the compatibility of objective probability with determinism. This companion note evaluates each of these explananda through the apparatus of Quantum-Geometry Dynamics (QGD) and the Minimally Physically Derivable Theories (MPDT) programme. The central result is that QGD dissolves the reversibility problem at its root. Because QGD’s fundamental dynamics is categorically asymmetric rather than time-symmetric, the inference that the Past Hypothesis and Statistical Postulate are jointly designed to block never arises. The thermodynamic arrow at cosmological scale follows as a direct theorem of preonic binding; the Statistical Postulate is shown to be a modelling residual rather than a further fundamental law. QGD’s own isotropic initial state S0 S_0 S0 remains necessary but is immune to the continuum fine-tuning charge that the Mentaculus literature itself acknowledges for the Past Hypothesis. The epistemic arrow receives an independent non-statistical account of records, while a practical residual for finite observers is openly acknowledged. The causal, counterfactual and influence arrows are treated only as a sketch; a developed counterfactual semantics remains an open item. Objective probability under determinism is sidestepped rather than solved, because QGD treats probability as strictly epistemic. This is an ungrounded companion note to the QGD / MPDT Problem Register and is not assigned a portfolio (P-series) number.

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The ability of modern quantum simulators--both digital and analogue--to generate large ensembles of single-shot projective "snapshots" has opened a data-rich avenue for the study of quantum many-body systems. Unsupervised machine learning analysis of such snapshots has gained traction, with numerous works reconstructing phase diagrams by learning and clustering low-dimensional representations of quantum states. Here, we forgo such representation learning in favour of distance learning: we infer the pairwise distances between quantum states--already sufficient for clustering--directly from snapshots. Specifically, we use a single neural discriminator to estimate Csiszar f-divergences--statistical distances between distributions--in an unsupervised manner. The resulting clusters reveal regimes with different dominant correlations, often coinciding with, but not limited to, conventionally defined phases of matter. Beyond phase-diagram exploration, we connect the infinitesimal limit of the inferred divergences to the Fisher information metric and analyse its finite-size scaling. This yields critical exponents of the discovered transitions and enables snapshot-based analysis of universality classes. We apply distance learning to a diverse set of systems characterised by conventional local order parameters (1D transverse-field and 2D classical Ising models), non-local topological order (extended toric code), and higher-order correlations (fermionic t-J model on a triangular lattice). In all cases, we correctly recover boundaries between distinct correlation regimes and, where applicable, quantitatively match established critical behaviour. Finally, we show that distances to suitably chosen reference snapshot distributions help identify the dominant correlations within the discovered clusters, positioning distance learning as a versatile information-geometric probe of quantum many-body physics.

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Statistical Inference in Quantum Mechanics

Jean-Paul Marchand

Springer eBooks•Jan 1, 1981•6 citations
This chapter is a step towards understanding why quantum nonlocalityis a misleading concept. Metaphorically speaking, quantum nonlocality isJanus faced. One face is an apparent nonlocality of the state update basedon the Luders projection postulate. It can be referred as intrinsic quantumnonlocality. And the other face is subquantumnonlocality: by introducing a special model with hidden variables onederives the Bell inequality and claims that its violation implies the existenceof mysterious instantaneous influences between distant physical systems(Bell nonlocality). According to the Luders projection postulate, aquantum measurement performed on one of the two distant entangled physicalsystems, say on S1, modifies instantaneously the state of S2. Therefore, ifthe quantum state is considered to be an attribute of the individual physicalsystem (Copenhagen interpretation) and if one assumes thatexperimental outcomes are produced in a random way one arrives at the contradiction. It is a primary source of speculation about aspooky action at the distance. But Einstein had already pointed out that the quantum paradoxes disappear, ifone adopts the statistical interpretation.
Albert and Loewer’s Mentaculus is an ambitious unification project that aims to account for the reversibility problem, the thermodynamic arrow of time, the epistemic arrow, the influence and causal/counterfactual arrows, the applicability of statistical reasoning to macroscopic regularities, and the compatibility of objective probability with determinism. This companion note evaluates each of these explananda through the apparatus of Quantum-Geometry Dynamics (QGD) and the Minimally Physically Derivable Theories (MPDT) programme. The central result is that QGD dissolves the reversibility problem at its root. Because QGD’s fundamental dynamics is categorically asymmetric rather than time-symmetric, the inference that the Past Hypothesis and Statistical Postulate are jointly designed to block never arises. The thermodynamic arrow at cosmological scale follows as a direct theorem of preonic binding; the Statistical Postulate is shown to be a modelling residual rather than a further fundamental law. QGD’s own isotropic initial state S0 S_0 S0 remains necessary but is immune to the continuum fine-tuning charge that the Mentaculus literature itself acknowledges for the Past Hypothesis. The epistemic arrow receives an independent non-statistical account of records, while a practical residual for finite observers is openly acknowledged. The causal, counterfactual and influence arrows are treated only as a sketch; a developed counterfactual semantics remains an open item. Objective probability under determinism is sidestepped rather than solved, because QGD treats probability as strictly epistemic. This is an ungrounded companion note to the QGD / MPDT Problem Register and is not assigned a portfolio (P-series) number.
The ability of modern quantum simulators--both digital and analogue--to generate large ensembles of single-shot projective "snapshots" has opened a data-rich avenue for the study of quantum many-body systems. Unsupervised machine learning analysis of such snapshots has gained traction, with numerous works reconstructing phase diagrams by learning and clustering low-dimensional representations of quantum states. Here, we forgo such representation learning in favour of distance learning: we infer the pairwise distances between quantum states--already sufficient for clustering--directly from snapshots. Specifically, we use a single neural discriminator to estimate Csiszar f-divergences--statistical distances between distributions--in an unsupervised manner. The resulting clusters reveal regimes with different dominant correlations, often coinciding with, but not limited to, conventionally defined phases of matter. Beyond phase-diagram exploration, we connect the infinitesimal limit of the inferred divergences to the Fisher information metric and analyse its finite-size scaling. This yields critical exponents of the discovered transitions and enables snapshot-based analysis of universality classes. We apply distance learning to a diverse set of systems characterised by conventional local order parameters (1D transverse-field and 2D classical Ising models), non-local topological order (extended toric code), and higher-order correlations (fermionic t-J model on a triangular lattice). In all cases, we correctly recover boundaries between distinct correlation regimes and, where applicable, quantitatively match established critical behaviour. Finally, we show that distances to suitably chosen reference snapshot distributions help identify the dominant correlations within the discovered clusters, positioning distance learning as a versatile information-geometric probe of quantum many-body physics.