Extending the falsification-based macroscopicity measure of quantum superposition tests: classicalizing modifications of QED for photonic experiments, general likelihood reduction for many-body systems, and separation of heating-only from coherence evidence

Schrinski, Nimmrichter, Stickler and Hornberger (Phys. Rev. A 100, 032111 (2019)) introduced an operational macroscopicity measure that ranks quantum superposition experiments by how strongly their raw data rule out a minimal, Galilean-invariant class of classicalizing modifications of quantum mechanics, using Bayesian hypothesis testing with Jeffreys' prior. The authors themselves state three limitations of the framework, which together constitute one open problem: the measure requires a mechanical (massive) degree of freedom, so purely photonic superposition experiments cannot be ranked until a minimal, physically consistent class of classicalizing modifications of quantum electrodynamics (QED) is constructed; the likelihoods for complex many-body or solid-state systems demand system-by-system reductions of the many-particle modification generator to the few relevant modes, with ad hoc treatments of modification-induced particle loss and binding-energy thresholds; and because classicalizing modifications necessarily heat, a finite macroscopicity can arise from heating-only constraints unless the likelihood is conditioned on the heating-only observables, a step whose prescription must currently be identified experiment by experiment. Later work applies the measure only case by case — most prominently to high-overtone bulk-acoustic-wave resonators (collective vibrational modes of microgram-scale solid crystals) and to nanoparticle near-field (Talbot) interferometry — and none of the three limitations is resolved: no classicalizing modification of QED has been constructed, no generalizable likelihood-reduction technique exists, and the heating-versus-coherence separation remains a per-experiment prescription. The problem is open.

Background

A natural way to quantify how 'macroscopic' a quantum superposition experiment is, is not to count particles or mass, but to ask how strongly the experiment rules out hypothetical classicalizing modifications of quantum mechanics: minimal, maximally benign extensions that restore macrorealism (the view that macroscopic quantities possess definite values at all times) by suppressing large superpositions while reproducing standard quantum mechanics microscopically. The best-known example is continuous spontaneous localization (CSL), a stochastic collapse model in which the wave function is weakly and randomly localized in space; such models are conventionally written as Lindblad master equations, i.e. Markovian dissipative extensions of the Schrödinger evolution of the density operator $\rho_t$ . Nimmrichter and Hornberger (Phys. Rev. Lett. 110, 160403 (2013)) turned this idea into a quantitative macroscopicity measure for mechanical superpositions. They parametrized a minimal class of classicalizing modifications by requiring Galilean invariance, particle-exchange symmetry, and related consistency principles, obtaining a dissipator $\mathcal{M}_{\boldsymbol\sigma}$ built from Gaussian distributions of phase-space kicks of widths $\boldsymbol\sigma=(\sigma_q,\sigma_s)$ acting on all $N$ constituent particles with mass-weighted amplitudes $m_\alpha/m_e$ (with $m_e$ the electron mass), so that the modified evolution reads $\partial_t\rho_t=\mathcal{L}\rho_t+\mathcal{M}_{\boldsymbol\sigma}\rho_t/\tau_e$ . Here $\mathcal{L}$ is the ordinary quantum dynamics including environmental decoherence, and the coherence time $\tau_e$ sets the modification strength: larger $\tau_e$ means a weaker modification. Crucially, the mass weighting amplifies the modification for heavy composites while making it negligible for microscopic systems, which is what allows macrorealism to emerge on macroscopic scales without disturbing verified quantum physics. The macroscopicity of an experiment is the base-10 logarithm of the largest classicalization time it excludes. Schrinski, Nimmrichter, Stickler and Hornberger (Phys. Rev. A 100, 032111 (2019)) made this measure fully empirical and experiment-agnostic: the raw measurement data $d$ of any superposition test determine a likelihood $P(d|\tau_e,\boldsymbol\sigma,I)$ (with $I$ the background information modeling the experiment, including Hamiltonian, environmental decoherence, and measurement protocol); Bayesian updating with Jeffreys' prior — the least-informative, reparametrization-invariant prior that keeps the rating unbiased — yields a posterior for $\tau_e$ whose lowest five-percent quantile $\tau_m(\boldsymbol\sigma)$ defines $\mu_m=\max_{\boldsymbol\sigma}\log_{10}(\tau_m(\boldsymbol\sigma)/1\,\mathrm{s})$ . The framework was demonstrated on three very different tests: double-well interference of Bose-Einstein condensates ( $\mu_m=8.5$ ), Leggett-Garg tests (temporal-correlation tests of macrorealism, the time analogue of Bell tests) with atomic random walks ( $\mu_m=7.1$ ), and entangled nanomechanical beams ( $\mu_m=7.8$ ). In its conclusion the paper states the limitations that define this problem. First, the measure 'can be used for any superposition test, provided a mechanical degree of freedom is involved... As such it does not apply to quantum tests involving only spins or photons. It seems natural to generalize the macroscopicity measure to pure photon experiments by drawing on a minimal class of classicalizing modifications of QED, but it is still an open problem how to get hold of the latter.' The obstacle is that the consistency principles behind the massive-particle class (Galilean invariance, mass-weighted amplification) have no direct counterpart for the quantized electromagnetic field, which is massless, gauge-invariant, and relativistic. Second, the required likelihoods 'are in general considerably more difficult to determine' than specific coherences of the state: one must capture how the few experimentally relevant degrees of freedom (collective modes) are affected by the many-particle modification generator, including modification-induced particle loss when the deposited energy exceeds binding thresholds (treated in the paper by splitting the kick distribution at a critical momentum transfer $q_c$ set by the binding energy) — a reduction carried out by hand, separately, for each of the three experiments. Third, because classicalizing modifications are necessarily accompanied by heating, 'a naive application of the macroscopicity measure may yield a finite value even for experiments demonstrating no quantum superposition'; to be safe, the likelihood must be conditioned on the heating-only observables, $P(D|\tau_e,\boldsymbol\sigma,I,d_{\mathrm{heat}})=P(D,d_{\mathrm{heat}}|\tau_e,\boldsymbol\sigma,I)/P(d_{\mathrm{heat}}|\tau_e,\boldsymbol\sigma,I)$ , which requires identifying, per experiment, which observables carry only energy-gain information (e.g. for the condensate tests, the sum of the well populations constrains heating-induced loss, while their difference carries the interference signal).

Problem Statement

The common objective is to make the operational, falsification-based macroscopicity measure of Schrinski, Nimmrichter, Stickler and Hornberger (2019) applicable, without methodological artifacts, to quantum superposition tests beyond the three case studies analyzed in that paper — in particular to purely photonic platforms, to complex many-body and solid-state systems, and to protocols in which the constraints on classicalizing modifications are dominated by heating rather than by observed coherence. The problem, as left open by the source paper, comprises three linked deliverables. (1) Construct and characterize a minimal, physically consistent class of classicalizing modifications of quantum electrodynamics — the analogue, for the quantized electromagnetic field interacting with charged matter, of the Galilean-invariant mass-weighted Lindblad class that underlies the massive-particle measure — including an explicit identification of the amplification mechanism that keeps the modification negligible in ordinary microscopic QED while restoring macrorealism for superpositions of macroscopically distinct field states; verify the required consistency properties (preservation of linearity and the semigroup structure of the evolution, gauge invariance, compatibility with established precision tests, correct recovery of standard QED in the microscopic limit, and the relationship to the massive-particle class where both apply); and demonstrate how the Bayesian likelihood and the resulting macroscopicity of at least one purely photonic superposition test (for example, a superposition of macroscopically distinct coherent states of the cavity radiation field — quasi-classical light states — or a micro-macro entanglement test, i.e. one entangling a microscopic with a macroscopic subsystem) are computed under this class, with measurement uncertainties propagated as in the source framework. (2) Develop practical, generalizable reduction techniques and efficient numerical schemes for deriving such likelihoods for complex many-body or solid-state superposition tests: mapping the full many-particle modification generator onto the few experimentally relevant degrees of freedom (modes), together with a systematic treatment of modification-induced particle loss and binding-energy thresholds that goes beyond the per-experiment constructions given in the source paper and its successors. (3) Formulate robust, generally applicable criteria or procedures for separating heating-only evidence from genuine coherence evidence in arbitrary experimental protocols — generalizing the source paper's per-experiment likelihood conditioning on the heating-only observables — so that a reported macroscopicity always reflects the falsification of classicalizing modifications by quantum coherence, and not merely the absence of modification-induced heating. These three deliverables are jointly required for a complete resolution of the single objective stated above: each is an independently assessable sub-problem that, on its own, resolves only the corresponding limitation of the source framework, so a solution of the objective must address all three; progress on each remains meaningful exactly to the extent that it extends or safeguards the measure's applicability.

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