Macroscopicity of quantum mechanical superposition tests via hypothesis falsification

Abstract

We establish an objective scheme to determine the macroscopicity of quantum superposition tests with mechanical degrees of freedom. It is based on the Bayesian hypothesis falsification of a class of macrorealist modifications of quantum theory, such as the model of Continuous Spontaneous Localization. The measure uses the raw data gathered in an experiment, taking into account all measurement uncertainties, and can be used to directly assess any conceivable quantum mechanical test. We determine the resulting macroscopicity for three recent tests of quantum physics: doublewell interference of Bose-Einstein condensates, Leggett-Garg tests with atomic random walks, and entanglement generation and read-out of nanomechanical oscillators.

Quantum-classical hypothesis tests in macroscopic matter-wave interferometry

“…An experiment can thus be deemed as more macroscopic than another one if its measurement record falsifies a greater set of parameters. This empirical notion of macroscopicity can be cast into a quantitative measure using the concept of Bayesian hypothesis testing [22]. To this end, we consider the odds ratio…”

“…Here, H τ * e states that macrorealism holds and a MMM affects (1) with a time parameter τ e τ * e (at fixed σ ); H τ * e assumes weaker modifications, τ e > τ * e , or possibly none at all (τ e → ∞). With help of Bayes' theorem, we can express the odds ratio in terms of the likelihoods P(d|τ e , σ, I ) that the MMM model (1) predicts the observed data d based on the given parameters [22]…”

“…The prior (6) is the most objective choice when it comes to comparing different experiments. Jeffreys' prior is invariant under reparametrizations and results in the largest information gain between prior and posterior on average, as measured by the Kullback-Leibler divergence (or relative entropy) [17,27,28]. As a conservative criterion for rejecting the MMM hypothesis H τ * e , we require the odds ratio (5) to fall below 1:19.…”

“…They are then gauged with functionals such as the quantum Fisher information [16], that return large values for states intuitively deemed macroscopic. This approach may lead to inconclusive or unintuitive results [17]. As an alternative, one may adhere to an empirical notion of macroscopicity [18], which is based on how much the observation of quantum behavior constrains the hypothesis that quantum mechanics ceases to be valid on the macroscale.…”

“…In practice however, there are always unaccounted sources of noise and decoherence in the experiment so that both the quantum and the classical model are incomplete, and the measurement data will likely fit neither. One can alleviate this problem by instead considering a continuous hypothesis test against a set of minimal macrorealist modifications (MMM) of quantum mechanics [22]. These models augment the Schrödinger equation by a parametrized stochastic process that destroys superpositions above a certain size, time, and mass threshold, while preserving them on the microscopic scale and fulfilling minimal consistency requirements [23].…”