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arXiv:2602.19993

GAP Measures and Wave Function Collapse

Roderich Tumulka

quant-phmath-phSubmitted Feb 23, 2026• Announced Feb 24, 2026• 0 citations
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Abstract

GAP measures (also known as Scrooge measures) are a natural class of probability distributions on the unit sphere of a Hilbert space that come up in quantum statistical mechanics; for each density matrix $ρ$ there is a unique measure GAP$_ρ$. We describe and prove a property of these measures that was not recognized so far: If a wave function $Ψ$ is GAP$_ρ$ distributed and a collapse occurs, then the collapsed wave function $Ψ'$ is again GAP distributed (relative to the appropriate $ρ'$). This fact applies to collapses due to a quantum measurement carried out by an observer, as well as to spontaneous collapse theories such as CSL or GRW. More precisely, it is the conditional distribution of $Ψ'$, given the measurement outcome (respectively, the noise in CSL or the collapse history in GRW), that is GAP$_{ρ'}$.

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Consider a quantum system prepared in state ψ, a unit vector in a d -dimensional Hilbert space. Let b 1 , ..., b d be an orthonormal basis and suppose that, with some probability 0 < p < 1, ψ 'collapses', i.e., gets replaced by b k (possibly times a phase factor) with Born's probability |〈 b k |ψ〉| 2 . The question we investigate is: how well can any quantum experiment on the system determine afterwards whether a collapse has occurred? The answer depends on how much is known about the initial vector ψ. We provide a number of different results addressing several variants of the question. In each case, no experiment can provide more than rather limited probabilistic information. In case ψ is drawn randomly with uniform distribution over the unit sphere in Hilbert space, no experiment performs better than a blind guess without measurement; that is, no experiment provides any useful information.
Whether wave function collapses or not is a major remaining question in the theory of quantum measurement. This difficulty stems from following two facts. First, it has not been recognized that single-particle quantum mechanics and many-particle quantum mechanics must be treated separately. Second, quantum jump (QJ) and wave function collapse (WFC) need clearer definitions. We define a QJ as a process of selecting a set of system eigenvalues (SEVs) of an observable and a WFC as a process of determining the probability distribution (PD) of SEVs, both from a single measurement. The goal of quantum observation is to obtain the PD, which is determined from an ensemble of SEVs. The wave function becomes an observable when the PD is determined. In single-particle quantum mechanics, a single measurement results in only one set of SEVs and the PD is not observable. Therefore the WFC does not happen. In many-particle quantum mechanics, we focus on the occupation number of a singe quantum state. The wave function does not collapse in general, but there are exceptions. The occupation number can be huge and macroscopic for photons or for Bose-Einstein condensates. In such a case, the PD is determined from a single measurement of a real ensemble and the WFC occurs. We call it a macroscopic quantum jump, which effectively is a measurement of a classical observable.
We are concerned with the problem of detecting with high probability whether a wave function has collapsed or not, in the following framework: A quantum system with a d-dimensional Hilbert space is initially in state ψ; with probability 0 < p < 1, the state collapses relative to the orthonormal basis b1, …, bd. That is, the final state ψ′ is random, it is ψ with probability 1 − p and bk (up to a phase) with p times Born’s probability bk|ψ2. Now an experiment on the system in state ψ′ is desired that provides information about whether or not a collapse has occurred. Elsewhere [C. W. Cowan and R. Tumulka, J. Phys. A: Math. Theor. 47, 195303 (2014)], we identify and discuss the optimal experiment in case that ψ is either known or random with a known probability distribution. Here, we present results about the case that no a priori information about ψ is available, while we regard p and b1, …, bd as known. For certain values of p, we show that the set of ψs for which any experiment E is more reliable than blind guessing is at most half the unit sphere; thus, in this regime, any experiment is of questionable use, if any at all. Remarkably, however, there are other values of p and experiments E such that the set of ψs for which E is more reliable than blind guessing has measure greater than half the sphere, though with a conjectured maximum of 64% of the sphere.
Whether the wave function collapses or not is a major remaining question in the theory of quantum measurement. This difficulty stems from the following two facts. First, it has not been recognized that single-particle quantum mechanics and many-particle quantum mechanics must be treated separately. Second, quantum jump (QJ) and wave function collapse (WFC) need clearer definitions. We define a QJ as a process of selecting a set of system eigenvalues (SEVs) of an observable and a WFC as a process of determining the probability distribution (PD) of SEVs, both from a single measurement. The goal of quantum observation is to obtain the PD, which is determined from an ensemble of SEVs. The wave function becomes an observable when the PD is determined. In single-particle quantum mechanics, a single measurement results in only one set of SEVs, and the PD is not observable. Therefore, the WFC does not happen. In many-particle quantum mechanics, we focus on the occupation number of a single quantum state. The wave function does not collapse in general, but there are exceptions. The occupation number can be huge and macroscopic for photons or for Bose-Einstein condensates. In such a case, the PD is determined from a single measurement of a real ensemble, and the WFC occurs. We call it a macroscopic quantum jump, which effectively is a measurement of a classical observable.
Whether the wave function collapses or not is a major remaining question in the theory of quantum measurement. This difficulty stems from the following two facts. First, it has not been recognized that single-particle quantum mechanics and many-particle quantum mechanics must be treated separately. Second, quantum jump (QJ) and wave function collapse (WFC) need clearer definitions. We define a QJ as a process of selecting a set of system eigenvalues (SEVs) of an observable and a WFC as a process of determining the probability distribution (PD) of SEVs, both from a single measurement. The goal of quantum observation is to obtain the PD, which is determined from an ensemble of SEVs. The wave function becomes an observable when the PD is determined. In single-particle quantum mechanics, a single measurement results in only one set of SEVs, and the PD is not observable. Therefore, the WFC does not happen. In many-particle quantum mechanics, we focus on the occupation number of a single quantum state. The wave function does not collapse in general, but there are exceptions. The occupation number can be huge and macroscopic for photons or for Bose-Einstein condensates. In such a case, the PD is determined from a single measurement of a real ensemble, and the WFC occurs. We call it a macroscopic quantum jump, which effectively is a measurement of a classical observable.
An open quantum system (i.e., one that interacts with its environment) is almost always entangled with its environment; it is therefore usually not attributed a wave function but only a reduced density matrix ρ. Nevertheless, there is a precise way of attributing to it a wave function ψ1, called its conditional wave function, which is a random wave function of the system whose probability distribution µ1 depends on the entangled wave function ψ ∈ H1 ⊗ H2 in the Hilbert space of system and environment together. We prove several universality (or typicality) results about µ1; they show that if the environment is sufficiently large then µ1 does not depend much on the details of ψ and is approximately given by one of the so-called GAP measures. Specifically, for most entangled states ψ with given reduced density matrix ρ1, µ1 is close to GAP (ρ1). We also show that, if the coupling between the system and the environment is weak, then for most entangled states ψ from a microcanonical subspace corresponding to energies in a narrow interval [E, E + δE] (and most bases of H2), µ1 is close to GAP (ρβ) with ρβ the canonical density matrix on H1 at inverse temperature β = β(E). This provides the mathematical justification of the claim that GAP (ρβ) is the thermal equilibrium distribution of ψ1.