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$\psi$-epistemic models are exponentially bad at explaining the distinguishability of quantum states
M. S. Leifer
TL;DR
The paper asks whether overlap between ontic-state distributions can quantitatively explain quantum-state indistinguishability. Using state constructions based on Hadamard states, orthogonality graphs, and the Frankl–Rödl theorem, it proves that the overlap-to-indistinguishability ratio is exponentially small in dimensions divisible by 4, making the ψ-epistemic explanation increasingly implausible.
Problem
ψ-epistemic models require overlapping probability measures to account for the indistinguishability of nonorthogonal quantum states, but the overlap must be comparable to quantum indistinguishability for that explanation to work.
Source: p. 1Method
The paper constructs finite state sets and orthogonality graphs using Hadamard states, then applies the Frankl–Rödl theorem to bound their independence numbers.
Source: pp. 3–4Results
For dimensions d divisible by 4, the ratio of classical to quantum overlaps is bounded by 2de^(-cd), with c positive, yielding exponential improvement over the previous 4/(d −1) scaling.
Source: pp. 1, 4Takeaways & Limitations
The portion of quantum indistinguishability explained by probability-measure overlap decreases rapidly as Hilbert-space dimension increases.
Source: pp. 2, 4The stated exponential bound applies to Hilbert-space dimensions divisible by 4.
Source: p. 4
Abstract
from arXiv · showhide
The status of the quantum state is perhaps the most controversial issue in the foundations of quantum theory. Is it an epistemic state (state of knowledge) or an ontic state (state of reality)? In realist models of quantum theory, the epistemic view asserts that nonorthogonal quantum states correspond to overlapping probability measures over the true ontic states. This naturally accounts for a large number of otherwise puzzling quantum phenomena. For example, the indistinguishability of nonorthogonal states is explained by the fact that the ontic state sometimes lies in the overlap region, in which case there is nothing in reality that could distinguish the two states. For this to work, the amount of overlap of the probability measures should be comparable to the indistinguishability of the quantum states. In this letter, I exhibit a family of states for which the ratio of these two quantities must be $\leq 2de^{-cd}$ in Hilbert spaces of dimension $d$ that are divisible by $4$. This implies that, for large Hilbert space dimension, the epistemic explanation of indistinguishability becomes implausible at an exponential rate as the Hilbert space dimension increases.