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No-go theorems for \psi-epistemic models based on a continuity assumption
M. K. Patra, S. Pironio, S. Massar
TL;DR
The paper investigates whether ψ-epistemic models can remain compatible with quantum predictions under continuity, separability, and ontic-indifference assumptions. It shows that shared ontic states propagate across the Hilbert space, yielding no-go results for such models.
Problem
The paper addresses whether ψ-epistemic models can reproduce quantum predictions when distinct quantum states share underlying ontic states.
Source: Appendix B · pp. 5–6Method
The paper applies ontic-indifference arguments and separability to propagate a shared ontic state across transformed and increasingly broad sets of quantum states.
Source: Appendix B · p. 6Results
If distinct states share an ontic state, ontic indifference implies that all Hilbert-space states share it, contradicting quantum predictions; restricted ontic indifference also yields a no-go theorem with separability.
Source: Appendix B · p. 6Takeaways & Limitations
Ontic indifference is incompatible with ψ-epistemic models reproducing quantum theory, while restricted ontic indifference remains subject to no-go results under additional assumptions.
Source: Appendix B · p. 6
Abstract
from arXiv · showhide
The quantum state \psi is a mathematical object used to determine the probabilities of different outcomes when measuring a physical system. Its fundamental nature has been the subject of discussions since the inception of quantum theory: is it ontic, that is, does it correspond to a real property of the physical system? Or is it epistemic, that is, does it merely represent our knowledge about the system? Assuming a natural continuity assumption and a weak separability assumption, we show here that epistemic interpretations of the quantum state are in contradiction with quantum theory. Our argument is different from the recent proof of Pusey, Barrett, and Rudolph and it already yields a non-trivial constraint on \psi-epistemic models using a single copy of the system in question.
Appendix A: General formulation of the no-go theorems in the context of measurable spaces
The appendix extends the no-go theorems to measurable ontic-state spaces, formalizing continuity and separability for probability measures and treating absolutely continuous cases. Under δ-continuity, the proof constructs a shared ontic region that conflicts with quantum predictions.
Measure-theoretic framework
- The measurable-space formulation decomposes preparation measures into singular and absolutely continuous parts, while the main results are proved for absolutely continuous measures and can extend to mixed discrete-singular cases.The results do not apply when the preparation measures contain only a singular continuous part.Source: Appendix A · p. 4
Definitions and assumptions
- A ψ-epistemic model is defined by overlapping ontic supports for distinct quantum states, while δ-continuity requires a positive-measure set shared across all sufficiently nearby preparations.The continuity condition is motivated by uniform continuity of preparation distributions under compactness and continuous dependence on preparation parameters.Source: Appendix A · p. 5
Theorem 1
- For δ ≥ 1 − sqrt((d −1)/d), the proof identifies a positive-measure ontic subset common to d non-orthogonal states and derives a contradiction with quantum theory.The measure-theoretic argument uses threshold sets to show that the common region has strictly positive measure, contradicting the quantum prediction that its probability must vanish.Source: Appendix A · p. 5
Separability
- Separability requires that any nonzero-probability ontic subset for one preparation yields a nonzero-probability product subset for repeated identical preparations.The ontic state space for multiple preparations may be larger than a Cartesian product, without affecting this definition.Source: Appendix A · p. 5
Appendix B: Relation between our continuity assumption and ontic indifference
The appendix relates continuity-based no-go results to ontic indifference. It shows that unrestricted ontic indifference propagates a shared ontic state to every quantum state, while restricted ontic indifference plus separability is also incompatible with quantum predictions.
Appendix B: Relation between our continuity assumption and ontic indifference
- Restricted ontic indifference avoids the immediate all-state overlap problem of unrestricted ontic indifference but breaks unitary invariance by privileging one state.The appendix therefore supplies a separate no-go theorem rather than treating restricted ontic indifference as equivalent to the unrestricted condition.Source: Appendix B · p. 6
- Ontic indifference implies that if two distinct states share an ontic state, unitary transformations preserving one state propagate that ontic state to transformed versions of the other.Iterating this construction produces sets of states whose size grows exponentially, eventually covering the Bloch sphere; the argument extends beyond two dimensions because the relevant span is the whole Hilbert space.Source: Appendix B · p. 6
- Under ontic indifference, any shared ontic state of two distinct quantum states becomes shared by all states in the Hilbert space.This would make even orthogonal states share a real state, directly conflicting with quantum predictions.Source: Appendix B · p. 6
- The paper applies its proof strategy to restricted ontic indifference, showing that this assumption plus separability is incompatible with quantum predictions when a specified state shares an ontic state with another state.The construction adds an ancilla and generates states sharing the same ontic state before reproducing the contradiction from Theorem 1.Source: Appendix B · p. 6