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From Joint to Single-System Psi-Onticity Without Preparation Independence

Shan Gao

TL;DR

The paper addresses whether ψ-onticity for composite systems, proven by PBR under PIP, implies ψ-onticity for individual subsystems even when PIP is dropped. Using a representation theorem, it shows that joint ψ-onticity enforces a delta-function encoding of the ψ-label in the ontic state, and tensor-product structure then splits this label into subsystem components, yielding μ_{ψ1}(λ_{ψ1}) = δ(λ_{ψ1} - ψ1) and μ_{ψ2}(λ_{ψ2}) = δ(λ_{ψ2} - ψ2). Thus μ_{ψ1⊗ψ2}(λ) = δ(λ_ψ - ψ1⊗ψ2) ν_{ψ1⊗ψ2}(η) implies the subsystems carry uniquely defined ψ-labels, ensuring subsystem ψ-onticity independent of hidden-variable correlations. The result closes the common loophole that relaxing PIP could preserve ψ-epistemicity for subsystems, reinforcing ψ-ontology as a property of individual quantum systems and clarifying the role of PIP in foundational theorems.

Abstract

The Pusey-Barrett-Rudolph (PBR) theorem establishes $ψ$-onticity for individual quantum systems, but its standard formulation relies on the Preparation Independence Postulate (PIP). This has led to a prevalent view that rejecting PIP leaves open the possibility of $ψ$-epistemic models for individual systems. In this work, we show that this understanding is incomplete: once the PBR theorem establishes $ψ$-onticity for composite systems prepared in product states, the $ψ$-onticity of the individual subsystems follows directly from the tensor-product structure of quantum mechanics, without invoking PIP or any further auxiliary assumptions. This result removes a key auxiliary assumption from the PBR theorem, closes a persistent loophole for preserving $ψ$-epistemic models, and strengthens the conceptual foundations of $ψ$-ontology.

From Joint to Single-System Psi-Onticity Without Preparation Independence

TL;DR

The paper addresses whether ψ-onticity for composite systems, proven by PBR under PIP, implies ψ-onticity for individual subsystems even when PIP is dropped. Using a representation theorem, it shows that joint ψ-onticity enforces a delta-function encoding of the ψ-label in the ontic state, and tensor-product structure then splits this label into subsystem components, yielding μ_{ψ1}(λ_{ψ1}) = δ(λ_{ψ1} - ψ1) and μ_{ψ2}(λ_{ψ2}) = δ(λ_{ψ2} - ψ2). Thus μ_{ψ1⊗ψ2}(λ) = δ(λ_ψ - ψ1⊗ψ2) ν_{ψ1⊗ψ2}(η) implies the subsystems carry uniquely defined ψ-labels, ensuring subsystem ψ-onticity independent of hidden-variable correlations. The result closes the common loophole that relaxing PIP could preserve ψ-epistemicity for subsystems, reinforcing ψ-ontology as a property of individual quantum systems and clarifying the role of PIP in foundational theorems.

Abstract

The Pusey-Barrett-Rudolph (PBR) theorem establishes -onticity for individual quantum systems, but its standard formulation relies on the Preparation Independence Postulate (PIP). This has led to a prevalent view that rejecting PIP leaves open the possibility of -epistemic models for individual systems. In this work, we show that this understanding is incomplete: once the PBR theorem establishes -onticity for composite systems prepared in product states, the -onticity of the individual subsystems follows directly from the tensor-product structure of quantum mechanics, without invoking PIP or any further auxiliary assumptions. This result removes a key auxiliary assumption from the PBR theorem, closes a persistent loophole for preserving -epistemic models, and strengthens the conceptual foundations of -ontology.
Paper Structure (21 sections, 2 theorems, 43 equations)

This paper contains 21 sections, 2 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. The PBR Theorem: Joint $\psi$-Onticity and PIP
  3. Structural Decomposition of $\psi$-Ontic Models
  4. Joint $\psi$-Onticity in the PBR Scenario
  5. The Role of PIP
  6. Our Contribution: Single-System $\psi$-Onticity from Joint $\psi$-Onticity without PIP
  7. A Direct Proof of Single-System $\psi$-Ontology
  8. Consequence of joint $\psi$-onticity
  9. Tensor decomposition of the $\psi$-label
  10. Extraction of the subsystem distributions
  11. Single-system $\psi$-ontology
  12. Limitations of $\psi$-Epistemic Models
  13. Lewis et al. Model
  14. Aaronson et al. Model
  15. Criticisms and Responses
  16. ...and 6 more sections

Key Result

Theorem 1

Let $(\Lambda, \Sigma)$ be a measurable space and let $\Psi = \mathbb{CP}^{n-1}$ be the projective Hilbert space equipped with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{CP}^{n-1})$. Suppose $\{\mu_\psi\}_{\psi\in\Psi}$ is a family of probability measures on $(\Lambda,\Sigma)$ with pairwise dis Then there exists a $\Sigma$-measurable function such that, for every $\psi\in\Psi$, Equivalently

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Corollary 2: Joint $\psi$-onticity for product states