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arXiv:1909.06771 · Quantum Reports

Quantum PBR Theorem as a Monty Hall Game

Del Rajan, Matt Visser

Published Dec 31, 2019• 6 citations
arXivHTMLPDF

Abstract

The quantum Pusey–Barrett–Rudolph (PBR) theorem addresses the question of whether the quantum state corresponds to a ψ-ontic model (system’s physical state) or to a ψ-epistemic model (observer’s knowledge about the system). We reformulate the PBR theorem as a Monty Hall game and show that winning probabilities, for switching doors in the game, depend on whether it is a ψ-ontic or ψ-epistemic game. For certain cases of the latter, switching doors provides no advantage. We also apply the concepts involved in quantum teleportation, in particular for improving reliability.

DOI
10.3390/quantum2010003

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The quantum Pusey–Barrett–Rudolph (PBR) theorem addresses the question of whether the quantum state corresponds to a ψ-ontic model (system’s physical state) or to a ψ-epistemic model (observer’s knowledge about the system). We reformulate the PBR theorem as a Monty Hall game and show that winning probabilities, for switching doors in the game, depend on whether it is a ψ-ontic or ψ-epistemic game. For certain cases of the latter, switching doors provides no advantage. We also apply the concepts involved in quantum teleportation, in particular for improving reliability.
The Pusey-Barrett-Rudolph (PBR) no-go theorem provides an argument for the reality of the quantum state by ruling out $\ensuremath{\psi}$-epistemic ontological theories, in which the quantum state is of a statistical nature. It applies under an assumption of preparation independence, the validity of which has been subject to debate. We propose two plausible and less restrictive alternatives: a weaker notion allowing for classical correlations, and an even weaker, physically motivated notion of independence, which merely prohibits the possibility of superluminal causal influences in the preparation process. The latter is a minimal requirement for enabling a reasonable treatment of subsystems in any theory. It is demonstrated by means of an explicit $\ensuremath{\psi}$-epistemic ontological model that the argument of PBR becomes invalid under the alternative notions of independence. As an intermediate step, we recover a result which is valid in the presence of classical correlations. Finally, we obtain a theorem which holds under the minimal requirement, approximating the result of PBR. For this, we consider experiments involving randomly sampled preparations and derive bounds on the degree of $\ensuremath{\psi}$ epistemicity that is consistent with the quantum-mechanical predictions. The approximation is exact in the limit as the sample space of preparations becomes infinite.
The detection-efficiency loophole poses a significant problem for experimental tests of Bell inequalities. The recently discovered Pusey-Barrett-Rudolph (PBR) theorem suffers from the same vulnerability. In this paper we calculate the critical detection efficiency, below which the PBR argument for the ontic nature of quantum state is inconclusive. This is done for the maximally $\ensuremath{\psi}$-epistemic models. We use two different definitions of this property. The optimal number of parties for which the critical detection efficiency is the lowest is given. We also approach the problem from the opposite direction. We provide a function that enables us to specify which epistemic models are ruled out by the results of an experiment with a given detection efficiency.
In this paper, a proof similar to the Pusey-Barrett-Rudolph (PBR) theorem is given to prove that time in quantum theory is not epistemic but has ontological reality. For this, we first discuss how time can be encoded as an intrinsic property of a quantum system. We prepare a quantum state containing time as an intrinsic property and show by a PBR-type proof that time cannot be an epistemic notion. In fact, the PBR theorem implies an ontic notion of time. Indeed, if the quantum state is a physical property, then quantum state reduction is a physical process involving information destruction. In this case, quantum probabilities are intrinsic probabilities without epistemic origin, and they generate a genuinely new sequence of events. This novelty introduced by quantum probabilities can be interpreted as time.
The Pusey--Barrett--Rudolph (PBR) theorem is commonly taken to establish that a pure quantum state represents a physical property of the system to which it is assigned, rather than merely encoding knowledge. In this work, I distinguish between state-first and dynamics-first physical theories, and argue that this reading presupposes the former. These two types of theories support different approaches to causal explanation, and one of PBR's stated assumptions depends on the state-first approach. Using a recent interpretation of quantum theory as a concrete example, I show how a dynamics-first theory can explain the PBR correlations while maintaining a broadly epistemic reading of quantum states. I conclude that the PBR theorem supports a dilemma: either the quantum state is part of reality, or reality is not fundamentally built out of states.
Towards the end of 2011, Pusey, Barrett and Rudolph derived a theorem that aimed to show that the quantum state must be ontic (a state of reality) in a broad class of realist approaches to quantum theory. This result attracted a lot of attention and controversy. The aim of this review article is to review the background to the Pusey–Barrett–Rudolph Theorem, to provide a clear presentation of the theorem itself, and to review related work that has appeared since the publication of the Pusey–Barrett–Rudolph paper. In particular, this review: Explains what it means for the quantum state to be ontic or epistemic (a state of knowledge); Reviews arguments for and against an ontic interpretation of the quantum state as they existed prior to the Pusey–Barrett–Rudolph Theorem; Explains why proving the reality of the quantum state is a very strong constraint on realist theories in that it would imply many of the known no-go theorems, such as Bell's Theorem and the need for an exponentially large ontic state space; Provides a comprehensive presentation of the Pusey–Barrett–Rudolph Theorem itself, along with subsequent improvements and criticisms of its assumptions; Reviews two other arguments for the reality of the quantum state: the first due to Hardy and the second due to Colbeck and Renner, and explains why their assumptions are less compelling than those of the Pusey–Barrett–Rudolph Theorem; Reviews subsequent work aimed at ruling out stronger notions of what it means for the quantum state to be epistemic and points out open questions in this area. The overall aim is not only to provide the background needed for the novice in this area to understand the current status, but also to discuss often overlooked subtleties that should be of interest to the experts. Quanta 2014; 3: 67–155.
The Pusey-Barrett-Rudolph (PBR) theorem establishes ψψψ-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 ψψψ-epistemic models for individual systems. In this work, we show that this understanding is incomplete: once the PBR theorem establishes ψψψ-onticity for composite systems prepared in product states, the ψψψ-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 ψψψ-epistemic models, and strengthens the conceptual foundations of ψψψ-ontology.