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arXiv:1608.05034 · Journal of Physics A Mathematical and Theoretical

Not-Post-Peierls compatibility under noisy channels

Andrés F. Ducuara, Cristian E. Susa, John Henry Reina

Published May 15, 2017• 3 citations
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Abstract

Abstract The Pusey–Barrett–Rudolph (PBR) theorem deals with the realism of the quantum states. It establishes that every pure quantum state is real, in the context of quantum ontological models. Specifically, by guaranteeing the property of not-Post-Peierls ( ¬ PP) compatibility (or antidistinguishability) for a particular set of states P , together with the ad hoc postulate known as preparation independence postulate (PIP), the theorem establishes that these two properties imply the ψ -onticity (realism) of the set of all pure states. This PBR result has triggered two particular lines of research: On the one hand, it has been possible to derive similar results without the use of the PIP, although at the expense of implying weaker properties than ψ -onticity. On the other hand, it has also been proven that the property of ¬ PP compatibility alone is an explicit witness of usefulness for the task known as conclusive exclusion of states. In this work, we explore the ¬ PP compatibility of the set of states P , when P is under the interaction of some noisy channels, which would consequently let us identify some noisy scenarios where it is still possible to perform the task of conclusive exclusion of states. Specifically, we consider the set P of n -qubit states in interaction with an environment by means of (i) individual and (ii) collective couplings. In both cases, we analytically show that the phenomenon of achieving ¬ PP compatibility, although reduced, it is still present. Searching for an optimisation of this phenomenon, we report numerical experiments up to n = 4 qubits. For individual qubit-noise coupling, be it a bit, a phase, or a bit–phase flip noise, the numerical search exhibits the same response to all the different noisy channels, without improving the analytical expectation. In contrast, the collective qubit-noise coupling leads to a more efficient numerical display of the phenomenon, and a variety of noise channel-dependent behaviours emerge. Furthermore, by adopting a slight modification of the definition of ψ -onticity, from being a property exclusively of sets of pure states to be a property of general sets of states (now ρ -onticity), in particular of the considered noisy set P , our results can also be seen as addressing the ρ -ontic realism of the set of n -qubit states P under the considered noisy channels.

DOI
10.1088/1751-8121/aa72fb

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Abstract The Pusey–Barrett–Rudolph (PBR) theorem deals with the realism of the quantum states. It establishes that every pure quantum state is real, in the context of quantum ontological models. Specifically, by guaranteeing the property of not-Post-Peierls ( ¬ PP) compatibility (or antidistinguishability) for a particular set of states P , together with the ad hoc postulate known as preparation independence postulate (PIP), the theorem establishes that these two properties imply the ψ -onticity (realism) of the set of all pure states. This PBR result has triggered two particular lines of research: On the one hand, it has been possible to derive similar results without the use of the PIP, although at the expense of implying weaker properties than ψ -onticity. On the other hand, it has also been proven that the property of ¬ PP compatibility alone is an explicit witness of usefulness for the task known as conclusive exclusion of states. In this work, we explore the ¬ PP compatibility of the set of states P , when P is under the interaction of some noisy channels, which would consequently let us identify some noisy scenarios where it is still possible to perform the task of conclusive exclusion of states. Specifically, we consider the set P of n -qubit states in interaction with an environment by means of (i) individual and (ii) collective couplings. In both cases, we analytically show that the phenomenon of achieving ¬ PP compatibility, although reduced, it is still present. Searching for an optimisation of this phenomenon, we report numerical experiments up to n = 4 qubits. For individual qubit-noise coupling, be it a bit, a phase, or a bit–phase flip noise, the numerical search exhibits the same response to all the different noisy channels, without improving the analytical expectation. In contrast, the collective qubit-noise coupling leads to a more efficient numerical display of the phenomenon, and a variety of noise channel-dependent behaviours emerge. Furthermore, by adopting a slight modification of the definition of ψ -onticity, from being a property exclusively of sets of pure states to be a property of general sets of states (now ρ -onticity), in particular of the considered noisy set P , our results can also be seen as addressing the ρ -ontic realism of the set of n -qubit states P under the considered noisy channels.
The Pusey-Barrett-Rudolph (PBR) theorem deals with the realism of the quantum state. It establishes that every pure quantum state is real, in the context of ψ\psiψ-ontic quantum ontological models. In order to achieve this, PBR consider an ad hoc postulate termed the Preparation Independence Postulate (PIP). Here we argue that even without the use of the PIP, the PBR argument leads to a link between scalability and realism of quantum states whereby the latter can be achieved through the former. Further, we consider a set of n-qubit registers in interaction with an environment by means of i) individual and ii) collective couplings. In both cases, we analytically show that the phenomenon of achieving realism through scalability is still present, even though this now occurs at a slower rate. Searching for an optimisation of this phenomenon, we report numerical experiments up to n = 4 qubits: for individual qubit-noise coupling, be it a bit, a phase, or a bit-phase flip noise, the numerical search exhibits the same response to all the different noisy channels, without improving the analytical expectation. In contrast, the collective qubit-noise coupling leads to a more efficient display of the phenomenon, and a variety of noise channel-dependent behaviours emerge.
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.
The task of conclusive exclusion for a set of quantum states is to find a measurement such that for each state in the set, there is an outcome that allows one to conclude with certainty that the state in question was not prepared. Defining classicality of statistics as realizability by a generalized-noncontextual ontological model, we show that there is a quantum-over-classical advantage for how well one can achieve conclusive exclusion. This is achieved in an experimental scenario motivated by the construction appearing in the Pusey-Barrett-Rudolph theorem. We derive noise-robust noncontextuality inequalities bounding the conclusiveness of exclusion, and describe a quantum violation of these. Finally, we show that this bound also constitutes a classical causal compatibility inequality within the bilocality scenario, and that its violation in quantum theory yields a novel possibilistic proof of a quantum-classical gap in that scenario.
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 Pusey–Barrett–Rudolph (PBR) theorem constrains ontological models in which distinct pure quantum states correspond to overlapping ontic preparation distributions and independently prepared systems satisfy preparation independence. Applying that result to Constraint-Surface Dynamics (CSD) requires care because CSD calls the projective quantum state epistemic in a different sense: the ray [ψ] is an incomplete, many-to-one image of an underlying ontic state x ∈ Σ. This paper separates these notions at the level of preparation measures. A general CSD preparation is a Borel probability measure on the ontic space Σ. We first prove the general measure-theoretic result that mutual singularity of two pushforward measures under a measurable map implies mutual singularity of the original measures. Hence distinct exact sharp preparations are mutually singular. The canonical exact CSD preparation interface is therefore ψ-ontic in the technical Harrigan–Spekkens sense and already satisfies the disjointness conclusion associated with the PBR theorem. We then distinguish this exact interface from finite-resolution region preparations. A region of positive Liouville measure has a conditional preparation law absolutely continuous with respect to Liouville measure, whose projective pushforward admits a Radon–Nikodym density with respect to Fubini–Study measure. On the concrete Kähler witness, individual projective fibres are Liouville-null while an exact sharp preparation assigns unit probability to its own fibre. Exact sharp measures are consequently singular with respect to Liouville measure, and no positive-volume region-conditioned preparation law can equal an exact sharp preparation law. Suitable finite-resolution region preparations may nevertheless overlap. These results explain how CSD can regard the quantum state as epistemic in the sense of being ontically incomplete while remaining ψ-ontic under the Harrigan–Spekkens classification of exact preparations. They also remove the need for the earlier claim that global symplectic non-factorisation obstructs PBR preparation independence; that inference is withdrawn. Preparation independence remains a separate compositional question. The preparation results used here are machine-checked in Lean 4. Programme background: https://www.constraintsurfacedynamics.com — Glossary: https://glossary.constraintsurfacedynamics.com
The main question under investigation in this thesis was the reality of the quantum state. We approached this problem by studying a recent no-go result by Pusey, Barrett and Rudolph, namely the PBR theorem. In Section 1 we provided a short introduction to quantum theory. A few relevant examples were covered, namely decompositions of quantum states, impossibility of the ignorance interpretation and the distinguishability of pure quantum states. Section 2 developed measure theoretic tools needed in the framework of ontological models. The Radon-Nikodym theorem was proven in order to justify the use of an alternative expression of the variational distance of probability measures. In Section 3 we developed a general framework for ontological models. In short, the ontological models framework can be considered the most general framework currently available for hidden variable theories and no-go results. We used the measure theoretic tools of Section 2 to make the formalism rigorous. Original research included in Section 3 was the characterization of a property called antidistinguishability for qubit states. A necessary and sufficient condition for antidistinguishability was given. It was then proven that any set of pure qubit states that is not antidistinguishable can be made antidistinguishable by adding a single carefully chosen state into the set. In the end of Section 3, some original observations were presented. Namely, a specific connection with contextuality was proven for sometimes ψ-ontic and never ψ-ontic ontological models. However, it was then noted that these observations were not enough to exclude never ψ-ontic ontological models, a feat that would have provided a new no-go theorem on its own. Contextuality in the ontological models framework was studied in Section 4. The main results were establishing preparation contextuality from the failure of maximal ψ-epistemicity and a proof of Bell's theorem based on preparation contextuality established this way. We concluded that ψ-ontic models cannot be Bell local based on these results. Finally in Section 5 the implications of ψ-ontology were discussed. By careful analysis of the results, it was noted that accepting the reality of the quantum state exposes explanatory gaps in the formalism. Possible solutions to the challenges introduced by these explanatory gaps were discussed through interpretations of quantum mechanics and the so-called measurement problem.