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.
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5+8 pages, 4 figures. A calculational error and some typos corrected. Comments welcome!
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.
In the task of quantum state exclusion, we consider a quantum system prepared in a state chosen from a known set. The aim is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures cannot have taken place. We ask what conditions the set of states must obey in order for this to be possible and how well we can complete the task when it is not. The task of quantum state discrimination forms a subclass of this set of problems. Within this paper, we formulate the general problem as a semidefinite program (SDP), enabling us to derive sufficient and necessary conditions for a measurement to be optimal. Furthermore, we obtain a necessary condition on the set of states for exclusion to be achievable with certainty, and we give a construction for a lower bound on the probability of error. This task of conclusively excluding states has gained importance in the context of the foundations of quantum mechanics due to a result from Pusey, Barrett, and Rudolph (PBR). Motivated by this, we use our SDP to derive a bound on how well a class of hidden variable models can perform at a particular task, proving an analog of Tsirelson's bound for the PBR experiment and the optimality of a measurement given by PBR in the process. We also introduce variations of conclusive exclusion, including unambiguous state exclusion, and state exclusion with worst-case error.
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.
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.
When it isn't possible to tell two distinct experimental procedures apart purely from their input/output statistics, then it seems a plausible hypothesis that the two procedures must be physically identical. We call such a hypothesis "noncontextuality", an instance of Leibniz's principle of the identity of indiscernibles. Read in the contrapositive, this hypothesis entails that any physical distinctions between two experimental procedures must necessarily lead to a difference in their operational statistics. The results I present in this thesis concern the failure of this hypothesis -- a failure dubbed "contextuality" -- when one tries to embed an operational theory (such as quantum theory) in the ontological models framework. The Kochen-Specker theorem demonstrates the failure of noncontextuality for deterministic ontological models of quantum theory, i.e., those ontological models where the ontic/physical state of the system fixes the outcome of any projective measurement on the system in a deterministic manner. This thesis goes beyond the Kochen-Specker (KS) theorem by asking what operational facts must be verified in experiments to conclude that Nature does not admit noncontextual ontological models, not even indeterministic ones. This leads to noncontextuality inequalities that are robust to noise in the preparations and measurements. In the particular case of quantum theory, these inequalities are meaningful even when unsharp measurements (or POVMs) are allowed, a feature not shared by the traditional approach to KS-noncontextuality where unsharp measurements are excluded by fiat: allowing them renders even trivial POVMs (proportional to identity) maximally KS-contextual. The sense in which trivial POVMs are indeed "trivial" (or "noncontextual") is clear in our approach: they are simply too noisy to lead to violation of our noncontextuality inequalities.
In this paper we relate notions of nonclassicality in what is known as the simplest nontrivial scenario (a prepare and measure scenario composed of four preparations and two binary-outcome tomographically complete measurements). Specifically, we relate the established method developed by Pusey [M. F. Pusey, Phys. Rev. A 98, 022112 (2018)] to witness a violation of preparation noncontextuality, that is not suitable in experiments where the operational equivalences to be tested are specified in advance, with an approach based on the notion of bounded ontological distinctness for preparations, defined by Chaturvedi and Saha [A. Chaturvedi and D. Saha, Quantum 4, 345 (2020)]. In our approach, we test bounded ontological distinctness for two particular preparations that are relevant in certain information processing tasks in that they are associated with the even and odd parity of the bits to communicate. When there exists an ontological model where this distance is preserved we talk of parity preservation. Our main result provides a noise threshold under which violating parity preservation (and so bounded ontological distinctness) agrees with the established method for witnessing preparation contextuality in the simplest nontrivial scenario. This is achieved by first relating the violation of parity preservation to the quantification of contextuality in terms of inaccessible information as developed by Marvian (I. Marvian, arXiv:2003.05984.), that we also show, given the way we quantify noise, to be more robust in witnessing contextuality than Pusey's noncontextuality inequality. As an application of our findings, we treat the case of two-bit parity-oblivious multiplexing in the presence of noise. In particular, given that we have a noise threshold below which preparation contextuality holds, we use it to establish a condition for which preparation contextuality is present in the case where the probability of success exceeds that achieved by any classical strategy.
Pusey, Barrett and Rudolph (PBR) have recently given a completely novel argument that restricts the class of possible models for quantum phenomena (arXiv:1111.3328). In these notes the assumptions used by PBR are considerably weakened, to further restrict the class of possible models. The `factorisability' assumption used by PBR is replaced by a far weaker `compatibility' assumption for uncorrelated quantum subsystems which, moreover, does not require the assignation of separate underlying properties to each subsystem (i.e, reductionism). Further, it is shown that an assumption of measurement independence may be dropped to obtain a related result having the same experimental significance (at the expense of a weaker conceptual significance). The latter is a remarkable feature of the PBR approach, given that Bell inequalities, steering inequalities and Kochen-Specker theorems all require an assumption of this type.