We introduce a hierarchy of semidefinite relaxations of the set of quantum correlations in generalized contextuality scenarios.This constitutes a simple and versatile tool for bounding the magnitude of quantum contextuality.To illustrate its utility, we use it to determine the maximal quantum violation of several noncontextuality inequalities whose maximum violations were previously unknown.We then go further and use it to prove that certain preparation-contextual correlations cannot be explained with pure states, thereby showing that mixed states are an indispensable resource for contextuality.In the second part of the paper, we turn our attention to the simulation of preparation-contextual correlations in general operational theories.We introduce the information cost of simulating preparation contextuality, which quantifies the additional, otherwise forbidden, information required to simulate contextual correlations in either classical or quantum models.In both cases, we show that the simulation cost can be efficiently bounded using a variant of our hierarchy of semidefinite relaxations, and we calculate it exactly in the simplest contextuality scenario of parity-oblivious multiplexing.
The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the particular extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In the first part of this work, we formally define contextuality scenarios via prepare-and-measure experiments, along with the polytope of general contextual behaviours containing the set of quantum contextual behaviours. This framework allows us to recover several properties of set of quantum behaviours in these scenarios, including contextuality scenarios and associated noncontextuality inequalities that require for their violation the individual quantum preparation and measurement procedures to be mixed states and unsharp measurements. With the framework in place, we formulate novel semidefinite programming relaxations for bounding these sets of quantum contextual behaviours. Most significantly, to circumvent the inadequacy of pure states and projective measurements in contextuality scenarios, we present a novel unitary operator based semidefinite relaxation technique. We demonstrate the efficacy of these relaxations by obtaining tight upper bounds on the quantum violation of several noncontextuality inequalities and identifying novel maximally contextual quantum strategies. To further illustrate the versatility of these relaxations, we demonstrate monogamy of preparation contextuality in a tripartite setting, and present a secure semi-device independent quantum key distribution scheme powered by quantum advantage in parity oblivious random access codes.
The quantum ontological feature of contextuality apart from being central to foundations of quantum theory forms the basis of quantum advantage in a multitude of information processing tasks. In particular, the contextuality of preparation procedures was shown to power a particular two-party information processing task “parity oblivious multiplexing” (Spekkens et al. Phys Rev Lett 102:010401 (2009). Specifically, it was shown that there exists a limit to how well any preparation noncontextual theory can perform in this task. This limit constitutes a noncontextuality inequality. Moreover, the authors demonstrated quantum violation of this inequality along with preparation contextuality associated with the ontic description underlying two-level completely mixed quantum state. In this work, we extend these arguments to apply to arbitrary dimensions by introducing a class of two-party information processing tasks, namely d-level parity oblivious random access codes. We analytically obtain classical (or equivalently preparation noncontextual) bounds on the success probability for these tasks for arbitrary d. For each value of d, this bound constitutes a unique noncontextuality inequality. Remarkably, these bounds are independent of the amount of communication. Furthermore, we find a classical protocol utilizing a
dc=d -dimensional classical message which saturates this bound. In order to establish nontriviality of these inequalities, we provide evidence of significant quantum violations. Specifically, by numerical techniques, we show that for
d=3,…,10 , the noncontextuality bound is violated by quantum theory. (1) We provide explicit quantum protocols which violate the associated noncontextuality inequality for
d=3,4,5 employing
dq=d -leveled quantum systems. (2) Using see-saw semi-definite programming (SDP) technique, we find evidence (lower bounds) of significant quantum violation of these inequalities for
d=3,…,10 . (3) With the help of state-of-the-art (NPA-hierarchy like) SDP technique, we provide upper bounds (independent of the dimension of the involved quantum systems) on quantum violation for
d=3,…,10 . The introduced class of information tasks, thus, provides for operational depiction of preparation contextuality of the ontic description underlying mixed higher dimensional quantum systems.
Communication games are collaborative information processing tasks involving a number of players with limited communication. Such games are useful tools for studying physical theories. A physical theory exhibits preparation contextuality whenever observed behaviour cannot be explained by a preparation noncontextual model. Here we show that there is a fundamental connection between the performance in communication games and the degree of preparation (non)contextuality. For this purpose, we present a general framework that allows us to construct communication games such that the game performance corresponds to a measure of preparation (non)contextuality. We illustrate the power of this framework by, 1) deriving many examples of tests of preparation contextuality, 2) showing that quantum violations of Bell inequalities can be derived from quantum violations of preparation noncontextuality, 3) qualitatively and quantitatively explaining related previous results on preparation contextuality in quantum communication games, and 4) solving the open problem of revealing the preparation contextuality of the maximally mixed quantum state in any dimension.
In [Phys. Rev. Lett. 102, 010401 (2009)], Spekkens et al. have shown that quantum preparation contextuality can power the parity-oblivious multiplexing (POM) task. The bound on the optimal success probability of an
n-bit POM task performed with the classical resources was shown to be the same as in a preparation noncontextual theory. This noncontextual bound is violated if the task is performed with quantum resources. While in the 2-bit POM task the optimal quantum success probability is achieved, in the 3-bit case optimality was left as an open question. In this paper, we show that the quantum success probability of an
n-bit POM task is solely dependent on a suitable Bell expression and the optimal value of it optimizes the success probability of the said POM task. Further, we discuss how the degree of quantum preparation contextuality restricts the amount of quantum violations of Bell's inequalities, and consequently, the success probability of a POM task.
In this work we present a hierarchy of generalized contextuality. It refines the traditional binary distinction between contextual and noncontextual theories, and facilitates their comparison based on how contextual they are. Our approach focuses on the contextuality of prepare-and-measure scenarios, described by general probabilistic theories (GPTs). To motivate the hierarchy, we define it as the resource ordering of a novel resource theory of GPT-contextuality. The building blocks of its free operations are classical systems and univalent simulations between GPTs. These simulations preserve operational equivalences and thus cannot generate contextuality. Noncontextual theories can be recovered as least elements in the hierarchy. We then define a new contextuality monotone, called classical excess, given by the minimal error of embedding a GPT within an infinite classical system. In addition, we show that the optimal success probability in the parity oblivious multiplexing game also defines a monotone in our resource theory. Finally, we discuss whether the non-free operations can be understood as implementing information erasure and thus explaining the fine-tuning aspect of contextuality.
A communication game consists of distributed parties attempting to jointly
complete a task with restricted communication. Such games are useful tools for
studying limitations of physical theories. A theory exhibits preparation
contextuality whenever its predictions cannot be explained by a preparation
noncontextual model. Here, we show that the ability of an operational theory to
perform communication games is a measure of the degree of preparation
contextuality in that theory. For statistics obtained in a paritcular family of
communication games, we show a one-to-one correspondance with correlations in
space-like separated events obeying the no-signaling principle. Using this, we
prove that all mixed quantum states of any finite dimension are preparation
contextual. We present an experimental realization of a communication game
involving three-level quantum systems from which we observe a strong violation
of the constraints of preparation noncontextuality.