Abstract The parity-oblivious random-access-code (PORAC) is a class of communication games involving a sender (Alice) and a receiver (Bob). In such games, Alice’s amount of communication to Bob is constraint by the parity-oblivious (PO) conditions, so that the parity information of her inputs remains oblivious to Bob. The PO condition in an operational theory is equivalently represented in an ontological model that satisfies the preparation noncontextuality. In this paper, we provide a nontrivial generalization of the existing two-level PORAC and derive the winning probability of the game in the preparation noncontextual ontological model. We demonstrate that the quantum theory outperforms the preparation noncontextual model by predicting higher winning probability in our generalized PORAC.
Abstract The parity-oblivious random-access-code (PORAC) is a class of communication games involving a sender (Alice) and a receiver (Bob). In such games, Alice’s amount of communication to Bob is constraint by the parity-oblivious (PO) conditions, so that the parity information of her inputs remains oblivious to Bob. The PO condition in an operational theory is equivalently represented in an ontological model that satisfies the preparation noncontextuality. In this paper, we provide a nontrivial generalization of the existing two-level PORAC and derive the winning probability of the game in the preparation noncontextual ontological model. We demonstrate that the quantum theory outperforms the preparation noncontextual model by predicting higher winning probability in our generalized PORAC.
The optimal success probability of a communication game sets fundamental limitations on an operational theory. The quantum advantage of a parity oblivious random access code (PORAC), a communication game, over classical resources reveals the preparation contextuality of quantum theory [R. W. Spekkens et al., Phys. Rev. Lett. 102, 010401 (2009)]. The optimal quantum advantage in the N-dit PORAC game for finite dimensions is an open problem. Here, we show that the degree of uncertainty allowed in an operational theory determines the amount of preparation contextuality. We connect the upper bound of the fine-grained uncertainty relation to the success probability of a PORAC game played with the quantum resource. Subsequently, we find the optimal success probability for a 2-dit PORAC game using mutually unbiased bases for the decoding strategy. Finally, we also derive an upper bound on the quantum advantage for the N-dit PORAC game.
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.
We provide an interesting two-party parity-oblivious communication game whose success probability is solely determined by the Bell expression. The parity-oblivious condition in an operational quantum theory implies the preparation noncontextuality in an ontological model of it. We find that the aforementioned Bell expression has two upper bounds in an ontological model: the usual local bound and a nontrivial preparation noncontextual bound arising from the nontrivial parity-oblivious condition, which is smaller that the local bound. We first demonstrate the communication game when both Alice and Bob perform three measurements of dichotomic observables in their respective sites. The optimal quantum value of the Bell expression in this scenario enables us to device-independently self-test the maximally entangled state and trine set of observables, three-outcome qubit positive-operator-valued measures and 1.58 bits of local randomness. Further, we generalize the above communication game in that both Alice and Bob perform the same but arbitrary (odd) number (n>3) of measurements. Based on the optimal quantum value of the relevant Bell expression for any arbitrary n, we also demonstrate device-independent self-testing of the state and measurements.
Preparation contextuality is often manifested operationally through a quantitative advantage over preparation-noncontextual models in information-processing tasks, yet quantum theory typically falls short of perfect success. Here we introduce a parity-oblivious Hidden Matching task that instead yields an all-vs-nothing manifestation of preparation contextuality. Alice encodes an n-bit string so that her message reveals no information about any input parity other than the two-bit parities. Bob, given a perfect matching on the input positions, must output an edge of the matching together with the parity of its two bits. A quantum protocol using a single ⌈log2n⌉-qubit message satisfies the parity-obliviousness constraint and succeeds with certainty. We prove that no preparation-noncontextual model achieves perfect success for any even n≥6. The result holds for arbitrary ontic state spaces and assumes neither outcome determinism nor measurement noncontextuality. For n=6 and n=8, the optimal noncontextual success probabilities are 4/5 and 3/4, respectively, while asymptotically the optimal noncontextual success probability is 1/2+Θ(1/n). Our asymptotic upper bound follows from a Fourier-analytic sum rule derived via hypercontractivity on the Boolean hypercube. The resulting separation is qualitative rather than merely quantitative: quantum theory achieves perfect success, whereas preparation-noncontextual models cannot. We aslo show that at n=8 the noncontextual bound remains 3/4 even when Bob is restricted to just suitable 4 of the 105 possible perfect matchings, bringing the effect within experimental reach.
Replacing the task of retrieval with exclusion changes how preparation contextuality manifests operationally under parity-oblivious constraints, with exclusion showing a quantum advantage where retrieval does not. We introduce the parity-oblivious random exclusion code (POREC) and show that for prime symbol size m, classical and preparation-noncontextual encodings provide a tight noncontextual bound. For the first nontrivial case (two digits, three symbols), our derived exact qubit optimum violates this bound, in contrast to parity-oblivious retrieval, which displays no quantum advantage. This characteristic difference is absent without parity constraints. For general prime m, qubit strategies achieve a quantum-to-noncontextual gap that grows linearly relative to the random exclusion code (REC) gap, exceeding both parity-oblivious retrieval and standard REC. The exact qubit bound yields a sharp semi-device-independent certification of dimension d≥3. Our analysis of noise robustness demonstrates POREC to be amenable for experimental implementation on existing prepare-and-measure platforms, establishing parity-oblivious exclusion as a distinct operational probe of preparation contextuality, as well as a practical information processing protocol with wide applications.
An ontological model of an operational theory is considered to be universally noncontextual if both preparation and measurement noncontextuality assumptions are satisfied in that model. In this report, we first generalize the logical proofs of quantum preparation and measurement contextuality for qubit system for any odd number of preparations and measurements. Based on the logical proof, we derive testable universally non-contextual inequalities violated by quantum theory. We then propose a class of two-party communication games and show that the average success probability of winning such games is solely linked to suitable Bell expression whose local bound is greater than universal non-contextual bound. Thus, for a given state, even if quantum theory does not exhibit non-locality, it may still reveal non-classicality by violating the universal non-contextual bound. Further, we consider a different communication game to demonstrate that for a given choices of observables in quantum theory, even if there is no logical proof of preparation and measurement contextuality exist, the universal quantum contextuality can be revealed through that communication game. Such a game thus test a weaker form of universal non-contextuality with minimal assumption.