Parity oblivious d-level random access codes and class of noncontextuality inequalities

Abstract

One of the fundamental results in quantum foundations is the Kochen-Specker no-go theorem. For the quantum theory, the no-go theorem excludes the possibility of a class of hidden variable models where value attribution is context independent. Recently, the notion of contextuality has been generalized for different operational procedures and it has been shown that preparation contextuality of mixed quantum states can be a useful resource in an information-processing task called parityoblivious multiplexing. Here, we introduce a new class of information processing tasks, namely d-level parity oblivious random access codes and obtain bounds on the success probabilities of performing such tasks in any preparation noncontextual theory. These bounds constitute noncontextuality inequalities for any value of d. For d = 3, using a set of mutually asymmetric biased bases we show that the corresponding noncontextual bound is violated by quantum theory. We also show quantum violation of the inequalities for some other higher values of d. This reveals operational usefulness of preparation contextuality of higher level quantum systems.

State independent contextuality advances one-way communication

“…We point out that the classical bound for an OC task is also the optimal bound in all theories that satisfy the notion of preparation noncontextuality [9]. This generalizes the result obtained for parity oblivious multiplexing [13,[39][40][41][42][43]. As we have shown KS sets yield advantage in OC, one may infer the consistency with previous result [9] that KS contextuality for sharp measurements implies preparation contextuality.…”

“…Third, the quantum advantage in OC: by proposing the broad framework of OC, we provide a generalization to the previously studied parity oblivious multiplexing tasks [13,[40][41][42][43], and a methodology to obtain the optimal strategy in classical communication. Unlike the communication scenario with bounded dimension, the quantum advantage in OC is unconditional on the amount of classical resource.…”

“…We remark that the OC problem is a generalization of oblivious transfer which serves as a primitive for several classical and quantum cryptographic protocols [34][35][36][37] and privacy-preserving computation [38]. It has been shown that preparation contextuality is useful in parity oblivious random access code [13,[39][40][41][42][43]. Parity oblivious multiplexing can be interpreted as a particular case of the above general framework, wherein the oblivious variable w, that is, the parity of a set of input bits, is a function of the input x.…”

Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen–Specker theorem: the Peres–Mermin square

“…Therefore, the noncontextual correlation polytope is the convex hull of the correlations one obtains for all possible pairings , k k¢ ( ) of a vertex κ from the noncontextual measurement-assignment polytope and a vertex 12 In particular, the noncontextuality inequalities we derive here are not isomorphic to the Bell inequality derived in [38] (and experimentally tested in [39]) even though the latter is inspired by a consideration of the Peres-Mermin construction (one such construction on each wing of the Bell experiment). This is because the inequality of [38] is based on the assumption of local causality alone.…”

“…It has been shown that violations of noncontextuality inequalities defined in terms of this notion can imply advantages for information processing which are independent of the validity of quantum theory. For example, they imply an advantage for the cryptographic task of parity-oblivious random access codes [10][11][12]. Such inequalities also hold promise for making the results on quantum computational advantages discussed above robust to noise and for expressing the origin of the advantage in a manner that is independent of the validity of quantum theory.…”