Quantum contextuality provides communication complexity advantage

Abstract

Despite the conceptual importance of contextuality in quantum mechanics, there is a hitherto limited number of applications requiring contextuality but not entanglement. Here, we show that for any quantum state and observables of sufficiently small dimension producing contextuality, there exists a communication task with quantum advantage. Conversely, any quantum advantage in this task admits a proof of contextuality whenever an additional condition holds. We further show that given any set of observables allowing for quantum state-independent contextuality, there exists a class of communication tasks wherein the difference between classical and quantum communication complexities increases as the number of inputs grows. Finally, we show how to convert each of these communication tasks into a semi-device independent protocol for quantum key distribution.

Optimal Measurement Structures for Contextuality Applications

“…Contextuality can serve as the basis for randomness (or key) generation, either via stand-alone protocols that test for the violation of a non-contextuality inequality [29] (where one assumes that the measurements conform to the specific orthogonality graph), or through the conversion of a single-party contextuality test into a two-party Bell inequality [11] or through the conversion of a non-contextuality inequality to a prepare-and-measure protocol [12]. A common step in all such protocols [30][31][32][33][34][35][36][37] is the identification of a suitable measurement in the (contextuality) test that yields the highest possible randomness or key generation rate.…”

“…where I(P A|X ) is a non-contextuality inequality evaluated on the observed conditional probability distributions P A|X , I * ∈ (I c , I q ] with classical and quantum values given by I c and I q respectively, and Q denotes the set of conditional distributions (boxes) achievable by performing measurements (compatible with the test structure on Alice's side) on quantum states shared between Alice and adversary Eve, P guess (A i |E) = ∑ e P(e)P e (a = e|i) is the guessing probability of Alice's outcome by an adversary E. By an optimal rigid contextuality test in dimension d we mean one in which there exist a measurement bases x * such that P A|X (a|x * ) = 1/d for all outcomes a ∈ [d] when the maximum value I q is observed. It is an open question to derive such a rigid class of contextuality tests for arbitrary dimension d (see for example [12] where the guessing probability was calculated for the well known 5-cycle non-contextuality inequality [40]).…”

“…We identify how these vector sets can be found inside general KS proofs, and show how these can be applied as building blocks for constructing novel KS proofs. We then show how gadgets are the optimal measurement structures in many applications of contextuality [10][11][12][13][14] . We present an application of definite-prediction sets in providing an experimentally feasible test of a recent result showing that quantum correlations cannot be reproduced by fundamentally binary theories.…”

Systematic construction of quantum contextuality scenarios with rank advantage

“…For every of the examples, we determine a lower bound on the SIC ratio by using the PR obtained from the Kronecker product of the rank-one PR of the graph and the PR of AR(r) in Eq. (9). We then use the see-saw algorithm to exclude a PR with rank r − 1 in dimension d < (r − 1)χ f (G).…”

“…Quantum contextuality is a fundamental feature of quantum measurements [1] closely connected to the incompatibility of observables [2] and quantum nonlocality [3,4]. It has been proved to be vital for quantum advantage like in quantum computation [5,6], quantum communication [7,8,9], randomness generation [10], quantum cryptography [11] and quantum key distribution [12]. At the heart of quantum contextually are projective quantum measurements for which one cannot construct a noncontextual hidden variable model.…”