Informationally restricted quantum correlations

Abstract

Quantum communication leads to strong correlations, that can outperform classical ones. Complementary to previous works in this area, we investigate correlations in prepare-and-measure scenarios assuming a bound on the information content of the quantum communication, rather than on its Hilbert-space dimension. Specifically, we explore the extent of classical and quantum correlations given an upper bound on the one-shot accessible information. We provide a characterisation of the set of classical correlations and show that quantum correlations are stronger than classical ones. We also show that limiting information rather than dimension leads to stronger quantum correlations. Moreover, we present device-independent tests for placing lower bounds on the information given observed correlations. Finally, we show that quantum communication carrying log⁡d bits of information is at least as strong a resource as d-dimensional classical communication assisted by pre-shared entanglement.

Device-independent quantum key distribution from computational assumptions

“…As we noted when we stated theorem 1, the constant 1 128 is a consequence of fixing certain parameters in the protocol to 1 2 . Specifically, from the proof, we see that this constant arises from the probability 1 32 of being in a generation round, and the probability 1 Unlike previous approaches to weakening the no-communication assumption [19][20][21], which required an a priori device-dependent upper bound on the amount of information exchanged between different parts of the device, the LWE assumption is a general assumption about any computationally bounded quantum device, and our belief in it does not require us to inspect the specific device at hand in detail.…”

“…Given the difficulty of perfectly shielding components of a device from one another, recent works have aimed at formulating Bell inequalities that tolerate some limited amount of communication between the two components of the device [19][20][21]. For a key distribution scheme based on such a Bell inequality to be secure, one needs to assume an a priori bound on the amount of communication; this bound cannot be verified during the protocol.…”

Stronger correlations than dense coding with the same quantum resources

“…However, when (potentially unbounded) entanglement is added, this picture breaks down. Instead, upper bounds on S can be determined using the hierarchy of semidefinite programming relaxations developed in [26], which uses the concept of informationally-restricted quantum correlations [ 28,29]. Using this method, and matching it with an explicit entanglement-based strategy with classical communication, we find S ent+bit ≈ 3.799.…”

Criteria for nonclassicality in the prepare-and-measure scenario

“…Notice that unless we bound the dimension of M (the number of possible values it can assume) to be strictly smaller than the cardinality of X , the problem becomes trivial since any distribution p(b|x, y) can be generated by such causal model. Thus, the dimension of the physical system being prepared is typically bounded (see [34,35,43] for alternative constraints not bounding the dimension of the prepared states). In a classical description, without loss of generality, any randomness present in the probabilities can be absorbed into latent/hidden variables [1].…”