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Nonsignaling quantum random access code boxes

A well known cryptographic primitive is so called random access code. Namely, Alice is to send to Bob one of two bits, so that Bob has the choice which bit he wants to learn about. However at any time Alice should not learn Bob's choice, and Bob should learn only the bit of his choice. The task is impossible to accomplish by means of either classical or quantum communication. On the other hand, a concept of correlations stronger than quantum ones, exhibited by so called Popescu- Rohrlich box, was introduced and widely studied. In particular, it is known that Popescu-Rohrlich box enables simulation of the random access code with the support of one bit of communication. Here, we propose a quantum analogue of this phenomenon. Namely, we define an analogue of a random access code, where instead of classical bits, one encodes qubits. We provide a quantum non-signaling box that if supported with two classical bits, allows to simulate a quantum version of random access code. We point out that two bits are necessary. We also show that a quantum random access code cannot be fully quantum: when Bob inputs superposition of two choices, the output will be in a mixed state rather than in a superposition of required states.