Two instances of random access code in the quantum regime

We consider two classes of quantum generalisations of random access code (RAC) and study the bounds for probabilities of success for such tasks. It provides a useful framework for the study of certain information processing tasks with constrained resources. The first class is based on a RAC with quantum inputs and output known as the no-signalling quantum RAC (NS-QRAC) box (Grudka et al 2015 Phys. Rev. A 92 052312), where unbounded entanglement and constrained classical communication are allowed. We show that it can be seen as quantum teleportation with constrained classical communication and provide a lower quantum bound for the success probability. We consider two modifications to the NS-QRAC scenario: the first, where unbounded entanglement and constrained quantum communication is allowed and the second, where bounded entanglement and unconstrained classical communication is allowed. We find a monogamy relation for the transmission fidelities, which—in contrast to the usual communication schemes—involves multiple senders and a single receiver. We provide an upper bounds for the latter and a lower one for the former. The second class is based on a RAC with a quantum channel and shared entanglement (Tavakoli et al 2021 PRX Quantum 2 040357). We study the set of tasks where two inputs made of two digits of d -base are encoded over a qudit and a maximally entangled state. We show that such tasks can be seen as quantum dense-coding with constrained quantum communication and explicit protocols, which give lower quantum bounds for the tasks’ efficiency, in dimensions d = 2 , 3 , 4 . The employed encoding utilises Gray codes.