A random access code (RAC) is a communication task in which the sender encodes a random message into a shorter one to be decoded by the receiver so that a randomly chosen character of the original message is recovered with some probability. Both the message and the character to be recovered are assumed to be uniformly distributed. In this paper, we extend this protocol by allowing more general distributions of these inputs, which alters the encoding and decoding strategies optimizing the protocol performance, with either classical or quantum resources. We approach the problem of optimizing the performance of these biased RACs with both numerical and analytical tools. On the numerical front, we present algorithms that allow a numerical evaluation of the optimal performance over both classical and quantum strategies and provide a Python package designed to implement them, called RAC-tools. We then use this numerical tool to investigate single-parameter families of biased RACs in the ${n}^{2}\ensuremath{\mapsto}1$ and ${2}^{d}\ensuremath{\mapsto}1$ scenarios. For RACs in the ${n}^{2}\ensuremath{\mapsto}1$ scenario, we derive a general upper bound for the cases in which the inputs are not correlated, which coincides with the quantum value for $n=2$ and in some cases for $n=3$. Moreover, it is shown that attaining this upper bound self-tests pairs or triples of rank-1 projective measurements, respectively. An analogous upper bound is derived for the value of RACs in the ${2}^{d}\ensuremath{\mapsto}1$ scenario, which is shown to be always attainable using mutually unbiased measurements if the distribution of input strings is unbiased.
A random access code (RAC) is a communication task in which the sender encodes a random message into a shorter one to be decoded by the receiver so that a randomly chosen character of the original message is recovered with some probability. Both the message and the character to be recovered are assumed to be uniformly distributed. In this paper, we extend this protocol by allowing more general distributions of these inputs, which alters the encoding and decoding strategies optimizing the protocol performance, with either classical or quantum resources. We approach the problem of optimizing the performance of these biased RACs with both numerical and analytical tools. On the numerical front, we present algorithms that allow a numerical evaluation of the optimal performance over both classical and quantum strategies and provide a Python package designed to implement them, called RAC-tools. We then use this numerical tool to investigate single-parameter families of biased RACs in the n2\ensuremath↦1 and 2d\ensuremath↦1 scenarios. For RACs in the n2\ensuremath↦1 scenario, we derive a general upper bound for the cases in which the inputs are not correlated, which coincides with the quantum value for n=2 and in some cases for n=3. Moreover, it is shown that attaining this upper bound self-tests pairs or triples of rank-1 projective measurements, respectively. An analogous upper bound is derived for the value of RACs in the 2d\ensuremath↦1 scenario, which is shown to be always attainable using mutually unbiased measurements if the distribution of input strings is unbiased.
Quantum resources and protocols are known to outperform their classical counterparts in a variety of communication and information processing tasks. Random access codes (RACs) are one such cryptographically significant family of bipartite communication tasks, wherein the sender encodes a data set (typically a string of input bits) onto a physical system of bounded dimension and transmits it to the receiver, who then attempts to guess a randomly chosen part of the sender's data set (typically one of the sender's input bits). In this work, we introduce a generalization of this task wherein the receiver, in addition to the individual input bits, aims to retrieve randomly chosen functions of the sender's input string. Specifically, we employ sets of mutually unbiased balanced functions, such that perfect knowledge of any one of the constituent functions yields no knowledge about the others. We investigate and bound the performance of (i) classical, (ii) quantum prepare and measure, and (iii) entanglement assisted classical communication (EACC) protocols for the resultant generalized RACs (GRACs). Finally, we detail the case of GRACs with three input bits and find maximal success probabilities for classical, quantum, and EACC protocols, along with the effect of noisy quantum channels on the performance of quantum protocols. Moreover, with the help of this case study, we reveal several characteristic properties of the GRACs which deviate from the standard RACs.
A random access code (RAC) encodes an L-bit string into a k-bit message, L>k, so that any requested bit can be recovered with high probability; a quantum RAC (QRAC) uses k qubits instead. We give a geometric characterization of optimal classical (L,k)-RACs under average and worst-case decoding criteria. The average criterion is reduced to choosing 2k representatives in {0,1}L, while the worst-case criterion is reduced to a minimax problem over 2k points in [0,1]L with a distance-like objective. This framework proves optimality for several parameter families, with many optimal constructions arising from standard infinite families of binary linear codes. It also yields two explicit classical--quantum separations. First, for every L>1, we construct a (L,1)-QRAC whose average decoding success probability strictly exceeds the optimal classical value. Second, for the family (2k−1,k), we prove worst-case optimality of a classical RAC and construct a QRAC with strictly larger worst-case success probability. For the family (L,L−1), the framework identifies a classical RAC that is average-case optimal and, under a stated conjecture, also worst-case optimal. The same viewpoint further recovers explicit (L,L−1)-QRACs attaining a previously conjectured upper-bound value.
Random access code (RAC) is a strategy to access remote data even if one has limited information about it. We derive here the conditions for the optimal performance of two most basic RAC protocols implemented using two-qubit states. The conditions derived are valid for any set of decoding operations. This allows a meaningful comparison between quantum and classical resources in RAC and leads to identifying the condition for quantum advantage. It turns out that the necessary resource is quantum discord of the assisting state, but, importantly, the non-vanishing value of quantum discord in itself does not guarantee quantum advantage for any choice of decoding operations. We argue that simultaneous correlation in mutually unbiased bases as well as a particular notion of nonclassicality based on quantum steering can capture efficiency of the optimal RAC for orthogonal decoding operations.
A (quantum) random access code ((Q)RAC) is a scheme that encodes n bits into m (qu)bits such that any of the n bits can be recovered with a worst case probability p>21. We generalize (Q)RACs to a scheme encoding nd-levels into m (quantum) d-levels such that any d-level can be recovered with the probability for every wrong outcome value being less than d1. We construct explicit solutions for all $n\ensuremath{\le}\frac{{d}^{2m}\ensuremath{-}1}{d\ensuremath{-}1}$. For d=2, the constructions coincide with those previously known. We show that the (Q)RACs are d-parity oblivious, generalizing ordinary parity obliviousness. We further investigate optimization of the success probabilities. For d=2, we use the measure operators of the previously best-known solutions, but improve the encoding states to give a higher success probability. We conjecture that for maximal $(n={4}^{m}\ensuremath{-}1,m,p)$ QRACs, $p=\frac{1}{2}{1+{[{(\sqrt{3}+1)}^{m}\ensuremath{-}1]}^{\ensuremath{-}1}}$ is possible, and show that it is an upper bound for the measure operators that we use. We then compare (n,m,pq) QRACs with classical (n,2m,pc) RACs. We can always find ${p}_{q}\ensuremath{\ge}{p}_{c}$, but the classical code gives information about every input bit simultaneously, while the QRAC only gives information about a subset. For several different (n,2,p) QRACs, we see the same trade-off, as the best p values are obtained when the number of bits that can be obtained simultaneously is as small as possible. The trade-off is connected to parity obliviousness, since high certainty information about several bits can be used to calculate probabilities for parities of subsets.
Quantum random access codes (QRACs) ask how well N classical bits can be encoded into M qubits while allowing any single bit to be recovered. Although the Nayak bound remains the standard general upper bound on the decoding probability, numerical evidence suggests a stronger upper bound in the small-qubit regime.
In this work, we formulate the optimal decoding probability in terms of decoding measurements, reformulating QRAC design as a spectral problem for noncommuting measurements. Using this formulation, we give an elementary proof of the Nayak bound by simplifying the Chernoff-bound argument. Moreover, we refine the argument to obtain upper bounds that improve over Nayak's bound in the entire finite-size regime.
The equality conditions of our bounds justify defining mutually unbiased projector-valued measurements (MUPVMs), a generalization of mutually unbiased bases. We show that decoding measurement of any two-qubit QRAC attaining the conjectured bound must form MUPVMs. We also show that any MUPVM, assisted by one ancillary qubit, yields a QRAC with optimal N-scaling decoding probability. Finally, we propose a new MUPVM-based construction for the (M+2,M)-QRAC family attaining the conjectured bound.
An n↦pm random access code (RAC) is an encoding of n bits into m bits such that any initial bit can be recovered with probability at least p , while in a quantum RAC (QRAC), the n bits are encoded into m qubits. Since its proposal, the idea of RACs was generalized in many different ways, e.g. allowing the use of shared entanglement (called entanglement-assisted random access code, or simply EARAC) or recovering multiple bits instead of one. In this paper we generalize the idea of RACs to recovering the value of a given Boolean function f on any subset of fixed size of the initial bits, which we call f -random access codes. We study and give protocols for f -random access codes with classical ( f -RAC) and quantum ( f -QRAC) encoding, together with many different resources, e.g. private or shared randomness, shared entanglement ( f -EARAC) and Popescu-Rohrlich boxes ( f -PRRAC). The success probability of our protocols is characterized by the noise stability of the Boolean function f . Moreover, we give an upper bound on the success probability of any f -QRAC with shared randomness that matches its success probability up to a multiplicative constant (and f -RACs by extension), meaning that quantum protocols can only achieve a limited advantage over their classical counterparts.