An $(n,m,p)$ random access code (RAC) makes it possible to encode $n$ bits in an $m$-bit message in such a way that a receiver of the message can guess any of the original $n$ bits with probability $p$ greater than $\frac{1}{2}$. In quantum RACs (QRACs), one transmits $n$ qubits. The full set of primitive entanglement-assisted random access codes (EARACs) is introduced, in which parties are allowed to share a two-qubit singlet. It is shown that via a concatenation of these, one can build for any $n$ an $(n,1,p)$ EARAC. QRACs for $n>3$ exist only if parties also share classical randomness. We show that EARACs outperform the best of known QRACs not only in the success probabilities but also in the amount of communication needed in the preparatory stage of the protocol. Upper bounds on the performance of EARACs are given and shown to limit also QRACs.
An (n,m,p) random access code (RAC) makes it possible to encode n bits in an m-bit message in such a way that a receiver of the message can guess any of the original n bits with probability p greater than 21. In quantum RACs (QRACs), one transmits n qubits. The full set of primitive entanglement-assisted random access codes (EARACs) is introduced, in which parties are allowed to share a two-qubit singlet. It is shown that via a concatenation of these, one can build for any n an (n,1,p) EARAC. QRACs for n>3 exist only if parties also share classical randomness. We show that EARACs outperform the best of known QRACs not only in the success probabilities but also in the amount of communication needed in the preparatory stage of the protocol. Upper bounds on the performance of EARACs are given and shown to limit also QRACs.
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.
$n\ensuremath{\rightarrow}m$ random access code (RAC) is an information task to encode an n-bit random message into a shorter m-bit string. Resulting from the compression loss from n to m bits, there exists success probabilities for the string receiver to recover any bits of the initial message. It has been shown that success probabilities of bit recovery can be enhanced by employing quantum resources. Here we present experimental realizations of $2\ensuremath{\rightarrow}1$ and $3\ensuremath{\rightarrow}1$ distributed entanglement-assisted random access codes (EARACs) with a two-photon entangled source. The average success probabilities obtained in our experiment are 0.8491(41) for a $2\ensuremath{\rightarrow}1$ EARAC and 0.7724(34) for a $3\ensuremath{\rightarrow}1$ EARAC, exceeding the classical counterparts, respectively. Moreover, through the existing experimental data, success probabilities for n varying from 4 to 10 (m=1) are estimated.
We develop a family of semi-device-independent (SDI) entanglement-assisted prepare-measure (PM) communication games involving two parties, within the n→l random-access code (RAC) framework where the sender Alice holds a n-bit string and communicates l<n bits or qubits to the receiver Bob. In contrast to the standard quantum PMRAC, here the parties share a prior entanglement, and Alice applies quantum operations on her sub-system to encode her inputs and sends to Bob. We first consider the 4→l entanglement-assisted PMRAC with l=1 and 2 and derive the optimal quantum success probabilities using an elegant analytical technique. We demonstrate quantum supremacy over both classical RACs and conventional quantum PMRACs. Moreover, we exhibit that the optimal quantum advantage allows one to certify Alice's unitary operations. We then derive an upper bound on the quantum success probabilities for 5→l entanglement-assisted PMRAC with l=1,2 and 3. Further, we extend the demonstration of quantum advantage for n→n−2 case where n is arbitrary.
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. Such a code is denoted by the triple (n,m,p). It is known that n<4m for all QRACs and n<2m for classical RACs. These bounds are also known to be tight, as explicit constructions exist for n=4m−1 and n=2m−1 for quantum and classical codes respectively. We generalize (Q)RACs to a scheme encoding nd-levels into m (qu)-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≤d−1d2m−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=4m−1,m,p) QRACs, p=21+(3+1)m−11 is possible and show that it is an upper bound for the measure operators that we use. When we compare (n,m,pq) QRACs with classical (n,2m,pc) RACs, we can always find pq≥pc, 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.
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.
We consider a communication method, where the sender encodes n classical bits into 1 qubit and sends it to the receiver who performs a certain measurement depending on which of the initial bits must be recovered. This procedure is called (n,1,p) quantum random access code (QRAC) where p > 1/2 is its success probability. It is known that (2,1,0.85) and (3,1,0.79) QRACs (with no classical counterparts) exist and that (4,1,p) QRAC with p > 1/2 is not possible. We extend this model with shared randomness (SR) that is accessible to both parties. Then (n,1,p) QRAC with SR and p > 1/2 exists for any n > 0. We give an upper bound on its success probability (the known (2,1,0.85) and (3,1,0.79) QRACs match this upper bound). We discuss some particular constructions for several small values of n. We also study the classical counterpart of this model where n bits are encoded into 1 bit instead of 1 qubit and SR is used. We give an optimal construction for such codes and find their success probability exactly--it is less than in the quantum case. Interactive 3D quantum random access codes are available on-line at http://home.lanet.lv/~sd20008/racs .