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arXiv:1912.09900 · Physical Review A

Sufficient conditions for quantum advantage in random access code protocols with two-qubit states

Som Kanjilal, Chellasamy Jebarathinam, Tomasz Paterek, Dipankar Home

Published Jul 19, 2023• 5 citations
arXivHTMLPDF

Abstract

A random access code (RAC) is an important communication protocol to obtain information about a randomly specified substring of an $n$-bit string, while only having limited information about the $n$-bit string. Quantum RACs usually utilize either communication of quantum bits or a shared-in-advance quantum state used in conjunction with classical communication. Here we consider the latter version of the quantum protocols under the constraint of single-bit communication and with a shared arbitrary state of two qubits. Taking the worst-case success probability as the figure of merit, we demonstrate that any state with an invertible correlation matrix can be used to outperform the best classical RAC for $n=3$. We derive an additional condition sufficient to beat the best classical performance in the case of $n=2$. In particular, separable states turn out to be a useful resource behind the quantum advantage for $n=2,3$. For $n\ensuremath{\ge}4$, RACs assisted with a single copy of a quantum state do not outperform the classical RACs.

DOI
10.1103/physreva.108.012617

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A random access code (RAC) is an important communication protocol to obtain information about a randomly specified substring of an nnn-bit string, while only having limited information about the nnn-bit string. Quantum RACs usually utilize either communication of quantum bits or a shared-in-advance quantum state used in conjunction with classical communication. Here we consider the latter version of the quantum protocols under the constraint of single-bit communication and with a shared arbitrary state of two qubits. Taking the worst-case success probability as the figure of merit, we demonstrate that any state with an invertible correlation matrix can be used to outperform the best classical RAC for n=3n=3n=3. We derive an additional condition sufficient to beat the best classical performance in the case of n=2n=2n=2. In particular, separable states turn out to be a useful resource behind the quantum advantage for n=2,3n=2,3n=2,3. For n\ensuremath≥4n\ensuremath{\ge}4n\ensuremath≥4, RACs assisted with a single copy of a quantum state do not outperform the classical RACs.
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.
We develop a family of semi-device-independent (SDI) entanglement-assisted prepare-measure (PM) communication games involving two parties, within the n→ln\rightarrow ln→l random-access code (RAC) framework where the sender Alice holds a n-bit string and communicates l<nl<nl<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→l4\rightarrow l4→l entanglement-assisted PMRAC with l=1l=1l=1 and 222 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→l5\rightarrow l5→l entanglement-assisted PMRAC with l=1,2l=1,2l=1,2 and 333. Further, we extend the demonstration of quantum advantage for n→n−2n\rightarrow n-2n→n−2 case where n is arbitrary.
Abstract 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.
Random access code (RAC), a primitive for many information processing protocols, enables one party to encode a n -bit string into one bit of message such that another party can retrieve partial information of that string. We introduce the multiparty version of RAC in which the n -bit string is distributed among many parties. For this task, we consider two distinct quantum communication scenarios: one allows shared quantum entanglement among the parties with classical communication, and the other allows communication through a quantum channel. We present several multiparty quantum RAC protocols that outclass its classical counterpart in both the aforementioned scenarios.
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.
A random access code (RAC) encodes an LLL-bit string into a kkk-bit message, L>kL>kL>k, so that any requested bit can be recovered with high probability; a quantum RAC (QRAC) uses kkk qubits instead. We give a geometric characterization of optimal classical (L,k)(L,k)(L,k)-RACs under average and worst-case decoding criteria. The average criterion is reduced to choosing 2k2^k2k representatives in {0,1}L\{0,1\}^L{0,1}L, while the worst-case criterion is reduced to a minimax problem over 2k2^k2k points in [0,1]L[0,1]^L[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>1L>1L>1, we construct a (L,1)(L,1)(L,1)-QRAC whose average decoding success probability strictly exceeds the optimal classical value. Second, for the family (2k−1,k)(2^k-1,k)(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)(L,L-1)(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)(L,L-1)(L,L−1)-QRACs attaining a previously conjectured upper-bound value.