Quantum Advantages in (n,d)->1 Random Access Codes

TL;DR

The paper investigates quantum advantages in (n,d)→1 random access codes (RACs). It rigorously characterizes optimal classical strategies, proving MEID is optimal, and shows that a QRAC using two incompatible measurements can outperform classical strategies, while a single-measurement QRAC cannot. The authors introduce a two-measurement QRAC protocol based on mutually unbiased bases, derive closed-form quantum success probabilities for small n (n=2,3,4) and arbitrary d, and provide a general algorithm to compute quantum values for broader (n,d); they report positive quantum advantages in extensive tested regimes and conjecture a general advantage for all (n,d). The work advances understanding of minimal-resource quantum advantages in preparation-and-measurement tasks and informs design of QRAC protocols in quantum communication. It also situates its findings within broader bounds and related results, highlighting both the potential and the open questions for higher measurement settings.

Abstract

A random access code (RAC), corresponding to a communication primitive with various applications in quantum information theory, is an instance of a preparation-and-measurement scenario. In this work, we consider (n,d)-RACs constituting an "n"-length string, constructed from a "d" size set of letters, and send an encoding of the string in a single d-level physical system and present their quantum advantages. We first characterize optimal classical RACs, proving that the well-known classical strategy known as majority-encoding-identity-decoding is indeed optimal. We then construct a quantum protocol by exploiting only two incompatible measurements, the minimal requirement, and show the advantages beyond the classical one. We also discuss the generality of our results and whether quantum advantages are valid for all types of (n, d)->1 RACs.

Figures (7)

• Figure 1: In a $(n,d)\!\mapsto\! 1$ RAC: Alice receives a uniformly-random $n$-letter word $x=x_1x_2...x_n \in X^n$ where $X=\{0,1,...,d\!\!-\!\!1\}$. Bob receives uniformly at random an index $j\in \{1,2,...,n\}$ and his task is to guess the letter $x_j$ in Alice's word. Alice can encode her word and communicate to Bob only a single $d$-dimentional quantum or classical system $\mathcal{E}(x)$. The success of the task is measured by the average probability of a correct guess. A joint strategy for the task consists of an encoding $\mathcal{E}(x)$ by Alice and a decoding $\mathcal{D}(\mathcal{E}(x),j)$ by Bob.
• Figure 2: A revision of a decoding strategy is attempted by replacing elements, $f \mapsto \widetilde{f}$.
• Figure 3: Partitions of the set of all words $X^n$. The effect of revising the decoding $f \rightarrow \widetilde{f}$ are as follows: (i) does not affect the parts $\overline{W}_f$, $W^{\overline{\{a,b\}}}_f$, and $\overline{A}_f$ (regions shaded grey), (ii) the value of each word in $W^b_f$ is increased by one unit (region shaded green), (iii) the value of each word in $A_f$ is decreased by one unit (region shaded red), (iv) there is an injective map $A_f \mapsto W^b_f$ implying $\vert W^b_f\vert \geq \vert A_f\vert$.
• Figure 4: Quantum advantages in $(3,d)\!\mapsto\! 1$ RACs for $d\leq 100$ are shown. The proposed quantum protocol applies two MUBs and shows a success probability strictly higher than classical optimal ones.
• Figure 5: Quantum advantages in $(4,d)\!\mapsto\! 1$ RACs for $d\leq 100$ are shown. The quantum protocol with two MUBs shows a success probability strictly higher than classical optimal ones.

Theorems & Definitions (20)

• Definition 1: Similarity between two strings
• Definition 2: Value of a word $x\in X^n$ for strategy $f$ and value of the strategy
• Lemma 1
• proof : Proof
• Lemma 2
• proof : Proof
• Lemma 3
• proof : Proof
• Theorem 1
• Theorem 2