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PreprintCC BY 4.0arXiv

Optimal Average Success Probabilities of Binary (n,n−1) and (n,n−2) Quantum Random Access Codes

  • Shuo Tan
  • Syed A. Jafar
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Abstract

A binary (n,m) quantum random access code (QRAC) compresses an n-bit classical string into an m-qubit quantum state, from which a decoder attempts to recover a randomly selected target bit. Of particular interest is the optimal average probability of success, PQ,avg,optn,m, which was numerically conjectured to satisfy the bound PQ,avg,optn,m≤ 1/2+1/2√(m/n). Recent constructions of (n,n−1) QRACs by Suzuki and (n,n−2) QRACs by Akibue et al. meet this bound exactly, raising the question of their strict optimality. In this work, we settle this question by proving the conjectured upper bound for m∈{n−1,n−2}, thereby precisely determining PQ,avg,optn,n−1 and PQ,avg,optn,n−2. The proof utilizes a translation recently studied by Lin and de Wolf from local to global reconstruction via pretty good measurement, along with dimensional and positive-semidefinite constraints on an induced channel.

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Cite

BibTeXRIS
BibTeX
@misc{tan2026optimal,
  title = {{Optimal Average Success Probabilities of Binary $(n,n-1)$ and $(n,n-2)$ Quantum Random Access Codes}},
  author = {Shuo Tan and Syed A. Jafar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.10414},
  eprint = {2607.10414},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.10414},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Optimal Average Success Probabilities of Binary (n,n−1) and (n,n−2) Quantum Random Access Codes
AU  - Shuo Tan
AU  - Syed A. Jafar
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2607.10414
UR  - https://arxiv.org/abs/2607.10414
AB  - A binary (n,m) quantum random access code (QRAC) compresses an n-bit classical string into an m-qubit quantum state, from which a decoder attempts to recover a randomly selected target bit. Of particular interest is the optimal average probability of success, P^Q,avg,opt_n,m, which was numerically conjectured to satisfy the bound P^Q,avg,opt_n,m≤ 1/2+1/2√(m/n). Recent constructions of (n,n−1) QRACs by Suzuki and (n,n−2) QRACs by Akibue et al. meet this bound exactly, raising the question of their strict optimality. In this work, we settle this question by proving the conjectured upper bound for m∈{n−1,n−2}, thereby precisely determining P^Q,avg,opt_n,n−1 and P^Q,avg,opt_n,n−2. The proof utilizes a translation recently studied by Lin and de Wolf from local to global reconstruction via pretty good measurement, along with dimensional and positive-semidefinite constraints on an induced channel.
ER  -