bibkey: liabotro2017improved authors: Ola Liabøtrø year: 2017 title: “Improved Classical and Quantum Random Access Codes” doi: 10.1103/PhysRevA.95.052315 url: https://arxiv.org/abs/1607.02667v2 claim: “For maximal (4^m - 1, m, p) quantum random access codes with the paper’s projective measurements, the worst-case success probability is governed by the most negative eigenvalue of the signed Pauli sums Sigma(beta) = sum_{k=1}^{4^m-1} (-1)^{beta(k)} sigma_{c_1(k)} x … x sigma_{c_m(k)}; the paper checks m = 1, 2 exhaustively and conjectures ||Sigma(beta)|| <= (sqrt(3) + 1)^m - 1 for all m and all beta (Eq. (78)).” strata_touched:

• D5/S3/Quantum/Information/SignedPauliSumNormRefutation license: citation-only triage: anchor

Improved Classical and Quantum Random Access Codes

O. Liabøtrø, arXiv:1607.02667 (v1 2016-07, v2 2016-11-28, the latest); Phys. Rev. A 95, 052315 (2017). Subject: quant-ph.

The paper constructs classical and quantum random access codes encoding n d-levels into m (qu)-d-levels and optimizes their success probabilities. For maximal (4^m − 1, m, p) qubit codes built from the Pauli measurement operators, the worst-case success probability is expressed through −λ, the most negative eigenvalue of the matrices

Σ(β) = Σ_{k=1}^{4^m−1} (−1)^{β(k)} ⊗_{i=1}^{m} σ_{c_i(k)},

where σ_0 = I, σ_1, σ_2, σ_3 are the Pauli matrices, c_i(k) is the i-th base-4 digit of k, and β is any function {1, …, 4^m − 1} → {0, 1}. The matrix I − (I + σ_x + σ_y + σ_z)^{⊗m} gives the bound p ≤ (1 + 1/((1+√3)^m − 1))/2, and the paper states:

We have checked numerically that no $\Sigma (\beta )$ has an eigenvalue less than $1-(1+\sqrt{(}3){)}^m$ for $m=1,2$. This means that we can obtain equality in the bound in these cases. We conjecture that this is the case for all $m$, i.e.: $||{\sum }_{k=1}^{4^m-1}(-1{)}^{\beta (k)}{⨂}_{i=1}^m{\sigma }_{c_i(k)}||\le (\sqrt{3}+1{)}^m-1$

Random searches up to m = 6 found no counterexample (the paper’s table of random eigenvalues).

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.95.052315 (the journal text was not read).
• URL: https://arxiv.org/abs/1607.02667v2 (source retrieved 2026-09-30): paper_v2.tex, the Pauli word notation (l. 266–268), the matrices Σ(β) (l. 619–621), the conjecture (l. 633–638, Eq. (78) of the PDF) and the random searches (l. 640 and the table randomeigenvalues).