Quantum capacity of higher-dimensional depolarizing channels
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Problem
What is the unassisted quantum capacity of a depolarizing channel in every finite dimension \(d\geq3\)? For \(0\leq\lambda\leq1\), define the channel on \(\mathcal L(\mathbb C^d)\) by
The capacity \(Q(\mathcal N)\) is the supremum of asymptotic qubit rates achievable with entanglement fidelity tending to one by arbitrary encoders and decoders over independent uses, without entanglement or classical-communication assistance. For a complementary channel \(\mathcal N^c\) from any Stinespring dilation and \(S(\sigma)=-\operatorname{Tr}\sigma\log_2\sigma\), it equals
Determine Eq. (2) for the family in Eq. (1), including the exact boundary between zero and positive capacity. The unresolved parameter regime is
An evaluation of one input state or one finite block length alone does not determine the supremum in Eq. (2) throughout Eq. (3).
Source
The supplied note derives this question from Hayashi’s treatment of quantum capacity. Etxezarreta Martinez, deMarti iOlius, and Crespo explicitly state the unresolved capacity problem for the finite-dimensional depolarizing family in Section I, p. 2, and define the channel in Section II, Eq. (3) [EMC23]. Their replacement-noise parameter is \(p=1-\lambda\).
Progress
The channel has zero capacity for \(1-\lambda\geq d/[2(d+1)]\) because it is antidegradable; at \(\lambda=1\) it has capacity \(\log_2 d\). Put \(p=1-\lambda\) and \(q=p(1-d^{-2})\). The maximally mixed input gives the hashing lower bound, while the no-cloning bound gives
\begin{equation} \max\{0,\log_2 d-h_2(q)-q\log_2(d^2-1)\}\leq Q(\mathcal N_{d,\lambda})\leq\max\!\left\{0,1-\frac{2(d+1)p}{d}\right\}\log_2 d, \tag{4} \end{equation}where \(h_2(q)=-q\log_2q-(1-q)\log_2(1-q)\). These bounds in Eq. (4) leave a gap in Eq. (3); see Section II, Eqs. (4)–(6) [EMC23].
Kianvash, Fanizza, and Giovannetti construct degradable flagged extensions giving improved upper bounds for qudit depolarizing channels. Their Section 5.4 and Figure 2 include a nontrivial dimension-four comparison; optimizing these extension bounds does not supply matching transmission codes [KFG22].
In the limit of increasing dimension, the no-cloning and hashing bounds meet after division by \(\log_2 d\): for fixed \(p\), the normalized capacity tends to \(\max\{0,1-2p\}\). Theorem 1 and Corollary 1 analyze this normalized gap. This asymptotic statement neither evaluates the capacity for a fixed finite \(d\geq3\) nor proves that the absolute gap in qubits vanishes [EMC23].
Comment
Audited on 2026-09-09 against the full cited papers and subsequent capacity literature. The existing qubit Pauli-capacity record already contains the dimension-two depolarizing problem and its July–August 2026 coding progress; the present record retains only the higher-dimensional remainder. No existing general qudit-Pauli-capacity statement was found. The depolarizing channel’s additive classical Holevo capacity is a different quantity and does not remove the coherent-information regularization in Eq. (2). The large-dimension result is in qudits per channel use, obtained by normalizing rates by \(\log_2 d\), and supplies no fixed-dimension solution.