Private capacity of the qubit depolarizing channel

Problem

What is the private classical capacity \(P(\Lambda_p)\) of the qubit depolarizing channel, defined for a mixing parameter \(p\in[0,1]\) by

\begin{equation} \Lambda_p(\rho):=(1-p)\,\rho+p\,\mathrm{Tr}[\rho]\,\frac{I}{2} =\bigl(1-\tfrac{3p}{4}\bigr)\rho +\tfrac{p}{4}\bigl(X\rho X+Y\rho Y+Z\rho Z\bigr), \tag{1} \end{equation}

where \(\rho\) ranges over qubit states, \(I\) is the qubit identity, and \(X,Y,Z\) are the Pauli matrices? The second expression in Eq. (1) exhibits the Pauli error probabilities \(\bigl(p_I,p_X,p_Y,p_Z\bigr)=\bigl(1-\tfrac{3p}{4},\tfrac{p}{4},\tfrac{p}{4}, \tfrac{p}{4}\bigr)\), with total Pauli error probability \(\tfrac{3p}{4}\). Fix a Stinespring isometry \(V:A\to BE\) of \(\Lambda_p\) and a finite ensemble \(\{q_x,\rho_x\}_x\) of qubit states, and set \(\omega^{XBE}:=\sum_x q_x\,|x\rangle\!\langle x|^X\otimes V\rho_xV^\dagger\). The one-shot private information and the private classical capacity of the channel are

\begin{equation} P^{(1)}(\Lambda_p):=\max_{\{q_x,\rho_x\}} \bigl[I(X;B)_\omega-I(X;E)_\omega\bigr], \qquad P(\Lambda_p):=\sup_{n\ge1}\frac{1}{n}\, P^{(1)}\bigl(\Lambda_p^{\otimes n}\bigr), \tag{2} \end{equation}

where \(I(\cdot\,;\cdot)_\omega\) is the quantum mutual information and all logarithms are base 2 [Dev05], [CWY04]. Determine the value of \(P(\Lambda_p)\) in Eq. (2) as an explicit function of \(p\) on \([0,1]\). For context, every channel satisfies \(P\ge\mathcal Q\), where \(\mathcal Q\) denotes the quantum capacity, the supremum of achievable coherent-information rates: for any input state \(\rho=\sum_x q_x\,\psi_x\) decomposed into an ensemble of pure states, each \(V\psi_xV^\dagger\) is pure on \(BE\), so its \(B\) and \(E\) marginals have equal entropy, the conditional-entropy terms cancel in \(I(X;B)_\omega-I(X;E)_\omega\), and the private information of the ensemble equals the coherent information \(S(\Lambda_p(\rho))-S(\Lambda_p^{c}(\rho))\) at \(\rho\), where \(S(\cdot)\) is the von Neumann entropy and \(\Lambda_p^{c}(\sigma):=\mathrm{Tr}_B[V\sigma V^\dagger]\) is the complementary channel; the identity holds verbatim for every tensor power, so the private-information maximum dominates the coherent-information maximum for the channel and all its powers, giving \(P\ge\mathcal Q\) [Dev05]. Whether \(P(\Lambda_p)\) and \(\mathcal Q(\Lambda_p)\) differ anywhere on \(0<p<\tfrac{1}{3}\), i.e. whether an ensemble with a nontrivial classical index register strictly beats every pure-state-ensemble rate on this channel, is part of what is open.

Source

The question is stated explicitly by Leditzky, Leung, and Smith, who note that it has been an open problem for more than 20 years to determine the capacities of some of these low-noise channels such as the depolarizing channel [LLS18]. The regularized private information defining the capacity is due to Devetak and, in the wiretap formulation, to Cai, Winter, and Yeung [Dev05], [CWY04].

Progress

The channel in Eq. (1) is antidegradable precisely for \(p\ge\tfrac{1}{3}\), and an antidegradable channel has vanishing private capacity because the environment can reconstruct everything the receiver obtains, so \(P(\Lambda_p)=0\) throughout \(p\ge\tfrac{1}{3}\). Kondra et al. prove for the quantum capacity an all-code exponential strong converse that, in the convention of Eq. (1), reads \(\mathcal Q(\Lambda_p)\le\max\{1-3p,0\}\) and vanishes at \(p=\tfrac{1}{3}\); their result concerns \(\mathcal Q\) and leaves \(P(\Lambda_p)\) undetermined for every \(p<\tfrac{1}{3}\) [KBK+26].
The capacity is bounded below by the quantum capacity and hence by the hashing rate: for the Pauli error vector of Eq. (1),

\begin{equation} P(\Lambda_p)\ \ge\ \mathcal Q(\Lambda_p)\ \ge \max\Bigl\{0,\;1-H_2\bigl(\tfrac{3p}{4}\bigr) -\tfrac{3p}{4}\log_2 3\Bigr\}, \qquad H_2(q):=-q\log_2 q-(1-q)\log_2(1-q), \tag{3} \end{equation}

where \(H_2\) is the binary entropy and the second inequality is the coherent information at the maximally mixed input, itself a private-information rate [Dev05]. The bracket in Eq. (3) is positive below the vanishing point \(\tfrac{3p}{4}\approx0.18929\) and negative above it.
Positivity beyond Eq. (3) is certified through the quantum capacity: Krohn-Grimberghe exhibits a rank-two, permutation-symmetric input across \(45\) channel uses whose coherent information is provably positive at per-Pauli error \(16239/250000\), so \(\mathcal Q(\Lambda_p)>0\), and therefore \(P(\Lambda_p)>0\), whenever \(\tfrac{p}{4}\le\tfrac{16239}{250000}\), that is \(p\le0.259824\) [KG26]. Whether \(P(\Lambda_p)>0\) throughout \((0.259824,\tfrac{1}{3})\) remains open.
For low noise both capacities are known to leading order: Leditzky, Leung, and Smith show that for channels close to the identity, including \(\Lambda_p\) at small \(p\), the private and quantum capacities each equal the single-letter coherent information \(\max_\rho I_c(\rho,\Lambda_p)\) to leading order in the distance to the identity channel, so shielding part of the data with the auxiliary classical register gains nothing to that order; the exact values are left open [LLS18].
The regularization in Eq. (2) cannot be dropped in general: the private capacity is not additive [LWZG09], the private information is superadditive for every number of channel uses [ES15], and for depolarizing channels of growing dimension the superadditivity of the quantum capacity provably weakens [EMC+23]. No blocklength \(n(p)\) at which the regularization stabilizes is known for this channel.
Since the channel is degradable only at \(p=0\) [CRS08], no single-letter degradability formula applies, and the strongest upper bounds come from flagged extensions. Kianvash, Fanizza, and Giovannetti construct degradable flag extensions that gave, at the time, the strongest upper bounds on the quantum and private capacities of the depolarizing channel [KFG22]; Poshtvan and Karimipour upper-bound both capacities of the SO(2)-covariant Pauli family containing the channel of Eq. (1) across its whole parameter space [PK22]; and Nourozi refines the flagged-extension technique into single-letter upper bounds on both capacities with numerical evaluation, describing the depolarizing channel as one whose capacity genuinely requires multi-letter optimization [Nour26]. No known upper bound meets a lower bound at any fixed \(p\in(0,\tfrac{1}{3})\).
An additive upper bound of a different kind comes from squashed entanglement: \(P(\Lambda_p)\le P_2(\Lambda_p)\le E_{\mathrm{sq}}(\Lambda_p)\), where \(P_2\) is the two-way-assisted private capacity and \(E_{\mathrm{sq}}\) the squashed entanglement of the channel, which is additive over tensor products [TGW14]. The exact value of \(E_{\mathrm{sq}}(\Lambda_p)\) is itself unknown; it is archived as a separate problem (see Comment).

Comment

Beyond the zero region \(p\ge\tfrac{1}{3}\), the certified positivity region \(p\le0.259824\), and the leading-order behavior as \(p\to0\), nothing exact is known about \(P(\Lambda_p)\): no upper bound meets a lower bound at any fixed \(p\in(0,\tfrac{1}{3})\); positivity on \((0.259824,\tfrac{1}{3})\) is open; it is open whether the regularization in Eq. (2) stabilizes at some finite blocklength \(n(p)\), the negative answer to the universal-truncation question for channel-independent integers (solved record 01M1Q787QR8FTR00QF4PMHHWPE) not excluding finite stabilization for this particular channel; and it is open whether \(P(\Lambda_p)>\mathcal Q(\Lambda_p)\) somewhere on \(0<p<\tfrac{1}{3}\). The sibling record 01M20868F4GEPX6FBJZF8T0GRJ asks for the squashed entanglement \(E_{\mathrm{sq}}(\Lambda_p)\) of the same channel: it upper-bounds the assisted capacity \(P_2(\Lambda_p)\ge P(\Lambda_p)\), but determining it would neither determine \(P(\Lambda_p)\) nor be determined by it. The record 01M1HME7809M71BG24CSMYKA8A asks for the quantum capacity of the same channel family. The inequality \(P\ge\mathcal Q\) is the only relation between the two values known for all \(p\); both capacities vanish on the antidegradable region \(p\ge\tfrac{1}{3}\) and both equal \(1\) at \(p=0\), so equality is settled there and unproven only on the nontrivial interval \(0<p<\tfrac{1}{3}\).

References

[Dev05]: I. Devetak, “The Private Classical Capacity and Quantum Capacity of a Quantum Channel,” IEEE Transactions on Information Theory 51, 44–55 (2005).DOI arXiv
[CWY04]: N. Cai, A. Winter, and R. W. Yeung, “Quantum privacy and quantum wiretap channels,” Problems of Information Transmission 40, 318–336 (2004).DOI
[CRS08]: T. S. Cubitt, M.-B. Ruskai, and G. Smith, “The structure of degradable quantum channels,” Journal of Mathematical Physics 49, 102104 (2008).DOI arXiv
[LLS18]: F. Leditzky, D. Leung, and G. Smith, “Quantum and private capacities of low-noise channels,” Physical Review Letters 120, 160503 (2018).DOI arXiv
[KFG22]: F. Kianvash, M. Fanizza, and V. Giovannetti, “Bounding the quantum capacity with flagged extensions,” Quantum 6, 647 (2022).DOI arXiv
[TGW14]: M. Takeoka, S. Guha, and M. M. Wilde, “The squashed entanglement of a quantum channel,” IEEE Transactions on Information Theory 60(8), 4987–4998 (2014).DOI arXiv
[PK22]: A. Poshtvan and V. Karimipour, “Capacities of the covariant Pauli channel,” Physical Review A 106, 062408 (2022).DOI arXiv
[LWZG09]: K. Li, A. Winter, X. Zou, and G.-C. Guo, “The private capacity of quantum channels is not additive,” Physical Review Letters 103, 120501 (2009).DOI arXiv
[ES15]: D. Elkouss and S. Strelchuk, “Superadditivity of private information for any number of uses of the channel,” Physical Review Letters 115, 040501 (2015).DOI arXiv
[EMC+23]: J. Etxezarreta Martinez, A. de Marti i Olius, and P. M. Crespo, “The superadditivity effects of quantum capacity decrease with the dimension for qudit depolarizing channels,” Physical Review A 108, 032602 (2023).DOI arXiv
[Nour26]: V. Nourozi, “Flagged Extensions and Numerical Simulations for Quantum Channel Capacity: Bridging Theory and Computation,” arXiv preprint (2026).arXiv
[KBK+26]: T. V. Kondra, R. Brinster, H. Kampermann, D. Bruß, and N. Wyderka, “Sharp Quantum Capacity Thresholds: Exponential Strong Converses for Degradable and Antidegradable Channels,” arXiv preprint (2026).arXiv
[KG26]: A. Krohn-Grimberghe, “A Certified Lower Bound on the Quantum-Capacity Threshold of the Depolarizing Channel,” arXiv preprint (2026).arXiv

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