Exactly solvable nondegradable quantum channels
- Field
- Topics
Problem
Does there exist a finite-dimensional quantum channel that is neither degradable nor antidegradable and whose unassisted quantum capacity is known exactly? For a channel \(\mathcal N_{A\to B}\) with complementary channel \(\mathcal N^c_{A\to E}\), degradability and antidegradability mean, respectively, that there is a channel \(\mathcal D\) or \(\mathcal A\) satisfying
The channel sought must satisfy neither identity in Eq. (1). Its quantum capacity is defined by the coherent-information regularization
The question asks for an explicit channel outside both classes in Eq. (1) together with an exact evaluation of Eq. (2).
Source
Wilde explicitly identified finding the quantum capacity of a nondegradable channel as an important open challenge [Wil17].
Progress
Chessa and Giovannetti supplied a simple explicit qutrit solution to the existential problem. For \(1/2<\gamma<1\), define a channel \(\mathcal N_\gamma\) by the Stinespring isometry
\begin{equation} \begin{aligned} V_\gamma|0\rangle_A&=|0\rangle_B|0\rangle_E,\\ V_\gamma|1\rangle_A &=\sqrt{1-\gamma}\,|1\rangle_B|0\rangle_E +\sqrt{\gamma}\,|0\rangle_B|1\rangle_E,\\ V_\gamma|2\rangle_A&=|2\rangle_B|0\rangle_E, \qquad \mathcal N_\gamma(\rho):= \operatorname{Tr}_E[V_\gamma\rho V_\gamma^\dagger]. \end{aligned} \tag{3} \end{equation}They proved that the channel in Eq. (3) is neither degradable nor antidegradable throughout this parameter range and nevertheless has
\begin{equation} Q(\mathcal N_\gamma)=1. \tag{4} \end{equation}The noiseless subspace \(\operatorname{span}\{|0\rangle,|2\rangle\}\) gives the achievable qubit in Eq. (4); their channel analysis supplies the matching upper bound [CG21].
Earlier work of D’Arrigo, Benenti, Falci, and Macchiavello determined the quantum capacity and studied the degradability properties of a fully correlated two-qubit amplitude-damping channel. It is a historical antecedent to the multilevel constructions, but the qutrit witness in Eq. (3) independently suffices for the existential question here [DBF+13].
Smith and Wu later constructed broader families of genuinely nondegradable channels with additive coherent information, hence with quantum capacities given by a one-use optimization. Their results provide further exactly solvable examples but postdate the family in Eq. (3) [SW25].
Comment
The family in Eq. (3) solves the literal existence problem. It does not provide a general single-letter formula or a classification of nondegradable channels with additive coherent information.