Problem 11: Continuity of the Quantum channel capacity

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There are different notions of capacity for a noisy quantum communication channel, e. g. for classical, private classical or quantum communications. Continuity of these is an important property, but from a mathematical point of view it is not at all obvious, because very similiar channels can differ a lot given many copies, and the capacity is operationally defined in terms of an asymptotic number of channel uses. Of course this is not a problem when a single-letter formula is avaible, so that one can reason about it directly, but it becomes quite a challenge, when only a multi-letter formula is avaible, or even non at all.


From a pratical point of view, we wish for continuity, because in real systems there is always some channel uncertainty. So if nearby channels had dramatically different capacities, the theory of quantum capacities would be of limited value. Listed below are definitions of different capacities

  • Classical Capacity: A rate R is ε-classically-achievable if there is an nε such that for all n \geq n_{\epsilon} there is a classical code  \{\rho_k \in A^{\otimes n}\}_{k=1}^{K_n} and a decoding operation D_n:B^{\otimes n} \rightarrow \{|k><k|\}_{k=1}^{K_n} such that for all k,  ||D_n(N^{\otimes n}(\rho_k))-|k><k|||_1 \leq \epsilon with log K_n \geq nR. A rate is classically-achievable if it is ε-classically-achievable for all ε > 0. The classical capacity of N, C(N), is the supremum over classically-achievable rates.
  • Quantum Capacity: A rate R is ε-achievable if there is an nε such that for all n \geq n_{\epsilon} there is a quantum code, C_n \subset A^{\otimes n} and a decoding operation D_n:B^{\otimes n} \rightarrow C_n such that for all \Psi \in B(C_n), ||D_n(N^{\otimes n}(\Psi))-\Psi||_1 \leq \epsilon and log dim H_{C_n} \geq nR. A rate is achievable if it is ε-achievable for all ε > 0. The quantum capacity of N, Q(N), is the supremum over achievable rates.
  • Private Capacity: The private capacity is the capacity of a channel for classical communication with the added requirement that an adversary with access to the environment of the channel is ignorant of the communication. More formally a rate R is ε-privately-achievable if there is an nε such that for all n \geq n_{\epsilon} there is a classical code \{\rho_k \in A^{\otimes n}\}_{k=1}^{K_n} with log K_n \geq nR and a decoding operation D_n:B^{\otimes n} \rightarrow \{|k><k|\}_{k=1}^{K_n} such that for all k, ||D_n(N^{\otimes n}(\rho_k))-|k><k|||_1 \leq \epsilon and ||\rho_{E^{\otimes n}}^k - \sigma_{E^{\otimes n}}||_1 \leq \epsilon. Here \rho_{E^{\otimes n}}= \hat{N}^{\otimes n}(\rho_k) , where \hat{N}(\rho) = tr_B U \rho U^{\dagger}, with  U: H_A \rightarrow H_B \otimes H_E an isometric extension of N, and \sigma_{E^{\otimes n}} is a fixed state on E^{\otimes n} . A rate is privately-achievable if it is ε-privately-achievable for all ε > 0. The private capacity of N, Cp(N), is the supremum over privately-achievable rates.

Partial Results

On the way to prove continuity of channel capacities, various intermediate results could be achieved. These contain results for capacities itself or quantities related to it, i. e. entropic quantities and entanglement measures.

  • In [BVS] the continuity of the quantum channel capacity was assumed to upper bound the capacity of the quantum erasure channel, which was rigoroursly justified later in [BST].
  • In [KW] it was shown, that the quantum channel capacity is lower semi-continuous.
  • In [S2] the continuity of the Holevo information was considered (which is connected to the classical capacity through regularization) and it was shown that it is continuous for finite dimensional outputs and lower semi-continuous in general.
  • In [F] a tight bound on the variation of von Neumann entropy of finite dimensional states was found.
  • In [N] the latter was used to study the entanglement of formation.
  • In [DH] the continuity of the relative entropy of entanglement was proven.
  • In [V] asymptotic entanglement measures were proven to be continuous in any open set of distallable states.
  • In [AF] this continuity result was generalized to conditional entropy, which was used in [CW] to prove the continuity of squashed entanglement.


The problem was essentially solved in [LS]. There it was shown that the classical capacity, the quantum capacity and the private classical capacity are contiuous (with respect to the diamond norm), where the variation on arguments ε apart is bounded by a simple function of ε and the channel's output dimension. Furthermore for quantum capacities in the presence of free backward or two-way classical communication, conitinuity was proven on the interior of the set of non-zero capacity channels.


[BVS] C. H. Bennett, D. P. DiVincenzo, and J. A. Smolin, Phys. Rev. Lett. 78, 3217 (1997).
[BST] H. Barnum, A. J. Smolin, and B. Terhal, Phys. Rev. A 58, 3496 (1998).
[KW] M. Keyl and R. Werner, Lecture Notes in Physics 611, 263 (2002).
[S2] M. Shirokov, Comm. Math. Phys. 262, 137 (2006).
[F] M. Fannes, Comm. Math. Phys. 31, 291 (1973).
[N] M. A. Nielsen, Phys. Rev. A 61, 064301 (2000), arxiv:quant-ph/9808086.
[DH] M. Donald and M. Horodecki, Phys. Lett. A 264, 257 (1999), arxiv:quant-ph/9910002.
[V] G. Vidal, arxiv:quant-ph/0203107.
[AF] R. Alicki and M. Fannes, J. Phys. A: Math. Gen. 37, L55 (2004).
[CW] M. Christandl and A. Winter, J. Math. Phys. 45, 829 (2004), arxiv:quant-ph/0308088.
[LS] D. Leung and G. Smith, Commun. Math. Phys. 292, 201–215 (2009).