Stronger submultiplicativity for the diamond norm

Cite this problem as Problem 37.


Does there exist an absolute constant $\alpha<1$ such that the submultiplicativity statement

    \begin{equation*} \|\Theta\circ({\rm id}-T)\|_\diamond\leq\alpha\|\Theta\|_\diamond\|{\rm id}-T\|_\diamond \end{equation*}

holds for all quantum channels $T$ over any finite dimensional Hilbert space? Here $\Theta$ denotes the transposition map, so $\|\Theta\|_\diamond$ equals the Hilbert space dimension.


It has been shown in [1] that the $\varepsilon$-quantum capacity of any quantum channel $C$ satisfies $Q_\varepsilon(C)\leq\log\|\Theta\circ C\|_\diamond$, when the error $\|{\rm id}_2^{\otimes n}-D\circ C^{\otimes m}\circ E\|_\diamond/2$ of the coding scheme is to be at most $\varepsilon<1/2$. The proof uses ordinary submultiplicativity of the diamond norm, corresponding to $\alpha=1$. If submultiplicativity held with some $\alpha\leq1$ uniformly for all channels, the derivation from [1] would establish the same capacity bound for a larger range of errors $\varepsilon\in[0,1/2\alpha)$. The original hope was to show that submultiplicativity holds with $\alpha=1/2$, which would have given a \emph{strong converse rate} via [1].

For channels $T$ in any <em>fixed</em> dimension $d$, a certain constant $\alpha=\alpha(d)<1$ suffices for submultiplicativity by continuity arguments, since the transpose map $\Theta$ attains its diamond norm essentially only at the maximally mixed state whereas the output of $({\rm id}-T)$ is always traceless.

Partial results

For qubit-qubit channels, $\|{\rm id}-T\|_\diamond=\|\Theta\circ({\rm id}-T)\|_\diamond$ holds with <em>equality</em>, i.e.\ $\alpha(2)=1/2$ is the best and exact constant. Furthermore, for channels that are symmetric w.r.t.\ the orthogonal group O(d) or any representations of SU(2) at the in- and output, the submultiplicativity inequality holds with $\alpha=1/2$.

By random testing in dimensions at least three and evaluating the diamond norms [2], one finds channels that need $\alpha>1/2$ [3]. For unitarily implemented channels $T(\rho)=U\rho U^*$, analytic lower bounds in terms of the spectrum of the unitary give that $\alpha>0.63$ is required as the Hilbert space the dimension grows [3]. Larger required values of $\alpha$ have not been observed so far.

The aforementioned strong converse rate result for the quantum capacity (even with free two-way classical communication) has been established by different means in [4].


[1] A. S. Holevo and R. F. Werner, “Evaluating capacities of bosonic Gaussian channels”, Phys.\ Rev.\ A 63, 032312, or arXiv:quant-ph/9912067.

[2] J. Watrous, “Simpler semidefinite programs for completely bounded norms”, Chicago Journal of Theoretical Computer Science 2013, 08, 1-19, or arXiv:1207.5726.

[3] D. Fischer, “Quantum capacity bounds via semidefinite programming”, Bachelor’s Thesis, Leibniz University of Hannover (Germany), 2016.

[4] A. Mueller-Hermes, D. Reeb, M. Wolf, “Positivity of linear maps under tensor powers”, J. Math. Phys. 57, 015202 (2016), or arXiv:1502.05630.