# Stronger submultiplicativity for the diamond norm

Cite this problem as Problem 37.

### Problem

Does there exist an absolute constant such that the submultiplicativity statement

holds for all quantum channels over any finite dimensional Hilbert space? Here denotes the transposition map, so equals the Hilbert space dimension.

### Background

It has been shown in [1] that the -quantum capacity of any quantum channel satisfies , when the error of the coding scheme is to be at most . The proof uses ordinary submultiplicativity of the diamond norm, corresponding to . If submultiplicativity held with some uniformly for all channels, the derivation from [1] would establish the same capacity bound for a larger range of errors . The original hope was to show that submultiplicativity holds with , which would have given a \emph{strong converse rate} via [1].

For channels in any <em>fixed</em> dimension , a certain constant suffices for submultiplicativity by continuity arguments, since the transpose map attains its diamond norm essentially only at the maximally mixed state whereas the output of is always traceless.

### Partial results

For qubit-qubit channels, holds with <em>equality</em>, i.e.\ 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 .

By random testing in dimensions at least three and evaluating the diamond norms [2], one finds channels that need [3]. For unitarily implemented channels , analytic lower bounds in terms of the spectrum of the unitary give that is required as the Hilbert space the dimension grows [3]. Larger required values of 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].

### References

[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.