Multiplicativity for polarized Werner–Holevo channels

Unsolved ID op_ad05396ff490713c Last edited 4 September 2026

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Field: Quantum Communication
Topics: Additivity and regularization
Quantum channel structure
Matrix and entropy inequalities

Problem

For every integer \(d\geq3\), every \(x\in(0,1)\), and every \(1<p<2\), is the maximal output Schatten \(p\)-norm of the polarized Werner–Holevo channel multiplicative on two identical copies? Define the Werner–Holevo channel and its polarized interpolation with the identity channel by

\begin{equation} \mathcal W_d(X):=\frac{\operatorname{Tr}(X)I_d-X^{\mathsf T}}{d-1}, \qquad \Phi_{x,d}:=x\,\operatorname{id}_d+(1-x)\mathcal W_d, \tag{1} \end{equation}

where the transpose in Eq. (1) is taken in a fixed basis. For a channel \(\Phi\) with \(d\)-dimensional input, set

\begin{equation} \lVert A\rVert_p:=\bigl(\operatorname{Tr}\lvert A\rvert^p\bigr)^{1/p}, \qquad \nu_p(\Phi):=\max_{\rho\in\mathcal D(\mathbb C^d)} \lVert\Phi(\rho)\rVert_p. \tag{2} \end{equation}

With the convention in Eq. (2), determine whether

\begin{equation} \nu_p(\Phi_{x,d}\otimes\Phi_{x,d}) =\nu_p(\Phi_{x,d})^2 \tag{3} \end{equation}

holds throughout the stated parameter range.

Source

Ruskai explicitly asked for Eq. (3) for \(x\in[0,1]\) and \(1\leq p\leq2\) [Rus07]. The formulation above removes all regimes settled by the results below.

Progress

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For \(d=2\), the channel in Eq. (1) is a unital qubit channel, so King’s theorem gives multiplicativity with an arbitrary companion for every \(p\geq1\). The cases \(p=1\) and \(x=1\) are also immediate from trace preservation and the identity channel, respectively [Kin02].
At \(x=0\), Datta proved multiplicativity for two Werner–Holevo channels of arbitrary dimensions throughout \(1\leq p\leq2\). This settles the unpolarized endpoint but not any \(x\in(0,1)\) [Dat04].
Michalakis proved Eq. (3) at \(p=2\) for every \(d\geq2\) and every \(x\in[0,1]\). The proof is specific to the output \(2\)-norm and leaves \(1<p<2\) open [Mic07].

Comment

The only unresolved regime of the source problem is precisely \(d\geq3\), \(x\in(0,1)\), and \(1<p<2\), with two identical copies as in Eq. (3).

References

[Rus07]: M. B. Ruskai, “Open Problems in Quantum Information Theory,” arXiv preprint arXiv:0708.1902 (2007).DOI arXiv
[Kin02]: C. King, “Additivity for Unital Qubit Channels,” Journal of Mathematical Physics 43, 4641–4653 (2002).DOI arXiv
[Dat04]: N. Datta, “Multiplicativity of Maximal \(p\)-Norms in Werner–Holevo Channels for \(1\leq p\leq2\),” arXiv preprint quant-ph/0410063 (2004).arXiv
[Mic07]: S. Michalakis, “Multiplicativity of the Maximal Output \(2\)-Norm for Depolarized Werner–Holevo Channels,” Journal of Mathematical Physics 48, 122102 (2007).DOI arXiv

Multiplicativity for polarized Werner–Holevo channels

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