Equal-weight low-Choi-rank decompositions of quantum channels

Problem

Can every finite-dimensional quantum channel be written as the uniform mixture of \(d_B\) channels whose Choi ranks are at most the input dimension? Let \(A\) and \(B\) have dimensions \(d_A\) and \(d_B\), respectively, and let \(\Phi:\mathcal L(A)\to\mathcal L(B)\) be completely positive and trace preserving. In a fixed orthonormal basis of \(A\), define its Choi operator by

\begin{equation} J(\Phi) :=\sum_{i,j=1}^{d_A} \lvert i\rangle\!\langle j\rvert_A\otimes \Phi\!\left(\lvert i\rangle\!\langle j\rvert_A\right), \tag{1} \end{equation}

The rank of the operator in Eq. (1) is independent of the chosen basis. Determine whether every \(\Phi\) admits completely positive trace-preserving maps \(\Phi_1,\ldots,\Phi_{d_B}:\mathcal L(A)\to\mathcal L(B)\) satisfying

\begin{equation} \Phi=\frac1{d_B}\sum_{r=1}^{d_B}\Phi_r, \qquad \operatorname{rank}J(\Phi_r)\leq d_A \quad\text{for every }r\in\{1,\ldots,d_B\}. \tag{2} \end{equation}

Thus Eq. (2) requires both the prescribed input-dimension rank bound and exactly equal mixing weights.

Source

Ruskai records the Audenaert–Ruskai channel-decomposition conjecture and its unital and positive-block-matrix formulations explicitly [Rus07]. The Choi-rank bound \(d_A\) in Eq. (2) is the audited correction of the inconsistent \(d_B\) bound printed in the source’s channel formulation; it agrees with the companion formulations and the modern statement studied by Kumar and Wolf [KW26].

Progress

The stronger positive-block-matrix formulation is known when \(d_B=2\): a contraction can be expressed as the midpoint of two unitaries, yielding two positive summands of rank at most \(d_A\) with the prescribed diagonal-block sum. Because that formulation implies the channel decomposition, Eq. (2) holds for every qubit-output channel [RSW02], [Rus07].
Kumar and Wolf prove the equal-weight decomposition in Eq. (2) for every qubit-input channel, for all classical-to-quantum and quantum-to-classical channels, and for a nonzero-measure family in every pair of dimensions. They also prove an unequal-weight version for \(3\to3\) channels; that weaker result does not establish the equal weights required here. Their first-version preprint therefore settles substantial subclasses but not arbitrary \((d_A,d_B)\) [KW26].

Comment

The remaining problem is the equal-weight decomposition in Eq. (2) for arbitrary input and output dimensions outside the proved subclasses. At the source audit cutoff, the broad 2026 partial result was an unrefereed first-version preprint.

References

[Rus07]: M. B. Ruskai, “Open Problems in Quantum Information Theory,” arXiv preprint (2007), Conjectures 2–5, pp. 4–6.DOI arXiv
[RSW02]: M. B. Ruskai, S. Szarek, and E. Werner, “An Analysis of Completely Positive Trace-Preserving Maps on \(M_2\),” Linear Algebra and its Applications 347, 159–187 (2002).DOI arXiv
[KW26]: N. Kumar and M. M. Wolf, “The Ruskai–Audenaert Conjecture & Equipartitions of Positive Operators,” arXiv preprint (2026), version 1.arXiv

Equal-weight low-Choi-rank decompositions of quantum channels

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