Quantum-Channel Interpolation Characterization

Definitions and mathematical statement

Under the conventional complex-matrix interpretation, let $n,m\geq 1$ be integers, let $M_n=M_n({\mathbb{C}})$, and write $M_n^{\mathrm{sa}}=\{X\in M_n:X^*=X\}$. Let \[ A=(A_1,\ldots,A_m),\qquad B=(B_1,\ldots,B_m)\in(M_n^{\mathrm{sa}})^m. \] The scalar field and the ranges of $n,m$ are implicit in the supplied statement; the conventions above make its usual matrix interpretation explicit.

A complex-linear map $\Phi:M_n\to M_n$ is completely positive if, for every integer $r\geq 1$, the map $\operatorname{id}_{M_r}\otimes\Phi$ sends positive semidefinite elements of $M_r\otimes M_n$ to positive semidefinite elements. It is unital if $\Phi(I_n)=I_n$, and trace preserving if $\operatorname{Tr}(\Phi(X))=\operatorname{Tr}(X)$ for every $X\in M_n$, where $\operatorname{Tr}$ denotes the ordinary matrix trace.

The supplied interpolation question asks for necessary and sufficient conditions on $A$ and $B$ for the existence of a unital quantum channel $\Phi:M_n\to M_n$ satisfying \[ \Phi(B_k)=A_k\qquad\text{for every }k\in\{1,\ldots,m\}. \] The term "unital quantum channel" is not defined in the supplied excerpt. Under the convention that a quantum channel is a completely positive, trace-preserving map, the question is to characterize when there exists a completely positive complex-linear map satisfying \[ \Phi(I_n)=I_n,\qquad \operatorname{Tr}(\Phi(X))=\operatorname{Tr}(X)\quad(X\in M_n),\qquad \Phi(B_k)=A_k\quad(1\leq k\leq m). \] If instead the report uses "quantum channel" in the observable-picture sense of a unital completely positive map, the trace-preservation condition is absent. These are distinct formulations; the supplied evidence does not settle which convention the report intends. No commutativity or other structural hypothesis on either tuple is stated.

The intended unresolved target also requires clarification: the supplied verification note says that the report follows this question with a necessary-and-sufficient characterization in Theorem 5, whereas the supplied dated assessment labels the problem partially solved and says a general classification for arbitrary self-adjoint tuples is not known. The statement and scope of Theorem 5 are not supplied. Consequently, this draft records the interpolation question without asserting that its existence characterization remains open or specifying an unsupported further classification problem.

Short English statement

Given two equally sized tuples of self-adjoint matrices, characterize when one unital quantum channel sends every $B_k$ to the corresponding $A_k$. The source excerpt leaves the channel convention unclear, particularly whether trace preservation is required. A supplied note reports an existing necessary-and-sufficient theorem, so the precise remaining open target needs review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005920$, exact source problem code catalogue entry, titled "Quantum-Channel Interpolation Characterization." The record belongs to the Oberwolfach Reports Open Problems collection and attributes the problem to the 2024 report; no individual proposer is supplied.
• Non-commutative Function Theory and Free Probability (2024), Oberwolfach Reports, pp. 1227--1284, DOI 10.4171/owr/2024/22: https://doi.org/10.4171/owr/2024/22. The supplied problem excerpt carries the page marker 1257. The background describes these reports as documenting mathematical workshops and their open-problem discussions. This report citation is supplied provenance and has not been independently consulted here.
• Li--Poon: https://arxiv.org/abs/1012.1675.
• Li--Poon full paper: https://people.wm.edu/~cklixx/cpafinal.pdf.
• Literature assessment dated exactly 2026-08-22: supplied dated curation derived from the OWR record, not independently verified. It labels the problem "partially solved" and reports Choi-matrix and operator-system criteria, substantial results for structured or commuting families, and no known general classification for arbitrary self-adjoint tuples. Its time-sensitive status and cited claims require independent verification.
• The supplied statement-verification note says that the official report poses the interpolation question and then gives a necessary-and-sufficient characterization as Theorem 5. The registry nevertheless marks statement_status as unrecoverable; no theorem text or explanation of the remaining target is included.