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arXiv:2603.16962

Exactness of the doubly nonnegative relaxation for qubit-output quantum channels

Hyunho Cha, Jungwoo Lee

quant-phSubmitted Mar 17, 2026• Announced Mar 18, 2026• 0 citations• Revised Sep 1, 2026 (v2)

v1 · Mar 17, 2026 v2 · Sep 1, 2026

Abstract

The resource theory for nonnegativity of quantum amplitudes distinguishes completely positive completely positive (CPCP) quantum channels from the larger class of completely positive doubly nonnegative (CPDNN) quantum channels. Johnston and Sikora showed that all qubit-to-qubit quantum channels that are CPDNN are also CPCP. However, they left open the question of whether a qutrit-to-qubit quantum channel exists that is CPDNN but not CPCP. We prove that no such channel exists and, more generally, that every CPDNN quantum channel with qubit output is CPCP. Thus the doubly nonnegative relaxation is exact for all qubit-output quantum channels. Our argument yields an explicit structural picture in which, after a canonical permutation of the Choi matrix, every qubit-output CPDNN channel is determined by a nonnegative vector and a nonnegative matrix, subject to a single positive-semidefinite block constraint. This gives a complete binary-output normal form, an explicit formula for the action of the channel, and quantitative population-coherence tradeoff inequalities governing the unique output off-diagonal mode. In this sense, qubit output is the nontrivial binary regime in which trace preservation and double nonnegativity collapse the free-channel cone to a fully describable geometry.

Comments: 6 pages

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The resource theory for nonnegativity of quantum amplitudes distinguishes completely positive completely positive (CPCP) quantum channels from the larger class of completely positive doubly nonnegative (CPDNN) quantum channels. Johnston and Sikora showed that all qubit-to-qubit quantum channels that are CPDNN are also CPCP. However, they left open the question of whether a qutrit-to-qubit quantum channel exists that is CPDNN but not CPCP. We prove that no such channel exists and, more generally, that every CPDNN quantum channel with qubit output is CPCP. Thus the doubly nonnegative relaxation is exact for all qubit-output quantum channels. Our argument yields an explicit structural picture in which, after a canonical permutation of the Choi matrix, every qubit-output CPDNN channel is determined by a nonnegative vector and a nonnegative matrix, subject to a single positive-semidefinite block constraint. This gives a complete binary-output normal form, an explicit formula for the action of the channel, and quantitative population-coherence tradeoff inequalities governing the unique output off-diagonal mode. In this sense, qubit output is the nontrivial binary regime in which trace preservation and double nonnegativity collapse the free-channel cone to a fully describable geometry.

Completely positive (CP) matrices are ubiquitous in modern science and technology with applications in optimization, graph theory, and quantum entanglement. Recently, Johnston \emph{et al.} [Linear Algebra and its Applications, 2022] have cast CP matrices into the framework of quantum resource theories, where CP states serve as free states and CP-preserving channels act as free operations. This work addresses several questions raised in their work. Specifically, we provide the necessary and sufficient conditions of CP-preserving channels in small dimensions, which are necessary in higher dimensions, and discuss the resource quantification via the trace distance of non-negativity. By constructing an explicit counterexample, we demonstrate that the trace-distance measure of non-negativity violates strong monotonicity. We also provide an alternative proof that every CPDNN channel $Φ:\MM_n\to \MM_2$ is CPCP. Additionally, we show that any unital CPDNN map $Φ:\MM_2\to \MM_n$ is also CPCP.

In this work we examine quantum states which have non-negative amplitudes (in a fixed basis) and the channels which preserve them. These states include the ground states of stoquastic Hamiltonians and they are of interest since they avoid the Sign Problem and can thus be efficiently simulated. In optimization theory, the convex cone generated by such states is called the set of completely positive (CP) matrices (not be confused with completely positive superoperators). We introduce quantum channels which preserve these states and call them completely positive completely positive. To study these states and channels, we use the framework of resource theories and investigate how to measure and quantify this resource.

Abstract In this note we study completely positive, trace-preserving (CPT) maps M 4 ( C ) ↦ M 4 ( C ) , in particular the class of examples given by quantum Markov chains (QMCs) describing the evolution of one qubit on a 2-vertex graph. For this class of channels, and regarding its convex structure, we examine the problem of finding the extreme points both with respect to CPT maps and with respect to the set of QMCs. This is done by considering the class of maps whose image is contained in the set of block-diagonal matrices. Motivated by previous results in the literature, a practical criterion for extremality is obtained via the Choi–Jamiołkowski isomorphism and appropriate direct-sum subspaces. The study of extreme maps is natural from a structural point of view, as these play the role of indecomposable operations and provide fundamental information about the geometry of the set of quantum channels.

The complete positivity vs positivity correspondence in the Choi-Jamiołkowski-Kraus-Sudarshan quantum channel-state isomorphism depends on the choice of basis. Instead of the "canonical" basis, if we use, e.g., the Pauli spin matrices along with the identity as the basis for the space of bounded operators on the two-dimensional complex Hilbert space, this correspondence breaks down. A sufficient condition on the basis for validity of this correspondence is provided in the work of Paulsen and Shult~\cite{Paulsen}, which was later proven to be necessary by Kye~\cite{Kye}. A correspondence is also present between the space of super-maps and the tensor product of the spaces of the inputs and outputs of the same. In particular, a super-map is completely CP-preserving if and only if its Choi-type representation is completely positive (CP). This correspondence also depends on a specific choice of basis. In this work, we find the necessary and sufficient condition on a basis such that this correspondence holds true.

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