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arXiv:2609.02799

Noncommutative majorization

Matthew Kennedy, Paul Skoufranis

Published Sep 2, 2026• 0 citations
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Abstract

We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.

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•Mar 9, 2020•0 citations
We introduce a theory of noncommutative majorization that extends the classical majorization theory introduced by Hardy, Littlewood and Pólya to tuples of self-adjoint matrices that do not necessarily commute. We define and characterize a noncommutative majorization order that extends the classical majorization order. The definition is in terms of convex noncommutative functions, and we utilize the noncommutative convexity theory and noncommutative Choquet theory recently introduced by Davidson and the first author. As an application, we obtain a new necessary and sufficient condition for the existence of a trace-preserving completely positive map, i.e. a quantum channel, that interpolates between two finite sets of matrices. We give examples demonstrating that it is not always possible for this map to be chosen mixed unitary, even locally. We also address the computational difficulty of verifying the noncommutative majorization order. Our results further apply beyond the tracial case, and we obtain more general results characterizing the existence of unital completely positive maps that preserve an arbitrary faithful state or even unital completely positive map.
Abstract We establish an operator algebra generalization of Watrous’ theorem (Watrous 2009 Quantum Inf. Comput. 9 403–413) on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.
Uhlmann showed that there exists a positive, unital and trace-preserving map transforming a Hermitian matrix AAA into another BBB if and only if the vector of eigenvalues of AAA majorizes that of BBB. In this work I characterize the existence of such a transformation when one of the conditions of unitality or trace preservation is dropped. This induces two possible preorders in the set of Hermitian matrices and I argue how this can be used to construct measures quantifying the lack of positive semidefiniteness of any given Hermitian matrix with relevant monotonicity properties. It turns out that the measures in each of the two formalisms are essentially unique.
Unital quantum channels are the noncommutative analogues of doubly stochastic matrices. In 1959, Marcus and Ree showed that the squared Frobenius norm of a doubly stochastic matrix is at most its maximal diagonal sum. We prove a quantum version of this inequality for mixed unitary channels on Md\mathbb{M}_dMd​ and give an example showing that it fails for general unital channels. As in the classical case, we call the channels attaining equality as Erdős channels. We show that the trace-preserving conditional expectation onto any unital C∗C^*C∗-subalgebra of Md\mathbb{M}_dMd​ is an Erdős channel. It is also locally isolated among channels of the same Choi rank. Next, we introduce quantum RCDS channels. These are the noncommutative analogue of doubly stochastic matrices whose supported permutations all have the same diagonal sum. We prove a structure theorem for them and use it to characterize the quantum RCDS Erdős channels. We give an explicit form, up to unitary equivalence, for the Erdős channels of Choi rank 222, and further show that there are uncountably many unitary equivalence classes of Erdős channels unlike the classical case. We also characterize the Erdős channels among mixed Weyl-unitary channels of any Choi rank. For qubits, we prove that there are exactly four unitary equivalence classes of Erdős channels and they are represented by the identity channel and uniform averages of Pauli channels. Finally, restricting to the diagonal channels, our results recover the Marcus--Ree inequality, the structure of RCDS doubly stochastic matrices, and the characterization of Erdős.
Motivated by quantum thermodynamics, we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action on any full-rank state, and that the image of non-strictly positive maps lives inside a lower-dimensional subalgebra. This implies that the distance of such maps to the identity channel is lower bounded by one. The notion of strict positivity comes in handy when generalizing the majorization ordering on real vectors with respect to a positive vector d to majorization on square matrices with respect to a positive definite matrix D. For the two-dimensional case, we give a characterization of this ordering via finitely many trace norm inequalities and, moreover, investigate some of its order properties. In particular it admits a unique minimal and a maximal element. The latter is unique as well if and only if minimal eigenvalue of D has multiplicity one.
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.
Motivated by reachability questions in coherently controlled open quantum systems coupled to a thermal bath, as well as recent progress in the field of thermo-/vector-ddd-majorization we generalize classical majorization from unital quantum channels to channels with an arbitrary fixed point DDD of full rank. Such channels preserve some Gibbs-state and thus play an important role in the resource theory of quantum thermodynamics, in particular in thermo-majorization. Based on this we investigate DDD-majorization on matrices in terms of its topological and order properties, such as existence of unique maximal and minimal elements, etc. Moreover we characterize D-majorization in the qubit case via the trace norm and elaborate on why this is a challenging task when going beyond two dimensions.