Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.02294

Noncommutative Marcus-Ree inequality and Erdős channels

Repana Devendra, Chaman Kumar Sahu

Published Oct 1, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $\mathbb{M}_d$ 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 $C^*$-subalgebra of $\mathbb{M}_d$ 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 $2$, 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.

DOI
10.48550/arxiv.2610.02294

Related papers

Noncommutative majorization

Matthew Kennedy, Paul Skoufranis

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.

arXiv:2609.02799•Sep 2, 2026•0 citations

Operator algebra generalization of a theorem of Watrous and mixed unitary quantum channels

David W. Kribs, Jeremy Levick, Rajesh Pereira +1 more

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.

arXiv:2306.04077•Journal of Physics A Mathematical and Theoretical•Feb 23, 2024•2 citations

On unital qubit channels

Chi-Kwong Li, Man-Duen Choi

A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients $p_1, \dots, p_m$ as long as $2(p_1, \dots, p_m)$ is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.

arXiv:2301.01358•Jan 3, 2023•0 citations

Quantum de Finetti theorems for states and channels in any distance measure

Liuhang Ye, Bjarne Bergh, Nilanjana Datta

Standard finite quantum de Finetti theorems approximate the $k$-system marginals of permutation-invariant states of $n$-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our $k/n$ error bound in max-relative entropy improves on the previously best known $k^2/n$ scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels $N^{(n)}:A^{\otimes n}\to B^{\otimes n}$, where $d_A=\dim A$, we prove that the $k$-system reduced channel is CP-dominated by a mixture of tensor-power channels with error $O(k/\sqrt{n})$ and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of $r$ defects, the representation error is at most $\mathrm{poly}(n)\bigl(2d_A^4k^3/(nr^2)\bigr)^{(r+1)/2}$ and decays exponentially in $n$ for a suitable choice of parameters $r$ and $k$.

quant-pharXiv:2609.40343•Oct 1, 2026•0 citations

A sharp norm inequality for entanglement-breaking channels

Eran Kopel

For a channel $Φ$ on $M_d$ written in the generalised Bloch parameterisation $r \mapsto Ar+c$, we prove that entanglement breaking implies $\|A\|_*^2 + \frac{d(d-1)}{2}\,|c|^2 \;\le\; (d-1)^2 ,$ where $\|\cdot\|_*$ denotes the nuclear norm. Read on the Choi state of the channel, the inequality is an instance of the covariance-matrix criterion of Gittsovich et al.; what is added here is an elementary proof in Holevo form and a complete description of the equality cases. Apart from the constant channel onto a pure state, which shows that the coefficient $d(d-1)/2$ cannot be increased, equality holds exactly for the measure-and-prepare channels with rank-one effects $α_k|e_k\rangle\langle e_k|$ and pure outputs $|ψ_k\rangle\langleψ_k|$ whose overlaps are an affine image of those of the measurement directions, $|\langleψ_j|ψ_k\rangle|^2 = P + (1-P)\,(d|\langle e_j|e_k\rangle|^2 - 1)/(d-1)$ for a common $P = \mathrm{Tr}\,Φ(\mathbf{1}/d)^2$. These include the completely dephasing channel, the re-preparation channel of every rank-one POVM, a one-parameter family interpolating between dephasing and the constant channel, and non-unital non-projective examples already at $d = 2$. The qubit case reads $\|A\|_*^2+|c|^2\le1$ and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound $\|A\|_*\le1$ by retaining the POVM completeness relation $\sum_k f_k=0$, which her argument discards; this recentres $A$ from a second moment into a cross-covariance, and is the entire content of the improvement.

quant-pharXiv:2609.27906•Aug 24, 2026•0 citations

On unital qubit channels

Man-Duen Choi, Chi-Kwong Li

A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients $p_1, \dots, p_m$ as long as $2(p_1, \dots, p_m)$ is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.

Quantum Information and Computation•May 1, 2023•2 citations
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.
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.
A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients p1,…,pmp_1, \dots, p_mp1​,…,pm​ as long as 2(p1,…,pm)2(p_1, \dots, p_m)2(p1​,…,pm​) is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.
Standard finite quantum de Finetti theorems approximate the kkk-system marginals of permutation-invariant states of nnn-systems by mixtures of independent and identically distributed (iid) states, usually in trace distance. We prove both standard and Renner's exponential de Finetti theorems in the stronger form of operator inequalities, implying bounds in every Schatten norm and for every quantum Rényi divergence satisfying data processing. In the standard case, for fixed local dimension, our k/nk/nk/n error bound in max-relative entropy improves on the previously best known k2/nk^2/nk2/n scaling, even in the classical setting. The operator-inequality approach is particularly suited to study channel de Finetti representations because operator order between Choi states is equivalent to completely positive (CP) order between the underlying channels. For permutation-covariant channels N(n):A⊗n→B⊗nN^{(n)}:A^{\otimes n}\to B^{\otimes n}N(n):A⊗n→B⊗n, where dA=dim⁡Ad_A=\dim AdA​=dimA, we prove that the kkk-system reduced channel is CP-dominated by a mixture of tensor-power channels with error O(k/n)O(k/\sqrt{n})O(k/n​) and polynomial dependence on the local dimensions, addressing a question raised by Berta et al. [Math. Program. 194, 781-829 (2022)]. Under the no-signalling condition, we also prove an exponential channel de Finetti theorem where the approximating mixture consists of Choi-almost-iid channels, whose normalized Choi states are almost-iid in the sense of Mazzola-Sutter-Renner. In the case of rrr defects, the representation error is at most poly(n)(2dA4k3/(nr2))(r+1)/2\mathrm{poly}(n)\bigl(2d_A^4k^3/(nr^2)\bigr)^{(r+1)/2}poly(n)(2dA4​k3/(nr2))(r+1)/2 and decays exponentially in nnn for a suitable choice of parameters rrr and kkk.
For a channel ΦΦΦ on MdM_dMd​ written in the generalised Bloch parameterisation r↦Ar+cr \mapsto Ar+cr↦Ar+c, we prove that entanglement breaking implies ∥A∥∗2+d(d−1)2 ∣c∣2  ≤  (d−1)2,\|A\|_*^2 + \frac{d(d-1)}{2}\,|c|^2 \;\le\; (d-1)^2 ,∥A∥∗2​+2d(d−1)​∣c∣2≤(d−1)2, where ∥⋅∥∗\|\cdot\|_*∥⋅∥∗​ denotes the nuclear norm. Read on the Choi state of the channel, the inequality is an instance of the covariance-matrix criterion of Gittsovich et al.; what is added here is an elementary proof in Holevo form and a complete description of the equality cases. Apart from the constant channel onto a pure state, which shows that the coefficient d(d−1)/2d(d-1)/2d(d−1)/2 cannot be increased, equality holds exactly for the measure-and-prepare channels with rank-one effects αk∣ek⟩⟨ek∣α_k|e_k\rangle\langle e_k|αk​∣ek​⟩⟨ek​∣ and pure outputs ∣ψk⟩⟨ψk∣|ψ_k\rangle\langleψ_k|∣ψk​⟩⟨ψk​∣ whose overlaps are an affine image of those of the measurement directions, ∣⟨ψj∣ψk⟩∣2=P+(1−P) (d∣⟨ej∣ek⟩∣2−1)/(d−1)|\langleψ_j|ψ_k\rangle|^2 = P + (1-P)\,(d|\langle e_j|e_k\rangle|^2 - 1)/(d-1)∣⟨ψj​∣ψk​⟩∣2=P+(1−P)(d∣⟨ej​∣ek​⟩∣2−1)/(d−1) for a common P=Tr Φ(1/d)2P = \mathrm{Tr}\,Φ(\mathbf{1}/d)^2P=TrΦ(1/d)2. These include the completely dephasing channel, the re-preparation channel of every rank-one POVM, a one-parameter family interpolating between dephasing and the constant channel, and non-unital non-projective examples already at d=2d = 2d=2. The qubit case reads ∥A∥∗2+∣c∣2≤1\|A\|_*^2+|c|^2\le1∥A∥∗2​+∣c∣2≤1 and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound ∥A∥∗≤1\|A\|_*\le1∥A∥∗​≤1 by retaining the POVM completeness relation ∑kfk=0\sum_k f_k=0∑k​fk​=0, which her argument discards; this recentres AAA from a second moment into a cross-covariance, and is the entire content of the improvement.
A canonical form for unital qubit channels under local unitary transforms is obtained. In particular, it is shown that the eigenvalues of the Choi matrix of a unital quantum channel form a complete set of invariants of the canonical form. It follows immediately that every unital qubit channel is the average of four unitary channels. More generally, a unital qubit channel can be expressed as the convex combination of unitary channels with convex coefficients p1,…,pmp_1, \dots, p_mp1​,…,pm​ as long as 2(p1,…,pm)2(p_1, \dots, p_m)2(p1​,…,pm​) is majorized by the vector of eigenvalues of the Choi matrix of the channel. A unital qubit channel in the canonical form will transform the Bloch sphere onto an ellipsoid. We look into the detailed structure of the natural linear maps sending the Bloch sphere onto a corresponding ellipsoid.