It is established that both discrete and continuous semigroups of unital quantum channels are eventually mixed unitary. This result is novel even for the subclass of Schur maps and stands in sharp contrast to the resolution of the asymptotic quantum Birkhoff conjecture by Haagerup and Musat, who demonstrated that tensor powers of some unital quantum channels maintain a persistent positive distance from the set of mixed unitary channels. Remarkably, our results show that this gap vanishes in finite time when considering ordinary powers within a semigroup. Building on this, we define the mixed unitary index of a unital quantum channel as the minimum time (or power) beyond which all subsequent maps become mixed unitary. We demonstrate that for any fixed dimension $d \geq 3$ , there is no universal upper bound for this index. Furthermore, we observe that if a continuous semigroup is not mixed unitary at some $t > 0$ , it remains non-mixed unitary for all $t$ sufficiently close to the origin. Finally, we investigate quantum dynamical semigroups where mixed unitarity is restricted to specific families, such as Weyl or diagonal unitaries. We show that Schur semigroups of correlation matrices eventually become mixtures of rank-one correlation matrices, and we characterize the generators of Schur semigroups that remain within this set for all $t \geq 0$ .It is established that both discrete and continuous semigroups of unital quantum channels are eventually mixed unitary. This result is novel even for the subclass of Schur maps and stands in sharp contrast to the resolution of the asymptotic quantum Birkhoff conjecture by Haagerup and Musat, who demonstrated that tensor powers of some unital quantum channels maintain a persistent positive distance from the set of mixed unitary channels. Remarkably, our results show that this gap vanishes in finite time when considering ordinary powers within a semigroup. Building on this, we define the mixed unitary index of a unital quantum channel as the minimum time (or power) beyond which all subsequent maps become mixed unitary. We demonstrate that for any fixed dimension $d \geq 3$, there is no universal upper bound for this index. Furthermore, we observe that if a continuous semigroup is not mixed unitary at some $t > 0$, it remains non-mixed unitary for all $t$ sufficiently close to the origin. Finally, we investigate quantum dynamical semigroups where mixed unitarity is restricted to specific families, such as Weyl or diagonal unitaries. We show that Schur semigroups of correlation matrices eventually become mixtures of rank-one correlation matrices, and we characterize the generators of Schur semigroups that remain within this set for all $t \geq 0$.

arXiv:2512.23598Dec 29, 20250 citations

ON CAPACITIES OF QUANTUM CHANNELS

Masanori Ohya, Dėnes Petz, Noboru Watanabe

WORLD SCIENTIFIC eBooksFeb 1, 200836 citations

Quantum Advantage in Learning Mixed Unitary Channels

Yue Tu, Liang Jiang

We study the task of learning mixed unitary channels using Fisher information, under different quantum resource assumptions including ancilla and concatenation. Our result shows that the asymptotic sample complexity scales as $\frac{r}{d\varepsilon^2}$ , where $r$ is the rank of the channel (i.e.\ the number of different unitaries), $d$ is the dimension of the system, and $\varepsilon^2$ is the mean-square error. Thus the critical resource is the ancilla, which mirrors the result in~\cite{chen2022quantum} but in a more precise form, as we point out that $r$ is also important. Additionally, we demonstrate the practical potential of mixed unitary channels by showing that random mixed unitary channels are easy to learn.We study the task of learning mixed unitary channels using Fisher information, under different quantum resource assumptions including ancilla and concatenation. Our result shows that the asymptotic sample complexity scales as $\frac{r}{d\varepsilon^2}$, where $r$ is the rank of the channel (i.e.\ the number of different unitaries), $d$ is the dimension of the system, and $\varepsilon^2$ is the mean-square error. Thus the critical resource is the ancilla, which mirrors the result in~\cite{chen2022quantum} but in a more precise form, as we point out that $r$ is also important. Additionally, we demonstrate the practical potential of mixed unitary channels by showing that random mixed unitary channels are easy to learn.

quant-pharXiv:2511.13683Nov 18, 20250 citations

Noncommutative Marcus-Ree inequality and Erdős channels

Repana Devendra, Chaman Kumar Sahu

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.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.

arXiv:2610.02294Oct 1, 20260 citations

On diagonal quantum channels

Amir R. Arab

arXiv:2011.00237Reports on Mathematical PhysicsAug 1, 20212 citations

d \geq 3

, there is no universal upper bound for this index. Furthermore, we observe that if a continuous semigroup is not mixed unitary at some

t > 0

, it remains non-mixed unitary for all

t

sufficiently close to the origin. Finally, we investigate quantum dynamical semigroups where mixed unitarity is restricted to specific families, such as Weyl or diagonal unitaries. We show that Schur semigroups of correlation matrices eventually become mixtures of rank-one correlation matrices, and we characterize the generators of Schur semigroups that remain within this set for all

t \geq 0

\frac{r}{d\varepsilon^2}

, where

r

is the rank of the channel (i.e.\ the number of different unitaries),

d

is the dimension of the system, and

\varepsilon^2

is the mean-square error. Thus the critical resource is the ancilla, which mirrors the result in~\cite{chen2022quantum} but in a more precise form, as we point out that

r

is also important. Additionally, we demonstrate the practical potential of mixed unitary channels by showing that random mixed unitary channels are easy to learn.

\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.