A generalization of the Choi-Jamiolkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in 3 dimensions.A generalization of the Choi-Jamiolkowski isomorphism for completely positive maps between operator algebras is introduced. Particular emphasis is placed on the case of normal unital completely positive maps defined between von Neumann algebras. This generalization is applied especially to the study of maps which are covariant under actions of a symmetry group. We highlight with the example of, e.g., phase-shift-covariant quantum channels the ease of this method in particular in the case of a compact symmetry group. We also discuss the case of channels which are covariant under actions of the Euclidean group of rigid motions in 3 dimensions.

arXiv:1906.11442Jun 27, 20191 citation

Compatibility of Covariant Quantum Channels with Emphasis on Weyl Symmetry

Erkka Haapasalo

arXiv:1901.06113Annales Henri PoincaréJul 19, 201915 citations

Entanglement Breaking Structure of Cartan-Covariant Quantum Channels

Sean Prudhoe

Cartan-covariant quantum channels are introduced and studied using the Choi-Jamiołkowski isomorphism. The channels are Cartan-covariant as the covariance groups considered form symmetric pairs with the special unitary group SU( $n$ ), and their associated Cartan involution provides a route to exactly compute the eigenspectrum of their Choi states and the partial transpose for any $n\in \mathbb{N}$ . These channels include previously studied SO $(n)$ -covariant channels, and further include Sp $(\frac{n}{2})$ -covariant channels (when $n$ is even) and S(U( $p$ ) $\times$ U( $q$ ))-covariant channels (where $n\!=\!p\!+\!q$ ). We show that all Cartan-covariant channels satisfy the PPT $^{2}$ -conjecture and further demonstrate the nontrivial nature of this result for the class of Sp $(\frac{n}{2})$ -covariant and ${\rm S}({\rm U}(p) \!\times\!{\rm U}(q))$ -covariant channelsCartan-covariant quantum channels are introduced and studied using the Choi-Jamiołkowski isomorphism. The channels are Cartan-covariant as the covariance groups considered form symmetric pairs with the special unitary group SU($n$), and their associated Cartan involution provides a route to exactly compute the eigenspectrum of their Choi states and the partial transpose for any $n\in \mathbb{N}$. These channels include previously studied SO$(n)$-covariant channels, and further include Sp$(\frac{n}{2})$-covariant channels (when $n$ is even) and S(U($p$) $\times$ U($q$))-covariant channels (where $n\!=\!p\!+\!q$). We show that all Cartan-covariant channels satisfy the PPT$^{2}$-conjecture and further demonstrate the nontrivial nature of this result for the class of Sp$(\frac{n}{2})$-covariant and ${\rm S}({\rm U}(p) \!\times\!{\rm U}(q))$-covariant channels

arXiv:2501.03959Jan 7, 20250 citations

Orbit-injective covariant quantum channels

Kai Liu, Chuangxun Cheng, Deguang Han

Linear Algebra and its ApplicationsMar 22, 20232 citations

Channel-state duality

Min Jiang, Shunlong Luo, Shuangshuang Fu

Channel-state duality refers to the correspondence between quantum channels and bipartite states, and it is extremely useful and fruitful in quantum information theory. It is often called the Jamio\l{}kowski isomorphism, the Jamio\l{}kowski-Choi isomorphism, or the Choi-Jamio\l{}kowski isomorphism. We trace the original roots of this duality from a historic perspective, clarify the somewhat misleading term ``isomorphism,'' reveal the underlying subtle nature, and highlight the seemingly similar but actually different features of the original correspondences in the manner of Pillis, Jamio\l{}kowski, and Choi that lead to the duality. We further illustrate some fundamental properties and diverse applications.Channel-state duality refers to the correspondence between quantum channels and bipartite states, and it is extremely useful and fruitful in quantum information theory. It is often called the Jamio\l{}kowski isomorphism, the Jamio\l{}kowski-Choi isomorphism, or the Choi-Jamio\l{}kowski isomorphism. We trace the original roots of this duality from a historic perspective, clarify the somewhat misleading term ``isomorphism,'' reveal the underlying subtle nature, and highlight the seemingly similar but actually different features of the original correspondences in the manner of Pillis, Jamio\l{}kowski, and Choi that lead to the duality. We further illustrate some fundamental properties and diverse applications.

Physical Review AFeb 13, 2013136 citations

Quantum Channels with Quantum Group Symmetry

Hun Hee Lee, Sang-Gyun Youn

arXiv:2007.03901Communications in Mathematical PhysicsJan 6, 20227 citations

n

), and their associated Cartan involution provides a route to exactly compute the eigenspectrum of their Choi states and the partial transpose for any

n\in \mathbb{N}

. These channels include previously studied SO

(n)

-covariant channels, and further include Sp

(\frac{n}{2})

-covariant channels (when

n

is even) and S(U(

p

)

\times

q

))-covariant channels (where

n\!=\!p\!+\!q

). We show that all Cartan-covariant channels satisfy the PPT

^{2}

-conjecture and further demonstrate the nontrivial nature of this result for the class of Sp

(\frac{n}{2})

-covariant and

{\rm S}({\rm U}(p) \!\times\!{\rm U}(q))

-covariant channels