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:1909.13309

Channel-state duality and the separability problem

K. V. Antipin

Published Sep 29, 2019• 1 citation
arXivHTMLPDF

Abstract

Separability of quantum states is analyzed with the use of the Choi-Jamiolkowski isomorphism. Spectral separability criteria are derived. The presented approach is illustrated with various examples, among which a separable decomposition of 2 \otimes 2 isotropic states is obtained.
DOI
10.48550/arxiv.1909.13309

Related papers

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.

Physical Review A•Feb 13, 2013•136 citations

Truncated moment sequences and a solution to the channel separability problem

Nadia Milazzo, Daniel Braun, O. Giraud

We consider the problem of separability of quantum channels via the Choi matrix representation given by the Choi-Jamio\l{}kowski isomorphism. We explore three classes of separability across different cuts between systems and ancillae, and we provide a solution based on the mapping of the coordinates of the Choi state (in a fixed basis) to a truncated moment sequence (tms) $y$. This results in an algorithm which gives a separability certificate using semidefinite programming. The computational complexity and the performance of it depend on the number of variables $n$ in the tms and on the size of the moment matrix ${M}_{t}(y)$ of order $t$. We exploit the algorithm to numerically investigate separability of families of two-qubit and single-qutrit channels; in the latter case we can provide an answer for examples explored earlier through the criterion based on the negativity $N$, a criterion which remains inconclusive for Choi matrices with $N=0$.

arXiv:2006.15003•Physical Review A•Nov 6, 2020•5 citations

Choi-Jamiołkowski isomorphism(s) and channel/state duality, naturally!

Paul E. Lammert

Provided a maximally entangled state vector in ${\mathcal{H}}\otimes{\mathcal{H}}$, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of ${\mathsf CP}({\mathcal{H}},{\mathcal{K}})$ (completely positive maps of density operators on ${\mathcal{H}}$ tothose on ${\mathcal{K}}$) onto the density operators on ${\mathcal{K}}\otimes{\mathcal{H}}$, which extends to a unitary map. Here, ${\mathcal{H}}$ and ${\mathcal{K}}$ are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both ${\mathcal{H}}$ and ${\mathcal{K}}$ which produces density operators on either ${\mathsf L}({\mathcal{H}},{\mathcal{K}})$ (Hilbert-Schmidt inner product) or ${\mathcal{K}}\otimes{\mathcal{H}}^*$ instead of ${\mathcal{K}}\otimes{\mathcal{H}}$. In particular, no entangled state input is required. There are no such binatural maps carried on ${\mathcal{K}}\otimes{\mathcal{H}}$, but those natural in only ${\mathcal{K}}$ are canonically associated with unitary maps ${\mathcal{H}}\rightarrow{\mathcal{H}}^*$ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.

quant-pharXiv:2610.27394•Oct 21, 2026•0 citations

Quantum channels

Paul Skrzypczyk, Daniel Cavalcanti

In chapter 7 we will study quantum channels. We will see how the so-called Choi–Jamiołkowski isomorphism—which links quantum channels with quantum states—allows for semidefinite programming techniques to be applied to quantum channels. We will then see how we can estimate an uncharacterised channel, calculate the diamond norm (which characterises the operational distinguishability between channels) and finally how to use duality to uncover a link between a channel optimisation problem and quantum entropies.

Institute of Physics eBooks•Mar 1, 2023•0 citations

Quantum state-channel duality for the calculation of Standard Model scattering amplitudes

Clelia Altomonte, Alan J. Barr

Recent instances of successful application of quantum information techniques to particle physics problems invite for an analysis of the mathematical details behind such connection. In this paper, we identify the Choi-Jamiolkowski isomorphism, or state-channel duality, as a theoretical principle enabling the application of the theory of quantum information to the scattering amplitudes associated with Standard Model processes.

arXiv:2312.02242•Physics Letters B•Nov 7, 2023•8 citations

Anti-isomorphisms and the naturality of channel--state duality

Marcin Marciniak, Michał Cholewiak

Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on $\mathcal{M}\bar{\otimes}\mathcal{N}$ but on the tensor product of $\mathcal{M}$ with the \emph{opposite} algebra of $\mathcal{N}$, or equivalently with a commutant; returning to $\mathcal{N}$ itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is $*$-anti-isomorphic to itself. Naturality cannot be dropped: whenever $\mathcal{M}$ is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.

math.OACross-listedarXiv:2608.13750•Entropy•Aug 17, 2026•0 citations
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.
We consider the problem of separability of quantum channels via the Choi matrix representation given by the Choi-Jamio\l{}kowski isomorphism. We explore three classes of separability across different cuts between systems and ancillae, and we provide a solution based on the mapping of the coordinates of the Choi state (in a fixed basis) to a truncated moment sequence (tms) yyy. This results in an algorithm which gives a separability certificate using semidefinite programming. The computational complexity and the performance of it depend on the number of variables nnn in the tms and on the size of the moment matrix Mt(y){M}_{t}(y)Mt​(y) of order ttt. We exploit the algorithm to numerically investigate separability of families of two-qubit and single-qutrit channels; in the latter case we can provide an answer for examples explored earlier through the criterion based on the negativity NNN, a criterion which remains inconclusive for Choi matrices with N=0N=0N=0.
Provided a maximally entangled state vector in H⊗H{\mathcal{H}}\otimes{\mathcal{H}}H⊗H, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of CP(H,K){\mathsf CP}({\mathcal{H}},{\mathcal{K}})CP(H,K) (completely positive maps of density operators on H{\mathcal{H}}H tothose on K{\mathcal{K}}K) onto the density operators on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, which extends to a unitary map. Here, H{\mathcal{H}}H and K{\mathcal{K}}K are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both H{\mathcal{H}}H and K{\mathcal{K}}K which produces density operators on either L(H,K){\mathsf L}({\mathcal{H}},{\mathcal{K}})L(H,K) (Hilbert-Schmidt inner product) or K⊗H∗{\mathcal{K}}\otimes{\mathcal{H}}^*K⊗H∗ instead of K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H. In particular, no entangled state input is required. There are no such binatural maps carried on K⊗H{\mathcal{K}}\otimes{\mathcal{H}}K⊗H, but those natural in only K{\mathcal{K}}K are canonically associated with unitary maps H→H∗{\mathcal{H}}\rightarrow{\mathcal{H}}^*H→H∗ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.
In chapter 7 we will study quantum channels. We will see how the so-called Choi–Jamiołkowski isomorphism—which links quantum channels with quantum states—allows for semidefinite programming techniques to be applied to quantum channels. We will then see how we can estimate an uncharacterised channel, calculate the diamond norm (which characterises the operational distinguishability between channels) and finally how to use duality to uncover a link between a channel optimisation problem and quantum entropies.
Channel--state duality identifies completely positive maps with bipartite states. Its generalisation to von Neumann algebras is known to produce states not on M⊗ˉN\mathcal{M}\bar{\otimes}\mathcal{N}M⊗ˉ​N but on the tensor product of M\mathcal{M}M with the \emph{opposite} algebra of N\mathcal{N}N, or equivalently with a commutant; returning to N\mathcal{N}N itself requires a transposition, and it has been observed that this step typically fails. We determine exactly when it does not fail. Our main result is that a Choi--Jamiołkowski type isomorphism which is natural in the first algebra exists if and only if the second algebra is ∗*∗-anti-isomorphic to itself. Naturality cannot be dropped: whenever M\mathcal{M}M is anti-isomorphic to itself such an isomorphism exists for trivial reasons, since the order structure of a predual is a Jordan invariant and cannot distinguish an algebra from its opposite. Consequently no such correspondence exists for the type III factors constructed by Connes. We show in addition that the domain admits no description in terms of properties of individual maps: the space of all normal maps with the operator norm is too large, and neither complete boundedness nor complete positivity cuts it down to the right size.