Let H and K be (finite- or infinite-dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from into is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied to give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is given which shows that a state ρ on H ⊗ K is separable if and only if (Φ⊗ I )ρ ⩾ 0 for all positive finite-rank elementary operators Φ. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.Let H and K be (finite- or infinite-dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from into is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied to give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is given which shows that a state ρ on H ⊗ K is separable if and only if (Φ⊗ I )ρ ⩾ 0 for all positive finite-rank elementary operators Φ. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.

arXiv:1007.0560Journal of Physics A Mathematical and TheoreticalAug 6, 201044 citations

Positive Mappings in Quantum Physics

Philippe Blanchard, Erwin Brüning

Progress in mathematical physicsJan 1, 20150 citations

The Mathematical Aspects of Quantum Maps

Mirko Degli Esposti, S. Graffi

Springer eBooksJan 1, 200356 citations

0 A CHARACTERIZATION OF POSITIVE LINEAR MAPS AND CRITERIA OF ENTANGLEMENT FOR QUANTUM STATES

Jinchuan Hou

Let $H$ and $K$ be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from ${\mathcal B}(H)$ into ${\mathcal B}(K)$ is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state $\rho$ on $H\otimes K$ is separable if and only if $(\Phi\otimes I)\rho\geq 0$ for all positive finite rank elementary operators $\Phi$ . Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.Let $H$ and $K$ be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from ${\mathcal B}(H)$ into ${\mathcal B}(K)$ is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state $\rho$ on $H\otimes K$ is separable if and only if $(\Phi\otimes I)\rho\geq 0$ for all positive finite rank elementary operators $\Phi$. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.

Aug 16, 201641 citations

Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach

Thea Li, Vladimir Zamdzhiev

We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schrödinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schrödinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces.We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schrödinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schrödinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces.

math.CTCross-listedarXiv:2601.15832Jan 23, 2026

Let

H

and

K

be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from

{\mathcal B}(H)

into

{\mathcal B}(K)

is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state

\rho

H\otimes K

is separable if and only if

(\Phi\otimes I)\rho\geq 0

for all positive finite rank elementary operators

\Phi

. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.