Completely Positive Maps

• Completely Positive Maps are linear maps that maintain positivity under tensoring with the identity, ensuring valid state and observable transformations.
• They are characterized by the Choi–Jamiołkowski isomorphism and Kraus decomposition, offering precise structural and operational insights.
• Their applications span quantum channel design, entanglement detection, and noncommutative probability, highlighting their pivotal role in quantum theory.

A completely positive (CP) map is a linear transformation between matrix algebras or, more generally, operator systems or $C^*$-algebras, whose ampliations by the identity map remain positive at all finite levels. This concept is foundational in operator algebras, quantum information theory, and the theory of noncommutative probability, where it encapsulates the structure-preserving transformations of quantum states and observables. The theory of CP maps combines spectral, convex, and duality methods and connects with entanglement theory, dilation theorems, mapping cones, and resource-theoretic frameworks.

1. Definitions, Characterizations, and Structural Properties

Let $M_m(\mathbb{C})$ and $M_n(\mathbb{C})$ denote $m \times m$ and $n \times n$ complex matrix algebras. A linear map $\Phi: M_m(\mathbb{C}) \to M_n(\mathbb{C})$ is said to be:

• Hermitian-preserving (Hermitian): $\Phi(A^*) = \Phi(A)^*$ for all $A$.
• Positive: $A \ge 0 \implies \Phi(A) \ge 0$.
• Completely Positive (CP): For every $k \ge 1$, $$I_k \otimes \Phi: M_k \otimes M_m \to M_k \otimes M_n$$ is positive.

The Choi–Jamiołkowski isomorphism associates to $\Phi$ its Choi matrix:

$$C_\Phi = \sum_{i,j=1}^m E_{ij} \otimes \Phi(E_{ij}) \in M_m \otimes M_n,$$

where $(E_{ij})$ are matrix units. Choi’s Theorem states that $\Phi$ is completely positive if and only if $C_\Phi \ge 0$ (Kian et al., 11 Jun 2025).

A map is trace-preserving (TP) if $\operatorname{Tr} \Phi(A) = \operatorname{Tr} A$; a quantum channel is a completely positive, trace-preserving map.

The Kraus decomposition expresses any CP map as

$\Phi(A) = \sum_k X_k A X_k^\dagger,$

with $\sum_k X_k^\dagger X_k = I_m$ in the TP case (Kian et al., 11 Jun 2025).

2. Canonical Decompositions: Jordan and Trace-Minus-CP Structures

Given a Hermitian map $\Phi: M_m \to M_n$ (i.e., $C_\Phi$ Hermitian), the Jordan decomposition yields unique CP maps $\Phi^+$, $\Phi^-$ such that

$\Phi = \Phi^+ - \Phi^-,$

with $C_\Phi = C_\Phi^+ - C_\Phi^-$, $C_\Phi^+, C_\Phi^- \ge 0$, and $C_\Phi^+ C_\Phi^- = 0$ (Kian et al., 11 Jun 2025). The negative part is tightly bounded: let $\lambda_\text{min} < 0$ be the smallest eigenvalue of $C_\Phi$ (with multiplicity $k$). Then, for any CP decomposition $\Phi = \Psi_1 - \Psi_2$, the Hilbert-Schmidt norm satisfies $\|\Psi_2\|_{HS} \ge \sqrt{k} |\lambda_\text{min}|$, with equality for the Jordan decomposition.

For positive but not CP maps on finite-dimensional Hilbert spaces, any $\varphi$ can be expressed as

$$\varphi(x) = \lambda \operatorname{Tr}(x) 1_H - \psi(x),$$

where $\lambda = \|C_\varphi\|$ and $\psi$ is CP, $C_\psi = \lambda I - C_\varphi \ge 0$ (Størmer, 2010). This formulation provides norm-based criteria for positivity, decomposability, and $k$-positivity.

3. Duality, Convexity, and Bipolarity in CP Map Cones

The convex structure of CP maps is governed by both classical and operator algebraic convexity. $C^*$-convexity is generated by summing over conjugations with elements in the target algebra:

$$\Psi(\cdot) = \sum_{i=1}^m a_i^* \Phi_i(\cdot) a_i, \quad \sum_i a_i^* a_i = 1.$$

A matrix duality pairs CP maps $\Phi: \mathscr{S} \to \mathscr{T}$ with matrix tests $(k, f, s)$, where $f$ is a state on $M_k(\mathscr{T})$ and $s \in M_k(\mathscr{S})$.

The matrix bipolar theorem asserts that for any subset $$K \subset \operatorname{CP}(\mathscr{S}, \mathscr{T})$$, the double polar with respect to this pairing reconstructs the $\tau$-closed $C^*$-convex hull of $K$ (Kian, 17 Nov 2025):

$$\left( K^{\circ_{C^*}} \right)^{\circ_{C^*}} = \overline{\mathrm{cconv}(K)}^\tau.$$

In finite dimensions, this reduces under the Jamiołkowski isomorphism to the bipolar theorem for positive semidefinite matrix cones.

4. Extensions, Liftings, and Approximation by CP Maps

Given a Hermitian or more general linear map $\Phi$, the question of embedding $\Phi$ into a CP map defined on a larger algebra is addressed via completely positive extensions:

$$\Phi(X) = \operatorname{Tr}_k \big[ \Psi(X \otimes I_k) (I_n \otimes Q) \big],$$

where $Q \in M_k$ is Hermitian, and $\Psi$ is a CP map on $M_m \otimes M_k \to M_n \otimes M_k$ (Kian et al., 11 Jun 2025). The minimal auxiliary dimension equals the rank of $C_\Phi$.

The optimal CP approximation of a Hermitian map (in Hilbert-Schmidt norm) is given by its positive part $\Phi^+$:

$$\min_{\Psi \:\mathrm{CP}} \| \Phi - \Psi \|_{HS} = \| \Phi^- \|_{HS},$$

where $\Phi^- = \Phi - \Phi^+$ is the negative component in the Jordan decomposition.

In $C^*$-algebraic frameworks, asymptotic lifting results guarantee the existence of continuous families of (asymptotically) CP lifts for CP maps into a quotient algebra, with full CP liftings possible if and only if the group action is amenable in the equivariant setting (Forough et al., 2021).

5. Classes, Examples, and Distinction from Merely Positive Maps

The set of CP maps is a proper subset of all positive maps. Quantitative results show that the “fraction” of positive maps which are CP decays as $O(\min(n, m)^{-1})$ in matrix size, with explicit real-algebraic-geometric construction of positive but not CP maps for $n, m \ge 3$ (Klep et al., 2016). These employ biquadratic forms which are nonnegative but not sums-of-squares; the Segre variety and Blekherman's convexity theory are crucial.

Constructive correspondences link positive block matrices with trace-preserving CP maps and provide explicit methods for generating decomposable and nondecomposable positive maps from block-structured Hermitian matrices (Guo et al., 2012).

Some classes of linear maps—characterized structurally by their block or Hill representations—obey “positivity implies complete positivity” as shown by surjectivity of suitable associated bilinear maps (Horst et al., 2021). In other cases (such as the transpose map for $n=2$), positivity and complete positivity diverge sharply.

6. Applications in Quantum Theory and Beyond

CP maps are central to quantum channel theory, entanglement detection (mapping cones and entanglement witnesses), and the formulation of quantum measurements (instruments, Kraus/Stinespring/KSGNS dilations) (Pellonpää, 2012). In direct-sum decompositions of state spaces for open quantum systems, families of initial system-environment states are identified such that the reduced dynamics is always CP, even permitting fixed entangled blocks (Liu et al., 2014).

Resource theories based on completely positive, completely positive (CPCP) maps analyze quantum operations that preserve cones of nonnegative amplitude states, yielding characterizations tied to entrywise nonnegativity and resource monotones such as robustness and trace-norm measures (Johnston et al., 2021).

Unital CP (UCP) maps admit structural decompositions into persistent (“boundary” $*$-automorphism) and transient parts, with peripherally automorphic UCP maps acting as $*$-automorphisms on their peripheral spectrum (Bhat et al., 2022).

7. Parametric Families and Operator-Theoretic Constructions

Families of CP maps arise from operator convex functions and monotone Riemannian metrics on positive definite matrices. The structure and CP property of mappings induced by such functions (typically via noncommutative multiplication or Schur multipliers) is dictated by positivity of associated kernels, Fourier transforms, and infinite divisibility (Hiai et al., 2012). The only symmetric operator convex function whose associated map and its inverse are both CP is $k(x) = x^{-1/2}$.

Completely positive, quasimultiplicative maps can also be constructed for group algebras, notably for imprimitive reflection groups, leading to representations on deformed Fock spaces and generalized statistics (Randriamaro, 2020).

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The theory of completely positive maps thus unifies spectral, convex, and operational analyses for linear maps on operator algebras, provides powerful dualities and decomposition theorems, and underlies fundamental constructions in quantum information, harmonic analysis, and noncommutative geometry. The subject continues to evolve, with investigations into structural classification, asymptotic lifting phenomena, mapping cone duality, and their implications for resource theories and quantum technologies (Kian et al., 11 Jun 2025, Kian, 17 Nov 2025, Størmer, 2010, Klep et al., 2016, Horst et al., 2021).