Inmathematics,Choi's theorem on completely positive maps is a result that classifiescompletely positive maps between finite-dimensional (matrix)C*-algebras. The theorem is due toMan-Duen Choi. An infinite-dimensional algebraic generalization of Choi's theorem is known asBelavkin's "Radon–Nikodym" theorem for completely positive maps.

Choi's theorem. Let${\displaystyle \mathrm{\Phi }:{\mathbb{C}}^{n\times n}\to {\mathbb{C}}^{m\times m}}$ be a linear map. The following are equivalent:

(i)Φ isn-positive (i.e.${\displaystyle \left({\mathrm{id}}_n\otimes \mathrm{\Phi }\right)(A)\in {\mathbb{C}}^{n\times n}\otimes {\mathbb{C}}^{m\times m}}$ is positive whenever${\displaystyle A\in {\mathbb{C}}^{n\times n}\otimes {\mathbb{C}}^{n\times n}}$ is positive).

(ii) The matrix with operator entries

${\displaystyle C_{\mathrm{\Phi }}=\left({\mathrm{id}}_n\otimes \mathrm{\Phi }\right)\left(\sum _{ij}{E}_{ij}\otimes E_{ij}\right)=\sum _{ij}{E}_{ij}\otimes \mathrm{\Phi }(E_{ij})\in {\mathbb{C}}^{nm\times nm}}$

is positive semi-definite (PSD), where${\displaystyle E_{ij}\in {\mathbb{C}}^{n\times n}}$ is the matrix with 1 in theij-th entry and 0s elsewhere. (The matrixC_Φ is sometimes called theChoi matrix ofΦ.)

(iii)Φ is completely positive.

We observe that if

${\displaystyle E=\sum _{ij}{E}_{ij}\otimes E_{ij},}$

thenE=E^* andE²=nE, soE=n⁻¹EE^* which is positive. ThereforeC_Φ =(I_n ⊗ Φ)(E) is positive by then-positivity of Φ.

This holds trivially.

This mainly involves chasing the different ways of looking atC^nm×nm:

${\displaystyle {\mathbb{C}}^{nm\times nm}\cong {\mathbb{C}}^{nm}\otimes ({\mathbb{C}}^{nm}{)}^{\ast }\cong {\mathbb{C}}^n\otimes {\mathbb{C}}^m\otimes ({\mathbb{C}}^n\otimes {\mathbb{C}}^m{)}^{\ast }\cong {\mathbb{C}}^n\otimes ({\mathbb{C}}^n{)}^{\ast }\otimes {\mathbb{C}}^m\otimes ({\mathbb{C}}^m{)}^{\ast }\cong {\mathbb{C}}^{n\times n}\otimes {\mathbb{C}}^{m\times m}.}$

Let the eigenvector decomposition ofC_Φ be

${\displaystyle C_{\mathrm{\Phi }}=\sum _{i=1}^{nm}{\lambda }_i{v}_i{v}_i^{\ast },}$

where the vectors${\displaystyle v_i}$ lie inC^nm . By assumption, each eigenvalue${\displaystyle {\lambda }_i}$ is non-negative so we can absorb the eigenvalues in the eigenvectors and redefine${\displaystyle v_i}$ so that

${\displaystyle C_{\mathrm{\Phi }}=\sum _{i=1}^{nm}{v}_i{v}_i^{\ast }.}$

The vector spaceC^nm can be viewed as the direct sum${\displaystyle {\oplus }_{i=1}^n{\mathbb{C}}^m}$ compatibly with the above identification${\displaystyle {\mathbb{C}}^{nm}\cong {\mathbb{C}}^n\otimes {\mathbb{C}}^m}$and the standard basis ofCⁿ.

IfP_k ∈C^{m ×nm} is projection onto thek-th copy ofC^m, thenP_k^* ∈C^nm×m is the inclusion ofC^m as thek-th summand of the direct sum and

${\displaystyle \mathrm{\Phi }(E_{kl})=P_k\cdot C_{\mathrm{\Phi }}\cdot P_l^{\ast }=\sum _{i=1}^{nm}{P}_k{v}_i(P_l{v}_i{)}^{\ast }.}$

Now if the operatorsV_i ∈C^m×n are defined on thek-th standardbasis vectore_k ofCⁿ by

${\displaystyle V_i{e}_k=P_k{v}_i,}$

then

${\displaystyle \mathrm{\Phi }(E_{kl})=\sum _{i=1}^{nm}{P}_k{v}_i(P_l{v}_i{)}^{\ast }=\sum _{i=1}^{nm}{V}_i{e}_k{e}_l^{\ast }{V}_i^{\ast }=\sum _{i=1}^{nm}{V}_i{E}_{kl}{V}_i^{\ast }.}$

Extending by linearity gives us

${\displaystyle \mathrm{\Phi }(A)=\sum _{i=1}^{nm}{V}_{i}A{V}_i^{\ast }}$

for anyA ∈C^n×n. Any map of this form is manifestly completely positive: the map${\displaystyle A\to V_{i}A{V}_i^{\ast }}$ is completely positive, and the sum (across${\displaystyle i}$) of completely positive operators is again completely positive. Thus${\displaystyle \mathrm{\Phi }}$ is completely positive, the desired result.

The above is essentially Choi's original proof. Alternative proofs have also been known.

In the context ofquantum information theory, the operators {V_i} are called theKraus operators (afterKarl Kraus) of Φ. Notice, given a completely positive Φ, its Kraus operators need not be unique. For example, any "square root" factorization of the Choi matrixC_Φ =B^∗B gives a set of Kraus operators.

Let

${\displaystyle B^{\ast }=[b_1,\dots ,b_{nm}],}$

whereb_i*'s are the row vectors ofB, then

${\displaystyle C_{\mathrm{\Phi }}=\sum _{i=1}^{nm}{b}_i{b}_i^{\ast }.}$

The corresponding Kraus operators can be obtained by exactly the same argument from the proof.

When the Kraus operators are obtained from the eigenvector decomposition of the Choi matrix, because the eigenvectors form an orthogonal set, the corresponding Kraus operators are also orthogonal in theHilbert–Schmidtinner product. This is not true in general for Kraus operators obtained from square root factorizations. (Positive semidefinite matrices do not generally have a unique square-root factorizations.)

If two sets of Kraus operators {A_i}₁^nm and {B_i}₁^nm represent the same completely positive map Φ, then there exists a unitaryoperator matrix

${\displaystyle \{U_{ij}{\}}_{ij}\in {\mathbb{C}}^{n{m}^2\times n{m}^2}\text{such that}{A}_i=\sum _{j=1}{U}_{ij}{B}_j.}$

This can be viewed as a special case of the result relating twominimal Stinespring representations.

Alternatively, there is an isometryscalar matrix {u_ij}_ij ∈C^{nm ×nm} such that

${\displaystyle A_i=\sum _{j=1}{u}_{ij}{B}_j.}$

This follows from the fact that for two square matricesM andN,M M* =N N* if and only ifM = N U for some unitaryU.

It follows immediately from Choi's theorem that Φ is completely copositive if and only if it is of the form

${\displaystyle \mathrm{\Phi }(A)=\sum _i{V}_i{A}^T{V}_i^{\ast }.}$

Choi's technique can be used to obtain a similar result for a more general class of maps. Φ is said to be Hermitian-preserving ifA is Hermitian implies Φ(A) is also Hermitian. One can show Φ is Hermitian-preserving if and only if it is of the form

${\displaystyle \mathrm{\Phi }(A)=\sum _{i=1}^{nm}{\lambda }_i{V}_{i}A{V}_i^{\ast }}$

where λ_i are real numbers, the eigenvalues ofC_Φ, and eachV_i corresponds to an eigenvector ofC_Φ. Unlike the completely positive case,C_Φ may fail to be positive. Since Hermitian matrices do not admit factorizations of the formB*B in general, the Kraus representation is no longer possible for a given Φ.

• Stinespring factorization theorem
• Quantum operation
• Holevo's theorem

• M.-D. Choi,Completely Positive Linear Maps on Complex Matrices, Linear Algebra and its Applications, 10, 285–290 (1975).
• V. P. Belavkin, P. Staszewski,Radon-Nikodym Theorem for Completely Positive Maps, Reports on Mathematical Physics, v.24, No 1, 49–55 (1986).
• J. de Pillis,Linear Transformations Which Preserve Hermitian and Positive Semidefinite Operators, Pacific Journal of Mathematics, 23, 129–137 (1967).

• Linear algebra
• Operator theory
• Theorems in functional analysis

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