Choi's theorem on completely positive maps

In mathematics, Choi's theorem on completely positive maps (after Man-Duen Choi) is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's "Radon–Nikodym" theorem for completely positive maps.

Some preliminary notions

Before stating Choi's result, we give the definition of a completely positive map and fix some notation. $C n \times n$ will denote the C*-algebra of $n \times n$ complex matrices. We will call $A \in C n \times n$ positive, or symbolically, $A \geq 0$ , if A is Hermitian and the spectrum of A is nonnegative. (This condition is also called positive semidefinite.)

A linear map Φ : C^n×n → C^m×m is said to be a positive map if Φ(A) ≥ 0 for all A ≥ 0. In other words, a map Φ is positive if it preserves Hermiticity and the cone of positive elements.

Any linear map Φ induces another map

I_k \otimes \Phi : \mathbb{C}^{k \times k} \otimes \mathbb{C}^{n \times n} \to \mathbb{C}^{k \times k} \otimes \mathbb{C}^{m \times m}

in a natural way: define

( I_k \otimes \Phi ) (M \otimes A) = M \otimes \Phi (A)

and extend by linearity. In matrix notation, a general element in

\mathbb{C} ^{k \times k} \otimes \mathbb{C} ^{n \times n}

can be expressed as a k × k operator matrix:

\begin{bmatrix} A_{11} & \cdots & A_{1k} \\ \vdots & \ddots & \vdots \\ A_{k1} & \cdots & A_{kk} \end{bmatrix},

and its image under the induced map is

\left (I_k \otimes \Phi \right ) \left (\begin{bmatrix} A_{11} & \cdots & A_{1k} \\ \vdots & \ddots & \vdots \\A_{k1} & \cdots & A_{kk} \end{bmatrix} \right ) = \begin{bmatrix} \Phi (A_{11}) & \cdots & \Phi( A_{1k} ) \\ \vdots & \ddots & \vdots \\ \Phi (A_{k1}) & \cdots & \Phi( A_{kk} ) \end{bmatrix}.

Writing out the individual elements in the above matrix-of-matrices amounts to the natural identification of algebras

\mathbb{C}^{k\times k}\otimes\mathbb{C}^{m\times m}\cong\mathbb{C}^{km\times km}.

We say that Φ is k-positive if $I_k \otimes \Phi$ , considered as an element of C^km×km, is a positive map, and Φ is called completely positive if Φ is k-positive for all k.

The transposition map is a standard example of a positive map that fails to be 2-positive. Let T denote this map on $C 2\times2$ . The following is a positive matrix in $C 2\times2 \otimes C 2\times2$ :

\begin{bmatrix} \begin{pmatrix}1&0\\0&0\end{pmatrix}& \begin{pmatrix}0&1\\0&0\end{pmatrix}\\ \begin{pmatrix}0&0\\1&0\end{pmatrix}& \begin{pmatrix}0&0\\0&1\end{pmatrix} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ \end{bmatrix}.

The image of this matrix under $I_2 \otimes T$ is

\begin{bmatrix} \begin{pmatrix}1&0\\0&0\end{pmatrix}^T& \begin{pmatrix}0&1\\0&0\end{pmatrix}^T\\ \begin{pmatrix}0&0\\1&0\end{pmatrix}^T& \begin{pmatrix}0&0\\0&1\end{pmatrix}^T \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} ,

which is clearly not positive, having determinant -1. Moreover, the eigenvalues of this matrix are 1,1,1 and -1.

Incidentally, a map Φ is said to be co-positive if the composition Φ $\circ$ T is positive. The transposition map itself is a co-positive map.

The above notions concerning positive maps extend naturally to maps between C*-algebras.

Choi's result

Statement of theorem

Choi's Theorem. Let

Φ : C n \times n \to C m \times m

be a positive map. The following are equivalent:

(i)

Φ

n

-positive.

(ii) The matrix with operator entries

C_\Phi= \left (I_n\otimes\Phi \right ) \left (\sum_{ij}E_{ij}\otimes E_{ij} \right ) = \sum_{ij}E_{ij}\otimes\Phi(E_{ij}) \in \mathbb{C} ^{nm \times nm}

is positive, where

E ij \in C n \times n

is the matrix with 1 in the

ij

-th entry and 0s elsewhere. (The matrix C_Φ is sometimes called the Choi matrix of

Φ

(iii)

Φ

is completely positive.

Proof

(i) implies (ii)

We observe that if

E=\sum_{ij}E_{ij}\otimes E_{ij},

then E=E^* and E²=nE, so E=n⁻¹EE^* which is positive and C_Φ =(I_n ⊗ Φ)(E) is positive by the n-positivity of Φ.

(iii) implies (i)

This holds trivially.

(ii) implies (iii)

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

\mathbb{C}^{nm\times nm} \cong\mathbb{C}^{nm}\otimes(\mathbb{C}^{nm})^* \cong\mathbb{C}^n\otimes\mathbb{C}^m\otimes(\mathbb{C}^n\otimes\mathbb{C}^m)^* \cong\mathbb{C}^n\otimes(\mathbb{C}^n)^*\otimes\mathbb{C}^m\otimes(\mathbb{C}^m)^* \cong\mathbb{C}^{n\times n}\otimes\mathbb{C}^{m\times m}.

Let the eigenvector decomposition of C_Φ be

C_\Phi = \sum _{i = 1} ^{nm} \lambda_i v_i v_i ^*,

where the vectors $v_i$ lie in C^nm . By assumption, each eigenvalue $\lambda_i$ is non-negative so we can absorb the eigenvalues in the eigenvectors and redefine $v_i$ so that

\; C_\Phi = \sum _{i = 1} ^{nm} v_i v_i ^* .

The vector space C^nm can be viewed as the direct sum $\textstyle \oplus_{i=1}^n \mathbb{C}^m$ compatibly with the above identification $\textstyle\mathbb{C}^{nm}\cong\mathbb{C}^n\otimes\mathbb{C}^m$ and the standard basis of Cⁿ.

If P_k ∈ C^{m × nm} is projection onto the k-th copy of C^m, then P_k^* ∈ C^nm×m is the inclusion of C^m as the k-th summand of the direct sum and

\; \Phi (E_{kl}) = P_k \cdot C_\Phi \cdot P_l^* = \sum _{i = 1} ^{nm} P_k v_i ( P_l v_i )^*.

Now if the operators V_i ∈ C^m×n are defined on the k-th standard basis vector e_k of Cⁿ by

\; V_i e_k = P_k v_i,

then

\Phi (E_{kl}) = \sum _{i = 1} ^{nm} P_k v_i ( P_l v_i )^* = \sum _{i = 1} ^{nm} V_i e_k e_l ^* V_i ^* = \sum _{i = 1} ^{nm} V_i E_{kl} V_i ^*.

Extending by linearity gives us

\Phi(A) = \sum_{i=1}^{nm} V_i A V_i^*

for any A ∈ C^n×n. Since any map of this form is manifestly completely positive, we have the desired result.

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

Consequences

Kraus operators

In the context of quantum information theory, the operators {V_i} are called the Kraus operators (after Karl 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 matrix $C Φ = B * B$ gives a set of Kraus operators. (Notice B need not be the unique positive square root of the Choi matrix.)

Let

B^* = [b_1, \ldots, b_{nm}],

where b_i*'s are the row vectors of B, then

C_\Phi = \sum _{i = 1} ^{nm} b_i b_i ^*.

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 the Hilbert–Schmidt inner 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 unitary operator matrix

\{U_{ij}\}_{ij} \in \mathbb{C}^{nm^2 \times nm^2} \quad \text{such that} \quad A_i = \sum _{i = 1} U_{ij} B_j.

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

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

A_i = \sum _{i = 1} u_{ij} B_j.

This follows from the fact that for two square matrices M and N, M M* = N N* if and only if M = N U for some unitary U.

Completely copositive maps

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

\Phi(A) = \sum _i V_i A^T V_i ^* .

Hermitian-preserving maps

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 if A is Hermitian implies Φ(A) is also Hermitian. One can show Φ is Hermitian-preserving if and only if it is of the form

\Phi (A) = \sum_{i=1} ^{nm} \lambda_i V_i A V_i ^*

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

References

M. Choi, Completely Positive Linear Maps on Complex matrices, Linear Algebra and Its Applications, 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, 129–137, 1967.