Choi's theorem on completely positive maps

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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[edit]

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

A linear map Φ : Cn × nCm × 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} \rightarrow \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


(I_k \otimes \Phi)
(\begin{bmatrix} A_{11} & \cdots & A_{1k} \\ \vdots & \ddots & \vdots \\A_{k1} & \cdots & A_{kk} \end{bmatrix})
=
\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 Ckm×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 × 2. The following is a positive matrix in  \mathbb{C} ^{2 \times 2} \otimes \mathbb{C}^{2 \times 2} :


\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[edit]

Statement of theorem[edit]

Choi's theorem reads as follows:

Let

\Phi : \mathbb{C} ^{n \times n} \rightarrow \mathbb{C} ^{m \times m}

be a positive map. The following are equivalent:

i) \Phi is n-positive.

ii) The matrix with operator entries

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

is positive, where E_{ij}\in\mathbb{C}^{n\times n} is the matrix with 1 in the ij-th entry and 0s elsewhere. (The matrix C_\Phi is sometimes called the Choi matrix of \Phi.)

iii) \Phi is completely positive.

Proof[edit]

To show 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 CΦ=(In⊗Φ)(E) is positive by the n-positivity of Φ.

If iii) holds, then so does i) trivially.

We now turn to the argument for ii) ⇒ iii). This mainly involves chasing the different ways of looking at Cnm×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 CΦ be


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

where the vectors v_i lie in Cnm . 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 Cnm 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 PkCm × nm is projection onto the k-th copy of Cm, then Pk*Cnm×m is the inclusion of Cm 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 ViCm×n are defined on the k-th standard basis vector ek 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 ACn × 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[edit]

Kraus operators[edit]

In the context of quantum information theory, the operators {Vi} 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_\Phi = 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 bi*'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 {Ai}1nm and {Bi}1nm 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 {uij}ijCnm × 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[edit]

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[edit]

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 λi are real numbers, the eigenvalues of CΦ, and each Vi corresponds to an eigenvector of CΦ. Unlike the completely positive case, CΦ 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.

See also[edit]

References[edit]

  • 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.