Choi's theorem on completely positive maps

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In mathematics, Choi's theorem on completely positive maps 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.

1 Some preliminary notions
2 Choi's result
- 2.1 Statement of theorem
- 2.2 Proof
3 Consequences
4 See also
5 References

[edit] Some preliminary notions

Before stating Choi's result, we give the definition of a completely positive map and fix some notation. C^{n × n} will denote the C*-algebra of n × n complex matrices. We will call A ∈ C^{n × 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 : 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 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

$\mathbb{C} ^{k \times k} \otimes \mathbb{C} ^{m \times m}$

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

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

We say that Φ is k-positive if $I_k \otimes \Phi$ 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 $\mathbb{C} ^{2 \times 2} \otimes \mathbb{C}^{2 \times 2}$ :

$\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ \end{bmatrix} .$

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

$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} ,$

which is clearly not positive.

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 definitions of k-copositivity and complete copositivity should be obvious.

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

[edit] Choi's result

[edit] Statement of theorem

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) Φ is n-positive.

ii) The matrix with operator entries

$\; (I \otimes \Phi)(E_{ij})_{ij} = ( \Phi (E_{ij}) )_{ij} \in \mathbb{C} ^{nm \times nm}$

is positive, where E_ij ∈ C^{n × n} is the matrix with 1 in the ij-th entry and 0's elsewhere. (The matrix ( Φ(E_ij))_ij is sometimes called the Choi matrix of Φ.)

iii) Φ is completely positive.

[edit] Proof

To show i) implies ii), we simply note that (E_ij)_ij ² = n · (E_ij)_ij and therefore (E_ij)_ij is positive.

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

We now turn to the argument for ii) ⇒ iii). Let the spectral resolution of (Φ(E_ij))_ij be

$\; ( \Phi (E_{ij}) )_{ij} = \sum _{i = 1} ^{nm} \lambda_i v_i v_i ^* .$

, where the vectors $v i$ lie in C^nm . By assumption, each eigenvalue $λ i$ is positive. So we can absorb the eigenvalues into the eigenvectors and put, as slight abuse of notation,

$\; ( \Phi (E_{ij}) )_{ij} = \sum _{i = 1} ^{nm} v_i v_i ^* .$

The vector space C^nm can be viewed as the direct sum

$\oplus _{i = 1} ^n \mathbb{C}^m .$

If P_i ∈ C^{nm × nm} is projection on the i-th copy of C^m, then

$\; \Phi (E_{lm}) = P_l \cdot ( \Phi (E_{ij}) )_{ij} \cdot P_m = \sum _{i = 1} ^{nm} P _l v_i ( v_i P_m )^*.$

Now it is clear how to obtain the operators V_i. Namely, define

$\; V_i e_l = P_l v_i .$

In other words, the matrix V_i is obtained by undoing the Vec operation on v_i. So

$\; \Phi (E_{lm}) = \sum _{i = 1} ^{nm} P _l v_i ( v_i P_m )^* = \sum _{i = 1} ^{nm} V_i e_l e_m ^* V_i ^* = \sum _{i = 1} ^{nm} V_i E_{lm} V_i ^*.$

Extending by linearity gives us Φ (A) = ∑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.

[edit] Consequences

[edit] Kraus operators

In the context of quantum information theory, the operators {V_i} are called the Kraus operators 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

$\; ( \Phi (E_{ij}) )_{ij} = B^* B.$

gives a set of Kraus operators. (Notice B need not be the unique positive square root of the Choi matrix.)

Let

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

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

$\; ( \Phi (E_{ij}) )_{ij} = \sum _{i = 1} ^{nm} b_i b_i ^*.$

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

When the Kraus operators are obtained from the spectral resolution 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. It might be relevant here to note that the square root factorizations for a positive semidefinite matrix are not unique in general.

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

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

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.

[edit] Completely copositive maps

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

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

[edit] 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 λ_i's are the eigenvalues of

$M_{\Phi} = \; ( \Phi (E_{ij}) )_{ij}$

and each V_i corresponds to an eigenvector of M_Φ. Unlike the completely positive case, M_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.

[edit] See also

[edit] 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.