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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*-algebra]]s. An infinite-dimensional algebraic generalization of Choi's theorem is known as [[Viacheslav Belavkin|Belavkin]]'s "[[Radon–Nikodym theorem|Radon–Nikodym]]" theorem for completely positive maps.
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| == Some preliminary notions ==
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| Before stating Choi's result, we give the definition of a completely positive map and fix some notation. '''C'''<sup>''n'' × ''n''</sup> will denote the C*-algebra of ''n'' × ''n'' complex matrices. We will call ''A'' ∈ '''C'''<sup>''n'' × ''n''</sup> '''positive''', or symbolically, ''A'' ≥ 0, if ''A'' is Hermitian and the [[spectrum of a matrix|spectrum]] of ''A'' is nonnegative. (This condition is also called '''positive semidefinite'''.)
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| A linear map Φ : '''C'''<sup>''n'' × ''n''</sup> → '''C'''<sup>''m'' × ''m''</sup> 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.
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| Any linear map Φ induces another map
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| :<math>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}</math>
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| in a natural way: define
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| :<math>
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| ( I_k \otimes \Phi ) (M \otimes A) = M \otimes \Phi (A)
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| </math>
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| and extend by linearity. In matrix notation, a general element in
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| :<math>\mathbb{C} ^{k \times k} \otimes \mathbb{C} ^{n \times n}</math>
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| can be expressed as a ''k'' × ''k'' operator matrix:
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| :<math>
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| \begin{bmatrix}
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| A_{11} & \cdots & A_{1k} \\
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| \vdots & \ddots & \vdots \\
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| A_{k1} & \cdots & A_{kk}
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| \end{bmatrix},
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| </math>
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| and its image under the induced map is
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| :<math>
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| (I_k \otimes \Phi)
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| (\begin{bmatrix} A_{11} & \cdots & A_{1k} \\ \vdots & \ddots & \vdots \\A_{k1} & \cdots & A_{kk} \end{bmatrix})
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| =
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| \begin{bmatrix}
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| \Phi (A_{11}) & \cdots & \Phi( A_{1k} ) \\
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| \vdots & \ddots & \vdots \\
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| \Phi (A_{k1}) & \cdots & \Phi( A_{kk} )
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| \end{bmatrix}.
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| </math>
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| Writing out the individual elements in the above matrix-of-matrices amounts to the natural identification of algebras
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| :<math>
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| \mathbb{C}^{k\times k}\otimes\mathbb{C}^{m\times m}\cong\mathbb{C}^{km\times km}.
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| </math>
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| We say that Φ is '''k-positive''' if <math>I_k \otimes \Phi</math>, considered as an element of '''C'''<sup>''km''×''km''</sup>, is a positive map, and Φ is called '''completely positive'''
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| if Φ is k-positive for all k.
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| The [[Transpose|transposition map]] is a standard example of a positive map that fails to be 2-positive. Let T denote this map on '''C''' <sup>2 × 2</sup>. The following is a positive matrix in <math> \mathbb{C} ^{2 \times 2} \otimes \mathbb{C}^{2 \times 2} </math>:
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| :<math>
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| \begin{bmatrix}
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| \begin{pmatrix}1&0\\0&0\end{pmatrix}&
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| \begin{pmatrix}0&1\\0&0\end{pmatrix}\\
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| \begin{pmatrix}0&0\\1&0\end{pmatrix}&
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| \begin{pmatrix}0&0\\0&1\end{pmatrix}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 0 & 0 & 1 \\
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| 0 & 0 & 0 & 0 \\
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| 0 & 0 & 0 & 0 \\
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| 1 & 0 & 0 & 1 \\
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| \end{bmatrix} .
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| </math>
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| The image of this matrix under <math>I_2 \otimes T</math> is
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| :<math>
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| \begin{bmatrix}
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| \begin{pmatrix}1&0\\0&0\end{pmatrix}^T&
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| \begin{pmatrix}0&1\\0&0\end{pmatrix}^T\\
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| \begin{pmatrix}0&0\\1&0\end{pmatrix}^T&
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| \begin{pmatrix}0&0\\0&1\end{pmatrix}^T
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & 0 & 0 & 0 \\
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| 0 & 0 & 1 & 0 \\
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| 0 & 1 & 0 & 0 \\
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| 0 & 0 & 0 & 1 \\
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| \end{bmatrix} ,
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| </math>
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| which is clearly not positive, having determinant -1.
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| Incidentally, a map Φ is said to be '''co-positive''' if the composition Φ <math>\circ</math> ''T'' is positive. The transposition map itself is a co-positive map.
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| The above notions concerning positive maps extend naturally to maps between C*-algebras.
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| == Choi's result ==
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| === Statement of theorem ===
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| Choi's theorem reads as follows:
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| Let
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| :<math>\Phi : \mathbb{C} ^{n \times n} \rightarrow \mathbb{C} ^{m \times m}</math>
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| be a positive map. The following are equivalent:
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| i) <math>\Phi</math> is ''n''-positive.
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| ii) The matrix with operator entries
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| :<math>\; 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}</math>
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| is positive, where <math>E_{ij}\in\mathbb{C}^{n\times n}</math> is the matrix with 1 in the <math>ij</math>-th entry and 0s elsewhere. (The matrix <math>C_\Phi</math> is sometimes called the ''Choi matrix'' of <math>\Phi</math>.)
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| iii) <math>\Phi</math> is completely positive.
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| === Proof ===
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| To show i) implies ii), we observe that if
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| :<math>
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| \; E=\sum_{ij}E_{ij}\otimes E_{ij},
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| </math>
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| then ''E''=''E''<sup>*</sup> and ''E''<sup>2</sup>=''nE'', so ''E''=''n''<sup>−1</sup>''EE''<sup>*</sup> which is positive and
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| ''C''<sub>Φ</sub>=(''I<sub>n</sub>''⊗Φ)(''E'') is positive by the ''n''-positivity of Φ.
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| If iii) holds, then so does i) trivially.
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| We now turn to the argument for ii) ⇒ iii). This mainly involves chasing the different ways of looking at '''C'''<sup>''nm''×''nm''</sup>:
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| :<math>
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| \mathbb{C}^{nm\times nm}
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| \cong\mathbb{C}^{nm}\otimes(\mathbb{C}^{nm})^*
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| \cong\mathbb{C}^n\otimes\mathbb{C}^m\otimes(\mathbb{C}^n\otimes\mathbb{C}^m)^*
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| \cong\mathbb{C}^n\otimes(\mathbb{C}^n)^*\otimes\mathbb{C}^m\otimes(\mathbb{C}^m)^*
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| \cong\mathbb{C}^{n\times n}\otimes\mathbb{C}^{m\times m}.
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| </math>
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| Let the eigenvector decomposition of ''C''<sub>Φ</sub> be
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| :<math>
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| \; C_\Phi = \sum _{i = 1} ^{nm} \lambda_i v_i v_i ^* ,
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| </math>
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| where the vectors <math>v_i</math> lie in '''C'''<sup>''nm''</sup> . By assumption, each eigenvalue <math>\lambda_i</math> is non-negative so we can absorb the eigenvalues in the eigenvectors and redefine <math>v_i</math> so that
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| :<math>
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| \; C_\Phi = \sum _{i = 1} ^{nm} v_i v_i ^* .
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| </math>
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| The vector space '''C'''<sup>''nm''</sup> can be viewed as the direct sum <math>\textstyle \oplus_{i=1}^n \mathbb{C}^m</math>
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| compatibly with the above identification <math>\textstyle\mathbb{C}^{nm}\cong\mathbb{C}^n\otimes\mathbb{C}^m</math>
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| and the standard basis of '''C'''<sup>''n''</sup>.
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| If ''P<sub>k</sub>'' ∈ '''C'''<sup>''m'' × ''nm''</sup> is projection onto the ''k''-th copy of '''C'''<sup>''m''</sup>, then ''P<sub>k</sub>''<sup>*</sup> ∈ '''C'''<sup>''nm''×''m''</sup>
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| is the inclusion of '''C'''<sup>''m''</sup> as the ''k''-th summand of the direct sum and
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| :<math>
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| \; \Phi (E_{kl}) = P_k \cdot C_\Phi \cdot P_l^* = \sum _{i = 1} ^{nm} P_k v_i ( P_l v_i )^*.
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| </math>
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| Now if the operators ''V<sub>i</sub>'' ∈ '''C'''<sup>''m''×''n''</sup> are defined on the ''k''-th standard
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| basis vector ''e<sub>k</sub>'' of '''C'''<sup>''n''</sup> by
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| :<math>\; V_i e_k = P_k v_i,</math>
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| then
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| :<math>
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| \; \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 ^*
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| = \sum _{i = 1} ^{nm} V_i E_{kl} V_i ^*.
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| </math>
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| Extending by linearity gives us
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| :<math>
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| \; \Phi(A) = \sum_{i=1}^{nm} V_i A V_i^*
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| </math>
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| for any ''A'' ∈ '''C'''<sup>''n'' × ''n''</sup>. Since any map of this form is manifestly completely positive, we have the desired result.
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| The above is essentially Choi's original proof. Alternative proofs have also been known.
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| == Consequences ==
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| === Kraus operators ===
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| In the context of [[quantum information theory]], the operators {''V<sub>i</sub>''} are called the ''[[Kraus operator]]s'' (after [[Karl Kraus (physicist)|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
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| :<math>
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| \; C_\Phi = B^* B.
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| </math>
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| gives a set of Kraus operators. (Notice ''B'' need not be the unique positive [[square root of a matrix|square root]] of the Choi matrix.)
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| Let
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| :<math>B^* = [b_1, \ldots, b_{nm}] ,</math>
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| where ''b''<sub>i</sub>*'s are the row vectors of ''B'', then
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| :<math>
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| \; C_\Phi = \sum _{i = 1} ^{nm} b_i b_i ^*.
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| </math>
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| The corresponding Kraus operators can be obtained by exactly the same argument from the proof.
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| 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 operator|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.)
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| If two sets of Kraus operators {''A<sub>i</sub>''}<sub>1</sub><sup>''nm''</sup> and {''B<sub>i</sub>''}<sub>1</sub><sup>''nm''</sup> represent the same completely positive map Φ, then there exists a unitary ''operator'' matrix
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| :<math>\{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.
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| </math>
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| This can be viewed as a special case of the result relating two [[Stinespring factorization theorem|minimal Stinespring representations]].
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| Alternatively, there is an isometry ''scalar'' matrix {''u<sub>ij</sub>''}<sub>''ij''</sub> ∈ '''C'''<sup>''nm'' × ''nm''</sup> such that
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| :<math>
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| \; A_i = \sum _{i = 1} u_{ij} B_j.
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| </math>
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| 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''.
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| === Completely copositive maps ===
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| It follows immediately from Choi's theorem that Φ is completely copositive if and only if it is of the form
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| :<math>\Phi(A) = \sum _i V_i A^T V_i ^* .</math>
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| === Hermitian-preserving maps ===
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| 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
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| :<math>\Phi (A) = \sum_{i=1} ^{nm} \lambda_i V_i A V_i ^*</math>
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| where λ<sub>''i''</sub> are real numbers, the eigenvalues of ''C''<sub>Φ</sub>,
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| and each ''V''<sub>''i''</sub> corresponds to an eigenvector of ''C''<sub>Φ</sub>. Unlike the completely positive case, ''C''<sub>Φ</sub> 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.
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| == See also ==
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| *[[Stinespring factorization theorem]]
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| *[[Quantum operation]]
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| *[[Holevo's theorem]]
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| == References ==
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| * M. Choi, ''Completely Positive Linear Maps on Complex matrices'', Linear Algebra and Its Applications, 285–290, 1975
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| * V. P. Belavkin, P. Staszewski, ''Radon-Nikodym Theorem for Completely Positive Maps,'' Reports on Mathematical Physics, v.24, No 1, 49–55, 1986.
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|
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| * J. de Pillis, ''Linear Transformations Which Preserve Hermitian and Positive Semidefinite Operators'', Pacific Journal of Mathematics, 129–137, 1967.
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| [[Category:Linear algebra]]
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| [[Category:Operator theory]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in functional analysis]]
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