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  "body": "\n## Statement\n\nLet $n, m \\ge 1$, and let $\\Phi \\colon M_n(\\C) \\to M_m(\\C)$ be a linear map between [[matrix rings|0FHVPRAE60]].\nLet $E_{ij} \\in M_n(\\C)$ be the matrix units, and let\n$$C_\\Phi = \\sum_{i,j=1}^{n} E_{ij} \\otimes \\Phi(E_{ij}) \\in M_n(\\C) \\otimes M_m(\\C) \\cong M_{nm}(\\C)$$\nbe the Choi matrix of $\\Phi$, where $\\otimes$ is the [[Kronecker product|0N7ENQRCXN]].\nThe following are equivalent:\n\n1. $\\Phi$ is [[completely positive|CGY299QMC3]];\n2. $\\Phi$ is $n$-positive;\n3. $C_\\Phi$ is [[positive semidefinite|01MMVW8CKC]];\n4. there are $m \\times n$ complex matrices $V_1, \\ldots, V_r$, with $r \\le nm$, such that $\\Phi(a) = \\sum_{k=1}^{r} V_k a V_k^*$ for all $a \\in M_n(\\C)$.\n\n## Remarks\n\nProof idea: $1 \\Rightarrow 2$ is clear, and $2 \\Rightarrow 3$ holds because $C_\\Phi = \\Phi_n(x)$ for the positive block matrix $x = [E_{ij}]_{i,j}$.\nFor $3 \\Rightarrow 4$, write $C_\\Phi = \\sum_k v_k v_k^*$ with at most $nm$ vectors $v_k \\in \\C^{nm}$, using the spectral decomposition, and reshape each $v_k$ into an $m \\times n$ matrix $V_k$.\nFor $4 \\Rightarrow 1$, each map $a \\mapsto V a V^*$ is completely positive, and sums of completely positive maps are completely positive.\n\nExample: for the transpose $\\Phi(a) = a^T$ on $M_2(\\C)$, the Choi matrix is the swap matrix on $\\C^2 \\otimes \\C^2$, which has eigenvalue $-1$.\nSo the transpose is not completely positive, although it is positive.\nThe matrices $V_k$ in clause 4 are called Kraus operators.\n\n## Explanations\n\n### L1 human\n\nComplete positivity is a condition about infinitely many matrix sizes, so it looks hard to check.\nChoi's theorem reduces it to a single test: build one $nm \\times nm$ matrix from $\\Phi$ and check that it has no negative eigenvalue.\nIt also gives a concrete formula for every completely positive map between matrix algebras.\n\n### L2 human\n\nThe theorem is the finite-dimensional form of the [[Stinespring dilation theorem|X8K0Z7484R]].\nIn quantum information, it gives the Kraus representation of quantum channels; the channel is trace-preserving exactly when $\\sum_k V_k^* V_k = I$.\nThe correspondence $\\Phi \\mapsto C_\\Phi$ between maps and matrices is the Choi–Jamiołkowski isomorphism, and condition 2 shows that $n$-positivity already implies complete positivity for maps defined on $M_n(\\C)$.\n",
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