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Fix a basis for the transposition map $T$. If $M\ge 0$, write $M=B{B}^{†}$. ThenThus transposition is a positive linear map and also preserves the trace.

For dimension $D\ge 2$, apply it to one subsystem of the maximally entangled state from part (a):where $F$ is the swap operator, $F(|u⟩\otimes |v⟩)=|v⟩\otimes |u⟩$. It satisfies $F^{†}=F$ and $F^2=I$, so its eigenvalues are $+1$ on symmetric vectors and $-1$ on antisymmetric vectors. In particular,A positive input has therefore acquired a negative expectation value after adjoining an identity channel. Transposition is positive but not completely positive in dimension at least two. The one-dimensional case is the identity map and is completely positive; the dimension qualification is necessary.

Use two precise finite-dimensional results. The positive-map separability criterion says that a state on ${\mathbb{C}}^2\otimes {\mathbb{C}}^3$ is separable exactly when $({\mathrm{id}}_2\otimes \Phi )(\sigma )\ge 0$ for every positive linear map $\Phi :M_3(\mathbb{C})\to M_2(\mathbb{C})$. The Størmer-Woronowicz decomposability theorem says that every such map is a decomposable positive map, sowith ${\Phi }_1,{\Phi }_2$ completely positive maps and $T$ the matrix transpose on $M_3$.

If $\sigma$ has positive partial transpose, both $\sigma$ and ${\sigma }^{T_B}$ are positive. HenceThe positive-map separability criterion now proves that $\sigma$ is a separable quantum state. The dimension-specific decomposability theorem is essential: the conclusion does not extend to arbitrary bipartite dimensions.

Let $\Phi :M_d(\mathbb{C})\to M_D(\mathbb{C})$ be a linear map. It is positive when $X\ge 0$ implies $\Phi (X)\ge 0$, and completely positive whenis positive for every ancillary dimension $r$. In finite dimensions it is enough to check $r=d$.

A finite family of Kraus operators $A^{\alpha }:{\mathbb{C}}^d\to {\mathbb{C}}^D$ defines the Kraus representationThis map is completely positive because, for every positive semidefinite matrix $Y$ on the enlarged space,

Conversely, use the unnormalized maximally entangled vector $|\Omega ⟩={\sum }_{j=1}^d|j⟩|j⟩$. Complete positivity makes the Choi matrixpositive semidefinite. By the spectral theorem for normal operators, $C_{\Phi }={\sum }_{\alpha =1}^R|v_{\alpha }⟩⟨v_{\alpha }|$, where $R=\mathrm{rank}{C}_{\Phi }\le dD$. Reshape each $v_{\alpha }\in {\mathbb{C}}^D\otimes {\mathbb{C}}^d$ into a matrix $A^{\alpha }$ by $|v_{\alpha }⟩={\sum }_{\mu j}{A}_{\mu j}^{\alpha }|\mu ⟩|j⟩$. The Choi matrix inversion formulathen gives $\Phi (X)={\sum }_{\alpha }{A}^{\alpha }X(A^{\alpha }{)}^{†}$. Thus finite Kraus representations characterize finite-dimensional completely positive maps.

The additional normalizationmakes $\Phi$ trace preserving and hence a quantum channel; ${\sum }_{\alpha }{A}^{\alpha }(A^{\alpha }{)}^{†}=I_D$ instead makes it unital.

A finite-dimensional bipartite density operator ${\rho }_{AB}$ is a separable quantum state exactly when $({\mathrm{id}}_A\otimes \Phi )({\rho }_{AB})\ge 0$ for every positive linear map from operators on $B$ to operators on $A$. In combination with the Størmer-Woronowicz decomposability theorem, it makes the positive partial transpose criterion sufficient for qubit-qutrit states. The criterion is established in the Horodeckis' separability paper.

Every positive linear map between $M_2(\mathbb{C})$ and $M_2(\mathbb{C})$, or between $M_2(\mathbb{C})$ and $M_3(\mathbb{C})$ in either direction, is a decomposable positive map. The low-dimensional result is the ingredient that makes the positive partial transpose criterion sufficient in two-qubit and qubit-qutrit systems. Woronowicz's low-dimensional positive-map paper establishes the $M_2$ to $M_3$ case; taking adjoints gives the reverse direction.