How does the partial transpose operation look like in matrix form? The link here gives a nice description of how partial trace looks in matrix notation. I want a similar explanation for the matrix partial-transposition. How does matrix partial-transposition operation look in the matrix form rather than in Dirac notation?
 A: The representaion you look for does not exist because partial transposition is not a complete positive map.
A: In bra-ket notatoin, if the matrix $M$ reads
$$M=\sum_{ijkl}M_{ij,k\ell} \lvert i,j\rangle\!\langle k,\ell\rvert\equiv \sum_{ijkl}M_{ij,k\ell} (\lvert i\rangle\!\langle k\rvert\otimes\lvert j\rangle\!\langle \ell\rvert),$$
then its partial transpose with respect to the second space is
$$M^{T_B}= \sum_{ijkl}M_{ij,k\ell} (\lvert i\rangle\!\langle k\rvert\otimes\lvert \ell\rangle\!\langle j\rvert).$$
Equivalently, the partial transpose $M^{T_B}$ is that matrix with components
$$\big(M^{T_B}\big)_{ij,k\ell}=M_{i\ell,kj}.$$
A: An operator $M$ acting on a vector space $V_A \otimes V_B$ can be decomposed as: $M = \sum_{ij} c_{ij} A^{(i)} \otimes B^{(i)}$ with $A^{(i)}$ acting on $V_A$ and $B^{(j)}$ acting on $V_B$. Using matrices to denote $M$, $A^{(i)}$ and $B^{(i)}$:
$$
M 
=
\sum_{ij} c_{ij}
\left(
\begin{array}{ccc}
a^{(i)}_{11} B^{(i)} & a^{(i)}_{12} B^{(i)} & \cdots \\
a^{(i)}_{21} B^{(i)} & a^{(i)}_{22} B^{(i)} & \cdots \\
\vdots & \vdots & \ddots \\
\end{array}
\right)
$$
Then, the partial transpositions are:
$$
M^{T_A}
=
\sum_{ij} c_{ij}
\left(
\begin{array}{ccc}
a^{(i)}_{11} B^{(i)} & a^{(i)}_{21} B^{(i)} & \cdots \\
a^{(i)}_{12} B^{(i)} & a^{(i)}_{22} B^{(i)} & \cdots \\
\vdots & \vdots & \ddots \\
\end{array}
\right)
\\
M^{T_B}
=
\sum_{ij} c_{ij}
\left(
\begin{array}{ccc}
a^{(i)}_{11} (B^{(i)})^T & a^{(i)}_{12} (B^{(i)})^T & \cdots \\
a^{(i)}_{21} (B^{(i)})^T & a^{(i)}_{22} (B^{(i)})^T & \cdots \\
\vdots & \vdots & \ddots \\
\end{array}
\right)
$$
