How does the partial transpose operation look like in matrix form?

Question

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?

Answer 1

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 $$\big(M^{T_B}\big)_{ij,k\ell}=M_{i\ell,kj}.$$

Answer 2

The representaion you look for does not exist because partial transposition is not a complete positive map.

Answer 3

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: $$ M^{T_A} = \sum_{ij} c_{ij} \left( \begin{array}{ccc} a^{(i)}_{11} B^{(i)} & a^{(i)}_{21} B^{(i)} & \cdots \\ a^{(i)}_{12} B^{(i)} & a^{(i)}_{22} B^{(i)} & \cdots \\ \vdots & \vdots & \ddots \\ \end{array} \right) \\ M^{T_B} = \sum_{ij} c_{ij} \left( \begin{array}{ccc} a^{(i)}_{11} (B^{(i)})^T & a^{(i)}_{12} (B^{(i)})^T & \cdots \\ a^{(i)}_{21} (B^{(i)})^T & a^{(i)}_{22} (B^{(i)})^T & \cdots \\ \vdots & \vdots & \ddots \\ \end{array} \right) $$