how to obtain partial transpose of a Tripartite operator?

Question

i know for a bipartite system with elements

|ij><kl|

elements of its partial transpose are

|kj><il|

now suppose a tripartite systems with elements

|abc><xyz|

what are elements of partial transpose of this tripartite systems versus component 1 and 2 and 3 separately?

please help me i need them for my calculation.

Answer 1

This is basically up to you: which elements are you transposing? If you're talking about transposing just the third system, then you'd be talking about $$ |abc\rangle\langle xyz|\mapsto |abz\rangle\langle xyc| $$ but you could do this on any of the three individual subsystems, or any of the three pairs of subsystems. Of course, if you're talking about doing the partial transpose on two subsystems, that's equivalent to the transpose over the whole system (which doesn't do anything in terms of entanglement detection, for example) plus the partial transpose of the third subsystem. So, generally, it's enough to consider the partial transposes of each of the three subsystems separately. \begin{align*} |abc\rangle\langle xyz|&\mapsto |xbc\rangle\langle ayz| \\ |abc\rangle\langle xyz|&\mapsto |ayc\rangle\langle xbz| \\ |abc\rangle\langle xyz|&\mapsto |abz\rangle\langle xyc| \end{align*}