# Partial transpose

PT (bit labels);
outputs the partial transpose with respect to the qubits specified by bit labels. There is also the command
PTb (bit labels);
This shows each vector of the output of the partial transpose in the format of binary digits.

To save the partial transpose into the file named file name, use the command:

PTsave (bit labels, file name);
The saved matrix can be loaded by mload;''. Further analysis with the saved matrix should be performed in another calculator such as Octave. This is possible when a user converts it to the matrix in another format (See Sec. 5.3).

To save the partial transpose into a temporary file, use the command

PTtmp (bit labels, file name);

Note: These four commands does not change the current state in calculation. They only show or save a partial transpose.

In general, the partial transpose of a density operator of a bipartite state (the state space is the Hilbert space ) with respect to the component system () is denoted as . Let us write the - element of as

 (12)

Then the - element of is given by

 (13)

In other words, the - element of , , is mapped to the - element of , , for all the possible integers , where are basis vectors of the component system , and are basis vectors of the component system .

For example, the partial transpose with respect to the left most qubit of is . For this purpose, you may just type PTb(1);'' after setting up the mixed state . Try the following example in the read-line mode:


=======================
> init-mixed 1.0 0.0 1.0;
> H(3); CN(3,1);
> showb;
0.5|010><010| + 0.5|010><111| + 0.5|111><010| + 0.5|111><111|
> PTb(1);
0.5|010><010| + 0.5|011><110| + 0.5|110><011| + 0.5|111><111|
> showb;
0.5|010><010| + 0.5|010><111| + 0.5|111><010| + 0.5|111><111|
>

root
2004-11-07