PartialTranspose

PartialTranspose
Computes the partial transpose of a matrix
Other toolboxes required	opt_args PermuteSystems
Related functions	PartialMap PartialTrace

PartialTranspose is a function that computes the partial transpose of a matrix. The transposition may be taken on any subset of the subsystems on which the matrix acts.

Syntax

• XPT = PartialTranspose(X)
• XPT = PartialTranspose(X,SYS)
• XPT = PartialTranspose(X,SYS,DIM)

Argument descriptions

• X: A matrix to have its partial transpose returned.
• SYS (optional, default 2): A scalar or vector containing the indices of the subsystems on which the transpose is to be applied.
• DIM (optional, by default has all subsystems of equal dimension): A specification of the dimensions of the subsystems that X lives on. DIM can be provided in one of three ways:
  • If DIM is a scalar, it is assumed that X lives on the tensor product of two spaces, the first of which has dimension DIM and the second of which has dimension length(X)/DIM.
  • If $X \in M_{n_1} \otimes \cdots \otimes M_{n_p}$ then DIM should be a row vector containing the dimensions (i.e., DIM = [n_1, ..., n_p]).
  • If the subsystems aren't square (i.e., $X \in M_{m_1, n_1} \otimes \cdots \otimes M_{m_p, n_p}$) then DIM should be a matrix with two rows. The first row of DIM should contain the row dimensions of the subsystems (i.e., the m_i's) and its second row should contain the column dimensions (i.e., the n_i's). In other words, you should set DIM = [m_1, ..., m_p; n_1, ..., n_p].

Examples

A bipartite square matrix

By default, the PartialTranspose function performs the transposition on the second subsystem:

>> X = reshape(1:16,4,4)'

X =

     1     2     3     4
     5     6     7     8
     9    10    11    12
    13    14    15    16

>> PartialTranspose(X)

ans =

     1     5     3     7
     2     6     4     8
     9    13    11    15
    10    14    12    16

By specifying the SYS argument, you can perform the transposition on the first subsystem instead:

>> PartialTranspose(X,1)

ans =

     1     2     9    10
     5     6    13    14
     3     4    11    12
     7     8    15    16

Applying the transpose to both the first and second subsystems results in the standard transpose of X:

>> norm(PartialTranspose(X,[1,2]) - X.')

ans =

     0