PartialTranspose

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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 of which in 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 mi's) and its second row should contain the column dimensions (i.e., the ni'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