PartialTranspose - Revision history

Nathaniel at 15:33, 29 September 2014

2014-09-29T15:33:30Z

Nathaniel: Uploaded v1.01 (fixed a bug when using symbolic matrices)

2012-12-28T13:33:22Z

Uploaded v1.01 (fixed a bug when using symbolic matrices)

Nathaniel: typos

2012-11-22T17:39:24Z

typos

Nathaniel: Created page with "{{Function |name=PartialTranspose |desc=Computes the partial transpose of a matrix |req=opt_args
PermuteSystems |rel=PartialMap
PartialTrace |upd..."

2012-11-19T14:54:30Z

Created page with "{{Function |name=PartialTranspose |desc=Computes the partial transpose of a matrix |req=opt_args PermuteSystems |rel=PartialMap PartialTrace |upd..."

New page

{{Function
|name=PartialTranspose
|desc=Computes the [[partial transpose]] of a matrix
|req=[[opt_args]] [[PermuteSystems]]
|rel=[[PartialMap]] [[PartialTrace]]
|upd=November 17, 2012
|v=1.00}}
<tt>'''PartialTranspose'''</tt> is a [[List of functions|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==
* <tt>XPT = PartialTranspose(X)</tt>
* <tt>XPT = PartialTranspose(X,SYS)</tt>
* <tt>XPT = PartialTranspose(X,SYS,DIM)</tt>

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

==Examples==
===A bipartite square matrix===
By default, the <tt>PartialTranspose</tt> function performs the transposition on the second subsystem:
<pre>
>> 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
</pre>

By specifying the <tt>SYS</tt> argument, you can perform the transposition on the first subsystem instead:
<pre>
>> PartialTranspose(X,1)

ans =

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

Applying the transpose to both the first and second subsystems results in the standard transpose of <tt>X</tt>:
<pre>
>> norm(PartialTranspose(X,[1,2]) - X.')

ans =

0
</pre>

@@ Line 2: / Line 2: @@
 |name=PartialTranspose
 |desc=Computes the [[partial transpose]] of a matrix
 |rel=[[PartialMap]]<br />[[PartialTrace]]
 |upd=December 28, 2012
-|v=1.01}}
+|v=0.50}}
 <tt>'''PartialTranspose'''</tt> is a [[List of functions|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.
@@ Line 24: / Line 24: @@
 ===A bipartite square matrix===
 By default, the <tt>PartialTranspose</tt> function performs the transposition on the second subsystem:
-<pre>
+<syntaxhighlight>
 >> X = reshape(1:16,4,4)'
@@ Line 42: / Line 42: @@
     13    11    15
     14    12    16
-</pre>
+</syntaxhighlight>
 By specifying the <tt>SYS</tt> argument, you can perform the transposition on the first subsystem instead:
-<pre>
+<syntaxhighlight>
 >> PartialTranspose(X,1)
@@ Line 54: / Line 54: @@
      4    11    12
      8    15    16
-</pre>
+</syntaxhighlight>
 Applying the transpose to both the first and second subsystems results in the standard transpose of <tt>X</tt>:
-<pre>
+<syntaxhighlight>
 >> norm(PartialTranspose(X,[1,2]) - X.')
@@ Line 63: / Line 63: @@
-</pre>
+</syntaxhighlight>

@@ Line 4: / Line 4: @@
 |req=[[opt_args]]<br />[[PermuteSystems]]
 |rel=[[PartialMap]]<br />[[PartialTrace]]
-|upd=November 17, 2012
+|upd=December 28, 2012
-|v=1.00}}
+|v=1.01}}
 <tt>'''PartialTranspose'''</tt> is a [[List of functions|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.

PartialTranspose - Revision history

Nathaniel at 15:33, 29 September 2014

Nathaniel: Uploaded v1.01 (fixed a bug when using symbolic matrices)

Nathaniel: typos

Nathaniel: Created page with "{{Function |name=PartialTranspose |desc=Computes the partial transpose of a matrix |req=opt_argsPermuteSystems |rel=PartialMapPartialTrace |upd..."

Nathaniel: Created page with "{{Function |name=PartialTranspose |desc=Computes the partial transpose of a matrix |req=opt_args
PermuteSystems |rel=PartialMap
PartialTrace |upd..."