AntisymmetricProjection - Revision history

Nathaniel: removing red links

2015-04-13T20:15:49Z

removing red links

Nathaniel at 19:17, 23 September 2014

2014-09-23T19:17:32Z

Nathaniel: /* Two subsystems */

2013-07-18T22:01:25Z

Two subsystems

Nathaniel: /* Argument descriptions */ typo fix

2012-11-27T16:53:22Z

Argument descriptions: typo fix

Nathaniel: Uploaded v1.01

2012-11-26T20:40:45Z

Uploaded v1.01

Nathaniel: /* When DIM is too small */

2012-11-22T01:08:45Z

When DIM is too small

Nathaniel: /* Two subsystems */

2012-11-21T21:11:06Z

Two subsystems

Nathaniel: Created page with "{{Function |name=AntisymmetricProjection |desc=Produces the projection onto the antisymmetric subspace |req=opt_args
perm_sign
[[PermutationOperator]..."

2012-11-21T20:27:53Z

Created page with "{{Function |name=AntisymmetricProjection |desc=Produces the projection onto the antisymmetric subspace |req=opt_args perm_sign [[PermutationOperator]..."

New page

{{Function
|name=AntisymmetricProjection
|desc=Produces the [[projection]] onto the [[antisymmetric subspace]]
|req=[[opt_args]] [[perm_sign]] [[PermutationOperator]] [[PermuteSystems]] [[sporth]]
|rel=[[SymmetricProjection]] [[SwapOperator]]
|upd=November 21, 2012
|v=1.00}}
<tt>'''AntisymmetricProjection'''</tt> is a [[List of functions|function]] that computes the [[orthogonal projection]] onto the [[antisymmetric subspace]] of two or more subsystems. The output of this function is always a sparse matrix.

==Syntax==
* <tt>PA = AntisymmetricProjection(DIM)</tt>
* <tt>PA = AntisymmetricProjection(DIM,P)</tt>
* <tt>PA = AntisymmetricProjection(DIM,P,PARTIAL)</tt>
* <tt>PA = AntisymmetricProjection(DIM,P,PARTIAL,MODE)</tt>

==Argument descriptions==
* <tt>DIM</tt>: The dimension of each of the subsystems.
* <tt>P</tt> (optional, default 2): The number of subsystems.
* <tt>PARTIAL</tt> (optional, default 0): If <tt>PARTIAL = 1</tt> then <tt>PA</tt> isn't the orthogonal projection itself, but rather a matrix whose columns form an orthonormal basis for the antisymmetric subspace (and hence <tt>PA*PA'</tt> is the orthogonal projection onto the antisymmetric subspace).
* <tt>MODE</tt> (optional, default -1): A flag that determines which of two algorithms is used to compute the antisymmetric projection. If <tt>MODE = -1</tt> then this script chooses which algorithm it thinks will be faster based on the values of <tt>DIM</tt> and <tt>P</tt>. If you wish to force the script to use a specific one of the algorithms (not recommended!), they are as follows:
** <tt>MODE = 0</tt>: Computes the antisymmetric projection by explicitly constructing an orthonormal basis of the symmetric subspace. The method for constructing such a basis is a modification of the procedure described for the [[symmetric subspace]] in <ref>John Watrous. [https://cs.uwaterloo.ca/~watrous/quant-info/lecture-notes/21.pdf Lecture 21: The quantum de Finetti theorem], ''Theory of Quantum Information Lecture Notes'', 2008.</ref>. This method is typically fast when <tt>DIM</tt> is small compared to <tt>P</tt>.
** <tt>MODE = 1</tt>: Computes the antisymmetric projection by averaging all <tt>P!</tt> permutation operators (in the sense of the <tt>[[PermutationOperator]]</tt> function). Permutation operators corresponding to an even permutation are given a weight of +1, while permutation operators corresponding to an odd permutation are given a weight of -1.<ref>R.B. Griffiths. [http://quantum.phys.cmu.edu/qm2/qmc161.pdf Systems of Identical Particles], ''33-756 Quantum Mechanics II Course Notes'', 2011.</ref> Because <tt>P!</tt> grows very quickly, this method is only practical when <tt>P</tt> is small.

==Examples==
===Two subsystems===
To compute the antisymmetric projection on two-qubit space, the following code suffices:
<pre>
>> AntisymmetricProjection(2)

ans =

(2,2) 0.5000
(3,2) -0.5000
(2,3) -0.5000
(3,3) 0.5000
</pre>
Note that the output of this function is always sparse. If you want a full matrix (not recommended for even moderately large <tt>DIM</tt> or <tt>P</tt>), you must explicitly convert it (as in the following example).

===More subsystems and <tt>PARTIAL</tt>===
To compute a matrix whose columns form an orthonormal basis for the symmetric subspace of three-[[qutrit]] space, set <tt>PARTIAL = 1</tt>:
<pre>
>> PA = AntisymmetricProjection(3,3,1)

PA =

(6,1) 0.4082
(8,1) -0.4082
(12,1) -0.4082
(16,1) 0.4082
(20,1) 0.4082
(22,1) -0.4082
</pre>

Note that <tt>PA</tt> is an isometry from the antisymmetric subspace (which is one-dimensional in this case) to the full three-qutrit space. In other words, <tt>PA'*PA</tt> is the identity matrix (i.e., the scalar 1 here) and <tt>PA*PA'</tt> is the orthogonal projection onto the antisymmetric subspace, which we can verify as follows:
<pre>
>> PA'*PA

ans =

(1,1) 1.0000

>> norm(PA*PA' - AntisymmetricProjection(3,3),'fro')

ans =

3.3307e-016
</pre>

===When <tt>DIM</tt> is too small===
When <tt>DIM < P</tt>, the antisymmetric subspace is zero-dimensional, as verified by the following line of code:
<pre>
>> AntisymmetricProjection(4,6)

ans =

All zero sparse: 4096-by-4096
</pre>

==References==
<references />

@@ Line 35: / Line 35: @@
     (3,3)       0.5000
 </pre>
-Note that the output of this function is always sparse. If you want a full matrix (not recommended for even moderately large <tt>DIM</tt> or <tt>P</tt>), you must explicitly convert it.
+Note that the output of this function is always sparse. If you want a full matrix (not recommended for even moderately large <tt>DIM</tt> or <tt>P</tt>), you must explicitly convert it using MATLAB's built-in [http://www.mathworks.com/help/matlab/ref/full.html full] function.
 ===More subsystems and <tt>PARTIAL</tt>===

@@ Line 4: / Line 4: @@
 |req=[[opt_args]]<br />[[perm_sign]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[sporth]]
 |rel=[[SymmetricProjection]]<br />[[SwapOperator]]
-|upd=November 21, 2012
+|upd=November 26, 2012
-|v=1.00}}
+|v=1.01}}
 <tt>'''AntisymmetricProjection'''</tt> is a [[List of functions|function]] that computes the [[orthogonal projection]] onto the [[antisymmetric subspace]] of two or more subsystems. The output of this function is always a sparse matrix.

@@ Line 1: / Line 1: @@
 {{Function
 |name=AntisymmetricProjection
-|desc=Produces the [[projection]] onto the [[antisymmetric subspace]]
+|desc=Produces the projection onto the antisymmetric subspace
 |rel=[[SymmetricProjection]]<br />[[SwapOperator]]
 |cat=[[List of functions#Permutations_and_symmetry_of_subsystems|Permutations and symmetry of subsystems]]
 |upd=November 26, 2012
 |v=0.50}}
-<tt>'''AntisymmetricProjection'''</tt> is a [[List of functions|function]] that computes the [[orthogonal projection]] onto the [[antisymmetric subspace]] of two or more subsystems. The output of this function is always a sparse matrix.
+<tt>'''AntisymmetricProjection'''</tt> is a [[List of functions|function]] that computes the orthogonal projection onto the antisymmetric subspace of two or more subsystems. The output of this function is always a sparse matrix.
 ==Syntax==
@@ Line 19: / Line 19: @@
 * <tt>PARTIAL</tt> (optional, default 0): If <tt>PARTIAL = 1</tt> then <tt>PA</tt> isn't the orthogonal projection itself, but rather a matrix whose columns form an orthonormal basis for the antisymmetric subspace (and hence <tt>PA*PA'</tt> is the orthogonal projection onto the antisymmetric subspace).
 * <tt>MODE</tt> (optional, default -1): A flag that determines which of two algorithms is used to compute the antisymmetric projection. If <tt>MODE = -1</tt> then this script chooses which algorithm it thinks will be faster based on the values of <tt>DIM</tt> and <tt>P</tt>. If you wish to force the script to use a specific one of the algorithms (not recommended!), they are as follows:
-** <tt>MODE = 0</tt>: Computes the antisymmetric projection by explicitly constructing an orthonormal basis of the antisymmetric subspace. The method for constructing such a basis is a modification of the procedure described for the [[symmetric subspace]] in <ref>John Watrous. [https://cs.uwaterloo.ca/~watrous/quant-info/lecture-notes/21.pdf Lecture 21: The quantum de Finetti theorem], ''Theory of Quantum Information Lecture Notes'', 2008.</ref>. This method is typically fast when <tt>DIM</tt> is small compared to <tt>P</tt>.
+** <tt>MODE = 0</tt>: Computes the antisymmetric projection by explicitly constructing an orthonormal basis of the antisymmetric subspace. The method for constructing such a basis is a modification of the procedure described for the symmetric subspace in <ref>John Watrous. [https://cs.uwaterloo.ca/~watrous/quant-info/lecture-notes/21.pdf Lecture 21: The quantum de Finetti theorem], ''Theory of Quantum Information Lecture Notes'', 2008.</ref>. This method is typically fast when <tt>DIM</tt> is small compared to <tt>P</tt>.
 ** <tt>MODE = 1</tt>: Computes the antisymmetric projection by averaging all <tt>P!</tt> permutation operators (in the sense of the <tt>[[PermutationOperator]]</tt> function). Permutation operators corresponding to an even permutation are given a weight of +1, while permutation operators corresponding to an odd permutation are given a weight of -1.<ref>R.B. Griffiths. [http://quantum.phys.cmu.edu/qm2/qmc161.pdf Systems of Identical Particles], ''33-756 Quantum Mechanics II Course Notes'', 2011.</ref> Because <tt>P!</tt> grows very quickly, this method is only practical when <tt>P</tt> is small.

@@ Line 2: / Line 2: @@
 |name=AntisymmetricProjection
 |desc=Produces the [[projection]] onto the [[antisymmetric subspace]]
 |rel=[[SymmetricProjection]]<br />[[SwapOperator]]
 |upd=November 26, 2012
-|v=1.01}}
+|v=0.50}}
 <tt>'''AntisymmetricProjection'''</tt> is a [[List of functions|function]] that computes the [[orthogonal projection]] onto the [[antisymmetric subspace]] of two or more subsystems. The output of this function is always a sparse matrix.
@@ Line 25: / Line 25: @@
 ===Two subsystems===
 To compute the antisymmetric projection on two-qubit space, the following code suffices:
-<pre>
+<syntaxhighlight>
 >> AntisymmetricProjection(2)
@@ Line 34: / Line 34: @@
     (2,3)      -0.5000
     (3,3)       0.5000
-</pre>
+</syntaxhighlight>
 Note that the output of this function is always sparse. If you want a full matrix (not recommended for even moderately large <tt>DIM</tt> or <tt>P</tt>), you must explicitly convert it using MATLAB's built-in [http://www.mathworks.com/help/matlab/ref/full.html full] function.
 ===More subsystems and <tt>PARTIAL</tt>===
 To compute a matrix whose columns form an orthonormal basis for the symmetric subspace of three-[[qutrit]] space, set <tt>PARTIAL = 1</tt>:
-<pre>
+<syntaxhighlight>
 >> PA = AntisymmetricProjection(3,3,1)
@@ Line 50: / Line 50: @@
    (20,1)       0.4082
    (22,1)      -0.4082
-</pre>
+</syntaxhighlight>
 Note that <tt>PA</tt> is an isometry from the antisymmetric subspace (which is one-dimensional in this case) to the full three-qutrit space. In other words, <tt>PA'*PA</tt> is the identity matrix (i.e., the scalar 1 here) and <tt>PA*PA'</tt> is the orthogonal projection onto the antisymmetric subspace, which we can verify as follows:
-<pre>
+<syntaxhighlight>
 >> PA'*PA
@@ Line 65: / Line 65: @@
 .3307e-016
-</pre>
+</syntaxhighlight>
 ===When <tt>DIM</tt> is too small===
 When <tt>DIM < P</tt>, the antisymmetric subspace is zero-dimensional, as verified in the <tt>DIM = 4, P = 6</tt> case by the following line of code:
-<pre>
+<syntaxhighlight>
 >> AntisymmetricProjection(4,6)
@@ Line 75: / Line 75: @@
     All zero sparse: 4096-by-4096
-</pre>
+</syntaxhighlight>
 ==References==
 <references />

AntisymmetricProjection - Revision history

Nathaniel: removing red links

Nathaniel at 19:17, 23 September 2014

Nathaniel: /* Two subsystems */

Nathaniel: /* Argument descriptions */ typo fix

Nathaniel: Uploaded v1.01

Nathaniel: /* When DIM is too small */

Nathaniel: /* Two subsystems */

Nathaniel: Created page with "{{Function |name=AntisymmetricProjection |desc=Produces the projection onto the antisymmetric subspace |req=opt_argsperm_sign[[PermutationOperator]..."

Nathaniel: Created page with "{{Function |name=AntisymmetricProjection |desc=Produces the projection onto the antisymmetric subspace |req=opt_args
perm_sign
[[PermutationOperator]..."