PermuteSystems - Revision history

Nathaniel at 19:15, 6 November 2014

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Nathaniel at 19:19, 23 September 2014

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Nathaniel at 20:42, 14 November 2012

2012-11-14T20:42:10Z

Nathaniel at 20:41, 14 November 2012

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Nathaniel at 20:10, 14 November 2012

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Nathaniel: Created page with "{{Function |name=PermuteSystems |desc=Permutes subsystems within a state or operator |req=opt_args |upd=November 14, 2012 |v=1.00|}} `'''PermuteSystems'''` ..."

2012-11-14T17:29:25Z

Created page with "{{Function |name=PermuteSystems |desc=Permutes subsystems within a state or operator |req=opt_args |upd=November 14, 2012 |v=1.00|}} <tt>'''PermuteSystems'''</tt> ..."

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{{Function
|name=PermuteSystems
|desc=Permutes subsystems within a [[state]] or [[operator]]
|req=[[opt_args]]
|upd=November 14, 2012
|v=1.00|}}
<tt>'''PermuteSystems'''</tt> is a function that allows the user to permute the order of the subsystems underlying a [[quantum state]] or operator that is defined on the tensor product of 2 or more subsystems. It works with full and sparse numeric matrices as well as symbolic matrices.

==Syntax==
* <tt>PX = PermuteSystems(X,PERM)</tt>
* <tt>PX = PermuteSystems(X,PERM,DIM)</tt>
* <tt>PX = PermuteSystems(X,PERM,DIM,ROW_ONLY)</tt>
* <tt>PX = PermuteSystems(X,PERM,DIM,ROW_ONLY,INV_PERM)</tt>

==Argument Descriptions==
* <tt>X</tt>: a vector (i.e., a [[pure quantum state]]) or a matrix to have its subsystems permuted
* <tt>PERM</tt>: a permutation vector
* <tt>DIM</tt> (optional, default has all subsystems of equal dimension):
* <tt>ROW_ONLY</tt> (optional, default <tt>0</tt>):
* <tt>INV_PERM</tt> (optional, default <tt>0</tt>):

==Examples==
===All subsystems of equal dimension===

In cases when all subsystems have the same dimension, most arguments can be omitted. Let's start with a matrix $X \in M_2 \otimes M_2$:

<pre>
>> X = [1 2 3 4;5 6 7 8;9 10 11 12;13 14 15 16]

X =

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

If we want to permute the two subsystems, we can call <tt>PermuteSystems</tt> with the permutation vector <tt>PERM = [2,1]</tt> (though if your needs are as simple as this, you may be better off using <tt>[[Swap.m]]</tt>):

<pre>
>> PermuteSystems(X,[2,1])

ans =

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

Similarly, the following code acts on a matrix $X \in M_A \otimes M_B \otimes M_C$ (where each subsystem has dimension 2) and outputs a matrix representation in the standard basis of $M_B \otimes M_C \otimes M_A$:

<pre>
>> X = reshape(1:64,8,8)'

X =

1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 62 63 64

>> PermuteSystems(X,[2,3,1])

ans =

1 5 2 6 3 7 4 8
33 37 34 38 35 39 36 40
9 13 10 14 11 15 12 16
41 45 42 46 43 47 44 48
17 21 18 22 19 23 20 24
49 53 50 54 51 55 52 56
25 29 26 30 27 31 28 32
57 61 58 62 59 63 60 64
</pre>

@@ Line 4: / Line 4: @@
 |rel=[[PermutationOperator]]<br />[[Swap]]<br />[[SwapOperator]]
 |cat=[[List of functions#Permutations_and_symmetry_of_subsystems|Permutations and symmetry of subsystems]]
-|upd=November 14, 2012
+|upd=November 6, 2014
 |v=0.50}}
 <tt>'''PermuteSystems'''</tt> is a function that allows the user to permute the order of the subsystems underlying a [[quantum state]] or operator that is defined on the tensor product of 2 or more subsystems. It works with full and sparse numeric matrices as well as symbolic matrices.

@@ Line 19: / Line 19: @@
 * <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 two ways:
 ** 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., <math>X \in M_{m_1, n_1} \otimes \cdots \otimes M_{m_p, n_p}</math>) 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 m<sub>i</sub>'s) and its second row should contain the column dimensions (i.e., the n<sub>i</sub>'s). In other words, you should set <tt>DIM = [m_1, ..., m_p; 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 m<sub>i</sub>'s) and its second row should contain the column dimensions (i.e., the n<sub>i</sub>'s). In other words, you should set <tt>DIM = [m_1, ..., m_p; n_1, ..., n_p]</tt>.
 * <tt>ROW_ONLY</tt> (optional, default <tt>0</tt>): If set equal to <tt>1</tt>, only the rows of <tt>X</tt> are permuted (this is equivalent to multiplying <tt>X</tt> on the left by <tt>[[PermutationOperator|PermutationOperator(DIM,PERM)]]</tt>). If equal to <tt>0</tt>, both the rows and columns of <tt>X</tt> are permuted (this is equivalent to multiplying <tt>X</tt> on both the left and right by the permutation operator).
 * <tt>INV_PERM</tt> (optional, default <tt>0</tt>): If equal to 0, this argument has no effect. If equal to 1, the subsystems are permuted according to the inverse of <tt>PERM</tt> rather than <tt>PERM</tt> itself.

@@ Line 19: / Line 19: @@
 * <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 two ways:
 ** 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 m<sub>i</sub>'s) and its second row should contain the column dimensions (i.e., the n<sub>i</sub>'s). In other words, you should set <tt>DIM = [m_1, ..., m_p; n_1, ..., n_p]</tt>.
+** If the subsystems aren't square (i.e., <math>X \in M_{m_1, n_1} \otimes \cdots \otimes M_{m_p, n_p}</math>) 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 m<sub>i</sub>'s) and its second row should contain the column dimensions (i.e., the n<sub>i</sub>'s). In other words, you should set <tt>DIM = [m_1, ..., m_p; n_1, ..., n_p]</tt>.
 * <tt>ROW_ONLY</tt> (optional, default <tt>0</tt>): If set equal to <tt>1</tt>, only the rows of <tt>X</tt> are permuted (this is equivalent to multiplying <tt>X</tt> on the left by <tt>[[PermutationOperator|PermutationOperator(DIM,PERM)]]</tt>). If equal to <tt>0</tt>, both the rows and columns of <tt>X</tt> are permuted (this is equivalent to multiplying <tt>X</tt> on both the left and right by the permutation operator).
 * <tt>INV_PERM</tt> (optional, default <tt>0</tt>): If equal to 0, this argument has no effect. If equal to 1, the subsystems are permuted according to the inverse of <tt>PERM</tt> rather than <tt>PERM</tt> itself.

@@ Line 16: / Line 16: @@
 ==Argument Descriptions==
 * <tt>X</tt>: a vector (e.g., a [[pure quantum state]]) or a matrix to have its subsystems permuted
-* <tt>PERM</tt>: a permutation vector
+* <tt>PERM</tt>: a permutation vector (i.e., a permutation of the vector <tt>1:n</tt>)
-* <tt>DIM</tt> (optional, default has all subsystems of equal dimension):
+* <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 two ways:
-* <tt>ROW_ONLY</tt> (optional, default <tt>0</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>).
-* <tt>INV_PERM</tt> (optional, default <tt>0</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 m<sub>i</sub>'s) and its second row should contain the column dimensions (i.e., the n<sub>i</sub>'s). In other words, you should set <tt>DIM = [m_1, ..., m_p; n_1, ..., n_p]</tt>.
 ==Examples==
@@ Line 37: / Line 39: @@
 </pre>
-If we want to permute the two subsystems, we can call <tt>PermuteSystems</tt> with the permutation vector <tt>PERM = [2,1]</tt> (though if your needs are as simple as this, you may be better off using <tt>[[Swap.m]]</tt>):
+If we want to permute the two subsystems, we can call <tt>PermuteSystems</tt> with the permutation vector <tt>PERM = [2,1]</tt> (though if your needs are as simple as this, you may be better off using the <tt>[[Swap]]</tt> function):
 <pre>

@@ Line 1: / Line 1: @@
 {{Function
 |name=PermuteSystems
-|desc=Permutes subsystems within a [[state]] or [[operator]]
+|desc=Permutes subsystems within a state or operator
 |rel=[[PermutationOperator]]<br />[[Swap]]<br />[[SwapOperator]]
 |upd=November 14, 2012
-|v=1.00|}}
+|v=0.50}}
 <tt>'''PermuteSystems'''</tt> is a function that allows the user to permute the order of the subsystems underlying a [[quantum state]] or operator that is defined on the tensor product of 2 or more subsystems. It works with full and sparse numeric matrices as well as symbolic matrices.
@@ Line 28: / Line 28: @@
 In cases when all subsystems have the same dimension, most arguments can be omitted. Let's start with a matrix $X \in M_2 \otimes M_2$:
-<pre>
+<syntaxhighlight>
 >> X = [1 2 3 4;5 6 7 8;9 10 11 12;13 14 15 16]
@@ Line 37: / Line 37: @@
     10    11    12
     14    15    16
-</pre>
+</syntaxhighlight>
 If we want to permute the two subsystems, we can call <tt>PermuteSystems</tt> with the permutation vector <tt>PERM = [2,1]</tt> (though if your needs are as simple as this, you may be better off using the <tt>[[Swap]]</tt> function):
-<pre>
+<syntaxhighlight>
 >> PermuteSystems(X,[2,1])
@@ Line 50: / Line 50: @@
      7     6     8
     15    14    16
-</pre>
+</syntaxhighlight>
 Similarly, the following code acts on a matrix $X \in M_A \otimes M_B \otimes M_C$ (where each subsystem has dimension 2) and outputs a matrix representation in the standard basis of $M_B \otimes M_C \otimes M_A$:
-<pre>
+<syntaxhighlight>
 >> X = reshape(1:64,8,8)'
@@ Line 80: / Line 80: @@
     29    26    30    27    31    28    32
     61    58    62    59    63    60    64
-</pre>
+</syntaxhighlight>