ComplementaryMap - Revision history

Nathaniel at 02:27, 28 November 2014

2014-11-28T02:27:05Z

Nathaniel at 18:00, 12 November 2014

2014-11-12T18:00:06Z

Nathaniel at 15:23, 29 September 2014

2014-09-29T15:23:47Z

Nathaniel: Created page with "{{Function |name=ComplementaryMap |desc=Computes the complementary map of a superoperator |rel=DualMap |upd=January 22, 2013 |v=1.00}} `'''ComplementaryMap'''` ..."

2013-01-22T18:19:05Z

Created page with "{{Function |name=ComplementaryMap |desc=Computes the complementary map of a superoperator |rel=DualMap |upd=January 22, 2013 |v=1.00}} <tt>'''ComplementaryMap'''</tt> ..."

New page

{{Function
|name=ComplementaryMap
|desc=Computes the [[complementary map]] of a superoperator
|rel=[[DualMap]]
|upd=January 22, 2013
|v=1.00}}
<tt>'''ComplementaryMap'''</tt> is a [[List of functions|function]] that computes the [[complementary map]] of a superoperator (in the sense that the output of this function describes the information leaked by the original superoperator to the environment).

==Syntax==
* <tt>PHIC = ComplementaryMap(PHI)</tt>
* <tt>PHIC = ComplementaryMap(PHI,DIM)</tt>

==Argument descriptions==
* <tt>PHI</tt>: A superoperator. Should be provided as either a [[Choi matrix]], or as a cell with either 1 or 2 columns (see the [[tutorial]] page for more details about specifying superoperators within QETLAB). <tt>PHIC</tt> will be a cell of Kraus operators if <tt>PHI</tt> is a cell of Kraus operators, and similarly <tt>PHIC</tt> will be a Choi matrix if <tt>PHI</tt> is a Choi matrix.
* <tt>DIM</tt> (optional, default has input and output spaces of equal dimension): A 1-by-2 vector containing the input and output dimensions of <tt>PHI</tt>, in that order (equivalently, these are the dimensions of the first and second subsystems of the Choi matrix <tt>PHI</tt>, in that order). If the input or output space is not square, then <tt>DIM</tt>'s first row should contain the input and output row dimensions, and its second row should contain its input and output column dimensions. <tt>DIM</tt> is required if and only if <tt>PHI</tt> has unequal input and output dimensions and is provided as a Choi matrix.

==Examples==
===Non-uniqueness===
Complementary maps are not unique, and hence different maps <tt>PHIC</tt> may be returned depending on the particular representation of the input map <tt>PHI</tt>. The particular complementary map that is returned by this function is the one that is obtained by placing all of the first rows of Kraus operators of <tt>PHI</tt> into the first Kraus operator of <tt>PHIC</tt>, all of the second rows of Kraus operators of <tt>PHI</tt> into the second Kraus operator of <tt>PHIC</tt>, and so on. The following code defines two families of Kraus operators <tt>Phi</tt> and <tt>Phi2</tt>, verifies that they represent the same map by showing that their [[Choi matrices]] are the same, and then shows that nonetheless the different Kraus representations lead to different complementary maps.
<pre<noinclude></noinclude>>
>> Phi = {[1 0;0 0] ; [0 1;0 0] ; [0 0;1 0] ; [0 0;0 1]};
>> Phi2 = {[1 0;0 1]/sqrt(2) ; [0 1;1 0]/sqrt(2) ; [0 -1i;1i 0]/sqrt(2) ; [1 0;0 -1]/sqrt(2)};
>> [[ChoiMatrix|ChoiMatrix(Phi)]]

ans =

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

>> [[ChoiMatrix|ChoiMatrix(Phi2)]]

ans =

1.0000 0 0 0
0 1.0000 0 0
0 0 1.0000 0
0 0 0 1.0000

>> celldisp(ComplementaryMap(Phi))

ans{1} =

1 0
0 1
0 0
0 0

ans{2} =

0 0
0 0
1 0
0 1

>> celldisp(ComplementaryMap(Phi2))

ans{1} =

0.7071 0
0 0.7071
0 0 - 0.7071i
0.7071 0

ans{2} =

0 0.7071
0.7071 0
0 + 0.7071i 0
0 -0.7071
</pre<noinclude></noinclude>>

@@ Line 4: / Line 4: @@
 |rel=[[DualMap]]
 |cat=[[List of functions#Superoperators|Superoperators]]
-|upd=January 22, 2013
+|upd=November 12, 2014
 |v=0.50}}
 <tt>'''ComplementaryMap'''</tt> is a [[List of functions|function]] that computes the [[complementary map]] of a superoperator (in the sense that the output of this function describes the information leaked by the original superoperator to the environment).

@@ Line 1: / Line 1: @@
 {{Function
 |name=ComplementaryMap
-|desc=Computes the [[complementary map]] of a superoperator
+|desc=Computes the complementary map of a superoperator
 |rel=[[DualMap]]
 |cat=[[List of functions#Superoperators|Superoperators]]
-|upd=November 12, 2014
+|upd=November 24, 2014}}
-|v=0.50}}
+<tt>'''ComplementaryMap'''</tt> is a [[List of functions|function]] that computes the complementary map of a superoperator (in the sense that the output of this function describes the information leaked by the original superoperator to the environment).
 ==Syntax==
@@ Line 13: / Line 12: @@
 ==Argument descriptions==
-* <tt>PHI</tt>: A superoperator. Should be provided as either a [[Choi matrix]], or as a cell with either 1 or 2 columns (see the [[tutorial]] page for more details about specifying superoperators within QETLAB). <tt>PHIC</tt> will be a cell of Kraus operators if <tt>PHI</tt> is a cell of Kraus operators, and similarly <tt>PHIC</tt> will be a Choi matrix if <tt>PHI</tt> is a Choi matrix.
+* <tt>PHI</tt>: A superoperator. Should be provided as either a Choi matrix, or as a cell with either 1 or 2 columns (see the [[tutorial]] page for more details about specifying superoperators within QETLAB). <tt>PHIC</tt> will be a cell of Kraus operators if <tt>PHI</tt> is a cell of Kraus operators, and similarly <tt>PHIC</tt> will be a Choi matrix if <tt>PHI</tt> is a Choi matrix.
 * <tt>DIM</tt> (optional, default has input and output spaces of equal dimension): A 1-by-2 vector containing the input and output dimensions of <tt>PHI</tt>, in that order (equivalently, these are the dimensions of the first and second subsystems of the Choi matrix <tt>PHI</tt>, in that order). If the input or output space is not square, then <tt>DIM</tt>'s first row should contain the input and output row dimensions, and its second row should contain its input and output column dimensions. <tt>DIM</tt> is required if and only if <tt>PHI</tt> has unequal input and output dimensions and is provided as a Choi matrix.
 ==Examples==
 ===Non-uniqueness===
-Complementary maps are not unique, and hence different maps <tt>PHIC</tt> may be returned depending on the particular representation of the input map <tt>PHI</tt>. The particular complementary map that is returned by this function is the one that is obtained by placing all of the first rows of Kraus operators of <tt>PHI</tt> into the first Kraus operator of <tt>PHIC</tt>, all of the second rows of Kraus operators of <tt>PHI</tt> into the second Kraus operator of <tt>PHIC</tt>, and so on. The following code defines two families of Kraus operators <tt>Phi</tt> and <tt>Phi2</tt>, verifies that they represent the same map by showing that their [[Choi matrices]] are the same, and then shows that nonetheless the different Kraus representations lead to different complementary maps.
+Complementary maps are not unique, and hence different maps <tt>PHIC</tt> may be returned depending on the particular representation of the input map <tt>PHI</tt>. The particular complementary map that is returned by this function is the one that is obtained by placing all of the first rows of Kraus operators of <tt>PHI</tt> into the first Kraus operator of <tt>PHIC</tt>, all of the second rows of Kraus operators of <tt>PHI</tt> into the second Kraus operator of <tt>PHIC</tt>, and so on. The following code defines two families of Kraus operators <tt>Phi</tt> and <tt>Phi2</tt>, verifies that they represent the same map by showing that their Choi matrices are the same, and then shows that nonetheless the different Kraus representations lead to different complementary maps.
 <syntaxhighlight>
 >> Phi = {[1 0;0 0] ; [0 1;0 0] ; [0 0;1 0] ; [0 0;0 1]};

@@ Line 3: / Line 3: @@
 |desc=Computes the [[complementary map]] of a superoperator
 |rel=[[DualMap]]
 |upd=January 22, 2013
-|v=1.00}}
+|v=0.50}}
 <tt>'''ComplementaryMap'''</tt> is a [[List of functions|function]] that computes the [[complementary map]] of a superoperator (in the sense that the output of this function describes the information leaked by the original superoperator to the environment).
@@ Line 18: / Line 19: @@
 ===Non-uniqueness===
 Complementary maps are not unique, and hence different maps <tt>PHIC</tt> may be returned depending on the particular representation of the input map <tt>PHI</tt>. The particular complementary map that is returned by this function is the one that is obtained by placing all of the first rows of Kraus operators of <tt>PHI</tt> into the first Kraus operator of <tt>PHIC</tt>, all of the second rows of Kraus operators of <tt>PHI</tt> into the second Kraus operator of <tt>PHIC</tt>, and so on. The following code defines two families of Kraus operators <tt>Phi</tt> and <tt>Phi2</tt>, verifies that they represent the same map by showing that their [[Choi matrices]] are the same, and then shows that nonetheless the different Kraus representations lead to different complementary maps.
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
 >> Phi = {[1 0;0 0] ; [0 1;0 0] ; [0 0;1 0] ; [0 0;0 1]};
 >> Phi2 = {[1 0;0 1]/sqrt(2) ; [0 1;1 0]/sqrt(2) ; [0 -1i;1i 0]/sqrt(2) ; [1 0;0 -1]/sqrt(2)};
->> [[ChoiMatrix|ChoiMatrix(Phi)]]
+>> ChoiMatrix(Phi)
 ans =
@@ Line 30: / Line 31: @@
      0     0     1
->> [[ChoiMatrix|ChoiMatrix(Phi2)]]
+>> ChoiMatrix(Phi2)
 ans =
@@ Line 75: / Line 76: @@
 + 0.7071i        0
             -0.7071
-</pre<noinclude></noinclude>>
+</syntaxhighlight>

ComplementaryMap - Revision history

Nathaniel at 02:27, 28 November 2014

Nathaniel at 18:00, 12 November 2014

Nathaniel at 15:23, 29 September 2014

Nathaniel: Created page with "{{Function |name=ComplementaryMap |desc=Computes the complementary map of a superoperator |rel=DualMap |upd=January 22, 2013 |v=1.00}} '''ComplementaryMap''' ..."

Nathaniel: Created page with "{{Function |name=ComplementaryMap |desc=Computes the complementary map of a superoperator |rel=DualMap |upd=January 22, 2013 |v=1.00}} `'''ComplementaryMap'''` ..."