ChoiMap - Revision history

Nathaniel at 17:22, 29 September 2014

2014-09-29T17:22:47Z

← Older revision		Revision as of 17:22, 29 September 2014
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	\|name=ChoiMap		\|name=ChoiMap
	\|desc=Produces the Choi map or one of its generalizations		\|desc=Produces the Choi map or one of its generalizations
			\|rel=[[ReductionMap]]
	\|cat=[[List of functions#Superoperators\|Superoperators]]		\|cat=[[List of functions#Superoperators\|Superoperators]]
	\|upd=August 5, 2013		\|upd=August 5, 2013
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	===The reduction map===		===The reduction map===
	The [[reduction map]] is the map $R$ defined by $R(X) = {\rm Tr}(X)I - X$, where $I$ is the identity operator. The reduction map is the Choi map that arises when $a = 0$, $b = c = 1$:		The [[ReductionMap\|reduction map]] is the map $R$ defined by $R(X) = {\rm Tr}(X)I - X$, where $I$ is the identity operator. The reduction map is the Choi map that arises when $a = 0$, $b = c = 1$:
	<syntaxhighlight>		<syntaxhighlight>
	>> ChoiMap(0,1,1)		>> ChoiMap(0,1,1)

			ans =

			0 0 0 0 -1 0 0 0 -1
			0 1 0 0 0 0 0 0 0
			0 0 1 0 0 0 0 0 0
			0 0 0 1 0 0 0 0 0
			-1 0 0 0 0 0 0 0 -1
			0 0 0 0 0 1 0 0 0
			0 0 0 0 0 0 1 0 0
			0 0 0 0 0 0 0 1 0
			-1 0 0 0 -1 0 0 0 0

			>> full(ReductionMap(3))

	ans =		ans =

Nathaniel at 15:22, 29 September 2014

2014-09-29T15:22:17Z

← Older revision		Revision as of 15:22, 29 September 2014
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	\|name=ChoiMap		\|name=ChoiMap
	\|desc=Produces the Choi map or one of its generalizations		\|desc=Produces the Choi map or one of its generalizations
	\|~~req~~=[[~~iden]]<br />[[MaxEntangled]]<br />[[opt_args~~]]		\|cat=[[List of functions#Superoperators\|Superoperators]]
	\|upd=August 5, 2013		\|upd=August 5, 2013
	\|v=1.00}}		\|v=0.50}}
	<tt>'''ChoiMap'''</tt> is a [[List of functions\|function]] that returns the Choi matrix of the linear map on $3 \times 3$ matrices that acts as follows:		<tt>'''ChoiMap'''</tt> is a [[List of functions\|function]] that returns the Choi matrix of the linear map on $3 \times 3$ matrices that acts as follows:

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	===The standard Choi map===		===The standard Choi map===
	The following code returns the Choi matrix of the Choi map and then verifies that the Choi map is indeed positive (i.e., verifies that its Choi matrix is [[block positive]]):		The following code returns the Choi matrix of the Choi map and then verifies that the Choi map is indeed positive (i.e., verifies that its Choi matrix is [[block positive]]):
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> C = ChoiMap()		>> C = ChoiMap()

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	-1 0 0 0 -1 0 0 0 1		-1 0 0 0 -1 0 0 0 1

	>> ~~[[IsBlockPositive\|~~IsBlockPositive(C)]] % verify that the Choi map is positive		>> IsBlockPositive(C) % verify that the Choi map is positive

	ans =		ans =

	1		1
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	===The reduction map===		===The reduction map===
	The [[reduction map]] is the map $R$ defined by $R(X) = {\rm Tr}(X)I - X$, where $I$ is the identity operator. The reduction map is the Choi map that arises when $a = 0$, $b = c = 1$:		The [[reduction map]] is the map $R$ defined by $R(X) = {\rm Tr}(X)I - X$, where $I$ is the identity operator. The reduction map is the Choi map that arises when $a = 0$, $b = c = 1$:
	<~~pre~~>		<syntaxhighlight>
	>> ChoiMap(0,1,1)		>> ChoiMap(0,1,1)

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	0 0 0 0 0 0 0 1 0		0 0 0 0 0 0 0 1 0
	-1 0 0 0 -1 0 0 0 0		-1 0 0 0 -1 0 0 0 0
	</~~pre~~>		</syntaxhighlight>

			{{SourceCode\|name=ChoiMap}}

	==References==		==References==
	<references />		<references />

Nathaniel at 17:44, 4 March 2014

2014-03-04T17:44:44Z

← Older revision		Revision as of 17:44, 4 March 2014
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	==Examples==		==Examples==
	===The standard Choi map===		===The standard Choi map===
	The following code returns the Choi matrix of the Choi map:		The following code returns the Choi matrix of the Choi map and then verifies that the Choi map is indeed positive (i.e., verifies that its Choi matrix is [[block positive]]):
	<pre>		<pre<noinclude></noinclude>>
	>> ChoiMap()		>> C = ChoiMap()

	~~ans~~ =		C =

	1 0 0 0 -1 0 0 0 -1		1 0 0 0 -1 0 0 0 -1
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	0 0 0 0 0 0 0 1 0		0 0 0 0 0 0 0 1 0
	-1 0 0 0 -1 0 0 0 1		-1 0 0 0 -1 0 0 0 1

			>> [[IsBlockPositive\|IsBlockPositive(C)]] % verify that the Choi map is positive

			ans =

			1
			</pre<noinclude></noinclude>>

			===The reduction map===
			The [[reduction map]] is the map $R$ defined by $R(X) = {\rm Tr}(X)I - X$, where $I$ is the identity operator. The reduction map is the Choi map that arises when $a = 0$, $b = c = 1$:
			<pre>
			>> ChoiMap(0,1,1)

			ans =

			0 0 0 0 -1 0 0 0 -1
			0 1 0 0 0 0 0 0 0
			0 0 1 0 0 0 0 0 0
			0 0 0 1 0 0 0 0 0
			-1 0 0 0 0 0 0 0 -1
			0 0 0 0 0 1 0 0 0
			0 0 0 0 0 0 1 0 0
			0 0 0 0 0 0 0 1 0
			-1 0 0 0 -1 0 0 0 0
	</pre>		</pre>

	==References==		==References==
	<references />		<references />

Nathaniel at 17:36, 4 March 2014

2014-03-04T17:36:53Z

← Older revision		Revision as of 17:36, 4 March 2014
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	: <math>\begin{bmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33}\end{bmatrix} \mapsto \begin{bmatrix}ax_{11}+bx_{22}+cx_{33} & -x_{12} & -x_{13} \\ -x_{21} & cx_{11}+ax_{22}+bx_{33} & -x_{23} \\ -x_{31} & -x_{32} & bx_{11}+cx_{22}+ax_{33}\end{bmatrix},</math>		: <math>\begin{bmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33}\end{bmatrix} \mapsto \begin{bmatrix}ax_{11}+bx_{22}+cx_{33} & -x_{12} & -x_{13} \\ -x_{21} & cx_{11}+ax_{22}+bx_{33} & -x_{23} \\ -x_{31} & -x_{32} & bx_{11}+cx_{22}+ax_{33}\end{bmatrix},</math>

	where $a,b,c$ are given real numbers. This map is positive if and only if $a \geq 1$, $a + b + c \geq 3$, and $bc \geq (2-a)^2$ whenever $1 \leq a \leq 2$<ref>S. J. Cho, S.-H. Kye, and S. G. Lee. Generalized Choi maps in three-dimensional matrix algebra. ''Linear Algebra Appl.'', 171:213, 1992.</ref>.		where $a,b,c$ are given real numbers. This map is positive if and only if $a \geq 0$, $a + b + c \geq 2$, and $bc \geq (1-a)^2$ whenever $0 \leq a \leq 1$<ref>S. J. Cho, S.-H. Kye, and S. G. Lee. Generalized Choi maps in three-dimensional matrix algebra. ''Linear Algebra Appl.'', 171:213, 1992.</ref>.

	==Syntax==		==Syntax==

Nathaniel at 17:33, 4 March 2014

2014-03-04T17:33:32Z

← Older revision		Revision as of 17:33, 4 March 2014
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	: <math>\begin{bmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33}\end{bmatrix} \mapsto \begin{bmatrix}ax_{11}+bx_{22}+cx_{33} & -x_{12} & -x_{13} \\ -x_{21} & cx_{11}+ax_{22}+bx_{33} & -x_{23} \\ -x_{31} & -x_{32} & bx_{11}+cx_{22}+ax_{33}\end{bmatrix},</math>		: <math>\begin{bmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33}\end{bmatrix} \mapsto \begin{bmatrix}ax_{11}+bx_{22}+cx_{33} & -x_{12} & -x_{13} \\ -x_{21} & cx_{11}+ax_{22}+bx_{33} & -x_{23} \\ -x_{31} & -x_{32} & bx_{11}+cx_{22}+ax_{33}\end{bmatrix},</math>

	where $a,b,c$ are given real numbers.		where $a,b,c$ are given real numbers. This map is positive if and only if $a \geq 1$, $a + b + c \geq 3$, and $bc \geq (2-a)^2$ whenever $1 \leq a \leq 2$<ref>S. J. Cho, S.-H. Kye, and S. G. Lee. Generalized Choi maps in three-dimensional matrix algebra. ''Linear Algebra Appl.'', 171:213, 1992.</ref>.

	==Syntax==		==Syntax==
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	-1 0 0 0 -1 0 0 0 1		-1 0 0 0 -1 0 0 0 1
	</pre>		</pre>

			==References==
			<references />

Nathaniel: Created page with "{{Function |name=ChoiMap |desc=Produces the Choi map or one of its generalizations |req=iden
MaxEntangled
opt_args |upd=August 5, 2013 |v=1.00}} `'''C..."`

2014-03-04T17:19:34Z Created page with "{{Function |name=ChoiMap |desc=Produces the Choi map or one of its generalizations |req=iden<br />MaxEntangled<br />opt_args |upd=August 5, 2013 |v=1.00}} <tt>'''C..." New page {{Function |name=ChoiMap |desc=Produces the Choi map or one of its generalizations |req=[[iden]]<br />[[MaxEntangled]]<br />[[opt_args]] |upd=August 5, 2013 |v=1.00}} <tt>'''ChoiMap'''</tt> is a [[List of functions|function]] that returns the Choi matrix of the linear map on $3 \times 3$ matrices that acts as follows: : <math>\begin{bmatrix}x_{11} & x_{12} & x_{13} \\ x_{21} & x_{22} & x_{23} \\ x_{31} & x_{32} & x_{33}\end{bmatrix} \mapsto \begin{bmatrix}ax_{11}+bx_{22}+cx_{33} & -x_{12} & -x_{13} \\ -x_{21} & cx_{11}+ax_{22}+bx_{33} & -x_{23} \\ -x_{31} & -x_{32} & bx_{11}+cx_{22}+ax_{33}\end{bmatrix},</math> where $a,b,c$ are given real numbers. ==Syntax== * <tt>C = ChoiMap()</tt> * <tt>C = ChoiMap(A,B,C)</tt> ==Argument descriptions== * <tt>A,B,C</tt>: Real parameters of the Choi map. If they are not provided, the default Choi map (with <tt>A = B = 1</tt> and <tt>C = 0</tt>) is returned. ==Examples== ===The standard Choi map=== The following code returns the Choi matrix of the Choi map: <pre> >> ChoiMap() ans = 1 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 -1 0 0 0 1 </pre>