Nathaniel: typo

2014-10-15T16:27:48Z

typo

← Older revision		Revision as of 16:27, 15 October 2014
Line 10:		Line 10:
	: <math>\Phi(X) := \mathrm{Tr}(X)I - X,</math>		: <math>\Phi(X) := \mathrm{Tr}(X)I - X,</math>

	where $I$ is the identity matrix. This map is ~~positive~~ positive.		where $I$ is the identity matrix. This map is positive.

	==Syntax==		==Syntax==

Nathaniel: Created page with "{{Function |name=ReductionMap |desc=Produces the reduction map |rel=ChoiMap |cat=Superoperators |upd=September 29, 2014 |v=0.50}} `..."`

2014-09-29T17:21:33Z Created page with "{{Function |name=ReductionMap |desc=Produces the reduction map |rel=ChoiMap |cat=Superoperators |upd=September 29, 2014 |v=0.50}} <tt>..." New page {{Function |name=ReductionMap |desc=Produces the reduction map |rel=[[ChoiMap]] |cat=[[List of functions#Superoperators|Superoperators]] |upd=September 29, 2014 |v=0.50}} <tt>'''ReductionMap'''</tt> is a [[List of functions|function]] that returns the Choi matrix of the linear map that acts as follows: : <math>\Phi(X) := \mathrm{Tr}(X)I - X,</math> where $I$ is the identity matrix. This map is positive positive. ==Syntax== * <tt>R = ReductionMap(DIM)</tt> * <tt>R = ReductionMap(DIM,K)</tt> ==Argument descriptions== * <tt>DIM</tt>: The dimension of the reduction map. That is, the size of the matrices that the reduction map acts on. * <tt>K</tt> (optional, default 1): If this positive integer is provided, the script will instead return the Choi matrix of the following linear map: : <math>\Phi(X) := K\cdot\mathrm{Tr}(X)I - X.</math> ==Examples== ===The reduction map is positive=== The following code returns the Choi matrix of the 3-dimensional reduction map and then verifies that the reduction map is indeed positive (i.e., verifies that its Choi matrix is block positive): <syntaxhighlight> >> R = ReductionMap(3) R = (5,1) -1 (9,1) -1 (2,2) 1 (3,3) 1 (4,4) 1 (1,5) -1 (9,5) -1 (6,6) 1 (7,7) 1 (8,8) 1 (1,9) -1 (5,9) -1 >> IsBlockPositive(R) % verify that the reduction map is positive ans = 1 </syntaxhighlight> ===Higher values of <tt>K</tt>=== It is known that the generalization of the reduction map that is provided by the optional argument <tt>K</tt> is always <tt>K</tt>-positive, but not (<tt>K+1</tt>)-positive. The following code verifies this in the <tt>K = 2</tt> case: <syntaxhighlight> >> R = ReductionMap(3,2) R = (1,1) 1 (5,1) -1 (9,1) -1 (2,2) 2 (3,3) 2 (4,4) 2 (1,5) -1 (5,5) 1 (9,5) -1 (6,6) 2 (7,7) 2 (8,8) 2 (1,9) -1 (5,9) -1 (9,9) 1 >> IsBlockPositive(R,1) % verify that this map is positive ans = 1 >> IsBlockPositive(R,2) % verify that this map is 2-positive ans = 1 >> IsBlockPositive(R,3) % see that this map is not 3-positive ans = 0 </syntaxhighlight> {{SourceCode|name=ReductionMap}}

ReductionMap - Revision history

Nathaniel: typo

Nathaniel: Created page with "{{Function |name=ReductionMap |desc=Produces the reduction map |rel=ChoiMap |cat=Superoperators |upd=September 29, 2014 |v=0.50}} ..."

Nathaniel: Created page with "{{Function |name=ReductionMap |desc=Produces the reduction map |rel=ChoiMap |cat=Superoperators |upd=September 29, 2014 |v=0.50}} `..."`