Nathaniel at 19:53, 6 January 2016

2016-01-06T19:53:48Z

Nathaniel at 19:03, 30 September 2014

2014-09-30T19:03:15Z

Nathaniel: Created page with "{{Function |name=RandomSuperoperator |desc=Generates a random superoperator |rel=RandomDensityMatrix
RandomStateVector
RandomUnitary |cat=List of funct..."

2014-09-30T19:02:29Z

Created page with "{{Function |name=RandomSuperoperator |desc=Generates a random superoperator |rel=RandomDensityMatrix<br />RandomStateVector<br />RandomUnitary |cat=List of funct..."

New page

{{Function
|name=RandomSuperoperator
|desc=Generates a random superoperator
|rel=[[RandomDensityMatrix]]<br />[[RandomStateVector]]<br />[[RandomUnitary]]
|cat=[[List of functions#Random_things|Random things]]
|upd=September 30, 2014
|v=0.50}}
<tt>'''RandomSuperoperator'''</tt> is a [[List of functions|function]] that generates a random completely positive map satisfying some user-specified properties (such as being trace-preserving, being unital, and/or having a specified number of Kraus operators).

==Syntax==
* <tt>PHI = RandomSuperoperator(DIM)</tt>
* <tt>PHI = RandomSuperoperator(DIM,TP)</tt>
* <tt>PHI = RandomSuperoperator(DIM,TP,UN)</tt>
* <tt>PHI = RandomSuperoperator(DIM,TP,UN,RE)</tt>
* <tt>PHI = RandomSuperoperator(DIM,TP,UN,RE,KR)</tt>

==Argument descriptions==
* <tt>DIM</tt>: Either a scalar, indicating that <tt>PHI</tt> should act on <tt>DIM</tt>-by-<tt>DIM</tt> matrices, or a 1-by-2 vector, indicating that <tt>PHI</tt> should take <tt>DIM(1)</tt>-by-<tt>DIM(1)</tt> matrices as input and return <tt>DIM(2)</tt>-by-<tt>DIM(2)</tt> matrices as output.
* <tt>TP</tt> (optional, default 1): A flag (either 1 or 0) indicating that <tt>PHI</tt> should or should not be trace-preserving.
* <tt>UN</tt> (optional, default 0): A flag (either 1 or 0) indicating that <tt>PHI</tt> should or should not be unital (i.e., satisfy $\Phi(I) = I$).
* <tt>RE</tt> (optional, default 0): A flag (either 1 or 0) indicating that the Choi matrix of <tt>PHI</tt> (equivalently, its Kraus operators) should only have real entries (<tt>RE = 1</tt>) or that it is allowed to have complex entries (<tt>RE = 0</tt>).
* <tt>KR</tt> (optional, default <tt>prod(DIM)</tt>): The maximal number of Kraus operators of the superoperator to be produced. With probability 1, it will have exactly <tt>KR</tt> Kraus operators (if <tt>KR ≤ prod(DIM)</tt>).

==Examples==
===Random qubit channel===
The following code generates a random qubit channel and verifies that it is indeed a qubit channel:
<syntaxhighlight>
>> Phi = RandomSuperoperator(2);
>> ApplyMap(eye(2),DualMap(Phi)) % the dual map is unital if and only if Phi is trace-preserving

ans =

1.0000 - 0.0000i -0.0000 - 0.0000i
-0.0000 + 0.0000i 1.0000 - 0.0000i

>> IsCP(Phi) % verify that Phi is completely positive

ans =

1
</syntaxhighlight>

===Random unital channel with specified number of Kraus operators===
You can request completely positive maps with any combination of the optional arguments as constraints. For example, the following code requests a random unital quantum channel that sends 3-by-3 matrices to 4-by-4 matrices and has 5 Kraus operators. Note that a warning is produced because, strictly speaking, no such map exists (it's impossible for a map to be both trace-preserving and unital unless the input and output dimensions are the same). Instead, the map is that is produced is trace-preserving and sends the identity matrix to a scalar multiple of the identity matrix.
<syntaxhighlight>
>> Phi = RandomSuperoperator([3,4],1,1,0,5);
Warning: There does not exist a unital, trace-preserving map in the case when the input and output
dimensions are unequal. The identity matrix will map to a *multiple* of the identity matrix.
> In RandomSuperoperator at 45

>> ApplyMap(eye(3),Phi) % verify that Phi is (up to scaling) unital

ans =

0.7500 + 0.0000i -0.0000 - 0.0000i -0.0000 - 0.0000i -0.0000 - 0.0000i
-0.0000 + 0.0000i 0.7500 + 0.0000i 0.0000 + 0.0000i -0.0000 - 0.0000i
-0.0000 + 0.0000i 0.0000 - 0.0000i 0.7500 + 0.0000i -0.0000 + 0.0000i
-0.0000 + 0.0000i -0.0000 + 0.0000i -0.0000 - 0.0000i 0.7500 + 0.0000i

>> ApplyMap(eye(4),DualMap(Phi,[3,4])) % verify that Phi is trace-preserving

ans =

1.0000 - 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i
0.0000 - 0.0000i 1.0000 - 0.0000i 0.0000 + 0.0000i
0.0000 - 0.0000i 0.0000 - 0.0000i 1.0000 - 0.0000i

>> IsCP(Phi) % verify that Phi is completely positive

ans =

1

>> KrausOperators(Phi,[3,4]) % verify that Phi has 5 Kraus operators

ans =

[4x3 double]
[4x3 double]
[4x3 double]
[4x3 double]
[4x3 double]
</syntaxhighlight>

==Notes==
The superoperator is not generated with regard to any well-known or well-studied distribution, but it can output any superoperator satisfying the specified constraints (unital, trace-preserving, number of Kraus operators), and it derives from uniform spherical measure in a fairly straightforward way.

{{SourceCode|name=RandomSuperoperator}}

@@ Line 2: / Line 2: @@
 |name=RandomSuperoperator
 |desc=Generates a random superoperator (completely positive map)
-|rel=[[RandomDensityMatrix]]<br />[[RandomStateVector]]<br />[[RandomUnitary]]
+|rel=[[RandomDensityMatrix]]<br />[[RandomPOVM]]<br />[[RandomStateVector]]<br />[[RandomUnitary]]
 |cat=[[List of functions#Random_things|Random things]]
 |upd=September 30, 2014

RandomSuperoperator - Revision history

Nathaniel at 19:53, 6 January 2016

Nathaniel at 19:03, 30 September 2014

Nathaniel: Created page with "{{Function |name=RandomSuperoperator |desc=Generates a random superoperator |rel=RandomDensityMatrixRandomStateVectorRandomUnitary |cat=List of funct..."

Nathaniel: Created page with "{{Function |name=RandomSuperoperator |desc=Generates a random superoperator |rel=RandomDensityMatrix
RandomStateVector
RandomUnitary |cat=List of funct..."