Nathaniel at 15:30, 29 September 2014

2014-09-29T15:30:53Z

Nathaniel: Created page with "{{Function |name=KrausOperators |desc=Computes a set of Kraus operators for a superoperator |rel=ChoiMatrix |upd=January 8, 2013 |v=1.00}} `'''KrausOperators'''` is..."

2013-01-21T17:08:43Z

Created page with "{{Function |name=KrausOperators |desc=Computes a set of Kraus operators for a superoperator |rel=ChoiMatrix |upd=January 8, 2013 |v=1.00}} <tt>'''KrausOperators'''</tt> is..."

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{{Function
|name=KrausOperators
|desc=Computes a set of Kraus operators for a superoperator
|rel=[[ChoiMatrix]]
|upd=January 8, 2013
|v=1.00}}
<tt>'''KrausOperators'''</tt> is a [[List of functions|function]] that computes a set of [[Kraus operators]] for a [[superoperator]] $\Phi$. These superoperators will always be canonical in the following ways:
# If $\Phi$ is [[completely positive]], only the left Kraus operators will be returned (and the right Kraus operators are the same). That is, a cell will be returned that contains operators $\{A_j\}$ so that $\Phi(X) = \sum_j A_j X A_j^*$.
# If $\Phi$ is [[Hermiticity preserving]], the right Kraus operators are the same as the left Kraus operators, up to sign. The pairs of Kraus operators that are equal are given first, followed by the pairs that are negatives of each other.
# The left Kraus operators form an orthogonal set in the [[Hilbert-Schmidt inner product]], and similarly for the right Kraus operators.

==Syntax==
* <tt>KO = KrausOperators(PHI)</tt>
* <tt>KO = KrausOperators(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>DIM</tt> (optional, default has input and output spaces of equal dimension): This argument should only be given if <tt>PHI</tt> is provided as a Choi matrix. In this case, <tt>DIM</tt> should be 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).

==Examples==
Since the Choi matrix of the transpose map is the [[swap operator]], the following code finds a family of Kraus operators for the transpose map on 2-by-2 matrices:
<pre<noinclude></noinclude>>
>> T = KrausOperators([[SwapOperator|SwapOperator(2)]]);
>> celldisp(T)

T{1,1} =

0 0.7071
0.7071 0

T{2,1} =

1 0
0 0

T{3,1} =

0 0
0 1

T{4,1} =

0 0.7071
-0.7071 0

T{1,2} =

0 0.7071
0.7071 0

T{2,2} =

1 0
0 0

T{3,2} =

0 0
0 1

T{4,2} =

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

In other words, the following holds for all 2-by-2 matrices $X$:
$$X^T = \frac{1}{2}\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}X\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} + \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}X\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix} + \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}X\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} + \frac{1}{2}\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}X\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}.$$

@@ Line 3: / Line 3: @@
 |desc=Computes a set of Kraus operators for a superoperator
 |rel=[[ChoiMatrix]]
 |upd=January 8, 2013
-|v=1.00}}
+|v=0.50}}
 <tt>'''KrausOperators'''</tt> is a [[List of functions|function]] that computes a set of [[Kraus operators]] for a [[superoperator]] $\Phi$. These superoperators will always be canonical in the following ways:
 # If $\Phi$ is [[completely positive]], only the left Kraus operators will be returned (and the right Kraus operators are the same). That is, a cell will be returned that contains operators $\{A_j\}$ so that $\Phi(X) = \sum_j A_j X A_j^*$.
@@ Line 20: / Line 21: @@
 ==Examples==
 Since the Choi matrix of the transpose map is the [[swap operator]], the following code finds a family of Kraus operators for the transpose map on 2-by-2 matrices:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
->> T = KrausOperators([[SwapOperator|SwapOperator(2)]]);
+>> T = KrausOperators(SwapOperator(2));
 >> celldisp(T)
@@ Line 77: / Line 78: @@
    -0.7071
 .7071         0
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 In other words, the following holds for all 2-by-2 matrices $X$:
 $$X^T = \frac{1}{2}\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}X\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} + \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}X\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix} + \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}X\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix} + \frac{1}{2}\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}X\begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}.$$

KrausOperators - Revision history

Nathaniel at 15:30, 29 September 2014

Nathaniel: Created page with "{{Function |name=KrausOperators |desc=Computes a set of Kraus operators for a superoperator |rel=ChoiMatrix |upd=January 8, 2013 |v=1.00}} '''KrausOperators''' is..."

Nathaniel: Created page with "{{Function |name=KrausOperators |desc=Computes a set of Kraus operators for a superoperator |rel=ChoiMatrix |upd=January 8, 2013 |v=1.00}} `'''KrausOperators'''` is..."