SchmidtDecomposition - Revision history

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Nathaniel: Created page with "{{Function |name=SchmidtDecomposition |desc=Computes the Schmidt decomposition of a bipartite vector |req=opt_args |rel=SchmidtRank
SchmidtNumber |upd..."

2012-11-19T14:12:15Z

Created page with "{{Function |name=SchmidtDecomposition |desc=Computes the Schmidt decomposition of a bipartite vector |req=opt_args |rel=SchmidtRank<br />SchmidtNumber |upd..."

New page

{{Function
|name=SchmidtDecomposition
|desc=Computes the [[Schmidt decomposition]] of a [[bipartite]] vector
|req=[[opt_args]]
|rel=[[SchmidtRank]]<br />[[SchmidtNumber]]
|upd=November 16, 2012
|v=1.00}}
<tt>'''SchmidtDecomposition'''</tt> is a [[List of functions|function]] that computes the [[Schmidt decomposition]] of a [[bipartite]] vector. The user may specify how many terms in the Schmidt decomposition they wish to be returned.

==Syntax==
* <tt>S = SchmidtDecomposition(VEC)</tt>
* <tt>S = SchmidtDecomposition(VEC,DIM)</tt>
* <tt>S = SchmidtDecomposition(VEC,DIM,K)</tt>
* <tt>[S,U,V] = SchmidtDecomposition(VEC,DIM,K)</tt>

==Argument descriptions==
===Input arguments===
* <tt>VEC</tt>: A [[bipartite]] vector (e.g., a [[pure quantum state]]) to have its [[Schmidt decomposition]] computed.
* <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that <tt>VEC</tt> lives on.
* <tt>K</tt> (optional, default 0): A flag that determines how many terms in the Schmidt decomposition should be computed. If <tt>K</tt> = 0 then all terms with non-zero [[Schmidt coefficient]]s are computed. If <tt>K</tt> = -1 then all terms (including zero Schmidt coefficients) are computed. If <tt>K</tt> > 0 then the <tt>K</tt> terms with largest Schmidt coefficients are computed.

===Output arguments===
* <tt>S</tt>: A vector containing the Schmidt coefficients of <tt>VEC</tt>.
* <tt>U</tt> (optional): The left Schmidt vectors of <tt>VEC</tt>.
* <tt>V</tt> (optional): The right Schmidt vectors of <tt>VEC</tt>.

==Examples==
Please add examples here.

@@ Line 2: / Line 2: @@
 |name=SchmidtDecomposition
 |desc=Computes the [[Schmidt decomposition]] of a [[bipartite]] vector
-|rel=[[IsProductVector]]<br />[[OperatorSchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SchmidtNumber]]
+|rel=[[IsProductVector]]<br />[[OperatorSchmidtDecomposition]]<br />[[SchmidtRank]]
 |cat=[[List of functions#Entanglement_and_separability|Entanglement&nbsp;and&nbsp;separability]]
 |upd=December 1, 2012

@@ Line 4: / Line 4: @@
 |req=[[opt_args]]
 |rel=[[IsProductVector]]<br />[[OperatorSchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SchmidtNumber]]
-|upd=November 23, 2012
+|upd=December 1, 2012
-|v=1.01}}
+|v=1.02}}
 <tt>'''SchmidtDecomposition'''</tt> is a [[List of functions|function]] that computes the [[Schmidt decomposition]] of a [[bipartite]] vector. The user may specify how many terms in the Schmidt decomposition they wish to be returned.

@@ Line 3: / Line 3: @@
 |desc=Computes the [[Schmidt decomposition]] of a [[bipartite]] vector
 |req=[[opt_args]]
-|rel=[[OperatorSchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SchmidtNumber]]
+|rel=[[IsProductVector]]<br />[[OperatorSchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SchmidtNumber]]
 |upd=November 23, 2012
 |v=1.01}}

@@ Line 3: / Line 3: @@
 |desc=Computes the [[Schmidt decomposition]] of a [[bipartite]] vector
 |req=[[opt_args]]
-|rel=[[SchmidtRank]]<br />[[SchmidtNumber]]
+|rel=[[OperatorSchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SchmidtNumber]]
 |upd=November 23, 2012
 |v=1.01}}

@@ Line 1: / Line 1: @@
 {{Function
 |name=SchmidtDecomposition
-|desc=Computes the [[Schmidt decomposition]] of a [[bipartite]] vector
+|desc=Computes the Schmidt decomposition of a bipartite vector
 |rel=[[IsProductVector]]<br />[[OperatorSchmidtDecomposition]]<br />[[SchmidtRank]]
 |cat=[[List of functions#Entanglement_and_separability|Entanglement&nbsp;and&nbsp;separability]]
 |upd=December 1, 2012
 |v=0.50}}
-<tt>'''SchmidtDecomposition'''</tt> is a [[List of functions|function]] that computes the [[Schmidt decomposition]] of a [[bipartite]] vector. The user may specify how many terms in the Schmidt decomposition they wish to be returned.
+<tt>'''SchmidtDecomposition'''</tt> is a [[List of functions|function]] that computes the Schmidt decomposition of a bipartite vector. The user may specify how many terms in the Schmidt decomposition they wish to be returned.
 ==Syntax==
@@ Line 16: / Line 16: @@
 ==Argument descriptions==
 ===Input arguments===
-* <tt>VEC</tt>: A [[bipartite]] vector (e.g., a [[pure quantum state]]) to have its [[Schmidt decomposition]] computed.
+* <tt>VEC</tt>: A bipartite vector (e.g., a pure quantum state) to have its Schmidt decomposition computed.
 * <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that <tt>VEC</tt> lives on.
 * <tt>K</tt> (optional, default 0): A flag that determines how many terms in the Schmidt decomposition should be computed. If <tt>K</tt>&nbsp;=&nbsp;0 then all terms with non-zero [[Schmidt coefficient]]s are computed. If <tt>K</tt>&nbsp;=&nbsp;-1 then all terms (including zero Schmidt coefficients) are computed. If <tt>K</tt>&nbsp;&gt;&nbsp;0 then the <tt>K</tt> terms with largest Schmidt coefficients are computed.

@@ Line 2: / Line 2: @@
 |name=SchmidtDecomposition
 |desc=Computes the [[Schmidt decomposition]] of a [[bipartite]] vector
 |rel=[[IsProductVector]]<br />[[OperatorSchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SchmidtNumber]]
 |upd=December 1, 2012
-|v=1.02}}
+|v=0.50}}
 <tt>'''SchmidtDecomposition'''</tt> is a [[List of functions|function]] that computes the [[Schmidt decomposition]] of a [[bipartite]] vector. The user may specify how many terms in the Schmidt decomposition they wish to be returned.
@@ Line 22: / Line 22: @@
 ===Output arguments===
 * <tt>S</tt>: A vector containing the Schmidt coefficients of <tt>VEC</tt>.
-* <tt>U</tt> (optional): The left Schmidt vectors of <tt>VEC</tt>.
+* <tt>U</tt> (optional): A matrix whose columns are the left Schmidt vectors of <tt>VEC</tt>.
-* <tt>V</tt> (optional): The right Schmidt vectors of <tt>VEC</tt>.
+* <tt>V</tt> (optional): A matrix whose columns are the right Schmidt vectors of <tt>VEC</tt>.
 ==Examples==
-Please add examples here.
+The following code returns the Schmidt decomposition of the [[MaxEntangled|standard maximally-entangled pure state]] $\frac{1}{\sqrt{d}}\sum_j|j\rangle\otimes|j\rangle \in \mathbb{C}^d \otimes \mathbb{C}^d$ in the $d = 3$ case: