OperatorSchmidtDecomposition - Revision history

Nathaniel: Updated note now that the Hermitian issue is fixed

2024-08-29T12:54:40Z

Updated note now that the Hermitian issue is fixed

← Older revision		Revision as of 12:54, 29 August 2024
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	</syntaxhighlight>		</syntaxhighlight>

	==~~Known Issues~~==		==Notes==
	~~If the~~ input matrix <tt>X</tt> is Hermitian, all of the matrices in its operator-Schmidt decomposition can be chosen to be Hermitian. If <tt>X</tt> has distinct operator-Schmidt coefficients then such an operator-Schmidt decomposition is indeed computed, but if there is a repeated coefficient then a non-Hermitian decomposition may (erroneously) be provided instead. See [https://github.com/nathanieljohnston/QETLAB/issues/14 Issue #14 on github].		The input matrix <tt>X</tt> does not need to be Hermitian. However, if it <em>is</em> Hermitian then all of the matrices in its operator-Schmidt decomposition can be chosen to be Hermitian, and this function enforces such a choice (i.e., the matrices in <tt>U</tt> and <tt>V</tt> are guaranteed to be Hermitian in this case).

	~~This should be fixed in a future version of QETLAB. For now, to get around this problem you can add a tiny amount of random (but Hermitian) noise to~~ <tt>X</tt> ~~so that its operator-Schmidt coefficients~~ are ~~distinct~~.

	{{SourceCode\|name=OperatorSchmidtDecomposition}}		{{SourceCode\|name=OperatorSchmidtDecomposition}}

Nathaniel at 22:38, 28 April 2020

2020-04-28T22:38:58Z

← Older revision		Revision as of 22:38, 28 April 2020
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	1.8449e-016		1.8449e-016
	</syntaxhighlight>		</syntaxhighlight>

			==Known Issues==
			If the input matrix <tt>X</tt> is Hermitian, all of the matrices in its operator-Schmidt decomposition can be chosen to be Hermitian. If <tt>X</tt> has distinct operator-Schmidt coefficients then such an operator-Schmidt decomposition is indeed computed, but if there is a repeated coefficient then a non-Hermitian decomposition may (erroneously) be provided instead. See [https://github.com/nathanieljohnston/QETLAB/issues/14 Issue #14 on github].

			This should be fixed in a future version of QETLAB. For now, to get around this problem you can add a tiny amount of random (but Hermitian) noise to <tt>X</tt> so that its operator-Schmidt coefficients are distinct.

	{{SourceCode\|name=OperatorSchmidtDecomposition}}		{{SourceCode\|name=OperatorSchmidtDecomposition}}

Nathaniel at 18:06, 12 November 2014

2014-11-12T18:06:28Z

← Older revision		Revision as of 18:06, 12 November 2014
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	\|name=OperatorSchmidtDecomposition		\|name=OperatorSchmidtDecomposition
	\|desc=Computes the operator Schmidt decomposition of a bipartite operator		\|desc=Computes the operator Schmidt decomposition of a bipartite operator
	\|rel=[[FilterNormalForm]]<br />[[IsProductOperator]]<br />[[OperatorSchmidtRank]]<br />[[SchmidtDecomposition~~]]<br />[[SchmidtNumber~~]]<br />[[TensorSum]]		\|rel=[[FilterNormalForm]]<br />[[IsProductOperator]]<br />[[OperatorSchmidtRank]]<br />[[SchmidtDecomposition]]<br />[[TensorSum]]
	\|cat=[[List of functions#Entanglement_and_separability\|Entanglement and separability]]		\|cat=[[List of functions#Entanglement_and_separability\|Entanglement and separability]]
	\|upd=November 12, 2014		\|upd=November 12, 2014

Nathaniel at 18:05, 12 November 2014

2014-11-12T18:05:57Z

← Older revision		Revision as of 18:05, 12 November 2014
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	\|rel=[[FilterNormalForm]]<br />[[IsProductOperator]]<br />[[OperatorSchmidtRank]]<br />[[SchmidtDecomposition]]<br />[[SchmidtNumber]]<br />[[TensorSum]]		\|rel=[[FilterNormalForm]]<br />[[IsProductOperator]]<br />[[OperatorSchmidtRank]]<br />[[SchmidtDecomposition]]<br />[[SchmidtNumber]]<br />[[TensorSum]]
	\|cat=[[List of functions#Entanglement_and_separability\|Entanglement and separability]]		\|cat=[[List of functions#Entanglement_and_separability\|Entanglement and separability]]
	\|upd=November 26, ~~2012~~		\|upd=November 12, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''OperatorSchmidtDecomposition'''</tt> is a [[List of functions\|function]] that computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator. The user may specify how many terms in the operator Schmidt decomposition they wish to be returned.		<tt>'''OperatorSchmidtDecomposition'''</tt> is a [[List of functions\|function]] that computes the operator Schmidt decomposition of a bipartite operator. The user may specify how many terms in the operator Schmidt decomposition they wish to be returned.

	==Syntax==		==Syntax==

Nathaniel at 15:17, 22 September 2014

2014-09-22T15:17:21Z

← Older revision		Revision as of 15:17, 22 September 2014
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	{{Function		{{Function
	\|name=OperatorSchmidtDecomposition		\|name=OperatorSchmidtDecomposition
	\|desc=Computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator		\|desc=Computes the operator Schmidt decomposition of a bipartite operator
	\|~~req~~=[[~~opt_args~~]]<br />[[~~PermuteSystems~~]]<br />[[~~SchmidtDecomposition~~]]<br />[[~~Swap]]~~		\|rel=[[FilterNormalForm]]<br />[[IsProductOperator]]<br />[[OperatorSchmidtRank]]<br />[[SchmidtDecomposition]]<br />[[SchmidtNumber]]<br />[[TensorSum]]
	~~\|rel=[[FilterNormalForm~~]]<br />[[~~IsProductOperator~~]]<br />[[~~OperatorSchmidtRank~~]]~~<br />~~[[~~SchmidtNumber]]<br />[[TensorSum~~]]		\|cat=[[List of functions#Entanglement_and_separability\|Entanglement and separability]]
	\|upd=November 26, 2012		\|upd=November 26, 2012
	\|v=1.00}}		\|v=0.50}}
	<tt>'''OperatorSchmidtDecomposition'''</tt> is a [[List of functions\|function]] that computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator. The user may specify how many terms in the operator Schmidt decomposition they wish to be returned.		<tt>'''OperatorSchmidtDecomposition'''</tt> is a [[List of functions\|function]] that computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator. The user may specify how many terms in the operator Schmidt decomposition they wish to be returned.

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	===A random example===		===A random example===
	The following code computes the operator Schmidt decomposition of a random [[density matrix]] in $M_2 \otimes M_4$ and verifies that it is indeed a valid tensor decomposition of that density matrix:		The following code computes the operator Schmidt decomposition of a random [[density matrix]] in $M_2 \otimes M_4$ and verifies that it is indeed a valid tensor decomposition of that density matrix:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> rho = ~~[[RandomDensityMatrix\|~~RandomDensityMatrix(8)]];		>> rho = RandomDensityMatrix(8);
	>> [s,U,V] = OperatorSchmidtDecomposition(rho,[2,4])		>> [s,U,V] = OperatorSchmidtDecomposition(rho,[2,4])

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	1.8449e-016		1.8449e-016
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	As an alternative (and easier) way to verify that we have a valid tensor decomposition of <tt>rho</tt>, we can use the <tt>[[TensorSum]]</tt> function:		As an alternative (and easier) way to verify that we have a valid tensor decomposition of <tt>rho</tt>, we can use the <tt>[[TensorSum]]</tt> function:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> norm(TensorSum(s,U,V) - rho)		>> norm(TensorSum(s,U,V) - rho)

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	1.8449e-016		1.8449e-016
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

			{{SourceCode\|name=OperatorSchmidtDecomposition}}

Nathaniel at 02:49, 25 June 2013

2013-06-25T02:49:22Z

← Older revision		Revision as of 02:49, 25 June 2013
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	\|desc=Computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator		\|desc=Computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator
	\|req=[[opt_args]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[Swap]]		\|req=[[opt_args]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[Swap]]
	\|rel=[[IsProductOperator]]<br />[[OperatorSchmidtRank]]<br />[[SchmidtNumber]]<br />[[TensorSum]]		\|rel=[[FilterNormalForm]]<br />[[IsProductOperator]]<br />[[OperatorSchmidtRank]]<br />[[SchmidtNumber]]<br />[[TensorSum]]
	\|upd=November 26, 2012		\|upd=November 26, 2012
	\|v=1.00}}		\|v=1.00}}

Nathaniel at 20:16, 28 November 2012

2012-11-28T20:16:48Z

← Older revision		Revision as of 20:16, 28 November 2012
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	\|desc=Computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator		\|desc=Computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator
	\|req=[[opt_args]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[Swap]]		\|req=[[opt_args]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[Swap]]
	\|rel=[[IsProductOperator]]<br />[[OperatorSchmidtRank]]<br />[[SchmidtNumber]]		\|rel=[[IsProductOperator]]<br />[[OperatorSchmidtRank]]<br />[[SchmidtNumber]]<br />[[TensorSum]]
	\|upd=November 26, 2012		\|upd=November 26, 2012
	\|v=1.00}}		\|v=1.00}}
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	>> norm(s(1)kron(U{1},V{1}) + s(2)kron(U{2},V{2}) + s(3)kron(U{3},V{3}) + s(4)kron(U{4},V{4}) - rho)		>> norm(s(1)kron(U{1},V{1}) + s(2)kron(U{2},V{2}) + s(3)kron(U{3},V{3}) + s(4)kron(U{4},V{4}) - rho)

			ans =

			1.8449e-016
			</pre<noinclude></noinclude>>

			As an alternative (and easier) way to verify that we have a valid tensor decomposition of <tt>rho</tt>, we can use the <tt>[[TensorSum]]</tt> function:
			<pre<noinclude></noinclude>>
			>> norm(TensorSum(s,U,V) - rho)

	ans =		ans =

Nathaniel: /* A random example */

2012-11-26T21:10:15Z

A random example

← Older revision		Revision as of 21:10, 26 November 2012
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	===A random example===		===A random example===
	The following code computes the operator Schmidt decomposition of a random [[density matrix]] in $M_2 \otimes M_4$ and verifies that it is indeed a valid tensor decomposition of that density matrix:		The following code computes the operator Schmidt decomposition of a random [[density matrix]] in $M_2 \otimes M_4$ and verifies that it is indeed a valid tensor decomposition of that density matrix:
	<pre>		<pre<noinclude></noinclude>>
	>> rho = RandomDensityMatrix(8);		>> rho = [[RandomDensityMatrix\|RandomDensityMatrix(8)]];
	>> [s,U,V] = OperatorSchmidtDecomposition(rho,[2,4])		>> [s,U,V] = OperatorSchmidtDecomposition(rho,[2,4])

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	1.8449e-016		1.8449e-016
	</pre>		</pre<noinclude></noinclude>>

Nathaniel: Created page with "{{Function |name=OperatorSchmidtDecomposition |desc=Computes the operator Schmidt decomposition of a bipartite operator |req=opt_args
PermuteSystems
..."

2012-11-26T20:58:24Z

Created page with "{{Function |name=OperatorSchmidtDecomposition |desc=Computes the operator Schmidt decomposition of a bipartite operator |req=opt_args PermuteSystems ..."

New page

{{Function
|name=OperatorSchmidtDecomposition
|desc=Computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator
|req=[[opt_args]] [[PermuteSystems]] [[SchmidtDecomposition]] [[Swap]]
|rel=[[IsProductOperator]] [[OperatorSchmidtRank]] [[SchmidtNumber]]
|upd=November 26, 2012
|v=1.00}}
<tt>'''OperatorSchmidtDecomposition'''</tt> is a [[List of functions|function]] that computes the [[operator Schmidt decomposition]] of a [[bipartite]] operator. The user may specify how many terms in the operator Schmidt decomposition they wish to be returned.

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

==Argument descriptions==
===Input arguments===
* <tt>X</tt>: A [[bipartite]] operator to have its [[operator Schmidt decomposition]] computed.
* <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A specification of the dimensions of the subsystems that <tt>X</tt> lives on. <tt>DIM</tt> can be provided in one of three ways:
** If <tt>DIM</tt> is a scalar, it is assumed that the first subsystem has dimension <tt>DIM</tt> and the second subsystem has dimension <tt>length(X)/DIM</tt>.
** If $X \in M_{n_1} \otimes M_{n_2}$ then <tt>DIM</tt> should be a row vector containing the dimensions (i.e., <tt>DIM = [n_1, n_2]</tt>).
** If the subsystems aren't square (i.e., $X \in M_{m_1, n_1} \otimes M_{m_2, n_2}$) then <tt>DIM</tt> should be a matrix with two rows. The first row of <tt>DIM</tt> should contain the row dimensions of the subsystems (i.e., the mi's) and its second row should contain the column dimensions (i.e., the ni's). In other words, you should set <tt>DIM = [m_1, m_2; n_1, n_2]</tt>.
* <tt>K</tt> (optional, default 0): A flag that determines how many terms in the operator Schmidt decomposition should be computed. If <tt>K</tt> = 0 then all terms with non-zero operator Schmidt coefficients are computed. If <tt>K</tt> = -1 then all terms (including zero terms) are computed. If <tt>K</tt> > 0 then the <tt>K</tt> terms with largest operator Schmidt coefficients are computed.

===Output arguments===
* <tt>S</tt>: A vector containing the operator Schmidt coefficients of <tt>X</tt>.
* <tt>U</tt> (optional): A cell containing the left Schmidt operators of <tt>X</tt>.
* <tt>V</tt> (optional): A cell containing the right Schmidt operators of <tt>X</tt>.

==Examples==
===A random example===
The following code computes the operator Schmidt decomposition of a random [[density matrix]] in $M_2 \otimes M_4$ and verifies that it is indeed a valid tensor decomposition of that density matrix:
<pre>
>> rho = RandomDensityMatrix(8);
>> [s,U,V] = OperatorSchmidtDecomposition(rho,[2,4])

s =

0.3986
0.2310
0.1637
0.1118

U =

[2x2 double] [2x2 double] [2x2 double] [2x2 double]

V =

[4x4 double] [4x4 double] [4x4 double] [4x4 double]

>> norm(s(1)*kron(U{1},V{1}) + s(2)*kron(U{2},V{2}) + s(3)*kron(U{3},V{3}) + s(4)*kron(U{4},V{4}) - rho)

ans =

1.8449e-016
</pre>