FilterNormalForm - Revision history

Nathaniel at 17:45, 3 October 2014

2014-10-03T17:45:41Z

Nathaniel at 19:26, 30 September 2014

2014-09-30T19:26:30Z

Nathaniel at 15:03, 22 September 2014

2014-09-22T15:03:36Z

Nathaniel at 19:55, 21 September 2014

2014-09-21T19:55:46Z

Nathaniel: Updated using GeSHi

2014-09-20T20:39:40Z

Updated using GeSHi

Nathaniel: Updated to v1.02 (bug fix when dimensions are unequal)

2013-09-17T17:50:16Z

Updated to v1.02 (bug fix when dimensions are unequal)

Nathaniel: Updated to v1.01 (fixed a bug that caused the output to sometimes incorrectly return the identity matrix in the GA and GB cells)

2013-09-05T18:48:29Z

Updated to v1.01 (fixed a bug that caused the output to sometimes incorrectly return the identity matrix in the GA and GB cells)

Nathaniel: /* Using the filter normal form to detect entanglement */

2013-06-24T21:32:22Z

Using the filter normal form to detect entanglement

Nathaniel at 21:31, 24 June 2013

2013-06-24T21:31:15Z

Nathaniel: Created page with "{{Function |name=FilterNormalForm |desc=Computes the filter normal form of an operator |req=OperatorSchmidtDecomposition
opt_args.m
PartialTrace

2013-06-24T21:29:42Z

Created page with "{{Function |name=FilterNormalForm |desc=Computes the filter normal form of an operator |req=OperatorSchmidtDecomposition opt_args.m PartialTrace [[opt_args.m]] [[PartialTrace]] [[PermuteSystems]] [[SchmidtDecomposition]] [[Swap]]
|rel=[[IsSeparable]] [[OperatorSchmidtDecomposition]]
|upd=June 24, 2013
|v=1.00}}
<tt>'''FilterNormalForm'''</tt> is a [[List of functions|function]] that computes the [[filter normal form]] of a bipartite operator <tt>RHO</tt> (see Section IV.D of <ref name="GGHE08">O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert. Unifying several separability conditions using the covariance matrix criterion. Phys. Rev. A, 78:052319, 2008. E-print: [http://arxiv.org/abs/0803.0757 arXiv:0803.0757] [quant-ph]</ref> for an introductory discussion of the filter normal form). Note that if <tt>RHO</tt> is not full rank, it may not have a filter normal form and hence an error may be produced by this function.

==Syntax==
* <tt>XI = FilterNormalForm(RHO)</tt>
* <tt>[XI,GA,GB] = FilterNormalForm(RHO)</tt>
* <tt>[XI,GA,GB,FA,FB] = FilterNormalForm(RHO,DIM)</tt>
* <tt>[XI,GA,GB,FA,FB] = FilterNormalForm(RHO,DIM,TOL)</tt>

==Argument descriptions==
===Input arguments===
* <tt>RHO</tt>: An positive semidefinite operator (typically a density matrix) that acts on a [[bipartite]] Hilbert space. This operator will have its filter normal form computed.
* <tt>DIM</tt> (optional, default has <tt>RHO</tt> acting on two subsystems of equal size): A vector containing the dimensions of the two subsystems on which <tt>RHO</tt> acts.
* <tt>TOL</tt> (optional, default <tt>sqrt(eps)</tt>): The numerical tolerance used when computing the filter normal form.

===Output arguments===
* <tt>XI</tt>: A vector containing the coefficients of the filter normal form of <tt>RHO</tt>, as defined in <ref name="GGHE08"></ref>.
* <tt>GA,GB</tt> (optional): Cells of mutually orthonormal matrices in the filter normal form of <tt>RHO</tt>.
* <tt>FA,FB</tt> (optional): The local filtering operations used to convert <tt>RHO</tt> into its filter normal form.

To be explicit, the relationship between <tt>RHO</tt> and the output arguments is that <tt>kron(FA,FB)RHOkron(FA,FB)' == (eye(length(RHO)) + [[TensorSum|TensorSum(XI,GA,GB)]])/length(RHO)</tt>. In usual math notation, this means that

<math>(F_A \otimes F_B)\rho(F_A \otimes F_B)^\dagger = \frac{1}{d_A d_B}\big(I + \displaystyle\sum_{k} \xi_k G_k^A \otimes G_k^B\big).</math>

==Examples==
===Finding and verifying the filter normal form===
The following code computes the filter normal form of a random density matrix. The fact that <tt>FA</tt> and <tt>FB</tt> implement a local filter to this form is then verified (within reasonable numerical error).
<pre<noinclude></noinclude>>
>> rho = [[RandomDensityMatrix|RandomDensityMatrix(9)]];
>> [xi,GA,GB,FA,FB] = FilterNormalForm(rho);
>> norm((eye(9) + [[TensorSum|TensorSum(xi,GA,GB)]])/9 - kron(FA,FB)rhokron(FA,FB)')

ans =

1.8807e-008
</pre<noinclude></noinclude>>

===Using the filter normal form to detect entanglement===
As noted in <ref name="GGHE08"></ref>, the coefficients <tt>XI</tt> of the filter normal form provide useful information about the entanglement of <tt>RHO</tt>. In particular, if both subsystems have the same dimension <tt>D</tt> then we can conclude that <tt>RHO</tt> is entangled if <tt>sum(XI) > D^2 - D</tt>. Thus the following code generates a random two-qutrit density matrix and then determines that it is entangled:
<pre<noinclude></noinclude>>
>> d = 3;
>> rho = [[RandomDensityMatrix|RandomDensityMatrix(d^2)]];
>> xi = FilterNormalForm(rho);
>> [sum(xi), d^2-d]

ans =

6.3611 6.0000
</pre<noinclude></noinclude>>

===Low-rank states may not have a filter normal form===
It is known<ref>J. M. Leinaas, J. Myrheim, and E. Ovrum. Geometrical aspects of entanglement. Phys. Rev. A, 74:012313, 2006.</ref> that all full-rank density matrices have a filter normal form. However, low-rank density matrices may not have a filter normal form – an error will be produced by this script in these cases. The following code generates a random rank-2 density matrix in <math>M_4 \otimes M_4</math> and then tries to compute its filter normal form.
<pre<noinclude></noinclude>>
>> rho = RandomDensityMatrix(16,0,2);
>> xi = FilterNormalForm(rho);
??? Error using ==> FilterNormalForm at 153
The state RHO can not be transformed into a filter normal form. This is often the
case if RHO is not of full rank.
</pre<noinclude></noinclude>>

==References==
<references />

@@ Line 4: / Line 4: @@
 |rel=[[IsSeparable]]<br />[[OperatorSchmidtDecomposition]]<br />[[OperatorSinkhorn]]
 |cat=[[List of functions#Entanglement_and_separability|Entanglement&nbsp;and&nbsp;separability]]
-|upd=September 30, 2014
+|upd=October 3, 2014
 |v=0.50}}
 <tt>'''FilterNormalForm'''</tt> is a [[List of functions|function]] that computes the [[filter normal form]] of a bipartite operator <tt>RHO</tt> (see Section IV.D of <ref name="GGHE08">O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert. Unifying several separability conditions using the covariance matrix criterion. <em>Phys. Rev. A</em>, 78:052319, 2008. E-print: [http://arxiv.org/abs/0803.0757 arXiv:0803.0757] [quant-ph]</ref> for an introductory discussion of the filter normal form). Note that if <tt>RHO</tt> is not full rank, it may not have a filter normal form and hence an error may be produced by this function.
@@ Line 39: / Line 39: @@
 ans =
-.8807e-008
+.8088e-016
 </syntaxhighlight>
@@ Line 56: / Line 56: @@
 ===Low-rank states may not have a filter normal form===
-It is known<ref>J. M. Leinaas, J. Myrheim, and E. Ovrum. Geometrical aspects of entanglement. <em>Phys. Rev. A</em>, 74:012313, 2006. E-print: [http://arxiv.org/abs/quant-ph/0605079 arXiv:quant-ph/0605079]</ref> that all full-rank density matrices have a filter normal form. However, low-rank density matrices may not have a filter normal form &ndash; an error will be produced by this script in these cases. The following code generates a random rank-2 density matrix in <math>M_4 \otimes M_4</math> and then tries to compute its filter normal form.
+It is known<ref>J. M. Leinaas, J. Myrheim, and E. Ovrum. Geometrical aspects of entanglement. <em>Phys. Rev. A</em>, 74:012313, 2006. E-print: [http://arxiv.org/abs/quant-ph/0605079 arXiv:quant-ph/0605079]</ref> that all full-rank density matrices have a filter normal form. However, low-rank density matrices may not have a filter normal form (although they usually do) &ndash; an error will be produced by this script when low-rank problems occur. The following code generates a random rank-2 density matrix in <math>M_3 \otimes M_3</math> just fine, but is unable to compute the filter normal form of a low-rank state that is constructed specifically so that it does not have a filter normal form:
 <syntaxhighlight>
->> rho = RandomDensityMatrix(16,0,2);
+>> xi = FilterNormalForm(RandomDensityMatrix(9,0,2)); % no errors will come from this
 >> xi = FilterNormalForm(rho);
-Error using ==> FilterNormalForm at 153
+Error using FilterNormalForm (line 68)
-The state RHO can not be transformed into a filter normal form. This is often the
+The state RHO can not be transformed into a filter normal form. This is often the case if RHO is not of full rank.
 </syntaxhighlight>

@@ Line 2: / Line 2: @@
 |name=FilterNormalForm
 |desc=Computes the [[filter normal form]] of an operator
-|rel=[[IsSeparable]]<br />[[OperatorSchmidtDecomposition]]
+|rel=[[IsSeparable]]<br />[[OperatorSchmidtDecomposition]]<br />[[OperatorSinkhorn]]
 |cat=[[List of functions#Entanglement_and_separability|Entanglement&nbsp;and&nbsp;separability]]
-|upd=September 17, 2013
+|upd=September 30, 2014
 |v=0.50}}
 <tt>'''FilterNormalForm'''</tt> is a [[List of functions|function]] that computes the [[filter normal form]] of a bipartite operator <tt>RHO</tt> (see Section IV.D of <ref name="GGHE08">O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert. Unifying several separability conditions using the covariance matrix criterion. <em>Phys. Rev. A</em>, 78:052319, 2008. E-print: [http://arxiv.org/abs/0803.0757 arXiv:0803.0757] [quant-ph]</ref> for an introductory discussion of the filter normal form). Note that if <tt>RHO</tt> is not full rank, it may not have a filter normal form and hence an error may be produced by this function.

@@ Line 3: / Line 3: @@
 |desc=Computes the [[filter normal form]] of an operator
 |rel=[[IsSeparable]]<br />[[OperatorSchmidtDecomposition]]
 |upd=September 17, 2013
-|v=1.02}}
+|v=0.50}}
 <tt>'''FilterNormalForm'''</tt> is a [[List of functions|function]] that computes the [[filter normal form]] of a bipartite operator <tt>RHO</tt> (see Section IV.D of <ref name="GGHE08">O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert. Unifying several separability conditions using the covariance matrix criterion. <em>Phys. Rev. A</em>, 78:052319, 2008. E-print: [http://arxiv.org/abs/0803.0757 arXiv:0803.0757] [quant-ph]</ref> for an introductory discussion of the filter normal form). Note that if <tt>RHO</tt> is not full rank, it may not have a filter normal form and hence an error may be produced by this function.
@@ Line 63: / Line 64: @@
 case if RHO is not of full rank.
 </syntaxhighlight>
 ==References==
 <references />

@@ Line 2: / Line 2: @@
 |name=FilterNormalForm
 |desc=Computes the [[filter normal form]] of an operator
 |rel=[[IsSeparable]]<br />[[OperatorSchmidtDecomposition]]
 |upd=September 17, 2013

@@ Line 4: / Line 4: @@
 |req=[[OperatorSchmidtDecomposition]]<br />[[opt_args]]<br />[[PartialTrace]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[Swap]]
 |rel=[[IsSeparable]]<br />[[OperatorSchmidtDecomposition]]
-|upd=September 5, 2013
+|upd=September 17, 2013
-|v=1.01}}
+|v=1.02}}
 <tt>'''FilterNormalForm'''</tt> is a [[List of functions|function]] that computes the [[filter normal form]] of a bipartite operator <tt>RHO</tt> (see Section IV.D of <ref name="GGHE08">O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert. Unifying several separability conditions using the covariance matrix criterion. <em>Phys. Rev. A</em>, 78:052319, 2008. E-print: [http://arxiv.org/abs/0803.0757 arXiv:0803.0757] [quant-ph]</ref> for an introductory discussion of the filter normal form). Note that if <tt>RHO</tt> is not full rank, it may not have a filter normal form and hence an error may be produced by this function.

@@ Line 32: / Line 32: @@
 ===Finding and verifying the filter normal form===
 The following code computes the filter normal form of a random density matrix. The fact that <tt>FA</tt> and <tt>FB</tt> implement a local filter to this form is then verified (within reasonable numerical error).
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
->> rho = [[RandomDensityMatrix|RandomDensityMatrix(9)]];
+>> rho = RandomDensityMatrix(9);
 >> [xi,GA,GB,FA,FB] = FilterNormalForm(rho);
->> norm((eye(9) + [[TensorSum|TensorSum(xi,GA,GB)]])/9 - kron(FA,FB)*rho*kron(FA,FB)')
+>> norm((eye(9) + TensorSum(xi,GA,GB))/9 - kron(FA,FB)*rho*kron(FA,FB)')
 ans =
 .8807e-008
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 ===Using the filter normal form to detect entanglement===
 As noted in <ref name="GGHE08"></ref>, the coefficients <tt>XI</tt> of the filter normal form provide useful information about the entanglement of <tt>RHO</tt>. For example, if both subsystems have the same dimension <tt>D</tt> then we can conclude that <tt>RHO</tt> is entangled if <tt>sum(XI) > D^2 - D</tt>. Thus the following code generates a random two-qutrit density matrix and then determines that it is entangled:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
 >> d = 3;
->> rho = [[RandomDensityMatrix|RandomDensityMatrix(d^2)]];
+>> rho = RandomDensityMatrix(d^2);
 >> xi = FilterNormalForm(rho);
 >> [sum(xi), d^2-d]
@@ Line 53: / Line 53: @@
 .3611    6.0000
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 ===Low-rank states may not have a filter normal form===
 It is known<ref>J. M. Leinaas, J. Myrheim, and E. Ovrum. Geometrical aspects of entanglement. <em>Phys. Rev. A</em>, 74:012313, 2006. E-print: [http://arxiv.org/abs/quant-ph/0605079 arXiv:quant-ph/0605079]</ref> that all full-rank density matrices have a filter normal form. However, low-rank density matrices may not have a filter normal form &ndash; an error will be produced by this script in these cases. The following code generates a random rank-2 density matrix in <math>M_4 \otimes M_4</math> and then tries to compute its filter normal form.
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
->> rho = [[RandomDensityMatrix|RandomDensityMatrix(16,0,2)]];
+>> rho = RandomDensityMatrix(16,0,2);
 >> xi = FilterNormalForm(rho);
-??? Error using ==> FilterNormalForm at 153
+Error using ==> FilterNormalForm at 153
 The state RHO can not be transformed into a filter normal form. This is often the
 case if RHO is not of full rank.
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 ==References==
 <references />