Difference between revisions of "FilterNormalForm"

	FilterNormalForm
	Computes the filter normal form of an operator
Other toolboxes required	OperatorSchmidtDecomposition; opt_args; PartialTrace; PermuteSystems; SchmidtDecomposition; Swap
Related functions	IsSeparable; OperatorSchmidtDecomposition

Revision as of 21:31, 24 June 2013

FilterNormalForm is a function that computes the filter normal form of a bipartite operator RHO (see Section IV.D of ^[1] for an introductory discussion of the filter normal form). Note that if RHO is not full rank, it may not have a filter normal form and hence an error may be produced by this function.

Syntax

XI = FilterNormalForm(RHO)
[XI,GA,GB] = FilterNormalForm(RHO)
[XI,GA,GB,FA,FB] = FilterNormalForm(RHO,DIM)
[XI,GA,GB,FA,FB] = FilterNormalForm(RHO,DIM,TOL)

Argument descriptions

Input arguments

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

Output arguments

XI: A vector containing the coefficients of the filter normal form of RHO, as defined in ^[1].
GA,GB (optional): Cells of mutually orthonormal matrices in the filter normal form of RHO.
FA,FB (optional): The local filtering operations used to convert RHO into its filter normal form.

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

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

Examples

Finding and verifying the filter normal form

The following code computes the filter normal form of a random density matrix. The fact that FA and FB implement a local filter to this form is then verified (within reasonable numerical error).

>> rho = RandomDensityMatrix(9);
>> [xi,GA,GB,FA,FB] = FilterNormalForm(rho);
>> norm((eye(9) + TensorSum(xi,GA,GB))/9 - kron(FA,FB)*rho*kron(FA,FB)')

ans =

  1.8807e-008

Using the filter normal form to detect entanglement

As noted in ^[1], the coefficients XI of the filter normal form provide useful information about the entanglement of RHO. In particular, if both subsystems have the same dimension D then we can conclude that RHO is entangled if sum(XI) > D^2 - D. Thus the following code generates a random two-qutrit density matrix and then determines that it is entangled:

>> d = 3;
>> rho = RandomDensityMatrix(d^2);
>> xi = FilterNormalForm(rho);
>> [sum(xi), d^2-d]

ans =

    6.3611    6.0000

Low-rank states may not have a filter normal form

It is known^[2] 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 \(M_4 \otimes M_4\) and then tries to compute its filter normal form.

>> 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.

References

↑ ^1.0 ^1.1 ^1.2 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: arXiv:0803.0757 [quant-ph]
↑ J. M. Leinaas, J. Myrheim, and E. Ovrum. Geometrical aspects of entanglement. Phys. Rev. A, 74:012313, 2006. E-print: arXiv:quant-ph/0605079

[GGHE08-1] 1.0 ^1.1 ^1.2 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: arXiv:0803.0757 [quant-ph]

[2] J. M. Leinaas, J. Myrheim, and E. Ovrum. Geometrical aspects of entanglement. Phys. Rev. A, 74:012313, 2006. E-print: arXiv:quant-ph/0605079

[1]

[2]

@@ Line 2: / Line 2: @@
 |name=FilterNormalForm
 |desc=Computes the [[filter normal form]] of an operator
-|req=[[OperatorSchmidtDecomposition]]<br />[[opt_args.m]]<br />[[PartialTrace]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[Swap]]
+|req=[[OperatorSchmidtDecomposition]]<br />[[opt_args]]<br />[[PartialTrace]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[Swap]]
 |rel=[[IsSeparable]]<br />[[OperatorSchmidtDecomposition]]
 |upd=June 24, 2013
@@ 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.</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 &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>>
->> rho = RandomDensityMatrix(16,0,2);
+>> rho = [[RandomDensityMatrix|RandomDensityMatrix(16,0,2)]];
 >> xi = FilterNormalForm(rho);
 ??? Error using ==> FilterNormalForm at 153