Diploma thesis

�� compute the Schmidt eigenmatrix �i.e., the matrix which is used to compute the eigenvalues
schmidtEM�F�, s�� :� Chop�Table�Sum�F��i,n���Conjugate�F��j,n���, �n,1,s��, �i,1,s�, �j,1,s��
�� Compute the l’th �zero based� Schmidt function for overlap matrix F.
s denotes the maximal number of basis functions. ��
schmidtFnOne�F�, s�, l�, x�� :� schmidtFnOne�F, s, l, x� � Module��K,v,e�,
K � schmidtEM�F,s�;
v �Chop�Eigenvectors�K���l�1���;
Return�Sum�v��m���normH�m�1,x�, �m,1,s���;
�;
schmidtFnTwo�F�, s�, l�, x�� :� schmidtFnTwo�F, s, l, x� � Module��K,v,e�,
K �schmidtEM�F,s�;
v �Chop�Eigenvectors�K���l�1���;
e � Eigenvalues�K�;
Return�1�Sqrt�e��l�1����Sum�Conjugate�v��m����F��m,n���normH�n�1,x�, �m,1,s�, �n,1,s���;
�;
�� Here s denotes how many elements of the old basis are used to compute the new basis, i.e.
old basis functions a Schmidt basis function is composed of.
d tells how many of the new basis functions will be used, i.e., how many Schmidt terms are used
schmidtFn�F�, s�, d�, x�, y�� :� schmidtFn�F,s,d,x,y� � Module��K�,
Return�Simplify�Sum�Sqrt�schmidtEigenval�F,s���n�1����schmidtFnOne�F,s,n,x��schmidtFnTwo�F,s,
�;
�� Return the eigenvalues used to construct the Schmidt basis.
This measures how entangled the state is since the quantity is
invariant under basis transforms. Note that the number of basis elements
does not in the least imply anything about the basis dimension. ��
schmidtEigenval�F�, s�� :� Module��K�,
K � Chop�Table�Sum�F��i,n���Conjugate�F��j,n���, �n,1,s��, �i,1,s�, �j,1,s���;
Return�Chop�Eigenvalues�K���;
�;
�� Distance measures between two functions f and g.
TODO: Adapt this to variable function domains, not only ��3,3�. ��
dist�f�, g�� :� NIntegrate�Abs�f�x,y� � g�x,y��^2, �x,�3,3�, �y,�3,3�,AccuracyGoal��4�;
distEV�f�,e�� :� 1��Sum�e��n��,�n,1,Length�e�����NIntegrate�Conjugate�f�x,y���f�x,y�, �x,�3,3
�� Some call it Shannon Entropy, others call it entropy of entanglement ��
S�l�� :� �Sum�l��n���Log�2,l��n���, �n,1,Length�l���;
Entanglement�F�, s�� :� S�Eigenvalues�Table�Sum�F��i,n���Conjugate�F��j,n���, �n,1,s��, �i,1,
�� f is the function in the new polynomial basis ��not� the Schmidt basis��. T is the overlap
matrix, k denotes which order to use. ��
doSum�x�, y�,T�,k�� :� doSum�x,y,T,k�� Simplify�Sum�T��m,n���normH�m�1,x��normH�n�1,y�, �m,1,
f�x�, y�,T�,k�� :� doSum�x,y, T,k�;
�� Some sample functions. Gauss is, well, a Gau ian distribution, while shiftedGauss contains
gauss�x�, y�� :� gauss�x,y� � 1��2�Pi��Exp��0.5x^2�0.5y^2�;
gaussShifted�x�, y�� :� gaussShifted�x,y� � 1��2�Pi��Exp��0.5�x�1�^2�0.5�y�1�^2��1��2�Pi��Exp
�� TODO: Allow optional arguments be passed on to overlap ��
�� This computes a size x size overlap table for a given function ��
overlapTable�func�, size�� :� overlapTable�func, size� � Table�m,n,func, �m,0,size,1�, �n,0,size
EndPackage��
wmschmidt.m 3