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su2 Namespace Reference

A collection of SU(2) utilities. More...


Classes

struct  polar
 4D polar coordinates. Not really used. More...

Functions

void mat_to_cart3 (const Matrix< cdouble > &U, double *c)
 Calculate cartesian form of the 3D Bloch vector for U.
void cart3_to_mat (double *c, Matrix< cdouble > &U)
 Construct a matrix U from the 3D cartesian Bloch vector.
void mat_to_polar3 (const Matrix< cdouble > &U, double *p)
 Calculate polar form of the 3D Bloch vector for U.
void polar3_to_mat (double *p, Matrix< cdouble > &U)
 Construct a matrix U from the 3D polar Bloch vector.
void polar_to_cart3 (double *p, double *c)
 Convert a polar 3D vector to a cartesian 3D vector.
void cart_to_polar3 (double *c, double *p)
 Convert a cartesian 3D vector to a polar 3D vector.
void x_factor (double *z, double *x, double *y)
 Given a 3D cartesian vector z, find vectors x,y s.t. $z = x \times y$.
void mat_to_cart4 (const Matrix< cdouble > &U, double *a)
 Calculate cartesian form of the 4D unit vector for U.
void cart4_to_mat (double *a, Matrix< cdouble > &U)
 Construct a matrix U from the 4D cartesian unit vector.
void mat_to_polar4 (const Matrix< cdouble > &, double *)
 Calculate the polar form of the 4D unit vector for U.
void polar4_to_mat (double *, Matrix< cdouble > &)
 Construct a matrix U from the 4D polar unit vector.
void polar_to_cart4 (double *, double *)
 Convert a polar form 4D vector to cartesian form.
void cart_to_polar4 (double *, double *)
 Convert a cartesian form 4D vector to polar form.
void polar_to_cart4 (polar &p, double *r)
 Legacy conversion from a 4D polar struct to cartesian form.
void polar4_to_mat (polar &, Matrix< cdouble > &)
 Legacy conversion from a 4D cartesian vector to a polar struct.
void eig_vals (Matrix< cdouble > &U, cdouble *evals)
 Quick and dirty calculation of the eigenvalues of an SU(2) matrix.
void eig_vecs (Matrix< cdouble > &U, cdouble *evecs)
 Quick and dirty calculation of the eigenvectors of an SU(2) matrix.
double trace_dist (const Matrix< cdouble > &U, const Matrix< cdouble > &V)
 Trace distance between SU(2) matrices U,V.
double trace_dist (const Matrix< cdouble > &U, double *coord4)
 Trace distance between SU(2) matrices U,V.
double proj_trace_dist (const Matrix< cdouble > &U, const Matrix< cdouble > &V)
 'Projective' trace distance between SU(2) matrices U,V
double proj_trace_dist (const Matrix< cdouble > &U, double *coord4)
 'Projective' trace distance between SU(2) matrices U,V
void group_factor (const Matrix< cdouble > &U, Matrix< cdouble > &V, Matrix< cdouble > &W)
 Find 'balanced' V,W in SU(2) such that U = [V,W].
void x_group_factor (const Matrix< cdouble > &, Matrix< cdouble > &, Matrix< cdouble > &)
 'Balanced' group commutator decomposition for U a rotation about the x-axis
void similarity_matrix (const Matrix< cdouble > &V, Matrix< cdouble > &W, Matrix< cdouble > &)
 Find an SU(2) matrix S such that $ V = S W S^\dagger $.
double norm (double *v, int n)
 Norm of a real vector v of length n.

Detailed Description

A collection of SU(2) utilities.

This provides:

The parameterizations are the 3D Bloch vector r from $U = e^{i \mathbf{r} . \mathbf{\sigma}}$ and the 4D unit vector from $U = aI + ibX + icY + idZ$. The naming convention is pretty self explanatory e.g. mat_to_cart3(Matrix<cdouble>& U, double* r) will calculate the cartesian form of the Block vector for U and place it in r, while polar4_to_mat(double* p, Matrix<cdouble>& U) will construct the matrix U from the polar form of a 4D unit vector.

Function Documentation

void su2::cart3_to_mat (  double *  c,
Matrix< cdouble > &  U
) 
 

Construct a matrix U from the 3D cartesian Bloch vector.

void su2::cart4_to_mat (  double *  a,
Matrix< cdouble > &  U
) 
 

Construct a matrix U from the 4D cartesian unit vector.

Examples:
example.cpp.

void su2::cart_to_polar3 (  double *  c,
double *  p
) 
 

Convert a cartesian 3D vector to a polar 3D vector.

void su2::cart_to_polar4 (  double *  ,
double * 
) 
 

Convert a cartesian form 4D vector to polar form.

void su2::eig_vals (  Matrix< cdouble > &  U,
cdouble *  evals
) 
 

Quick and dirty calculation of the eigenvalues of an SU(2) matrix.

void su2::eig_vecs (  Matrix< cdouble > &  U,
cdouble *  evecs
) 
 

Quick and dirty calculation of the eigenvectors of an SU(2) matrix.

void su2::group_factor (  const Matrix< cdouble > &  U,
Matrix< cdouble > &  V,
Matrix< cdouble > &  W
) 
 

Find 'balanced' V,W in SU(2) such that U = [V,W].

void su2::mat_to_cart3 (  const Matrix< cdouble > &  U,
double *  c
) 
 

Calculate cartesian form of the 3D Bloch vector for U.

void su2::mat_to_cart4 (  const Matrix< cdouble > &  U,
double *  a
) 
 

Calculate cartesian form of the 4D unit vector for U.

void su2::mat_to_polar3 (  const Matrix< cdouble > &  U,
double *  p
) 
 

Calculate polar form of the 3D Bloch vector for U.

void su2::mat_to_polar4 (  const Matrix< cdouble > &  ,
double * 
) 
 

Calculate the polar form of the 4D unit vector for U.

double su2::norm (  double *  v,
int  n
) 
 

Norm of a real vector v of length n.

void su2::polar3_to_mat (  double *  p,
Matrix< cdouble > &  U
) 
 

Construct a matrix U from the 3D polar Bloch vector.

void su2::polar4_to_mat (  polar &  ,
Matrix< cdouble > & 
) 
 

Legacy conversion from a 4D cartesian vector to a polar struct.

void su2::polar4_to_mat (  double *  ,
Matrix< cdouble > & 
) 
 

Construct a matrix U from the 4D polar unit vector.

void su2::polar_to_cart3 (  double *  p,
double *  c
) 
 

Convert a polar 3D vector to a cartesian 3D vector.

void su2::polar_to_cart4 (  polar &  p,
double *  r
) 
 

Legacy conversion from a 4D polar struct to cartesian form.

void su2::polar_to_cart4 (  double *  ,
double * 
) 
 

Convert a polar form 4D vector to cartesian form.

double su2::proj_trace_dist (  const Matrix< cdouble > &  U,
double *  coord4
) 
 

'Projective' trace distance between SU(2) matrices U,V

V in this case is given by its 4D cartesian coordinates

double su2::proj_trace_dist (  const Matrix< cdouble > &  U,
const Matrix< cdouble > &  V
) 
 

'Projective' trace distance between SU(2) matrices U,V

Examples:
example.cpp.

void su2::similarity_matrix (  const Matrix< cdouble > &  V,
Matrix< cdouble > &  W,
Matrix< cdouble > & 
) 
 

Find an SU(2) matrix S such that $ V = S W S^\dagger $.

(or it might be the other way around)

double su2::trace_dist (  const Matrix< cdouble > &  U,
double *  coord4
) 
 

Trace distance between SU(2) matrices U,V.

Except V in this case is given by its 4D cartesian coordinates

double su2::trace_dist (  const Matrix< cdouble > &  U,
const Matrix< cdouble > &  V
) 
 

Trace distance between SU(2) matrices U,V.

void su2::x_factor (  double *  z,
double *  x,
double *  y
) 
 

Given a 3D cartesian vector z, find vectors x,y s.t. $z = x \times y$.

void su2::x_group_factor (  const Matrix< cdouble > &  ,
Matrix< cdouble > &  ,
Matrix< cdouble > & 
) 
 

'Balanced' group commutator decomposition for U a rotation about the x-axis

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