SkVectorNorm

From QETLAB
Jump to navigation Jump to search
SkVectorNorm
Computes the s(k)-norm of a vector

Other toolboxes required none
Related functions KyFanNorm
SchmidtDecomposition
SkOperatorNorm
Function category Norms
Usable within CVX? no

SkVectorNorm is a function that computes the s(k)-norm of a vector (i.e., the Euclidean norm of the vector of its k largest Schmidt coefficients[1]).

Syntax

  • SkVectorNorm(VEC)
  • SkVectorNorm(VEC,K)
  • SkVectorNorm(VEC,K,DIM)

Argument descriptions

  • VEC: A vector living in bipartite space.
  • K (optional, default 1): A positive integer.
  • DIM (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that VEC lives on.

Examples

Sum of squares of eigenvalues of reduced density matrix

The square of the s(k)-vector norm is equal to the Ky Fan k-norm of the vector's reduced density matrix:

>> v = RandomStateVector(9);
>> [SkVectorNorm(v,1)^2, KyFanNorm(PartialTrace(v*v'),1)]

ans =

    0.7754    0.7754

>> [SkVectorNorm(v,2)^2, KyFanNorm(PartialTrace(v*v'),2)]

ans =

    0.9333    0.9333

>> [SkVectorNorm(v,3)^2, KyFanNorm(PartialTrace(v*v'),3)]

ans =

    1.0000    1.0000

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

References

  1. N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory. J. Math. Phys., 51:082202, 2010. E-print: arXiv:0909.3907 [quant-ph]