# SkVectorNorm

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 here to view this function's source code on github.

## References

- ↑ 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]