# Difference between revisions of "SkVectorNorm"

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{{Function | {{Function | ||

|name=SkVectorNorm | |name=SkVectorNorm | ||

− | |desc=Computes the | + | |desc=Computes the s(k)-norm of a vector |

|rel=[[KyFanNorm]]<br />[[SchmidtDecomposition]]<br />[[SkOperatorNorm]] | |rel=[[KyFanNorm]]<br />[[SchmidtDecomposition]]<br />[[SkOperatorNorm]] | ||

|cat=[[List of functions#Norms|Norms]] | |cat=[[List of functions#Norms|Norms]] | ||

|upd=December 2, 2012 | |upd=December 2, 2012 | ||

− | | | + | |cvx=no}} |

− | <tt>'''SkVectorNorm'''</tt> is a [[List of functions|function]] that computes the | + | <tt>'''SkVectorNorm'''</tt> is a [[List of functions|function]] that computes the s(k)-norm of a vector (i.e., the Euclidean norm of the vector of its k largest Schmidt coefficients<ref>N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory. ''J. Math. Phys.'', 51:082202, 2010. E-print: [http://arxiv.org/abs/0909.3907 arXiv:0909.3907] [quant-ph]</ref>). |

==Syntax== | ==Syntax== | ||

Line 14: | Line 14: | ||

==Argument descriptions== | ==Argument descriptions== | ||

− | * <tt>VEC</tt>: A vector living in | + | * <tt>VEC</tt>: A vector living in bipartite space. |

* <tt>K</tt> (optional, default 1): A positive integer. | * <tt>K</tt> (optional, default 1): A positive integer. | ||

* <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that <tt>VEC</tt> lives on. | * <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that <tt>VEC</tt> lives on. | ||

Line 20: | Line 20: | ||

==Examples== | ==Examples== | ||

===Sum of squares of eigenvalues of reduced density matrix=== | ===Sum of squares of eigenvalues of reduced density matrix=== | ||

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

<syntaxhighlight> | <syntaxhighlight> | ||

>> v = RandomStateVector(9); | >> v = RandomStateVector(9); |

## Latest revision as of 16:49, 24 December 2014

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

- ↑ 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]