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]] |
| − | + | |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> |
| − | >> v = | + | >> v = RandomStateVector(9); |
| − | >> [SkVectorNorm(v,1)^2, | + | >> [SkVectorNorm(v,1)^2, KyFanNorm(PartialTrace(v*v'),1)] |
ans = | ans = | ||
| Line 40: | Line 40: | ||
1.0000 1.0000 | 1.0000 1.0000 | ||
| − | </ | + | </syntaxhighlight> |
| + | |||
| + | {{SourceCode|name=SkVectorNorm}} | ||
==References== | ==References== | ||
<references /> | <references /> | ||
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 1.0000Source 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]