Difference between revisions of "SkVectorNorm"

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m (added CVX parameter)
 
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{{Function
 
{{Function
 
|name=SkVectorNorm
 
|name=SkVectorNorm
|desc=Computes the [[s(k)-vector norm|s(k)-norm of a vector]]
+
|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
|v=0.50}}
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|cvx=no}}
<tt>'''SkVectorNorm'''</tt> is a [[List of functions|function]] that computes the [[s(k)-vector norm|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>).
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<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==
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==Argument descriptions==
 
==Argument descriptions==
* <tt>VEC</tt>: A vector living in [[bipartite]] space.
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* <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.
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==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 [[Ky Fan norm|Ky Fan k-norm]] of the vector's reduced density matrix:
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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

  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]