Difference between revisions of "Sk iterate"
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|name=sk_iterate | |name=sk_iterate | ||
|desc=Computes a lower bound of the [[S(k)-norm]] of an operator | |desc=Computes a lower bound of the [[S(k)-norm]] of an operator | ||
| − | | | + | |rel=[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SkOperatorNorm]] |
| − | | | + | |cat=[[List of functions#Helper_functions|Helper functions]] |
| − | |upd= | + | |upd=November 14, 2014 |
| − | |v= | + | |v=0.60 |
|helper=1}} | |helper=1}} | ||
<tt>'''sk_iterate'''</tt> is a [[List of functions|function]] that iteratively computes a lower bound on the [[S(k)-operator norm|S(k)-norm of an operator]]<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><ref>N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory II. Quantum Information & Computation, 11(1 & 2):104–123, 2011. E-print: [http://arxiv.org/abs/1006.0898 arXiv:1006.0898] [quant-ph]</ref>: | <tt>'''sk_iterate'''</tt> is a [[List of functions|function]] that iteratively computes a lower bound on the [[S(k)-operator norm|S(k)-norm of an operator]]<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><ref>N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory II. Quantum Information & Computation, 11(1 & 2):104–123, 2011. E-print: [http://arxiv.org/abs/1006.0898 arXiv:1006.0898] [quant-ph]</ref>: | ||
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\|X\|_{S(k)} := \sup_{|v\rangle , |w\rangle } \Big\{ \big| \langle w| X |v \rangle \big| : SR(|v \rangle), SR(|v \rangle) \leq k, \big\||v \rangle\big\| = \big\||w \rangle\big\| = 1 \Big\}, | \|X\|_{S(k)} := \sup_{|v\rangle , |w\rangle } \Big\{ \big| \langle w| X |v \rangle \big| : SR(|v \rangle), SR(|v \rangle) \leq k, \big\||v \rangle\big\| = \big\||w \rangle\big\| = 1 \Big\}, | ||
$$ | $$ | ||
| − | where $SR(\cdot)$ refers to the [[Schmidt rank]] of a pure state. | + | where $SR(\cdot)$ refers to the [[Schmidt rank]] of a pure state. The method used to compute this lower bound is described [http://www.njohnston.ca/2016/01/how-to-compute-hard-to-compute-matrix-norms/ here]. |
==Syntax== | ==Syntax== | ||
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$$ | $$ | ||
has S(1)-norm equal to $(3+2\sqrt{2})/8 \approx 0.7286$. The following code shows that this quantity is indeed a lower bound of the S(1)-norm: | has S(1)-norm equal to $(3+2\sqrt{2})/8 \approx 0.7286$. The following code shows that this quantity is indeed a lower bound of the S(1)-norm: | ||
| − | < | + | <syntaxhighlight> |
>> rho = [5 1 1 1;1 1 1 1;1 1 1 1;1 1 1 1]/8; | >> rho = [5 1 1 1;1 1 1 1;1 1 1 1;1 1 1 1]/8; | ||
>> sk_iterate(rho) | >> sk_iterate(rho) | ||
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0.7286 | 0.7286 | ||
| − | </ | + | </syntaxhighlight> |
| + | |||
| + | {{SourceCode|name=sk_iterate|helper=1}} | ||
==References== | ==References== | ||
<references /> | <references /> | ||
Latest revision as of 02:41, 12 January 2016
| sk_iterate | |
| Computes a lower bound of the S(k)-norm of an operator | |
| Other toolboxes required | none |
|---|---|
| Related functions | SchmidtDecomposition SchmidtRank SkOperatorNorm |
| Function category | Helper functions |
| This is a helper function that only exists to aid other functions in QETLAB. If you are an end-user of QETLAB, you likely will never have a reason to use this function. |
sk_iterate is a function that iteratively computes a lower bound on the S(k)-norm of an operator[1][2]: $$ \|X\|_{S(k)} := \sup_{|v\rangle , |w\rangle } \Big\{ \big| \langle w| X |v \rangle \big| : SR(|v \rangle), SR(|v \rangle) \leq k, \big\||v \rangle\big\| = \big\||w \rangle\big\| = 1 \Big\}, $$ where $SR(\cdot)$ refers to the Schmidt rank of a pure state. The method used to compute this lower bound is described here.
Syntax
- SK = sk_iterate(X)
- SK = sk_iterate(X,K)
- SK = sk_iterate(X,K,DIM)
- SK = sk_iterate(X,K,DIM,TOL)
- SK = sk_iterate(X,K,DIM,TOL,V0)
- [SK,V] = sk_iterate(X,K,DIM,TOL,V0)
Argument descriptions
Input arguments
- X: A square positive semidefinite matrix to have its S(k)-norm bounded.
- K (optional, default 1): A positive integer, the Schmidt rank to optimize over.
- DIM (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the subsystems that X acts on.
- TOL (optional, default \(10^{-5}\)): The numerical tolerance used when determining whether or not the iterative procedure has converged.
- V0 (optional, default is randomly-generated): The vector to begin the iterative procedure from.
Output arguments
- V (optional): A vector with Schmidt rank at most K such that V'*X*V == SK.
Examples
A two-qubit example
In [3], it was shown that the density matrix $$ \rho = \frac{1}{8}\begin{bmatrix}5 & 1 & 1 & 1\\1 & 1 & 1 & 1\\1 & 1 & 1 & 1\\1 & 1 & 1 & 1\end{bmatrix} $$ has S(1)-norm equal to $(3+2\sqrt{2})/8 \approx 0.7286$. The following code shows that this quantity is indeed a lower bound of the S(1)-norm:
>> rho = [5 1 1 1;1 1 1 1;1 1 1 1;1 1 1 1]/8;
>> sk_iterate(rho)
ans =
0.7286Source 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]
- ↑ N. Johnston and D. W. Kribs. A Family of Norms With Applications in Quantum Information Theory II. Quantum Information & Computation, 11(1 & 2):104–123, 2011. E-print: arXiv:1006.0898 [quant-ph]
- ↑ N. Johnston. Norms and Cones in the Theory of Quantum Entanglement. PhD thesis, University of Guelph, 2012. E-print: arXiv:1207.1479 [quant-ph]