Sk iterate - Revision history

Nathaniel: external link

2016-01-12T02:41:39Z

external link

← Older revision		Revision as of 02:41, 12 January 2016
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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==

Nathaniel: Minor bugfix

2014-11-15T00:56:56Z

Minor bugfix

← Older revision		Revision as of 00:56, 15 November 2014
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	\|rel=[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SkOperatorNorm]]		\|rel=[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SkOperatorNorm]]
	\|cat=[[List of functions#Helper_functions\|Helper functions]]		\|cat=[[List of functions#Helper_functions\|Helper functions]]
	\|upd=~~September 10~~, ~~2013~~		\|upd=November 14, 2014
	\|v=0.50		\|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>:

Nathaniel at 16:28, 29 September 2014

2014-09-29T16:28:19Z

← Older revision		Revision as of 16:28, 29 September 2014
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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
	\|~~req~~=~~[[iden]]<br />[[MaxEntangled]]<br />[[normalize_cols]]<br />[[opt_args]]<br />[[PermuteSystems]]<br />~~[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[~~sporth]]<br />[[Swap~~]]		\|rel=[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SkOperatorNorm]]
	\|~~rel~~=[[~~SkOperatorNorm~~]]		\|cat=[[List of functions#Helper_functions\|Helper functions]]
	\|upd=September 10, 2013		\|upd=September 10, 2013
	\|v=1.03		\|v=0.50
	\|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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	$$		$$
	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:
	<~~pre~~>		<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
	</~~pre~~>		</syntaxhighlight>

			{{SourceCode\|name=sk_iterate\|helper=1}}

	==References==		==References==
	<references />		<references />

Nathaniel: Updated to v1.03 (bug fix)

2013-09-17T17:48:55Z

Updated to v1.03 (bug fix)

← Older revision		Revision as of 17:48, 17 September 2013
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	\|req=[[iden]]<br />[[MaxEntangled]]<br />[[normalize_cols]]<br />[[opt_args]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[sporth]]<br />[[Swap]]		\|req=[[iden]]<br />[[MaxEntangled]]<br />[[normalize_cols]]<br />[[opt_args]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[sporth]]<br />[[Swap]]
	\|rel=[[SkOperatorNorm]]		\|rel=[[SkOperatorNorm]]
	\|upd=~~June 26~~, 2013		\|upd=September 10, 2013
	\|v=1.02		\|v=1.03
	\|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>:

Nathaniel: Updated to v1.02

2013-06-26T21:57:05Z

Updated to v1.02

← Older revision		Revision as of 21:57, 26 June 2013
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	\|req=[[iden]]<br />[[MaxEntangled]]<br />[[normalize_cols]]<br />[[opt_args]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[sporth]]<br />[[Swap]]		\|req=[[iden]]<br />[[MaxEntangled]]<br />[[normalize_cols]]<br />[[opt_args]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[sporth]]<br />[[Swap]]
	\|rel=[[SkOperatorNorm]]		\|rel=[[SkOperatorNorm]]
	\|upd=~~March 11~~, 2013		\|upd=June 26, 2013
	\|v=1.01		\|v=1.02
	\|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>:

Nathaniel: Created page with "{{Function |name=sk_iterate |desc=Computes a lower bound of the S(k)-norm of an operator |req=iden
MaxEntangled
normalize_cols
opt_args

2013-06-25T03:18:45Z

Created page with "{{Function |name=sk_iterate |desc=Computes a lower bound of the S(k)-norm of an operator |req=iden MaxEntangled normalize_cols opt_args [[MaxEntangled]] [[normalize_cols]] [[opt_args]] [[PermuteSystems]] [[SchmidtDecomposition]] [[SchmidtRank]] [[sporth]] [[Swap]]
|rel=[[SkOperatorNorm]]
|upd=March 11, 2013
|v=1.01
|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>:
$$
\|X\|_{S(k)} := \sup_{|v\rangle , |w\rangle } \Big\{ \big| \langle w| X |v \rangle \big| : SR(|v \rangle), SR(|v \rangle) \leq k, \big\||v \rangle\big\| = \big\||w \rangle\big\| = 1 \Big\},
$$
where $SR(\cdot)$ refers to the [[Schmidt rank]] of a pure state.

==Syntax==
* <tt>SK = sk_iterate(X)</tt>
* <tt>SK = sk_iterate(X,K)</tt>
* <tt>SK = sk_iterate(X,K,DIM)</tt>
* <tt>SK = sk_iterate(X,K,DIM,TOL)</tt>
* <tt>SK = sk_iterate(X,K,DIM,TOL,V0)</tt>
* <tt>[SK,V] = sk_iterate(X,K,DIM,TOL,V0)</tt>

==Argument descriptions==
===Input arguments===
* <tt>X</tt>: A square positive semidefinite matrix to have its S(k)-norm bounded.
* <tt>K</tt> (optional, default 1): A positive integer, the [[Schmidt rank]] to optimize over.
* <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>X</tt> acts on.
* <tt>TOL</tt> (optional, default <math>10^{-5}</math>): The numerical tolerance used when determining whether or not the iterative procedure has converged.
* <tt>V0</tt> (optional, default is randomly-generated): The vector to begin the iterative procedure from.

===Output arguments===
* <tt>V</tt> (optional): A vector with [[Schmidt rank]] at most <tt>K</tt> such that <tt>V'XV == SK</tt>.

==Examples==
===A two-qubit example===
In <ref name="Jthesis12">N. Johnston. Norms and Cones in the Theory of Quantum Entanglement. PhD thesis, University of Guelph, 2012. E-print: [http://arxiv.org/abs/1207.1479 arXiv:1207.1479] [quant-ph]</ref>, 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:
<pre>
>> rho = [5 1 1 1;1 1 1 1;1 1 1 1;1 1 1 1]/8;
>> sk_iterate(rho)

ans =

0.7286
</pre>

==References==
<references />