SkOperatorNorm - Revision history

Rmmateri at 16:39, 13 June 2018

2018-06-13T16:39:39Z

← Older revision		Revision as of 16:39, 13 June 2018
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	\|name=SkOperatorNorm		\|name=SkOperatorNorm
	\|desc=Computes the S(k)-norm of an operator		\|desc=Computes the S(k)-norm of an operator
	\|req=[http://cvxr.com/cvx/ ~~cvx~~]		\|req=[http://cvxr.com/cvx/ CVX]
	\|rel=[[IsBlockPositive]]<br />[[SkVectorNorm]]		\|rel=[[IsBlockPositive]]<br />[[SkVectorNorm]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]

Nathaniel: added CVX parameter

2014-12-24T16:38:58Z

added CVX parameter

← Older revision		Revision as of 16:38, 24 December 2014
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	{{Function		{{Function
	\|name=SkOperatorNorm		\|name=SkOperatorNorm
	\|desc=Computes the ~~[[S(k)-operator norm\|~~S(k)-norm of an operator]]		\|desc=Computes the S(k)-norm of an operator
	\|req=[http://cvxr.com/cvx/ cvx]		\|req=[http://cvxr.com/cvx/ cvx]
	\|rel=[[IsBlockPositive]]<br />[[SkVectorNorm]]		\|rel=[[IsBlockPositive]]<br />[[SkVectorNorm]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=September 22, 2014		\|upd=September 22, 2014
	\|v=~~0.50~~}}		\|cvx=no}}
	<tt>'''SkOperatorNorm'''</tt> is a [[List of functions\|function]] that computes 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>'''SkOperatorNorm'''</tt> is a [[List of functions\|function]] that computes the 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\},		\\|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 [[SchmidtRank\|Schmidt rank]] of a pure state.

	==Syntax==		==Syntax==
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	==Argument descriptions==		==Argument descriptions==
	===Input arguments===		===Input arguments===
	* <tt>X</tt>: A square matrix acting on [[bipartite]] space. Generally, <tt>X</tt> should be positive semidefinite – the bounds produced otherwise are quite poor.		* <tt>X</tt>: A square matrix acting on bipartite space. Generally, <tt>X</tt> should be positive semidefinite – the bounds produced otherwise are quite poor.
	* <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>X</tt> acts 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>X</tt> acts on.

Nathaniel: Updated date

2014-09-22T18:46:04Z

Updated date

← Older revision		Revision as of 18:46, 22 September 2014
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	\|rel=[[IsBlockPositive]]<br />[[SkVectorNorm]]		\|rel=[[IsBlockPositive]]<br />[[SkVectorNorm]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=~~June 25~~, ~~2013~~		\|upd=September 22, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''SkOperatorNorm'''</tt> is a [[List of functions\|function]] that computes 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>'''SkOperatorNorm'''</tt> is a [[List of functions\|function]] that computes 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 18:45, 22 September 2014

2014-09-22T18:45:39Z

Show changes

Nathaniel: Updated to v2.01

2013-06-25T20:44:12Z

Updated to v2.01

← Older revision		Revision as of 20:44, 25 June 2013
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	\|name=SkOperatorNorm		\|name=SkOperatorNorm
	\|desc=Computes the [[S(k)-operator norm\|S(k)-norm of an operator]]		\|desc=Computes the [[S(k)-operator norm\|S(k)-norm of an operator]]
	\|req=[http://cvxr.com/cvx/ cvx]<br />[[iden]]<br />[[IsPSD]]<br />[[kpNorm]]<br />[[MaxEntangled]]<br />[[normalize_cols]]<br />[[opt_args]]<br />[[PartialMap]]<br />[[PartialTrace]]<br />[[PartialTranspose]]<br />[[PermuteSystems]]<br />[[Realignment]]<br />[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[SkVectorNorm]]<br />[[sporth]]<br />[[Swap]]		\|req=[http://cvxr.com/cvx/ cvx]<br />[[iden]]<br />[[IsPSD]]<br />[[kpNorm]]<br />[[MaxEntangled]]<br />[[normalize_cols]]<br />[[opt_args]]<br />[[PartialMap]]<br />[[PartialTrace]]<br />[[PartialTranspose]]<br />[[PermuteSystems]]<br />[[Realignment]]<br />[[SchmidtDecomposition]]<br />[[SchmidtRank]]<br />[[sk_iterate]]<br />[[SkVectorNorm]]<br />[[sporth]]<br />[[Swap]]
	\|rel=[[IsBlockPositive]]		\|rel=[[IsBlockPositive]]
	\|upd=~~January 2~~, 2013		\|upd=June 25, 2013
	\|v=2.00}}		\|v=2.01}}
	<tt>'''SkOperatorNorm'''</tt> is a [[List of functions\|function]] that computes 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>'''SkOperatorNorm'''</tt> is a [[List of functions\|function]] that computes 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: /* Input arguments */

2013-06-25T03:08:41Z

Input arguments

← Older revision		Revision as of 03:08, 25 June 2013
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	* <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>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>ITS</tt> (optional, default 10): The number of iterations used in an algorithm that computes the lower bound <tt>LB</tt>. Higher values of <tt>ITS</tt> result in better lower bounds but slower computations. <tt>ITS</tt> should be set to 0 if the user is only interested in <tt>UB</tt>.		* <tt>ITS</tt> (optional, default 10): The number of iterations used in an algorithm that computes the lower bound <tt>LB</tt>. Higher values of <tt>ITS</tt> result in better lower bounds but slower computations. <tt>ITS</tt> should be set to 0 if the user is only interested in <tt>UB</tt>.
	* <tt>~~ITS~~</tt> (optional, default is the transpose map if <tt>K = 1</tt> and the <tt>K</tt>-positive map $\Phi(X) = K{\rm Tr}(X)I - X$ if <tt>K > 1</tt>): A [[positive map\|<tt>K</tt>-positive]] (but not [[completely positive]]) map used in the computation of the upper bound <tt>UB</tt>. Different <tt>K</tt>-positive maps lead to different upper bounds. Maps that are not <tt>K</tt>-positive will not result in error messages, but may result in nonsensical or incorrect calculations, so it is the user's responsibility to ensure that <tt>POS_MAP</tt> really is <tt>K</tt>-positive. <tt>PHI</tt> should be provided either as a Choi matrix, or as a cell with 2 columns whose entries are its Kraus operators (see the [[tutorial]] to learn how to deal with superoperators in QETLAB).		* <tt>POS_MAP</tt> (optional, default is the transpose map if <tt>K = 1</tt> and the <tt>K</tt>-positive map $\Phi(X) = K{\rm Tr}(X)I - X$ if <tt>K > 1</tt>): A [[positive map\|<tt>K</tt>-positive]] (but not [[completely positive]]) map used in the computation of the upper bound <tt>UB</tt>. Different <tt>K</tt>-positive maps lead to different upper bounds. Maps that are not <tt>K</tt>-positive will not result in error messages, but may result in nonsensical or incorrect calculations, so it is the user's responsibility to ensure that <tt>POS_MAP</tt> really is <tt>K</tt>-positive. <tt>PHI</tt> should be provided either as a Choi matrix, or as a cell with 2 columns whose entries are its Kraus operators (see the [[tutorial]] to learn how to deal with superoperators in QETLAB).

	===Output arguments===		===Output arguments===

Nathaniel: /* Exact computation in small dimensions */

2013-01-02T22:24:12Z

Exact computation in small dimensions

← Older revision		Revision as of 22:24, 2 January 2013
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	</pre<noinclude></noinclude>>		</pre<noinclude></noinclude>>

	The fact that this computation is correct is illustrated in Example 5.2.11 of <ref name="nj_thesis"></ref>, where it was shown that the S(1)-norm is exactly $(3 + 2\sqrt(2))/8 \approx 0.7286$. However, if we were still unconvinced, we could request witnesses that verify that 0.7286 is both a lower bound and an upper bound of the S(1)-norm:		The fact that this computation is correct is illustrated in Example 5.2.11 of <ref name="nj_thesis"></ref>, where it was shown that the S(1)-norm is exactly $(3 + 2\sqrt{2})/8 \approx 0.7286$. However, if we were still unconvinced, we could request witnesses that verify that 0.7286 is both a lower bound and an upper bound of the S(1)-norm:
	<pre<noinclude></noinclude>>		<pre<noinclude></noinclude>>
	>> [lb,lwit,ub,uwit] = SkOperatorNorm(X)		>> [lb,lwit,ub,uwit] = SkOperatorNorm(X)

Nathaniel at 21:08, 2 January 2013

2013-01-02T21:08:54Z

← Older revision		Revision as of 21:08, 2 January 2013
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	<tt>'''SkOperatorNorm'''</tt> is a [[List of functions\|function]] that computes 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>'''SkOperatorNorm'''</tt> is a [[List of functions\|function]] that computes 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\| : \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.

	==Syntax==		==Syntax==

Nathaniel: /* Only interested in the lower and upper bounds; not the witnesses */

2013-01-02T21:03:29Z

Only interested in the lower and upper bounds; not the witnesses

← Older revision		Revision as of 21:03, 2 January 2013
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	===Only interested in the lower and upper bounds; not the witnesses===		===Only interested in the lower and upper bounds; not the witnesses===
	If all you want are the lower and upper bounds, but don't require the witnesses <tt>LWIT</tt> and <tt>UWIT</tt>, you can use code like the following. Note that in this case, $\\|X\\|_{S(1)}$ is computed exactly, as the lower and upper bound are equal. However, all we know about $\\|X\\|_{S(2)}$ is that it lies in the interval [0.3522, 0.3546]. It is unsurprising that $\\|X\\|_{S(3)}$ is the usual operator norm of <tt>X</tt>, since this is always the case when <tt>K >= min(DIM)</tt>.		If all you want are the lower and upper bounds, but don't require the witnesses <tt>LWIT</tt> and <tt>UWIT</tt>, you can use code like the following. Note that in this case, $\\|X\\|_{S(1)}$ is computed exactly, as the lower and upper bound are equal (though this will not happen for all <tt>X</tt>!). However, all we know about $\\|X\\|_{S(2)}$ is that it lies in the interval [0.3522, 0.3546]. It is unsurprising that $\\|X\\|_{S(3)}$ is the usual operator norm of <tt>X</tt>, since this is always the case when <tt>K >= min(DIM)</tt>.
	<pre<noinclude></noinclude>>		<pre<noinclude></noinclude>>
	>> X = [[RandomDensityMatrix\|RandomDensityMatrix(9)]];		>> X = [[RandomDensityMatrix\|RandomDensityMatrix(9)]];

Nathaniel: /* Exact computation in small dimensions */

2013-01-02T21:02:33Z

Exact computation in small dimensions

← Older revision		Revision as of 21:02, 2 January 2013
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	0.0004 - 0.0000i -0.0631 + 0.0000i -0.0631 + 0.0000i 0.3026		0.0004 - 0.0000i -0.0631 + 0.0000i -0.0631 + 0.0000i 0.3026

	>> lwit'Xlwit		>> lwit'Xlwit % verify that the lower bound is correct

	ans =		ans =
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	0.7286 + 0.0000i		0.7286 + 0.0000i

	>> norm(X + [[PartialTranspose\|PartialTranspose(uwit,2)]])		>> norm(X + [[PartialTranspose\|PartialTranspose(uwit,2)]]) % verify that the upper bound is correct

	ans =		ans =