SymmetricExtension - Revision history

Nathaniel at 10:49, 9 May 2022

2022-05-09T10:49:45Z

← Older revision		Revision as of 10:49, 9 May 2022
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	==Notes==		==Notes==
	If your goal is to detect entanglement in an operator, then it is always in your best interest to set the optional arguments <tt>PPT</tt> and <tt>BOS</tt> to be <tt>1</tt>. Setting <tt>BOS = 1</tt> increases the effectiveness of the entanglement test ~~without any computational overhead at all~~. Setting <tt>PPT = 1</tt> slows down the computation quite a bit, but increases the effectiveness as an entanglement test ''considerably''.		If your goal is to detect entanglement in an operator, then it is always in your best interest to set the optional arguments <tt>PPT</tt> and <tt>BOS</tt> to be <tt>1</tt>. Setting <tt>BOS = 1</tt> increases the effectiveness of the entanglement test and actually makes the computation ''faster''. Setting <tt>PPT = 1</tt> slows down the computation quite a bit, but increases the effectiveness as an entanglement test ''considerably''.

	{{SourceCode\|name=SymmetricExtension}}		{{SourceCode\|name=SymmetricExtension}}

Rmmateri at 16:36, 13 June 2018

2018-06-13T16:36:57Z

← Older revision		Revision as of 16:36, 13 June 2018
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	\|name=SymmetricExtension		\|name=SymmetricExtension
	\|desc=Determines whether or not an operator has a symmetric extension		\|desc=Determines whether or not an operator has a symmetric extension
	\|req=[http://cvxr.com/cvx/ ~~cvx~~]		\|req=[http://cvxr.com/cvx/ CVX]
	\|rel=[[IsPPT]]<br />[[IsSeparable]]<br />[[SymmetricInnerExtension]]<br />[[SymmetricProjection]]		\|rel=[[IsPPT]]<br />[[IsSeparable]]<br />[[SymmetricInnerExtension]]<br />[[SymmetricProjection]]
	\|cat=[[List of functions#Entanglement_and_separability\|Entanglement and separability]]		\|cat=[[List of functions#Entanglement_and_separability\|Entanglement and separability]]

Nathaniel: typo

2016-05-11T11:57:13Z

typo

← Older revision		Revision as of 11:57, 11 May 2016
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	\|upd=February 21, 2014		\|upd=February 21, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given positive semidefinite operator has a symmetric extension. This function is extremely useful for showing that quantum states are entangled (see the [[#Examples\|Examples]] section). Various types of symmetric extensions (such as Bosonic and/or PPT extensions) can be looked for ~~buy~~ specifying optional arguments in the function.		<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given positive semidefinite operator has a symmetric extension. This function is extremely useful for showing that quantum states are entangled (see the [[#Examples\|Examples]] section). Various types of symmetric extensions (such as Bosonic and/or PPT extensions) can be looked for by specifying optional arguments in the function.

	==Syntax==		==Syntax==

Nathaniel: updated header section (no longer PPT and Bosonic by default)

2015-06-11T12:52:24Z

updated header section (no longer PPT and Bosonic by default)

← Older revision		Revision as of 12:52, 11 June 2015
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	\|upd=February 21, 2014		\|upd=February 21, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given positive semidefinite operator has a symmetric extension. This function is extremely useful for showing that quantum states are entangled (see the [[#Examples\|Examples]] section). Various types of symmetric extensions can be looked for ~~– by default,~~ the ~~extension that is looked for must be Bosonic and have positive partial transpose, but both of these restrictions can be removed~~.		<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given positive semidefinite operator has a symmetric extension. This function is extremely useful for showing that quantum states are entangled (see the [[#Examples\|Examples]] section). Various types of symmetric extensions (such as Bosonic and/or PPT extensions) can be looked for buy specifying optional arguments in the function.

	==Syntax==		==Syntax==

Nathaniel: removed red links

2015-04-27T18:19:56Z

removed red links

← Older revision		Revision as of 18:19, 27 April 2015
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	{{Function		{{Function
	\|name=SymmetricExtension		\|name=SymmetricExtension
	\|desc=Determines whether or not an operator has a [[symmetric extension]]		\|desc=Determines whether or not an operator has a symmetric extension
	\|req=[http://cvxr.com/cvx/ cvx]		\|req=[http://cvxr.com/cvx/ cvx]
	\|rel=[[IsPPT]]<br />[[IsSeparable]]<br />[[SymmetricInnerExtension]]<br />[[SymmetricProjection]]		\|rel=[[IsPPT]]<br />[[IsSeparable]]<br />[[SymmetricInnerExtension]]<br />[[SymmetricProjection]]
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	\|upd=February 21, 2014		\|upd=February 21, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given [[positive semidefinite]] operator has a symmetric extension. This function is extremely useful for showing that [[quantum ~~state]]s~~ are [[entangled]] (see the [[#Examples\|Examples]] section). Various types of symmetric extensions can be looked for – by default, the extension that is looked for must be [[Bosonic]] and have [[positive partial transpose]], but both of these restrictions can be removed.		<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given positive semidefinite operator has a symmetric extension. This function is extremely useful for showing that quantum states are entangled (see the [[#Examples\|Examples]] section). Various types of symmetric extensions can be looked for – by default, the extension that is looked for must be Bosonic and have positive partial transpose, but both of these restrictions can be removed.

	==Syntax==		==Syntax==
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	* <tt>K</tt> (optional, default 2): The number of copies of the second subsystem in the desired symmetric extension.		* <tt>K</tt> (optional, default 2): The number of copies of the second subsystem in the desired symmetric extension.
	* <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the two 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 two subsystems that <tt>X</tt> acts on.
	* <tt>PPT</tt> (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must have [[positive partial transpose]].		* <tt>PPT</tt> (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must have positive partial transpose.
	* <tt>BOS</tt> (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must be [[Bosonic]] (i.e., have its range contained within the [[symmetric subspace]]).		* <tt>BOS</tt> (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must be Bosonic (i.e., have its range contained within the symmetric subspace).
	* <tt>TOL</tt> (optional, default <tt>eps^(1/4)</tt>): The numerical tolerance used throughout this script. It is recommended that this is left at the default value unless numerical problems arise and the script has difficulty determining whether or not <tt>X</tt> has a symmetric extension.		* <tt>TOL</tt> (optional, default <tt>eps^(1/4)</tt>): The numerical tolerance used throughout this script. It is recommended that this is left at the default value unless numerical problems arise and the script has difficulty determining whether or not <tt>X</tt> has a symmetric extension.

	===Output arguments===		===Output arguments===
	* <tt>EX</tt>: A flag (either 1 or 0) indicating that <tt>X</tt> does or does not have a symmetric extension of the desired type.		* <tt>EX</tt>: A flag (either 1 or 0) indicating that <tt>X</tt> does or does not have a symmetric extension of the desired type.
	* <tt>WIT</tt> (optional): A witness that verifies that the answer provided by <tt>EX</tt> is correct. If <tt>EX = 1</tt> (i.e., <tt>X</tt> has a symmetric extension) then <tt>WIT</tt> is such a symmetric extension. If <tt>EX = 0</tt> (i.e., no symmetric extension exists) then WIT is an [[entanglement witness]] with <tt>trace(WITX) = -1</tt> but <tt>trace(WITY) >= 0</tt> for all symmetrically extendable <tt>Y</tt>.		* <tt>WIT</tt> (optional): A witness that verifies that the answer provided by <tt>EX</tt> is correct. If <tt>EX = 1</tt> (i.e., <tt>X</tt> has a symmetric extension) then <tt>WIT</tt> is such a symmetric extension. If <tt>EX = 0</tt> (i.e., no symmetric extension exists) then WIT is an entanglement witness with <tt>trace(WITX) = -1</tt> but <tt>trace(WITY) >= 0</tt> for all symmetrically extendable <tt>Y</tt>.

	==Examples==		==Examples==

Nathaniel at 15:37, 22 September 2014

2014-09-22T15:37:59Z

← Older revision		Revision as of 15:37, 22 September 2014
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	* <tt>K</tt> (optional, default 2): The number of copies of the second subsystem in the desired symmetric extension.		* <tt>K</tt> (optional, default 2): The number of copies of the second subsystem in the desired symmetric extension.
	* <tt>DIM</tt> (optional, by default has both subsystems of equal dimension): A 1-by-2 vector containing the dimensions of the two 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 two subsystems that <tt>X</tt> acts on.
	* <tt>PPT</tt> (optional, default 1): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must have [[positive partial transpose]].		* <tt>PPT</tt> (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must have [[positive partial transpose]].
	* <tt>BOS</tt> (optional, default 1): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must be [[Bosonic]] (i.e., have its range contained within the [[symmetric subspace]]).		* <tt>BOS</tt> (optional, default 0): A flag (either 1 or 0) that indicates whether or not the desired symmetric extension must be [[Bosonic]] (i.e., have its range contained within the [[symmetric subspace]]).
	* <tt>TOL</tt> (optional, default <tt>eps^(1/4)</tt>): The numerical tolerance used throughout this script. It is recommended that this is left at the default value unless numerical problems arise and the script has difficulty determining whether or not <tt>X</tt> has a symmetric extension.		* <tt>TOL</tt> (optional, default <tt>eps^(1/4)</tt>): The numerical tolerance used throughout this script. It is recommended that this is left at the default value unless numerical problems arise and the script has difficulty determining whether or not <tt>X</tt> has a symmetric extension.

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	8.0000 6.5000		8.0000 6.5000

	>> SymmetricExtension(rho~~,2,[2,2],0~~) % verify that rho has a symmetric extension		>> SymmetricExtension(rho) % verify that rho has a symmetric extension

	ans =		ans =
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	1		1
	</syntaxhighlight>		</syntaxhighlight>

			==Notes==
			If your goal is to detect entanglement in an operator, then it is always in your best interest to set the optional arguments <tt>PPT</tt> and <tt>BOS</tt> to be <tt>1</tt>. Setting <tt>BOS = 1</tt> increases the effectiveness of the entanglement test without any computational overhead at all. Setting <tt>PPT = 1</tt> slows down the computation quite a bit, but increases the effectiveness as an entanglement test ''considerably''.

	{{SourceCode\|name=SymmetricExtension}}		{{SourceCode\|name=SymmetricExtension}}

Nathaniel at 15:27, 22 September 2014

2014-09-22T15:27:56Z

← Older revision		Revision as of 15:27, 22 September 2014
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	\|name=SymmetricExtension		\|name=SymmetricExtension
	\|desc=Determines whether or not an operator has a [[symmetric extension]]		\|desc=Determines whether or not an operator has a [[symmetric extension]]
	\|req=[http://cvxr.com/cvx/ cvx]~~<br />~~[[IsPPT]]<br />[[~~IsPSD~~]]<br />[[~~opt_args]]<br />[[PartialTrace]]<br />[[PartialTranspose]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[sporth~~]]<br />[[SymmetricProjection]]		\|req=[http://cvxr.com/cvx/ cvx]
	\|~~rel~~=[[~~SymmetricInnerExtension~~]]		\|rel=[[IsPPT]]<br />[[IsSeparable]]<br />[[SymmetricInnerExtension]]<br />[[SymmetricProjection]]
			\|cat=[[List of functions#Entanglement_and_separability\|Entanglement and separability]]
	\|upd=February 21, 2014		\|upd=February 21, 2014
	\|v=1.03}}		\|v=0.50}}
	<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given [[positive semidefinite]] operator has a symmetric extension. This function is extremely useful for showing that [[quantum state]]s are [[entangled]] (see the [[#Examples\|Examples]] section). Various types of symmetric extensions can be looked for – by default, the extension that is looked for must be [[Bosonic]] and have [[positive partial transpose]], but both of these restrictions can be removed.		<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given [[positive semidefinite]] operator has a symmetric extension. This function is extremely useful for showing that [[quantum state]]s are [[entangled]] (see the [[#Examples\|Examples]] section). Various types of symmetric extensions can be looked for – by default, the extension that is looked for must be [[Bosonic]] and have [[positive partial transpose]], but both of these restrictions can be removed.

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	===2-qubit symmetric extension===		===2-qubit symmetric extension===
	It is known<ref>J. Chen, Z. Ji, D. Kribs, N. Lütkenhaus, and B. Zeng. ''Symmetric extension of two-qubit states''. E-print: [http://arxiv.org/abs/1310.3530 arXiv:1310.3530] [quant-ph]</ref> that a 2-qubit state $\rho_{AB}$ has a (not necessarily PPT) symmetric extension if and only if ${\rm Tr}(\rho_B^2) \geq {\rm Tr}(\rho_{AB}^2) - 4\sqrt{\det(\rho_{AB})}$. The following code verifies that one such state does indeed have a symmetric extension.		It is known<ref>J. Chen, Z. Ji, D. Kribs, N. Lütkenhaus, and B. Zeng. ''Symmetric extension of two-qubit states''. E-print: [http://arxiv.org/abs/1310.3530 arXiv:1310.3530] [quant-ph]</ref> that a 2-qubit state $\rho_{AB}$ has a (not necessarily PPT) symmetric extension if and only if ${\rm Tr}(\rho_B^2) \geq {\rm Tr}(\rho_{AB}^2) - 4\sqrt{\det(\rho_{AB})}$. The following code verifies that one such state does indeed have a symmetric extension.
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> rho = [1 0 0 -1;0 1 1/2 0;0 1/2 1 0;-1 0 0 1];		>> rho = [1 0 0 -1;0 1 1/2 0;0 1/2 1 0;-1 0 0 1];
	>> [trace(~~[[PartialTrace\|~~PartialTrace(rho)]]^2), trace(rho^2) - 4*sqrt(det(rho))] % if the first number is >= the second number, rho has a symmetric extension		>> [trace(PartialTrace(rho)^2), trace(rho^2) - 4*sqrt(det(rho))] % if the first number is >= the second number, rho has a symmetric extension

	ans =		ans =
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	1		1
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

			{{SourceCode\|name=SymmetricExtension}}

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

Nathaniel: Updated to v1.03 and added an example

2014-03-04T18:44:00Z

Updated to v1.03 and added an example

@@ Line 4: / Line 4: @@
 |req=[http://cvxr.com/cvx/ cvx]<br />[[IsPPT]]<br />[[IsPSD]]<br />[[opt_args]]<br />[[PartialTrace]]<br />[[PartialTranspose]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[sporth]]<br />[[SymmetricProjection]]
 |rel=[[SymmetricInnerExtension]]
-|upd=September 17, 2013
+|upd=February 21, 2014
-|v=1.02}}
+|v=1.03}}
 <tt>'''SymmetricExtension'''</tt> is a [[List of functions|function]] that determines whether or not a given [[positive semidefinite]] operator has a symmetric extension. This function is extremely useful for showing that [[quantum state]]s are [[entangled]] (see the [[#Examples|Examples]] section). Various types of symmetric extensions can be looked for &ndash; by default, the extension that is looked for must be [[Bosonic]] and have [[positive partial transpose]], but both of these restrictions can be removed.
@@ Line 31: / Line 31: @@
 ==Examples==
-Please add some examples.
+===2-qubit symmetric extension===

Nathaniel: Updated to v1.02 (Bosonic extensions are now faster to determine)

2013-09-17T17:46:29Z

Updated to v1.02 (Bosonic extensions are now faster to determine)

← Older revision		Revision as of 17:46, 17 September 2013
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	\|req=[http://cvxr.com/cvx/ cvx]<br />[[IsPPT]]<br />[[IsPSD]]<br />[[opt_args]]<br />[[PartialTrace]]<br />[[PartialTranspose]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[sporth]]<br />[[SymmetricProjection]]		\|req=[http://cvxr.com/cvx/ cvx]<br />[[IsPPT]]<br />[[IsPSD]]<br />[[opt_args]]<br />[[PartialTrace]]<br />[[PartialTranspose]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[sporth]]<br />[[SymmetricProjection]]
	\|rel=[[SymmetricInnerExtension]]		\|rel=[[SymmetricInnerExtension]]
	\|upd=~~March 6~~, 2013		\|upd=September 17, 2013
	\|v=1.01}}		\|v=1.02}}
	<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given [[positive semidefinite]] operator has a symmetric extension. This function is extremely useful for showing that [[quantum state]]s are [[entangled]] (see the [[#Examples\|Examples]] section). Various types of symmetric extensions can be looked for – by default, the extension that is looked for must be [[Bosonic]] and have [[positive partial transpose]], but both of these restrictions can be removed.		<tt>'''SymmetricExtension'''</tt> is a [[List of functions\|function]] that determines whether or not a given [[positive semidefinite]] operator has a symmetric extension. This function is extremely useful for showing that [[quantum state]]s are [[entangled]] (see the [[#Examples\|Examples]] section). Various types of symmetric extensions can be looked for – by default, the extension that is looked for must be [[Bosonic]] and have [[positive partial transpose]], but both of these restrictions can be removed.

Nathaniel at 19:20, 25 June 2013

2013-06-25T19:20:09Z

← Older revision		Revision as of 19:20, 25 June 2013
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	\|desc=Determines whether or not an operator has a [[symmetric extension]]		\|desc=Determines whether or not an operator has a [[symmetric extension]]
	\|req=[http://cvxr.com/cvx/ cvx]<br />[[IsPPT]]<br />[[IsPSD]]<br />[[opt_args]]<br />[[PartialTrace]]<br />[[PartialTranspose]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[sporth]]<br />[[SymmetricProjection]]		\|req=[http://cvxr.com/cvx/ cvx]<br />[[IsPPT]]<br />[[IsPSD]]<br />[[opt_args]]<br />[[PartialTrace]]<br />[[PartialTranspose]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[sporth]]<br />[[SymmetricProjection]]
			\|rel=[[SymmetricInnerExtension]]
	\|upd=March 6, 2013		\|upd=March 6, 2013
	\|v=1.01}}		\|v=1.01}}