IsEntanglingGate - Revision history

Nathaniel at 15:04, 22 September 2014

2014-09-22T15:04:30Z

← Older revision		Revision as of 15:04, 22 September 2014
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	\|desc=Determines if a [[unitary]] is an [[entangling gate]]		\|desc=Determines if a [[unitary]] is an [[entangling gate]]
	\|rel=[[IsProductOperator]]<br />[[OperatorSchmidtDecomposition]]<br />[[OperatorSchmidtRank]]		\|rel=[[IsProductOperator]]<br />[[OperatorSchmidtDecomposition]]<br />[[OperatorSchmidtRank]]
			\|cat=[[List of functions#Entanglement_and_separability\|Entanglement and separability]]
	\|upd=November 28, 2012		\|upd=November 28, 2012
	\|v=1.00}}		\|v=0.50}}
	<tt>'''IsEntanglingGate'''</tt> is a [[List of functions\|function]] that determines if a [[bipartite]] or [[multipartite]] unitary is an [[entangling gate]] or not (i.e., whether or not there is a [[product vector]] that is no longer a product vector after the unitary is applied). If it is an entangling gate, a [[product vector]] that is entangled by the gate can be provided, and if it is not an entangling gate then a decomposition that proves that it is not entangling can be provided.		<tt>'''IsEntanglingGate'''</tt> is a [[List of functions\|function]] that determines if a [[bipartite]] or [[multipartite]] unitary is an [[entangling gate]] or not (i.e., whether or not there is a [[product vector]] that is no longer a product vector after the unitary is applied). If it is an entangling gate, a [[product vector]] that is entangled by the gate can be provided, and if it is not an entangling gate then a decomposition that proves that it is not entangling can be provided.

Nathaniel: Updated documentation using GeSHi

2014-09-21T19:57:12Z

Updated documentation using GeSHi

← Older revision		Revision as of 19:57, 21 September 2014
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	\|name=IsEntanglingGate		\|name=IsEntanglingGate
	\|desc=Determines if a [[unitary]] is an [[entangling gate]]		\|desc=Determines if a [[unitary]] is an [[entangling gate]]
	\|~~req~~=~~[[opt_args]]<br />~~[[IsProductOperator]]<br />~~[[IsProductVector]]<br />[[perm_inv]]<br />[[PermuteSystems]]<br />[[SchmidtDecomposition]]~~		\|rel=[[IsProductOperator]]<br />[[OperatorSchmidtDecomposition]]<br />[[OperatorSchmidtRank]]
	~~\|rel=~~[[OperatorSchmidtDecomposition]]<br />[[OperatorSchmidtRank]]
	\|upd=November 28, 2012		\|upd=November 28, 2012
	\|v=1.00}}		\|v=1.00}}
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	===A bipartite example: the CNOT gate===		===A bipartite example: the CNOT gate===
	The following code verifies that the [[CNOT]] gate is an entangling gate. The following code also requests a product pure state <tt>wit</tt> that is entangled by the CNOT gate. The product pure state returned in this case is $(\|0\rangle + \|1\rangle) \otimes \|0\rangle/\sqrt{2}$, which maps to the [[Bell state]] $(\|00\rangle + \|11\rangle)/\sqrt{2}$:		The following code verifies that the [[CNOT]] gate is an entangling gate. The following code also requests a product pure state <tt>wit</tt> that is entangled by the CNOT gate. The product pure state returned in this case is $(\|0\rangle + \|1\rangle) \otimes \|0\rangle/\sqrt{2}$, which maps to the [[Bell state]] $(\|00\rangle + \|11\rangle)/\sqrt{2}$:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> U = [1 0 0 0;0 1 0 0;0 0 0 1;0 0 1 0];		>> U = [1 0 0 0;0 1 0 0;0 0 0 1;0 0 1 0];
	>> [eg,wit] = IsEntanglingGate(U)		>> [eg,wit] = IsEntanglingGate(U)
Line 50:		Line 49:
	0		0
	0.7071		0.7071
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	==Notes==		==Notes==
	Despite the name and description of the function, it can be used on non-unitary (and even non-square) operators just fine. If <tt>U</tt> is not unitary, the function has the exact same interpretation: it determines whether or not there is a product vector that is mapped to a non-product vector.		Despite the name and description of the function, it can be used on non-unitary (and even non-square) operators just fine. If <tt>U</tt> is not unitary, the function has the exact same interpretation: it determines whether or not there is a product vector that is mapped to a non-product vector.

			{{SourceCode\|name=IsEntanglingGate}}

Nathaniel: /* A bipartite example: the CNOT gate */ typo

2012-11-28T16:33:18Z

A bipartite example: the CNOT gate: typo

← Older revision		Revision as of 16:33, 28 November 2012
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	==Examples==		==Examples==
	===A bipartite example: the CNOT gate===		===A bipartite example: the CNOT gate===
	The following code ~~verified~~ that the [[CNOT]] gate is an entangling gate. The following code also requests a product pure state <tt>wit</tt> that is entangled by the CNOT gate. The product pure state returned in this case is $(\|0\rangle + \|1\rangle) \otimes \|0\rangle/\sqrt{2}$, which maps to the [[Bell state]] $(\|00\rangle + \|11\rangle)/\sqrt{2}$:		The following code verifies that the [[CNOT]] gate is an entangling gate. The following code also requests a product pure state <tt>wit</tt> that is entangled by the CNOT gate. The product pure state returned in this case is $(\|0\rangle + \|1\rangle) \otimes \|0\rangle/\sqrt{2}$, which maps to the [[Bell state]] $(\|00\rangle + \|11\rangle)/\sqrt{2}$:
	<pre<noinclude></noinclude>>		<pre<noinclude></noinclude>>
	>> U = [1 0 0 0;0 1 0 0;0 0 0 1;0 0 1 0];		>> U = [1 0 0 0;0 1 0 0;0 0 0 1;0 0 1 0];

Nathaniel: Created page with "{{Function |name=IsEntanglingGate |desc=Determines if a unitary is an entangling gate |req=opt_args
IsProductOperator
IsProductVector
[[perm..."

2012-11-28T16:18:18Z

Created page with "{{Function |name=IsEntanglingGate |desc=Determines if a unitary is an entangling gate |req=opt_args IsProductOperator IsProductVector [[perm..."

New page

{{Function
|name=IsEntanglingGate
|desc=Determines if a [[unitary]] is an [[entangling gate]]
|req=[[opt_args]] [[IsProductOperator]] [[IsProductVector]] [[perm_inv]] [[PermuteSystems]] [[SchmidtDecomposition]]
|rel=[[OperatorSchmidtDecomposition]] [[OperatorSchmidtRank]]
|upd=November 28, 2012
|v=1.00}}
<tt>'''IsEntanglingGate'''</tt> is a [[List of functions|function]] that determines if a [[bipartite]] or [[multipartite]] unitary is an [[entangling gate]] or not (i.e., whether or not there is a [[product vector]] that is no longer a product vector after the unitary is applied). If it is an entangling gate, a [[product vector]] that is entangled by the gate can be provided, and if it is not an entangling gate then a decomposition that proves that it is not entangling can be provided.

==Syntax==
* <tt>EG = IsEntanglingGate(U)</tt>
* <tt>EG = IsEntanglingGate(U,DIM)</tt>
* <tt>[EG,WIT] = IsEntanglingGate(U,DIM)</tt>

==Argument descriptions==
===Input arguments===
* <tt>U</tt>: An operator (typically a unitary operator) that acts on a [[bipartite]] or [[multipartite]] Hilbert space.
* <tt>DIM</tt> (optional, by default has two subsystems of equal dimension): A specification of the dimensions of the subsystems that <tt>U</tt> lives on. <tt>DIM</tt> can be provided in one of three ways:
** If <tt>DIM</tt> is a scalar, it is assumed that <tt>U</tt> lives on the tensor product of two spaces, the first of which has dimension <tt>DIM</tt> and the second of which has dimension <tt>length(U)/DIM</tt>.
** If $U \in M_{n_1} \otimes \cdots \otimes M_{n_p}$ then <tt>DIM</tt> should be a row vector containing the dimensions (i.e., <tt>DIM = [n_1, ..., n_p]</tt>).
** If the subsystems aren't square (i.e., $U \in M_{m_1, n_1} \otimes \cdots \otimes M_{m_p, n_p}$) then <tt>DIM</tt> should be a matrix with two rows. The first row of <tt>DIM</tt> should contain the row dimensions of the subsystems (i.e., the mi's) and its second row should contain the column dimensions (i.e., the ni's). In other words, you should set <tt>DIM = [m_1, ..., m_p; n_1, ..., n_p]</tt>.

===Output arguments===
* <tt>EG</tt>: Either 1 or 0, indicating that <tt>U</tt> is or is not an entangling gate.
* <tt>WIT</tt> (optional): If <tt>EG = 0</tt> (i.e., <tt>U</tt> is not an entangling gate), then <tt>WIT</tt> is a struct with fields <tt>WIT.perm</tt> and <tt>WIT.dec</tt> such that <tt>[[PermutationOperator|PermutationOperator(DIM,WIT.perm)]]*[[Tensor|Tensor(WIT.dec)]]</tt> equals <tt>U</tt> and thus verifies that <tt>U</tt> is indeed not an entangling gate. If <tt>EG = 1</tt> (i.e., <tt>U</tt> is an entangling gate) then <tt>WIT</tt> is a (sparse) product pure state that is entangled by <tt>U</tt>. Furthermore, <tt>WIT</tt> will always have 2 or fewer non-zero entries in this case.

==Examples==
===A bipartite example: the CNOT gate===
The following code verified that the [[CNOT]] gate is an entangling gate. The following code also requests a product pure state <tt>wit</tt> that is entangled by the CNOT gate. The product pure state returned in this case is $(|0\rangle + |1\rangle) \otimes |0\rangle/\sqrt{2}$, which maps to the [[Bell state]] $(|00\rangle + |11\rangle)/\sqrt{2}$:
<pre<noinclude></noinclude>>
>> U = [1 0 0 0;0 1 0 0;0 0 0 1;0 0 1 0];
>> [eg,wit] = IsEntanglingGate(U)

eg =

1

wit =

(1,1) 0.7071
(3,1) 0.7071

>> U*wit

ans =

0.7071
0
0
0.7071
</pre<noinclude></noinclude>>

==Notes==
Despite the name and description of the function, it can be used on non-unitary (and even non-square) operators just fine. If <tt>U</tt> is not unitary, the function has the exact same interpretation: it determines whether or not there is a product vector that is mapped to a non-product vector.

IsEntanglingGate - Revision history

Nathaniel at 15:04, 22 September 2014

Nathaniel: Updated documentation using GeSHi

Nathaniel: /* A bipartite example: the CNOT gate */ typo

Nathaniel: Created page with "{{Function |name=IsEntanglingGate |desc=Determines if a unitary is an entangling gate |req=opt_argsIsProductOperatorIsProductVector[[perm..."

Nathaniel: Created page with "{{Function |name=IsEntanglingGate |desc=Determines if a unitary is an entangling gate |req=opt_args
IsProductOperator
IsProductVector
[[perm..."