WernerState - Revision history

Nathaniel: Corrected date

2014-11-12T17:54:47Z

Corrected date

← Older revision		Revision as of 17:54, 12 November 2014
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	\|rel=[[IsotropicState]]		\|rel=[[IsotropicState]]
	\|cat=[[List of functions#Special_states,_vectors,_and_operators\|Special states, vectors, and operators]]		\|cat=[[List of functions#Special_states,_vectors,_and_operators\|Special states, vectors, and operators]]
	\|upd=~~September~~ 12, 2014		\|upd=November 12, 2014
	\|v=0.51}}		\|v=0.51}}
	<tt>'''WernerState'''</tt> is a [[List of functions\|function]] that returns the Werner state (i.e., a state of the following form):<ref name="Wer89">R. F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. ''Phys. Rev. A'', 40(8):4277–4281.</ref>		<tt>'''WernerState'''</tt> is a [[List of functions\|function]] that returns the Werner state (i.e., a state of the following form):<ref name="Wer89">R. F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. ''Phys. Rev. A'', 40(8):4277–4281.</ref>

Nathaniel: Updated to v0.51 (since SwapOperator.m is changing)

2014-11-12T17:54:28Z

Updated to v0.51 (since SwapOperator.m is changing)

← Older revision		Revision as of 17:54, 12 November 2014
Line 4:		Line 4:
	\|rel=[[IsotropicState]]		\|rel=[[IsotropicState]]
	\|cat=[[List of functions#Special_states,_vectors,_and_operators\|Special states, vectors, and operators]]		\|cat=[[List of functions#Special_states,_vectors,_and_operators\|Special states, vectors, and operators]]
	\|upd=September 23, 2014		\|upd=September 12, 2014
	\|v=0.50}}		\|v=0.51}}
	<tt>'''WernerState'''</tt> is a [[List of functions\|function]] that returns the Werner state (i.e., a state of the following form):<ref name="Wer89">R. F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. ''Phys. Rev. A'', 40(8):4277–4281.</ref>		<tt>'''WernerState'''</tt> is a [[List of functions\|function]] that returns the Werner state (i.e., a state of the following form):<ref name="Wer89">R. F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. ''Phys. Rev. A'', 40(8):4277–4281.</ref>
	<center><math>\displaystyle\rho_\alpha := \frac{1}{d^2-d\alpha}\big(I \otimes I - \alpha S\big) \in M_d \otimes M_d,</math></center>		<center><math>\displaystyle\rho_\alpha := \frac{1}{d^2-d\alpha}\big(I \otimes I - \alpha S\big) \in M_d \otimes M_d,</math></center>

Nathaniel at 12:58, 23 September 2014

2014-09-23T12:58:27Z

← Older revision		Revision as of 12:58, 23 September 2014
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	\|name=WernerState		\|name=WernerState
	\|desc=Produces a [[Werner state]]		\|desc=Produces a [[Werner state]]
	~~\|req=[[iden]]<br />[[opt_args]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[Swap]]<br />[[SwapOperator]]~~
	\|rel=[[IsotropicState]]		\|rel=[[IsotropicState]]
	\|upd=~~November 28~~, ~~2012~~		\|cat=[[List of functions#Special_states,_vectors,_and_operators\|Special states, vectors, and operators]]
	\|v=1.01}}		\|upd=September 23, 2014
	<tt>'''WernerState'''</tt> is a [[List of functions\|function]] that returns the [[~~Werner state~~]] ~~requested by the user~~. The ~~Werner state may be bipartite or multipartite and either full or~~ sparse.		\|v=0.50}}
			<tt>'''WernerState'''</tt> is a [[List of functions\|function]] that returns the Werner state (i.e., a state of the following form):<ref name="Wer89">R. F. Werner. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. ''Phys. Rev. A'', 40(8):4277–4281.</ref>
			<center><math>\displaystyle\rho_\alpha := \frac{1}{d^2-d\alpha}\big(I \otimes I - \alpha S\big) \in M_d \otimes M_d,</math></center>
			where $S$ is the [[SwapOperator\|swap operator]]. This function is also capable of producing multipartite Werner states. The output of this function is always sparse.

	==Syntax==		==Syntax==
	* <tt>RHO = WernerState(DIM,ALPHA)</tt>		* <tt>RHO = WernerState(DIM,ALPHA)</tt>
	* <tt>RHO = WernerState(DIM,ALPHA,SP)</tt>

	==Argument descriptions==		==Argument descriptions==
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	** If <tt>ALPHA</tt> is a scalar, the Werner state returned is the normalization of <tt>I - ALPHA*S</tt>, where <tt>I</tt> is the identity matrix and <tt>S</tt> is the bipartite swap operator.		** If <tt>ALPHA</tt> is a scalar, the Werner state returned is the normalization of <tt>I - ALPHA*S</tt>, where <tt>I</tt> is the identity matrix and <tt>S</tt> is the bipartite swap operator.
	** If <tt>ALPHA</tt> is a vector of length p! - 1 for some integer p, the Werner state returned is a multipartite state acting on p copies of <tt>DIM</tt>-dimensional space. More explicitly, the state returned is the normalization of <tt>I - ALPHA(1)P(2) - ... - ALPHA(p!-1)P(p!)</tt>, where <tt>P(i)</tt> is the operator that permutes the p subsystems according to the i-th permutation (when the permutations are ordered in ascending lexicographical order).		** If <tt>ALPHA</tt> is a vector of length p! - 1 for some integer p, the Werner state returned is a multipartite state acting on p copies of <tt>DIM</tt>-dimensional space. More explicitly, the state returned is the normalization of <tt>I - ALPHA(1)P(2) - ... - ALPHA(p!-1)P(p!)</tt>, where <tt>P(i)</tt> is the operator that permutes the p subsystems according to the i-th permutation (when the permutations are ordered in ascending lexicographical order).
	* <tt>SP</tt> (optional, default 0): A flag (either 1 or 0) indicating that the Werner state produced should or should not be sparse.

	==Examples==		==Examples==
	===A qutrit Werner state===		===A qutrit Werner state===
	To generate the Werner state with parameter $\alpha = 1/2$, the following code suffices:		To generate the Werner state with parameter $\alpha = 1/2$, the following code suffices:
	<~~pre~~>		<syntaxhighlight>
	>> WernerState(3,1/2)		>> full(WernerState(3,1/2))

	ans =		ans =
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	0 0 0 0 0 -0.0667 0 0.1333 0		0 0 0 0 0 -0.0667 0 0.1333 0
	0 0 0 0 0 0 0 0 0.0667		0 0 0 0 0 0 0 0 0.0667
	</~~pre~~>		</syntaxhighlight>

	Werner states in general have a lot of zero entries, so you ~~will usually save~~ a ~~lot of memory by specifying <tt>SP = 1</tt>~~, ~~which causes the Werner state that is generated to be sparse~~:		Werner states in general have a lot of zero entries, so this function always returns a sparse matrix. If you want a full matrix (as above), use MATLAB's [http://www.mathworks.com/access/helpdesk/help/techdoc/ref/full.html full] function.
	<~~pre~~>		<syntaxhighlight>
	>> WernerState(3,1/2,1)		>> WernerState(3,1/2)

	ans =		ans =
Line 59:		Line 59:
	(8,8) 0.1333		(8,8) 0.1333
	(9,9) 0.0667		(9,9) 0.0667
	</~~pre~~>		</syntaxhighlight>

	===A multipartite Werner state===		===A multipartite Werner state===
	In the multipartite setting, the family of Werner states is specified by more than 1 parameter <tt>ALPHA</tt>, so we need to provide more than 1 parameter to the <tt>WernerState</tt> function. In the [[tripartite]] case, there are 3! - 1 = 5 parameters that we need to specify: one for each of the non-identity permutations of the systems. The lexicographical ordering of the permutations of three elements is: 123, 132, 213, 231, 312, 321. Thus the following code produces the Werner state that is the normalization of $I - 0.01 P_{1,3,2} - 0.02 P_{2,1,3} - 0.03 P_{2,3,1} - 0.04 P_{3,1,2} - 0.05 P_{3,2,1}$, where $P_{x,y,z}$ is the permutation operator that maps $\|v_1\rangle \otimes \|v_2\rangle \otimes \|v_3\rangle$ to $\|v_x\rangle \otimes \|v_y\rangle \otimes \|v_z\rangle$:		In the multipartite setting, the family of Werner states is specified by more than 1 parameter <tt>ALPHA</tt>, so we need to provide more than 1 parameter to the <tt>WernerState</tt> function. In the tripartite case, there are 3! - 1 = 5 parameters that we need to specify: one for each of the non-identity permutations of the systems. The lexicographical ordering of the permutations of three elements is: 123, 132, 213, 231, 312, 321. Thus the following code produces the Werner state that is the normalization of $I - 0.01 P_{1,3,2} - 0.02 P_{2,1,3} - 0.03 P_{2,3,1} - 0.04 P_{3,1,2} - 0.05 P_{3,2,1}$, where $P_{x,y,z}$ is the permutation operator that maps $\|v_1\rangle \otimes \|v_2\rangle \otimes \|v_3\rangle$ to $\|v_x\rangle \otimes \|v_y\rangle \otimes \|v_z\rangle$:
	<~~pre~~>		<syntaxhighlight>
	>> WernerState(2,[0.01,0.02,0.03,0.04,0.05])		>> full(WernerState(2,[0.01,0.02,0.03,0.04,0.05]))

	ans =		ans =
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	0 0 0 -0.0106 0 -0.0066 0.1300 0		0 0 0 -0.0106 0 -0.0066 0.1300 0
	0 0 0 0 0 0 0 0.1127		0 0 0 0 0 0 0 0.1127
	</~~pre~~>		</syntaxhighlight>

			{{SourceCode\|name=WernerState}}

			==References==
			<references />

Nathaniel: Uploaded v1.01

2012-11-28T19:04:52Z

Uploaded v1.01

← Older revision		Revision as of 19:04, 28 November 2012
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	\|name=WernerState		\|name=WernerState
	\|desc=Produces a [[Werner state]]		\|desc=Produces a [[Werner state]]
	\|req=[[opt_args]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[Swap]]<br />[[SwapOperator]]		\|req=[[iden]]<br />[[opt_args]]<br />[[PermutationOperator]]<br />[[PermuteSystems]]<br />[[Swap]]<br />[[SwapOperator]]
	\|rel=[[IsotropicState]]		\|rel=[[IsotropicState]]
	\|upd=November 17, 2012		\|upd=November 28, 2012
	\|v=1.00}}		\|v=1.01}}
	<tt>'''WernerState'''</tt> is a [[List of functions\|function]] that returns the [[Werner state]] requested by the user. The Werner state may be bipartite or multipartite and either full or sparse.		<tt>'''WernerState'''</tt> is a [[List of functions\|function]] that returns the [[Werner state]] requested by the user. The Werner state may be bipartite or multipartite and either full or sparse.

Nathaniel: Created page with "{{Function |name=WernerState |desc=Produces a Werner state |req=opt_args
PermutationOperator
PermuteSystems
Swap
SwapOperator |rel=..."

2012-11-23T16:41:26Z

Created page with "{{Function |name=WernerState |desc=Produces a Werner state |req=opt_args PermutationOperator PermuteSystems Swap SwapOperator |rel=..."

New page

{{Function
|name=WernerState
|desc=Produces a [[Werner state]]
|req=[[opt_args]] [[PermutationOperator]] [[PermuteSystems]] [[Swap]] [[SwapOperator]]
|rel=[[IsotropicState]]
|upd=November 17, 2012
|v=1.00}}
<tt>'''WernerState'''</tt> is a [[List of functions|function]] that returns the [[Werner state]] requested by the user. The Werner state may be bipartite or multipartite and either full or sparse.

==Syntax==
* <tt>RHO = WernerState(DIM,ALPHA)</tt>
* <tt>RHO = WernerState(DIM,ALPHA,SP)</tt>

==Argument descriptions==
* <tt>DIM</tt>: Dimension of the local subsystems on which <tt>RHO</tt> acts.
* <tt>ALPHA</tt>: A parameter that specifies which [[Werner state]] is to be returned as follows:
** If <tt>ALPHA</tt> is a scalar, the Werner state returned is the normalization of <tt>I - ALPHA*S</tt>, where <tt>I</tt> is the identity matrix and <tt>S</tt> is the bipartite swap operator.
** If <tt>ALPHA</tt> is a vector of length p! - 1 for some integer p, the Werner state returned is a multipartite state acting on p copies of <tt>DIM</tt>-dimensional space. More explicitly, the state returned is the normalization of <tt>I - ALPHA(1)*P(2) - ... - ALPHA(p!-1)*P(p!)</tt>, where <tt>P(i)</tt> is the operator that permutes the p subsystems according to the i-th permutation (when the permutations are ordered in ascending lexicographical order).
* <tt>SP</tt> (optional, default 0): A flag (either 1 or 0) indicating that the Werner state produced should or should not be sparse.

==Examples==
===A qutrit Werner state===
To generate the Werner state with parameter $\alpha = 1/2$, the following code suffices:
<pre>
>> WernerState(3,1/2)

ans =

0.0667 0 0 0 0 0 0 0 0
0 0.1333 0 -0.0667 0 0 0 0 0
0 0 0.1333 0 0 0 -0.0667 0 0
0 -0.0667 0 0.1333 0 0 0 0 0
0 0 0 0 0.0667 0 0 0 0
0 0 0 0 0 0.1333 0 -0.0667 0
0 0 -0.0667 0 0 0 0.1333 0 0
0 0 0 0 0 -0.0667 0 0.1333 0
0 0 0 0 0 0 0 0 0.0667
</pre>

Werner states in general have a lot of zero entries, so you will usually save a lot of memory by specifying <tt>SP = 1</tt>, which causes the Werner state that is generated to be sparse:
<pre>
>> WernerState(3,1/2,1)

ans =

(1,1) 0.0667
(2,2) 0.1333
(4,2) -0.0667
(3,3) 0.1333
(7,3) -0.0667
(2,4) -0.0667
(4,4) 0.1333
(5,5) 0.0667
(6,6) 0.1333
(8,6) -0.0667
(3,7) -0.0667
(7,7) 0.1333
(6,8) -0.0667
(8,8) 0.1333
(9,9) 0.0667
</pre>

===A multipartite Werner state===
In the multipartite setting, the family of Werner states is specified by more than 1 parameter <tt>ALPHA</tt>, so we need to provide more than 1 parameter to the <tt>WernerState</tt> function. In the [[tripartite]] case, there are 3! - 1 = 5 parameters that we need to specify: one for each of the non-identity permutations of the systems. The lexicographical ordering of the permutations of three elements is: 123, 132, 213, 231, 312, 321. Thus the following code produces the Werner state that is the normalization of $I - 0.01 P_{1,3,2} - 0.02 P_{2,1,3} - 0.03 P_{2,3,1} - 0.04 P_{3,1,2} - 0.05 P_{3,2,1}$, where $P_{x,y,z}$ is the permutation operator that maps $|v_1\rangle \otimes |v_2\rangle \otimes |v_3\rangle$ to $|v_x\rangle \otimes |v_y\rangle \otimes |v_z\rangle$:
<pre>
>> WernerState(2,[0.01,0.02,0.03,0.04,0.05])

ans =

0.1127 0 0 0 0 0 0 0
0 0.1300 -0.0066 0 -0.0106 0 0 0
0 -0.0053 0.1260 0 -0.0080 0 0 0
0 0 0 0.1313 0 -0.0066 -0.0119 0
0 -0.0119 -0.0066 0 0.1313 0 0 0
0 0 0 -0.0080 0 0.1260 -0.0053 0
0 0 0 -0.0106 0 -0.0066 0.1300 0
0 0 0 0 0 0 0 0.1127
</pre>

WernerState - Revision history

Nathaniel: Corrected date

Nathaniel: Updated to v0.51 (since SwapOperator.m is changing)

Nathaniel at 12:58, 23 September 2014

Nathaniel: Uploaded v1.01

Nathaniel: Created page with "{{Function |name=WernerState |desc=Produces a Werner state |req=opt_argsPermutationOperatorPermuteSystemsSwapSwapOperator |rel=..."

Nathaniel: Created page with "{{Function |name=WernerState |desc=Produces a Werner state |req=opt_args
PermutationOperator
PermuteSystems
Swap
SwapOperator |rel=..."