RandomStateVector - Revision history

Nathaniel at 21:16, 17 December 2014

2014-12-17T21:16:09Z

← Older revision		Revision as of 21:16, 17 December 2014
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	{{Function		{{Function
	\|name=RandomStateVector		\|name=RandomStateVector
	\|desc=Generates a random [[pure state]] vector		\|desc=Generates a random pure state vector
	\|rel=[[RandomDensityMatrix]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]		\|rel=[[RandomDensityMatrix]]<br />[[RandomProbabilities]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]
	\|cat=[[List of functions#Random_things\|Random things]]		\|cat=[[List of functions#Random_things\|Random things]]
	\|upd=November 12, 2014		\|upd=November 12, 2014

Nathaniel at 18:07, 12 November 2014

2014-11-12T18:07:58Z

← Older revision		Revision as of 18:07, 12 November 2014
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	\|rel=[[RandomDensityMatrix]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]		\|rel=[[RandomDensityMatrix]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]
	\|cat=[[List of functions#Random_things\|Random things]]		\|cat=[[List of functions#Random_things\|Random things]]
	\|upd=~~September 30~~, 2014		\|upd=November 12, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''RandomStateVector'''</tt> is a [[List of functions\|function]] that generates a random pure state vector, uniformly distributed on the unit hypersphere (sometimes said to be uniformly distributed according to Haar measure).		<tt>'''RandomStateVector'''</tt> is a [[List of functions\|function]] that generates a random pure state vector, uniformly distributed on the unit hypersphere (sometimes said to be uniformly distributed according to Haar measure).

Nathaniel at 16:43, 30 September 2014

2014-09-30T16:43:59Z

← Older revision		Revision as of 16:43, 30 September 2014
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	\|rel=[[RandomDensityMatrix]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]		\|rel=[[RandomDensityMatrix]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]
	\|cat=[[List of functions#Random_things\|Random things]]		\|cat=[[List of functions#Random_things\|Random things]]
	\|upd=~~November 23~~, ~~2012~~		\|upd=September 30, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''RandomStateVector'''</tt> is a [[List of functions\|function]] that generates a random pure state vector, uniformly distributed on the unit hypersphere (~~equivalently,~~ uniformly distributed according to [[Haar measure]]).		<tt>'''RandomStateVector'''</tt> is a [[List of functions\|function]] that generates a random pure state vector, uniformly distributed on the unit hypersphere (sometimes said to be uniformly distributed according to Haar measure).

	==Syntax==		==Syntax==
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	===Random states with fixed Schmidt rank===		===Random states with fixed Schmidt rank===
	To generate a random product [[qutrit]]-qutrit state and verify that it is indeed a product state, use the following code:		To generate a random product qutrit-qutrit state and verify that it is indeed a product state, use the following code:
	<syntaxhighlight>		<syntaxhighlight>
	>> v = RandomStateVector(3,0,1)		>> v = RandomStateVector(3,0,1)

Nathaniel at 15:35, 25 September 2014

2014-09-25T15:35:11Z

← Older revision		Revision as of 15:35, 25 September 2014
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	\|name=RandomStateVector		\|name=RandomStateVector
	\|desc=Generates a random [[pure state]] vector		\|desc=Generates a random [[pure state]] vector
	~~\|req=[[MaxEntangled]]<br />[[opt_args]]<br />[[PermuteSystems]]<br />[[Swap]]~~
	\|rel=[[RandomDensityMatrix]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]		\|rel=[[RandomDensityMatrix]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]
			\|cat=[[List of functions#Random_things\|Random things]]
	\|upd=November 23, 2012		\|upd=November 23, 2012
	\|v=1.00}}		\|v=0.50}}
	<tt>'''RandomStateVector'''</tt> is a [[List of functions\|function]] that generates a random [[pure state]] vector, uniformly distributed on the unit hypersphere (equivalently, uniformly distributed according to [[Haar measure]]).		<tt>'''RandomStateVector'''</tt> is a [[List of functions\|function]] that generates a random pure state vector, uniformly distributed on the unit hypersphere (equivalently, uniformly distributed according to [[Haar measure]]).

	==Syntax==		==Syntax==
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	* <tt>DIM</tt>: The dimension of the Hilbert space in which <tt>V</tt> lives. If <tt>K > 0</tt> (see optional arguments below) then <tt>DIM</tt> is the local dimension rather than the total dimension. If different local dimensions are desired, <tt>DIM</tt> should be a 1-by-2 vector containing the desired local dimensions.		* <tt>DIM</tt>: The dimension of the Hilbert space in which <tt>V</tt> lives. If <tt>K > 0</tt> (see optional arguments below) then <tt>DIM</tt> is the local dimension rather than the total dimension. If different local dimensions are desired, <tt>DIM</tt> should be a 1-by-2 vector containing the desired local dimensions.
	* <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>V</tt> should only have real entries (<tt>RE = 1</tt>) or that it is allowed to have complex entries (<tt>RE = 0</tt>).		* <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>V</tt> should only have real entries (<tt>RE = 1</tt>) or that it is allowed to have complex entries (<tt>RE = 0</tt>).
	* <tt>K</tt> (optional, default 0): If equal to 0 then <tt>V</tt> will be generated without considering its [[Schmidt rank]] ~~at all~~. If <tt>K > 0</tt> then a random pure state with Schmidt rank ≤ <tt>K</tt> will be generated (and with probability 1, its Schmidt rank will equal <tt>K</tt>). Note that when <tt>K = 1</tt> the states on the two subsystems are generated uniformly and independently according to Haar measure on those subsystems. When <tt>K = DIM</tt>, the usual Haar measure on the total space is used. When <tt>1 < K < DIM</tt>, a natural measure that interpolates between these two extremes is used (more specifically, the direct sum of the left (similarly, right) Schmidt vectors is chosen according to Haar measure on $\mathbb{C}^K \otimes \mathbb{C}^{DIM}$).		* <tt>K</tt> (optional, default 0): If equal to 0 then <tt>V</tt> will be generated without considering its [[SchmidtRank\|Schmidt rank]]. If <tt>K > 0</tt> then a random pure state with Schmidt rank ≤ <tt>K</tt> will be generated (and with probability 1, its Schmidt rank will equal <tt>K</tt>). Note that when <tt>K = 1</tt> the states on the two subsystems are generated uniformly and independently according to Haar measure on those subsystems. When <tt>K = DIM</tt>, the usual Haar measure on the total space is used. When <tt>1 < K < DIM</tt>, a natural measure that interpolates between these two extremes is used (more specifically, the direct sum of the left (similarly, right) Schmidt vectors is chosen according to Haar measure on $\mathbb{C}^K \otimes \mathbb{C}^{DIM}$).

	==Examples==		==Examples==
	===A random qubit===		===A random qubit===
	To generate a random qubit, use the following code:		To generate a random qubit, use the following code:
	<~~pre~~>		<syntaxhighlight>
	>> RandomStateVector(2)		>> RandomStateVector(2)

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	-0.1025 - 0.5498i		-0.1025 - 0.5498i
	-0.5518 + 0.6186i		-0.5518 + 0.6186i
	</~~pre~~>		</syntaxhighlight>

	If you want it to only have real entries, set <tt>RE = 1</tt>:		If you want it to only have real entries, set <tt>RE = 1</tt>:
	<~~pre~~>		<syntaxhighlight>
	>> RandomStateVector(2,1)		>> RandomStateVector(2,1)

Line 38:		Line 38:
	-0.4487		-0.4487
	0.8937		0.8937
	</~~pre~~>		</syntaxhighlight>

	===Random states with fixed Schmidt rank===		===Random states with fixed Schmidt rank===
	To generate a random product [[qutrit]]-qutrit state and verify that it is indeed a product state, use the following code:		To generate a random product [[qutrit]]-qutrit state and verify that it is indeed a product state, use the following code:
	<~~pre~~>		<syntaxhighlight>
	>> v = RandomStateVector(3,0,1)		>> v = RandomStateVector(3,0,1)

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	1		1
	</~~pre~~>		</syntaxhighlight>

	You could create a random pure state with Schmidt rank 2 in $\mathbb{C}^3 \otimes \mathbb{C}^4$, and verify its Schmidt rank, using the following lines of code:		You could create a random pure state with Schmidt rank 2 in $\mathbb{C}^3 \otimes \mathbb{C}^4$, and verify its Schmidt rank, using the following lines of code:
	<~~pre~~>		<syntaxhighlight>
	>> v = RandomStateVector([3,4],0,2)		>> v = RandomStateVector([3,4],0,2)

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	2		2
	</~~pre~~>		</syntaxhighlight>

			{{SourceCode\|name=RandomStateVector}}

Nathaniel: /* Argument descriptions */

2012-11-23T21:49:23Z

Argument descriptions

← Older revision		Revision as of 21:49, 23 November 2012
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	* <tt>DIM</tt>: The dimension of the Hilbert space in which <tt>V</tt> lives. If <tt>K > 0</tt> (see optional arguments below) then <tt>DIM</tt> is the local dimension rather than the total dimension. If different local dimensions are desired, <tt>DIM</tt> should be a 1-by-2 vector containing the desired local dimensions.		* <tt>DIM</tt>: The dimension of the Hilbert space in which <tt>V</tt> lives. If <tt>K > 0</tt> (see optional arguments below) then <tt>DIM</tt> is the local dimension rather than the total dimension. If different local dimensions are desired, <tt>DIM</tt> should be a 1-by-2 vector containing the desired local dimensions.
	* <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>V</tt> should only have real entries (<tt>RE = 1</tt>) or that it is allowed to have complex entries (<tt>RE = 0</tt>).		* <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>V</tt> should only have real entries (<tt>RE = 1</tt>) or that it is allowed to have complex entries (<tt>RE = 0</tt>).
	* <tt>K</tt> (optional, default 0): If equal to 0 then <tt>V</tt> will be generated without considering its [[Schmidt rank]] at all. If <tt>K > 0</tt> then a random pure state with Schmidt rank ≤ <tt>K</tt> will be generated (and with probability 1, its Schmidt rank will equal <tt>K</tt>).		* <tt>K</tt> (optional, default 0): If equal to 0 then <tt>V</tt> will be generated without considering its [[Schmidt rank]] at all. If <tt>K > 0</tt> then a random pure state with Schmidt rank ≤ <tt>K</tt> will be generated (and with probability 1, its Schmidt rank will equal <tt>K</tt>). Note that when <tt>K = 1</tt> the states on the two subsystems are generated uniformly and independently according to Haar measure on those subsystems. When <tt>K = DIM</tt>, the usual Haar measure on the total space is used. When <tt>1 < K < DIM</tt>, a natural measure that interpolates between these two extremes is used (more specifically, the direct sum of the left (similarly, right) Schmidt vectors is chosen according to Haar measure on $\mathbb{C}^K \otimes \mathbb{C}^{DIM}$).

	==Examples==		==Examples==

Nathaniel: Created page with "{{Function |name=RandomStateVector |desc=Generates a random pure state vector |req=MaxEntangled
opt_args
PermuteSystems
Swap |rel=[[RandomDe..."

2012-11-23T21:02:08Z

Created page with "{{Function |name=RandomStateVector |desc=Generates a random pure state vector |req=MaxEntangled opt_args PermuteSystems Swap |rel=[[RandomDe..."

New page

{{Function
|name=RandomStateVector
|desc=Generates a random [[pure state]] vector
|req=[[MaxEntangled]] [[opt_args]] [[PermuteSystems]] [[Swap]]
|rel=[[RandomDensityMatrix]] [[RandomSuperoperator]] [[RandomUnitary]]
|upd=November 23, 2012
|v=1.00}}
<tt>'''RandomStateVector'''</tt> is a [[List of functions|function]] that generates a random [[pure state]] vector, uniformly distributed on the unit hypersphere (equivalently, uniformly distributed according to [[Haar measure]]).

==Syntax==
* <tt>V = RandomStateVector(DIM)</tt>
* <tt>V = RandomStateVector(DIM,RE)</tt>
* <tt>V = RandomStateVector(DIM,RE,K)</tt>

==Argument descriptions==
* <tt>DIM</tt>: The dimension of the Hilbert space in which <tt>V</tt> lives. If <tt>K > 0</tt> (see optional arguments below) then <tt>DIM</tt> is the local dimension rather than the total dimension. If different local dimensions are desired, <tt>DIM</tt> should be a 1-by-2 vector containing the desired local dimensions.
* <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>V</tt> should only have real entries (<tt>RE = 1</tt>) or that it is allowed to have complex entries (<tt>RE = 0</tt>).
* <tt>K</tt> (optional, default 0): If equal to 0 then <tt>V</tt> will be generated without considering its [[Schmidt rank]] at all. If <tt>K > 0</tt> then a random pure state with Schmidt rank ≤ <tt>K</tt> will be generated (and with probability 1, its Schmidt rank will equal <tt>K</tt>).

==Examples==
===A random qubit===
To generate a random qubit, use the following code:
<pre>
>> RandomStateVector(2)

ans =

-0.1025 - 0.5498i
-0.5518 + 0.6186i
</pre>

If you want it to only have real entries, set <tt>RE = 1</tt>:
<pre>
>> RandomStateVector(2,1)

ans =

-0.4487
0.8937
</pre>

===Random states with fixed Schmidt rank===
To generate a random product [[qutrit]]-qutrit state and verify that it is indeed a product state, use the following code:
<pre>
>> v = RandomStateVector(3,0,1)

v =

0.0400 - 0.3648i
0.1169 - 0.0666i
0.0465 + 0.0016i
-0.1910 + 0.0524i
-0.0566 - 0.0455i
-0.0084 - 0.0236i
-0.4407 + 0.7079i
-0.3050 + 0.0214i
-0.0936 - 0.0489i

>> SchmidtRank(v)

ans =

1
</pre>

You could create a random pure state with Schmidt rank 2 in $\mathbb{C}^3 \otimes \mathbb{C}^4$, and verify its Schmidt rank, using the following lines of code:
<pre>
>> v = RandomStateVector([3,4],0,2)

v =

-0.2374 + 0.1984i
0.1643 + 0.0299i
-0.0499 + 0.0376i
-0.0689 - 0.0005i
0.7740 - 0.0448i
-0.1290 - 0.2224i
-0.0514 - 0.1565i
0.2195 + 0.2478i
-0.1636 + 0.1276i
0.0581 + 0.0608i
0.0482 - 0.0178i
-0.1050 + 0.0014i

>> SchmidtRank(v,[3,4])

ans =

2
</pre>