RandomDensityMatrix - Revision history

Nathaniel: Merged hs and haar

2014-10-15T15:09:28Z

Merged hs and haar

Nathaniel: Added purity example

2014-10-15T14:01:11Z

Added purity example

Nathaniel at 16:42, 30 September 2014

2014-09-30T16:42:06Z

Nathaniel at 13:58, 25 September 2014

2014-09-25T13:58:11Z

Nathaniel: /* Random mixed qubits */

2012-11-28T18:28:26Z

Random mixed qubits

Nathaniel: /* Argument descriptions */

2012-11-28T18:27:25Z

Argument descriptions

Nathaniel: /* Argument descriptions */

2012-11-28T18:26:48Z

Argument descriptions

Nathaniel: Created page with "{{Function |name=RandomDensityMatrix |desc=Generates a random density matrix |req=MaxEntangled
opt_args
PartialTrace
PermuteSystems
[[R..."

2012-11-24T04:37:43Z

Created page with "{{Function |name=RandomDensityMatrix |desc=Generates a random density matrix |req=MaxEntangled opt_args PartialTrace PermuteSystems [[R..."

New page

{{Function
|name=RandomDensityMatrix
|desc=Generates a random [[density matrix]]
|req=[[MaxEntangled]] [[opt_args]] [[PartialTrace]] [[PermuteSystems]] [[RandomStateVector]] [[RandomUnitary]] [[Swap]]
|rel=[[RandomSuperoperator]]
|upd=November 23, 2012
|v=1.00}}
<tt>'''RandomDensityMatrix'''</tt> is a [[List of functions|function]] that generates a random [[density matrix]], uniformly according to the [[Hilbert-Schmidt measure]], the [[Bures measure]], or a version of the [[Haar measure]].

==Syntax==
* <tt>RHO = RandomDensityMatrix(DIM)</tt>
* <tt>RHO = RandomDensityMatrix(DIM,RE)</tt>
* <tt>RHO = RandomDensityMatrix(DIM,RE,K)</tt>
* <tt>RHO = RandomDensityMatrix(DIM,RE,K,DIST)</tt>

==Argument descriptions==
* <tt>DIM</tt>: The number of rows (or equivalently, columns) that <tt>RHO</tt> will have.
* <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>RHO</tt> should only have real entries (<tt>RE = 1</tt>) or that it is allowed to have complex entries (<tt>RE = 1</tt>).
* <tt>K</tt> (optional, default <tt>DIM</tt>): The maximal rank of the density matrix to be produced. With probability 1, <tt>rank(RHO) = K</tt>.
* <tt>DIST</tt> (optional, default <tt>'hs'</tt>): A string indicating the desired distribution that <tt>RHO</tt> should be chosen from. It can take on one of three values:
** <tt>'hs'</tt>: The [[Hilbert-Schmidt measure]].
** <tt>'bures'</tt>: The [[Bures measure]].
** <tt>'haar'</tt>: The density matrix is generated by generating a [[Haar measure|Haar-uniform]] pure state in $\mathbb{C}^K \otimes \mathbb{C}^{DIM}$ and then [[partial trace|tracing out]] the first subsystem.

==Examples==
===Random mixed qubits===
The following code generates a random mixed state on a 2-level system:
<pre>
>> rho = RandomDensityMatrix(2)

rho =

0.1187 -0.0728 + 0.0409i
-0.0728 - 0.0409i 0.8813
</pre>

We can verify that this is indeed a valid density matrix as follows:
<pre>
>> trace(rho)

ans =

1

>> IsPSD(rho)

ans =

1
</pre>

The following code generates a density matrix with all real entries, chosen according to the [[Bures measure]]:
<pre>
>> RandomDensityMatrix(2,1,2,'bures')

ans =

0.1578 0.2259
0.2259 0.8422
</pre>

===A larger example of specified rank===
To generate a 6-by-6 density matrix with rank at most 4, you could use the following code:
<pre>
>> rho = RandomDensityMatrix(6,0,4)

rho =

0.1750 -0.0299 - 0.0103i -0.0304 - 0.0668i 0.0108 - 0.0176i -0.0294 - 0.0796i -0.0026 + 0.0705i
-0.0299 + 0.0103i 0.1461 -0.0483 + 0.0490i 0.0406 + 0.0422i -0.0064 + 0.1005i 0.0461 + 0.0225i
-0.0304 + 0.0668i -0.0483 - 0.0490i 0.1896 -0.0010 + 0.0652i 0.0156 + 0.0388i -0.0610 - 0.0002i
0.0108 + 0.0176i 0.0406 - 0.0422i -0.0010 - 0.0652i 0.1332 0.1221 + 0.0212i -0.0023 + 0.0264i
-0.0294 + 0.0796i -0.0064 - 0.1005i 0.0156 - 0.0388i 0.1221 - 0.0212i 0.2355 -0.0381 - 0.0789i
-0.0026 - 0.0705i 0.0461 - 0.0225i -0.0610 + 0.0002i -0.0023 - 0.0264i -0.0381 + 0.0789i 0.1206

>> rank(rho)

ans =

4
</pre>

@@ Line 4: / Line 4: @@
 |rel=[[RandomStateVector]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]
 |cat=[[List of functions#Random_things|Random things]]
-|upd=September 30, 2014
+|upd=October 10, 2014
 |v=0.50}}
-<tt>'''RandomDensityMatrix'''</tt> is a [[List of functions|function]] that generates a random density matrix, uniformly according to the Hilbert-Schmidt measure, the Bures measure, or a version of the Haar measure.
+<tt>'''RandomDensityMatrix'''</tt> is a [[List of functions|function]] that generates a random density matrix, uniformly according to the Hilbert-Schmidt measure (equivalently, by generating a pure state according to Haar measure on a larger system and then tracing out the ancillary space) or the Bures measure.
 ==Syntax==
@@ Line 18: / Line 18: @@
 * <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>RHO</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 <tt>DIM</tt>): The maximal rank of the density matrix to be produced. With probability 1, <tt>rank(RHO) = K</tt> (if <tt>K &le; DIM</tt>).
-* <tt>DIST</tt> (optional, default <tt>'hs'</tt>): A string indicating the desired distribution that <tt>RHO</tt> should be chosen from. It can take on one of three values:
+* <tt>DIST</tt> (optional, default <tt>'haar'</tt>): A string indicating the desired distribution that <tt>RHO</tt> should be chosen from. It can take on one of three values:
-** <tt>'hs'</tt>: The Hilbert-Schmidt measure.
+** <tt>'haar' or 'hs'</tt>: The density matrix is generated by generating a Haar-uniform pure state in $\mathbb{C}^K \otimes \mathbb{C}^{DIM}$ and then tracing out the first subsystem. In the special case when <tt>K = DIM</tt>, this is sometimes called the Hilbert-Schmidt measure.
 ** <tt>'bures'</tt>: The Bures measure.
 ==Examples==

← Older revision		Revision as of 14:01, 15 October 2014
Line 79:		Line 79:

	4		4
		+	</syntaxhighlight>
		+
		+	===Purity of random density matrices===
		+	It is known that the expected [[Purity\|purity]] of a random $n \times n$ density matrix, generated according to Haar measure with an ancillary space of dimension $m$, is $(n+m)/(nm+1)$. We can verify this numerically as follows:
		+	<syntaxhighlight>
		+	>> n = 3; m = 7;
		+	>> ct = 0; s = 100000;
		+	>> for j = 1:s
		+	ct = ct + Purity(RandomDensityMatrix(n,0,m));
		+	end
		+	ct/s % this is the numerically-generated average purity
		+
		+	ans =
		+
		+	0.4546
		+
		+	>> (n+m)/(n*m+1)
		+
		+	ans =
		+
		+	0.4545
	</syntaxhighlight>		</syntaxhighlight>

	{{SourceCode\|name=RandomDensityMatrix}}		{{SourceCode\|name=RandomDensityMatrix}}

@@ Line 4: / Line 4: @@
 |rel=[[RandomStateVector]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]
 |cat=[[List of functions#Random_things|Random things]]
-|upd=November 23, 2012
+|upd=September 30, 2014
 |v=0.50}}
-<tt>'''RandomDensityMatrix'''</tt> is a [[List of functions|function]] that generates a random [[density matrix]], uniformly according to the [[Hilbert-Schmidt measure]], the [[Bures measure]], or a version of the [[Haar measure]].
+<tt>'''RandomDensityMatrix'''</tt> is a [[List of functions|function]] that generates a random density matrix, uniformly according to the Hilbert-Schmidt measure, the Bures measure, or a version of the Haar measure.
 ==Syntax==
@@ Line 19: / Line 19: @@
 * <tt>K</tt> (optional, default <tt>DIM</tt>): The maximal rank of the density matrix to be produced. With probability 1, <tt>rank(RHO) = K</tt> (if <tt>K &le; DIM</tt>).
 * <tt>DIST</tt> (optional, default <tt>'hs'</tt>): A string indicating the desired distribution that <tt>RHO</tt> should be chosen from. It can take on one of three values:
-** <tt>'hs'</tt>: The [[Hilbert-Schmidt measure]].
+** <tt>'hs'</tt>: The Hilbert-Schmidt measure.
-** <tt>'bures'</tt>: The [[Bures measure]].
+** <tt>'bures'</tt>: The Bures measure.
-** <tt>'haar'</tt>: The density matrix is generated by generating a [[Haar measure|Haar-uniform]] pure state in $\mathbb{C}^K \otimes \mathbb{C}^{DIM}$ and then [[partial trace|tracing out]] the first subsystem.
+** <tt>'haar'</tt>: The density matrix is generated by generating a Haar-uniform pure state in $\mathbb{C}^K \otimes \mathbb{C}^{DIM}$ and then tracing out the first subsystem.
 ==Examples==

@@ Line 1: / Line 1: @@
 {{Function
 |name=RandomDensityMatrix
-|desc=Generates a random [[density matrix]]
+|desc=Generates a random density matrix
-|req=[[MaxEntangled]]<br />[[opt_args]]<br />[[PartialTrace]]<br />[[PermuteSystems]]<br />[[RandomStateVector]]<br />[[RandomUnitary]]<br />[[Swap]]
+|rel=[[RandomStateVector]]<br />[[RandomSuperoperator]]<br />[[RandomUnitary]]
-|rel=[[RandomSuperoperator]]
+|cat=[[List of functions#Random_things|Random things]]
 |upd=November 23, 2012
-|v=1.00}}
+|v=0.50}}
 <tt>'''RandomDensityMatrix'''</tt> is a [[List of functions|function]] that generates a random [[density matrix]], uniformly according to the [[Hilbert-Schmidt measure]], the [[Bures measure]], or a version of the [[Haar measure]].
@@ Line 26: / Line 26: @@
 ===Random mixed qubits===
 The following code generates a random mixed state on a 2-level system:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
 >> rho = RandomDensityMatrix(2)
@@ Line 33: / Line 33: @@
 .1187            -0.0728 + 0.0409i
    -0.0728 - 0.0409i   0.8813
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 We can verify that this is indeed a valid density matrix as follows:
-<pre<noinclude></noinclude>>
+<syntaxhighlight>
 >> trace(rho)
@@ Line 43: / Line 43: @@
->> [[IsPSD|IsPSD(rho)]]
+>> IsPSD(rho)
 ans =
-</pre<noinclude></noinclude>>
+</syntaxhighlight>
 The following code generates a density matrix with all real entries, chosen according to the [[Bures measure]]:
-<pre>
+<syntaxhighlight>
 >> RandomDensityMatrix(2,1,2,'bures')
@@ Line 58: / Line 58: @@
 .1578    0.2259
 .2259    0.8422
-</pre>
+</syntaxhighlight>
 ===A larger example of specified rank===
 To generate a 6-by-6 density matrix with rank at most 4, you could use the following code:
-<pre>
+<syntaxhighlight>
 >> rho = RandomDensityMatrix(6,0,4)
@@ Line 79: / Line 79: @@
-</pre>
+</syntaxhighlight>

@@ Line 16: / Line 16: @@
 ==Argument descriptions==
 * <tt>DIM</tt>: The number of rows (or equivalently, columns) that <tt>RHO</tt> will have.
-* <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>RHO</tt> should only have real entries (<tt>RE = 1</tt>) or that it is allowed to have complex entries (<tt>RE = 1</tt>).
+* <tt>RE</tt> (optional, default 0): A flag (either 0 or 1) indicating that <tt>RHO</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 <tt>DIM</tt>): The maximal rank of the density matrix to be produced. With probability 1, <tt>rank(RHO) = K</tt>.
 * <tt>DIST</tt> (optional, default <tt>'hs'</tt>): A string indicating the desired distribution that <tt>RHO</tt> should be chosen from. It can take on one of three values:

RandomDensityMatrix - Revision history

Nathaniel: Merged hs and haar

Nathaniel: Added purity example

Nathaniel at 16:42, 30 September 2014

Nathaniel at 13:58, 25 September 2014

Nathaniel: /* Random mixed qubits */

Nathaniel: /* Argument descriptions */

Nathaniel: /* Argument descriptions */

Nathaniel: Created page with "{{Function |name=RandomDensityMatrix |desc=Generates a random density matrix |req=MaxEntangledopt_argsPartialTracePermuteSystems[[R..."

Nathaniel: Created page with "{{Function |name=RandomDensityMatrix |desc=Generates a random density matrix |req=MaxEntangled
opt_args
PartialTrace
PermuteSystems
[[R..."