Difference between revisions of "RandomDensityMatrix"

	RandomDensityMatrix
	Generates a random density matrix
Other toolboxes required	none
Related functions	RandomStateVector; RandomSuperoperator; RandomUnitary
Function category	Random things

Revision as of 16:42, 30 September 2014

RandomDensityMatrix is a 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

RHO = RandomDensityMatrix(DIM)
RHO = RandomDensityMatrix(DIM,RE)
RHO = RandomDensityMatrix(DIM,RE,K)
RHO = RandomDensityMatrix(DIM,RE,K,DIST)

Argument descriptions

DIM: The number of rows (or equivalently, columns) that RHO will have.
RE (optional, default 0): A flag (either 0 or 1) indicating that RHO should only have real entries (RE = 1) or that it is allowed to have complex entries (RE = 0).
K (optional, default DIM): The maximal rank of the density matrix to be produced. With probability 1, rank(RHO) = K (if K ≤ DIM).
DIST (optional, default 'hs'): A string indicating the desired distribution that RHO should be chosen from. It can take on one of three values:
- 'hs': The Hilbert-Schmidt measure.
- 'bures': The Bures measure.
- 'haar': 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

Random mixed qubits

The following code generates a random mixed state on a 2-level system:

>> rho = RandomDensityMatrix(2)

rho =

   0.1187            -0.0728 + 0.0409i
  -0.0728 - 0.0409i   0.8813

We can verify that this is indeed a valid density matrix as follows:

>> trace(rho)

ans =

     1

>> IsPSD(rho)

ans =

     1

The following code generates a density matrix with all real entries, chosen according to the Bures measure:

>> RandomDensityMatrix(2,1,2,'bures')

ans =

    0.1578    0.2259
    0.2259    0.8422

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:

>> 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

Source code

Click here to view this function's source code on github.

@@ 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==