RandomDensityMatrix

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

RandomDensityMatrix is a 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

• 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 'haar'): A string indicating the desired distribution that RHO should be chosen from. It can take on one of three values:
  • 'haar' or 'hs': 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 K = DIM, this is sometimes called the Hilbert-Schmidt measure.
  • 'bures': The Bures measure.

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

Purity of random density matrices

It is known that the expected 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:

>> 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

Source code

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