QuTiP includes a collection of random state generators for simulations, theorem evaluation, and code testing:
| Function | Description |
|---|---|
| rand_ket | Random ket-vector |
| rand_dm | Random density matrix |
| rand_herm | Random Hermitian matrix |
| rand_unitary | Random Unitary matrix |
See the API documentation: Random Operators and States for details.
In all cases, these functions can be called with a single parameter \(N\) that indicates a \(NxN\) matrix (rand_dm, rand_herm, rand_unitary), or a \(Nx1\) vector (rand_ket), should be generated. For example:
In [2]: rand_ket(5)
Out[2]:
Quantum object: dims = [[5], [1]], shape = [5, 1], type = ket
Qobj data =
[[-0.62994785+0.1389789j ]
[-0.43288801+0.01284981j]
[-0.22400352-0.20321684j]
[-0.47922421-0.22806793j]
[-0.11912553-0.09464168j]]
or
In [3]: rand_herm(5)
Out[3]:
Quantum object: dims = [[5], [5]], shape = [5, 5], type = oper, isherm = True
Qobj data =
[[-0.58420310+0.j -0.43073690+0.2344248j 0.00000000+0.j
-0.27393962+0.12764388j -0.48896580+0.09691766j]
[-0.43073690-0.2344248j -0.52347888+0.j -0.13899371-0.06069369j
-0.35374472-0.22620005j 0.00000000+0.j ]
[ 0.00000000+0.j -0.13899371+0.06069369j -0.16089832+0.j
-0.65111568+0.25401902j -0.30001540-0.14135167j]
[-0.27393962-0.12764388j -0.35374472+0.22620005j -0.65111568-0.25401902j
-0.56096362+0.j -0.48167388+0.23237517j]
[-0.48896580-0.09691766j 0.00000000+0.j -0.30001540+0.14135167j
-0.48167388-0.23237517j 0.00000000+0.j ]]
In this previous example, we see that the generated Hermitian operator contains a fraction of elements that are identically equal to zero. The number of nonzero elements is called the density and can be controlled by calling any of the random state/operator generators with a second argument between 0 and 1. By default, the density for the operators is 0.75 where as ket vectors are completely dense (1). For example:
In [4]: rand_dm(5, 0.5)
Out[4]:
Quantum object: dims = [[5], [5]], shape = [5, 5], type = oper, isherm = True
Qobj data =
[[ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j
0.00000000+0.j 0.00000000+0.j ]
[ 0.00000000+0.j 0.32141161+0.j 0.00000000+0.j
0.19802585-0.11734087j 0.00000000+0.j ]
[ 0.00000000+0.j 0.00000000+0.j 0.01374332+0.j
0.00000000+0.j 0.01897778+0.04365046j]
[ 0.00000000+0.j 0.19802585+0.11734087j 0.00000000+0.j
0.17858839+0.j 0.00000000+0.j ]
[ 0.00000000+0.j 0.00000000+0.j 0.01897778-0.04365046j
0.00000000+0.j 0.48625668+0.j ]]
has rougly half nonzero elements, or equivalently a density of 0.5.
Important
In the case of a density matrix, setting the density too low will result in not enough diagonal elements to satisfy \(Tr(\rho)=1\).
In many cases, one is interested in generating random quantum objects that correspond to composite systems generated using the qutip.tensor.tensor function. Specifying the tensor structure of a quantum object is done using the dims keyword argument in the same fashion as one would do for a qutip.Qobj object:
In [5]: rand_dm(4, 0.5, dims=[[2,2], [2,2]])
Out[5]:
Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isherm = True
Qobj data =
[[ 0.39065157 0. 0. 0. ]
[ 0. 0.39065157 0. 0. ]
[ 0. 0. 0.10934843 0. ]
[ 0. 0. 0. 0.10934843]]