def rand_super_bcsz(N=2, enforce_tp=True, rank=None, dims=None):
"""
Returns a random superoperator drawn from the Bruzda
et al ensemble for CPTP maps [BCSZ08]_. Note that due to
finite numerical precision, for ranks less than full-rank,
zero eigenvalues may become slightly negative, such that the
returned operator is not actually completely positive.
Parameters
----------
N : int
Square root of the dimension of the superoperator to be returned.
enforce_tp : bool
If True, the trace-preserving condition of [BCSZ08]_ is enforced;
otherwise only complete positivity is enforced.
rank : int or None
Rank of the sampled superoperator. If None, a full-rank
superoperator is generated.
dims : list
Dimensions of quantum object. Used for specifying
tensor structure. Default is dims=[[[N],[N]], [[N],[N]]].
Returns
-------
rho : Qobj
A superoperator acting on vectorized dim × dim density operators,
sampled from the BCSZ distribution.
"""
if dims is not None:
# TODO: check!
pass
else:
dims = [[[N], [N]], [[N], [N]]]
if rank is None:
rank = N ** 2
if rank > N ** 2:
raise ValueError("Rank cannot exceed superoperator dimension.")
# We use mainly dense matrices here for speed in low
# dimensions. In the future, it would likely be better to switch off
# between sparse and dense matrices as the dimension grows.
# We start with a Ginibre uniform matrix X of the appropriate rank,
# and use it to construct a positive semidefinite matrix X X⁺.
X = randnz((N ** 2, rank), norm="ginibre")
# Precompute X X⁺, as we'll need it in two different places.
XXdag = np.dot(X, X.T.conj())
if enforce_tp:
# We do the partial trace over the first index by using dense reshape
# operations, so that we can avoid bouncing to a sparse representation
# and back.
Y = np.einsum("ijik->jk", XXdag.reshape((N, N, N, N)))
# Now we have the matrix 𝟙 ⊗ Y^{-1/2}, which we can find by doing
# the square root and the inverse separately. As a possible improvement,
# iterative methods exist to find inverse square root matrices directly,
# as this is important in statistics.
Z = np.kron(np.eye(N), sqrtm(la.inv(Y)))
# Finally, we dot everything together and pack it into a Qobj,
# marking the dimensions as that of a type=super (that is,
# with left and right compound indices, each representing
# left and right indices on the underlying Hilbert space).
D = Qobj(np.dot(Z, np.dot(XXdag, Z)))
else:
D = N * Qobj(XXdag / np.trace(XXdag))
D.dims = [
# Left dims
[[N], [N]],
# Right dims
[[N], [N]],
]
# Since [BCSZ08] gives a row-stacking Choi matrix, but QuTiP
# expects a column-stacking Choi matrix, we must permute the indices.
D = D.permute([[1], [0]])
D.dims = dims
# Mark that we've made a Choi matrix.
D.superrep = "choi"
return sr.to_super(D)
def rand_super_bcsz(N=2, enforce_tp=True, rank=None, dims=None):
"""
Returns a random superoperator drawn from the Bruzda
et al ensemble for CPTP maps [BCSZ08]_. Note that due to
finite numerical precision, for ranks less than full-rank,
zero eigenvalues may become slightly negative, such that the
returned operator is not actually completely positive.
Parameters
----------
N : int
Square root of the dimension of the superoperator to be returned.
enforce_tp : bool
If True, the trace-preserving condition of [BCSZ08]_ is enforced;
otherwise only complete positivity is enforced.
rank : int or None
Rank of the sampled superoperator. If None, a full-rank
superoperator is generated.
dims : list
Dimensions of quantum object. Used for specifying
tensor structure. Default is dims=[[[N],[N]], [[N],[N]]].
Returns
-------
rho : Qobj
A superoperator acting on vectorized dim × dim density operators,
sampled from the BCSZ distribution.
"""
if dims is not None:
# TODO: check!
pass
else:
dims = [[[N],[N]], [[N],[N]]]
if rank is None:
rank = N**2
if rank > N**2:
raise ValueError("Rank cannot exceed superoperator dimension.")
# We use mainly dense matrices here for speed in low
# dimensions. In the future, it would likely be better to switch off
# between sparse and dense matrices as the dimension grows.
# We start with a Ginibre uniform matrix X of the appropriate rank,
# and use it to construct a positive semidefinite matrix X X⁺.
X = randnz((N**2, rank), norm='ginibre')
# Precompute X X⁺, as we'll need it in two different places.
XXdag = np.dot(X, X.T.conj())
if enforce_tp:
# We do the partial trace over the first index by using dense reshape
# operations, so that we can avoid bouncing to a sparse representation
# and back.
Y = np.einsum('ijik->jk', XXdag.reshape((N, N, N, N)))
# Now we have the matrix 𝟙 ⊗ Y^{-1/2}, which we can find by doing
# the square root and the inverse separately. As a possible improvement,
# iterative methods exist to find inverse square root matrices directly,
# as this is important in statistics.
Z = np.kron(
np.eye(N),
sqrtm(la.inv(Y))
)
# Finally, we dot everything together and pack it into a Qobj,
# marking the dimensions as that of a type=super (that is,
# with left and right compound indices, each representing
# left and right indices on the underlying Hilbert space).
D = Qobj(np.dot(Z, np.dot(XXdag, Z)))
else:
D = N * Qobj(XXdag / np.trace(XXdag))
D.dims = [
# Left dims
[[N], [N]],
# Right dims
[[N], [N]]
]
# Since [BCSZ08] gives a row-stacking Choi matrix, but QuTiP
# expects a column-stacking Choi matrix, we must permute the indices.
D = D.permute([[1], [0]])
D.dims = dims
# Mark that we've made a Choi matrix.
D.superrep = 'choi'
return sr.to_super(D)
