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__all__ = ['entropy_vn', 'entropy_linear', 'entropy_mutual',
'concurrence', 'entropy_conditional', 'entangling_power']
from numpy import e, real, sort, sqrt
from scipy import log, log2
from qutip.qobj import ptrace
from qutip.states import ket2dm
from qutip.tensor import tensor
from qutip.operators import sigmay
from qutip.sparse import sp_eigs
from qutip.qip.gates import swap
[docs]def entropy_vn(rho, base=e, sparse=False):
"""
Von-Neumann entropy of density matrix
Parameters
----------
rho : qobj
Density matrix.
base : {e,2}
Base of logarithm.
sparse : {False,True}
Use sparse eigensolver.
Returns
-------
entropy : float
Von-Neumann entropy of `rho`.
Examples
--------
>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1)
>>> entropy_vn(rho,2)
1.0
"""
if rho.type == 'ket' or rho.type == 'bra':
rho = ket2dm(rho)
vals = sp_eigs(rho.data, rho.isherm, vecs=False, sparse=sparse)
nzvals = vals[vals != 0]
if base == 2:
logvals = log2(nzvals)
elif base == e:
logvals = log(nzvals)
else:
raise ValueError("Base must be 2 or e.")
return float(real(-sum(nzvals * logvals)))
[docs]def entropy_linear(rho):
"""
Linear entropy of a density matrix.
Parameters
----------
rho : qobj
sensity matrix or ket/bra vector.
Returns
-------
entropy : float
Linear entropy of rho.
Examples
--------
>>> rho=0.5*fock_dm(2,0)+0.5*fock_dm(2,1)
>>> entropy_linear(rho)
0.5
"""
if rho.type == 'ket' or rho.type == 'bra':
rho = ket2dm(rho)
return float(real(1.0 - (rho ** 2).tr()))
[docs]def concurrence(rho):
"""
Calculate the concurrence entanglement measure for a two-qubit state.
Parameters
----------
state : qobj
Ket, bra, or density matrix for a two-qubit state.
Returns
-------
concur : float
Concurrence
References
----------
.. [1] http://en.wikipedia.org/wiki/Concurrence_(quantum_computing)
"""
if rho.isket and rho.dims != [[2, 2], [1, 1]]:
raise Exception("Ket must be tensor product of two qubits.")
elif rho.isbra and rho.dims != [[1, 1], [2, 2]]:
raise Exception("Bra must be tensor product of two qubits.")
elif rho.isoper and rho.dims != [[2, 2], [2, 2]]:
raise Exception("Density matrix must be tensor product of two qubits.")
if rho.isket or rho.isbra:
rho = ket2dm(rho)
sysy = tensor(sigmay(), sigmay())
rho_tilde = (rho * sysy) * (rho.conj() * sysy)
evals = rho_tilde.eigenenergies()
# abs to avoid problems with sqrt for very small negative numbers
evals = abs(sort(real(evals)))
lsum = sqrt(evals[3]) - sqrt(evals[2]) - sqrt(evals[1]) - sqrt(evals[0])
return max(0, lsum)
[docs]def entropy_mutual(rho, selA, selB, base=e, sparse=False):
"""
Calculates the mutual information S(A:B) between selection
components of a system density matrix.
Parameters
----------
rho : qobj
Density matrix for composite quantum systems
selA : int/list
`int` or `list` of first selected density matrix components.
selB : int/list
`int` or `list` of second selected density matrix components.
base : {e,2}
Base of logarithm.
sparse : {False,True}
Use sparse eigensolver.
Returns
-------
ent_mut : float
Mutual information between selected components.
"""
if isinstance(selA, int):
selA = [selA]
if isinstance(selB, int):
selB = [selB]
if rho.type != 'oper':
raise TypeError("Input must be a density matrix.")
if (len(selA) + len(selB)) != len(rho.dims[0]):
raise TypeError("Number of selected components must match " +
"total number.")
rhoA = ptrace(rho, selA)
rhoB = ptrace(rho, selB)
out = (entropy_vn(rhoA, base, sparse=sparse) +
entropy_vn(rhoB, base, sparse=sparse) -
entropy_vn(rho, base, sparse=sparse))
return out
def _entropy_relative(rho, sigma, base=e, sparse=False):
"""
****NEEDS TO BE WORKED ON**** (after 2.0 release)
Calculates the relative entropy S(rho||sigma) between two density
matrices..
Parameters
----------
rho : qobj
First density matrix.
sigma : qobj
Second density matrix.
base : {e,2}
Base of logarithm.
Returns
-------
rel_ent : float
Value of relative entropy.
"""
if rho.type != 'oper' or sigma.type != 'oper':
raise TypeError("Inputs must be density matrices..")
# sigma terms
svals = sp_eigs(sigma.data, sigma.isherm, vecs=False, sparse=sparse)
snzvals = svals[svals != 0]
if base == 2:
slogvals = log2(snzvals)
elif base == e:
slogvals = log(snzvals)
else:
raise ValueError("Base must be 2 or e.")
# rho terms
rvals = sp_eigs(rho.data, rho.isherm, vecs=False, sparse=sparse)
rnzvals = rvals[rvals != 0]
# calculate tr(rho*log sigma)
rel_trace = float(real(sum(rnzvals * slogvals)))
return -entropy_vn(rho, base, sparse) - rel_trace
[docs]def entropy_conditional(rho, selB, base=e, sparse=False):
"""
Calculates the conditional entropy :math:`S(A|B)=S(A,B)-S(B)`
of a slected density matrix component.
Parameters
----------
rho : qobj
Density matrix of composite object
selB : int/list
Selected components for density matrix B
base : {e,2}
Base of logarithm.
sparse : {False,True}
Use sparse eigensolver.
Returns
-------
ent_cond : float
Value of conditional entropy
"""
if rho.type != 'oper':
raise TypeError("Input must be density matrix.")
if isinstance(selB, int):
selB = [selB]
B = ptrace(rho, selB)
out = (entropy_vn(rho, base, sparse=sparse) -
entropy_vn(B, base, sparse=sparse))
return out
def participation_ratio(rho):
"""
Returns the effective number of states for a density matrix.
The participation is unity for pure states, and maximally N,
where N is the Hilbert space dimensionality, for completely
mixed states.
Parameters
----------
rho : qobj
Density matrix
Returns
-------
pr : float
Effective number of states in the density matrix
"""
if rho.type == 'ket' or rho.type == 'bra':
return 1.0
else:
return 1.0 / (rho ** 2).tr()
def entangling_power(U):
"""
Calculate the entangling power of a two-qubit gate U, which
is zero of nonentangling gates and 1 and 2/9 for maximally
entangling gates.
Parameters
----------
U : qobj
Qobj instance representing a two-qubit gate.
Returns
-------
ep : float
The entanglement power of U (real number between 0 and 1)
References:
Explorations in Quantum Computing, Colin P. Williams (Springer, 2011)
"""
if not U.isoper:
raise Exception("U must be an operator.")
if U.dims != [[2, 2], [2, 2]]:
raise Exception("U must be a two-qubit gate.")
a = (tensor(U, U).dag() * swap(N=4, targets=[1, 3]) *
tensor(U, U) * swap(N=4, targets=[1, 3]))
b = (tensor(swap() * U, swap() * U).dag() * swap(N=4, targets=[1, 3]) *
tensor(swap() * U, swap() * U) * swap(N=4, targets=[1, 3]))
return 5.0/9 - 1.0/36 * (a.tr() + b.tr()).real