def entanglement_of_formation(state):
"""Calculate the entanglement of formation of quantum state.
The input quantum state must be either a bipartite state vector, or a
2-qubit density matrix.
Args:
state (Statevector or DensityMatrix): a 2-qubit quantum state.
Returns:
float: The entanglement of formation.
Raises:
QiskitError: if the input state is not a valid QuantumState.
QiskitError: if input is not a bipartite QuantumState.
QiskitError: if density matrix input is not a 2-qubit state.
"""
state = _format_state(state, validate=True)
if isinstance(state, Statevector):
# The entanglement of formation is given by the reduced state
# entropy
dims = state.dims()
if len(dims) != 2:
raise QiskitError("Input is not a bipartite quantum state.")
qargs = [0] if dims[0] > dims[1] else [1]
return entropy(partial_trace(state, qargs), base=2)
# If input is a density matrix it must be a 2-qubit state
if state.dim != 4:
raise QiskitError("Input density matrix must be a 2-qubit state.")
conc = concurrence(state)
val = (1 + np.sqrt(1 - (conc**2))) / 2
return shannon_entropy([val, 1 - val])
def entropy(state, base=2):
r"""Calculate the von-Neumann entropy of a quantum state.
The entropy :math:`S` is given by
.. math::
S(\rho) = - Tr[\rho \log(\rho)]
Args:
state (Statevector or DensityMatrix): a quantum state.
base (int): the base of the logarithm [Default: 2].
Returns:
float: The von-Neumann entropy S(rho).
Raises:
QiskitError: if the input state is not a valid QuantumState.
"""
import scipy.linalg as la
state = _format_state(state, validate=True)
if isinstance(state, Statevector):
return 0
# Density matrix case
evals = np.maximum(np.real(la.eigvals(state.data)), 0.0)
return shannon_entropy(evals, base=base)
