qml.math.vn_entanglement_entropy¶
- vn_entanglement_entropy(state, indices0, indices1, base=None, check_state=False, c_dtype='complex128')[source]¶
Compute the Von Neumann entanglement entropy between two subsystems in a given state.
\[S(\rho_A) = -\text{Tr}[\rho_A \log \rho_A] = -\text{Tr}[\rho_B \log \rho_B] = S(\rho_B)\]where \(S\) is the von Neumann entropy, and \(\rho_A = \text{Tr}_B [\rho_{AB}]\) and \(\rho_B = \text{Tr}_A [\rho_{AB}]\) are the reduced density matrices for each partition.
The Von Neumann entanglement entropy is a measure of the degree of quantum entanglement between two subsystems constituting a pure bipartite quantum state. The entropy of entanglement is the Von Neumann entropy of the reduced density matrix for any of the subsystems. If it is non-zero, it indicates the two subsystems are entangled.
Each state must be given as a density matrix. To find the mutual information given a pure state, call
dm_from_state_vector()first.- Parameters
state (tensor_like) –
(2**N, 2**N)or(batch_dim, 2**N, 2**N)density matrix.indices0 (list[int]) – Indices of the qubits in the first subsystem.
indices1 (list[int]) – Indices of the qubits in the second subsystem.
base (float) – Base for the logarithm. If
None, the natural logarithm is used.check_state (bool) – If True, the function will check the state validity (shape and norm).
c_dtype (str) – Complex floating point precision type.
- Returns
The von Neumann entanglement entropy of the bipartite state.
- Return type
float
Examples
The entanglement entropy between subsystems for a state vector can be returned as follows:
>>> x = np.array([0, -1, 1, 0]) / np.sqrt(2) >>> x = qml.math.dm_from_state_vector(x) >>> qml.math.vn_entanglement_entropy(x, indices0=[0], indices1=[1]) 0.6931471805599453
It is also possible to change the logarithm base:
>>> qml.math.vn_entanglement_entropy(x, indices0=[0], indices1=[1], base=2) 1
Similarly, the quantum state can be provided as a density matrix:
>>> y = np.array([[1, 1, -1, -1], [1, 1, -1, -1], [-1, -1, 1, 1], [-1, -1, 1, 1]]) * 0.25 >>> qml.math.vn_entanglement_entropy(y, indices0=[0], indices1=[1]) 0