toqito.state_props.von_neumann_entropy¶
Calculates the Von neumann entropy metric of a quantum state.
Module Contents¶
- toqito.state_props.von_neumann_entropy.von_neumann_entropy(rho)[source]¶
Compute the von Neumann entropy of a density matrix [@WikiUVonNeumann].
Let (P in text{Pos}(mathcal{X})) be a positive semidefinite operator, for a complex Euclidean space (mathcal{X}). Then one defines the von Neumann entropy as
- [
H(P) = H(lambda(P)),
]
where (lambda(P)) is the vector of eigenvalues of (P) and where the function (H(cdot)) is the Shannon entropy function defined as
- [
H(u) = -sum_{substack{a in Sigma \ u(a) > 0}} u(a) text{log}(u(a)),
]
where the (text{log}) function is assumed to be the base-2 logarithm, and where (Sigma) is an alphabet where (u in [0, infty)^{Sigma}) is a vector of nonnegative real numbers indexed by (Sigma).
Further information for computing the von Neumann entropy of a density matrix can be found in Section: “Definitions Of Quantum Entropic Functions” from [@Watrous_2018_TQI]).
Examples
Consider the following Bell state:
]
The corresponding density matrix of (u) may be calculated by:
- [
- rho = u u^* = frac{1}{2} begin{pmatrix}
1 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 1
end{pmatrix} in text{D}(mathcal{X}).
]
Calculating the von Neumann entropy of (rho) in |toqito⟩ can be done as follows.
```python exec=”1” source=”above” from toqito.state_props import von_neumann_entropy import numpy as np test_input_mat = np.array(
[[1 / 2, 0, 0, 1 / 2], [0, 0, 0, 0], [0, 0, 0, 0], [1 / 2, 0, 0, 1 / 2]]
)
print(von_neumann_entropy(test_input_mat)) ```
Consider the density operator corresponding to the maximally mixed state of dimension two
- [
rho = frac{1}{2} begin{pmatrix}
1 & 0 \ 0 & 1
end{pmatrix}.
]
As this state is maximally mixed, the von Neumann entropy of (rho) is equal to one. We can see this in |toqito⟩ as follows.
`python exec="1" source="above" from toqito.state_props import von_neumann_entropy import numpy as np rho = 1/2 * np.identity(2) print(von_neumann_entropy(rho)) `- Parameters:
rho (numpy.ndarray) – Density operator.
- Returns:
The von Neumann entropy of rho.
- Return type:
float