toqito.state_metrics.matsumoto_fidelity¶
Matsumoto fidelity is the maximum classical fidelity associated with a classical-to-quantum preparation procedure.
Module Contents¶
- toqito.state_metrics.matsumoto_fidelity.matsumoto_fidelity(rho, sigma)[source]¶
Compute the Matsumoto fidelity of two density matrices [@Matsumoto_2010_Reverse].
Calculate the Matsumoto fidelity between the two density matrices rho and sigma, defined by:
- [
mathrm{tr}(rho#sigma),
]
where (#) denotes the matrix geometric mean, which for invertible states is
- [
rho#sigma = rho^{1/2}sqrt{rho^{-1/2}sigmarho^{-1/2}}rho^{1/2}.
]
For singular states it is defined by the limit
- [
rho#sigma = lim_{epsilonto0}(rho+epsilonmathbb{I})#(+epsilonmathbb{I}).
]
The return is a value between (0) and (1), with (0) corresponding to matrices rho and sigma with orthogonal support, and (1) corresponding to the case rho = sigma. The Matsumoto fidelity is a lower bound for the fidelity.
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}).
]
In the event where we calculate the Matsumoto fidelity between states that are identical, we should obtain the value of (1). This can be observed in |toqito⟩ as follows.
```python exec=”1” source=”above” import numpy as np from toqito.state_metrics import matsumoto_fidelity
- rho = 1 / 2 * np.array(
- [[1, 0, 0, 1],
[0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 1]]
) sigma = rho
print(np.around(matsumoto_fidelity(rho, sigma), decimals=2)) ```
- Raises:
ValueError – If matrices are not of equal dimension.
- Parameters:
rho (numpy.ndarray) – Density operator.
sigma (numpy.ndarray) – Density operator.
- Returns:
The Matsumoto fidelity between rho and sigma.
- Return type:
float | numpy.floating