toqito.state_metrics.matsumoto_fidelity

toqito.state_metrics.matsumoto_fidelity(rho, sigma)

Compute the Matsumoto fidelity of two density matrices [Mat10].

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

\[\rho\#\sigma = \rho^{1/2}\sqrt{\rho^{-1/2}\sigma\rho^{-1/2}}\rho^{1/2}.\]

For singular states it is defined by the limit

\[\rho\#\sigma = \lim_{\epsilon\to0}(\rho+\epsilon\mathbb{I})\#(+\epsilon\mathbb{I}).\]

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

\[u = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) \in \mathcal{X}.\]

The corresponding density matrix of \(u\) may be calculated by:

\[\begin{split}\rho = u u^* = \frac{1}{2} \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix} \in \text{D}(\mathcal{X}).\end{split}\]

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.

>>> from toqito.state_metrics import matsumoto_fidelity
>>> import numpy as np
>>> rho = 1 / 2 * np.array(
>>>     [[1, 0, 0, 1],
>>>      [0, 0, 0, 0],
>>>      [0, 0, 0, 0],
>>>      [1, 0, 0, 1]]
>>> )
>>> sigma = rho
>>> matsumoto_fidelity(rho, sigma)
0.9999998585981018

References

[Mat10]

Keiji Matsumoto. “Reverse test and quantum analogue of classical fidelity and generalized fidelity” https://arxiv.org/abs/1006.0302

Raises:

ValueError – If matrices are not of equal dimension.

Parameters:

• rho – Density operator.

• sigma – Density operator.

Returns:

The Matsumoto fidelity between rho and sigma.