toqito.state_metrics.helstrom_holevo

Helstrom-Holevo metric gives the bst success probability to distinguish two mixed states.

Module Contents

toqito.state_metrics.helstrom_holevo.helstrom_holevo(rho, sigma)

Compute the Helstrom-Holevo distance between density matrices [@WikiHolevo].

In general, the best success probability to discriminate two mixed states represented by (rho) and (sigma) is given by [@WikiHolevo].

[

frac{1}{2}+frac{1}{2} left(frac{1}{2} left|rho - sigma right|_1right).

]

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:

[

rho = u u^* = 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 Helstrom-Holevo distance of states that are identical yield a value of (1/2). This can be verified in |toqito⟩ as follows.

```python exec=”1” source=”above” import numpy as np from toqito.states import basis from toqito.state_metrics import helstrom_holevo

e_0, e_1 = basis(2, 0), basis(2, 1) e_00 = np.kron(e_0, e_0) e_11 = np.kron(e_1, e_1)

u_vec = 1 / np.sqrt(2) * (e_00 + e_11) rho = u_vec @ u_vec.conj().T sigma = rho

print(helstrom_holevo(rho, sigma)) ```

Raises:

ValueError – If matrices are not density operators.

Parameters:

• rho (numpy.ndarray) – Density operator.

• sigma (numpy.ndarray) – Density operator.

Returns:

The Helstrom-Holevo distance between rho and sigma.

Return type:

float | numpy.floating