state_metrics.helstrom_holevo

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

Functions

helstrom_holevo(rho, sigma)

Compute the Helstrom-Holevo distance between density matrices [1].

Module Contents

state_metrics.helstrom_holevo.helstrom_holevo(rho, sigma)

Compute the Helstrom-Holevo distance between density matrices [1].

In general, the best success probability to discriminate two mixed states represented by \(\rho\) and \(\sigma\) is given by [1].

\[\frac{1}{2}+\frac{1}{2} \left(\frac{1}{2} \left|\rho - \sigma \right|_1\right).\]

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:

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

Calculating the Helstrom-Holevo distance of states that are identical yield a value of \(1/2\). This can be verified in toqito as follows.

>>> from toqito.states import basis
>>> from toqito.state_metrics import helstrom_holevo
>>> import numpy as np
>>> 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
>>>
>>> helstrom_holevo(rho, sigma)
np.float64(0.5)

References

[1] (1,2,3)

Wikipedia. Holevoś theorem. URL: https://en.wikipedia.org/wiki/Holevo%27s_theorem.

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