toqito.state_metrics.trace_distance

toqito.state_metrics.trace_distance(rho, sigma)[source]

Compute the trace distance between density operators rho and sigma.

The trace distance between \(\rho\) and \(\sigma\) is defined as

\[\delta(\rho, \sigma) = \frac{1}{2} \left( \text{Tr}(\left| \rho - \sigma \right| \right).\]

More information on the trace distance can be found in [WIKTD].

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:

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

The trace distance between \(\rho\) and another state \(\sigma\) is equal to \(0\) if any only if \(\rho = \sigma\). We can check this using the toqito package.

>>> from toqito.states import bell
>>> from toqito.state_metrics import trace_norm
>>> rho = bell(0) * bell(0).conj().T
>>> sigma = rho
>>> trace_distance(rho, sigma)
0.0

References

Raises:

ValueError – If matrices are not of density operators.

Parameters:

  • rho – An input matrix.

  • sigma – An input matrix.

Returns:

The trace distance between rho and sigma.