state_metrics.trace_distance

Trace distance metric gives a measure of distinguishability between two quantum states.

The trace distance is calculated via density matrices.

Functions

trace_distance(rho, sigma)

Compute the trace distance between density operators rho and sigma.

Module Contents

state_metrics.trace_distance.trace_distance(rho, sigma)

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 [1].

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_distance
>>> rho = bell(0) @ bell(0).conj().T
>>> sigma = rho
>>> trace_distance(rho, sigma)
np.float64(0.0)

References

[1]

Quantiki. Trace distance. URL: https://www.quantiki.org/wiki/trace-distance.

Raises:

ValueError – If matrices are not of density operators.

Parameters:

• rho (numpy.ndarray) – An input matrix.

• sigma (numpy.ndarray) – An input matrix.

Returns:

The trace distance between rho and sigma.

Return type:

float