toqito.state_metrics.trace_distance

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

The trace distance is calculated via density matrices.

Module Contents

toqito.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 [@Quantiki_TrDist].

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}).

]

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.

```python exec=”1” source=”above” from toqito.states import bell from toqito.state_metrics import trace_distance

rho = bell(0) @ bell(0).conj().T sigma = rho

print(trace_distance(rho, sigma)) ```

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 | numpy.floating