"""Trace distance metric."""
import numpy as np
from toqito.matrix_props import is_density
from toqito.state_metrics import trace_norm
[docs]
def trace_distance(rho: np.ndarray, sigma: np.ndarray) -> float:
r"""
Compute the trace distance between density operators `rho` and `sigma`.
The trace distance between :math:`\rho` and :math:`\sigma` is defined as
.. math::
\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
.. math::
u = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) \in \mathcal{X}.
The corresponding density matrix of :math:`u` may be calculated by:
.. math::
\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 :math:`\rho` and another state :math:`\sigma` is equal to :math:`0`
if any only if :math:`\rho = \sigma`. We can check this using the :code:`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
==========
.. [WIKTD] Quantiki: Trace distance
https://www.quantiki.org/wiki/trace-distance
:raises ValueError: If matrices are not of density operators.
:param rho: An input matrix.
:param sigma: An input matrix.
:return: The trace distance between :code:`rho` and :code:`sigma`.
"""
if not is_density(rho) or not is_density(sigma):
raise ValueError("Trace distance only defined for density matrices.")
return trace_norm(np.abs(rho - sigma)) / 2