toqito.state_metrics.sub_fidelity
=================================

.. py:module:: toqito.state_metrics.sub_fidelity

.. autoapi-nested-parse::

   Sub-fidelity metric is a lower bound for the fidelity.

   The sub-fidelity metric is a concave function and sub-multiplicative.





Module Contents
---------------

.. py:function:: sub_fidelity(rho, sigma)

   Compute the sub fidelity of two density matrices [@Miszczak_2008_Sub].

   The sub-fidelity is a measure of similarity between density operators. It is defined as

   \[
       E(\rho, \sigma) = \text{Tr}(\rho \sigma) +
       \sqrt{2 \left[ \text{Tr}(\rho \sigma)^2 - \text{Tr}(\rho \sigma \rho \sigma) \right]},
   \]

   where \(\sigma\) and \(\rho\) are density matrices. The sub-fidelity serves as an lower bound for the
   fidelity.

   .. rubric:: Examples

   Consider the following pair of states:

   \[
       \rho = \frac{3}{4}|0\rangle \langle 0| +
              \frac{1}{4}|1 \rangle \langle 1|
               \quad \text{and} \quad
       \sigma = \frac{1}{8}|0 \rangle \langle 0| +
                \frac{7}{8}|1 \rangle \langle 1|.
   \]

   Calculating the fidelity between the states \(\rho\) and \(\sigma\) as \(F(\rho, \sigma) \approx
   0.774\). This can be observed in `|toqito⟩` as

   ```python exec="1" source="above"
   from toqito.states import basis
   from toqito.state_metrics import fidelity

   e_0, e_1 = basis(2, 0), basis(2, 1)
   rho = 3 / 4 * e_0 @ e_0.conj().T + 1 / 4 * e_1 @ e_1.conj().T
   sigma = 1/8 * e_0 @ e_0.conj().T + 7/8 * e_1 @ e_1.conj().T

   print(fidelity(rho, sigma))
   ```

   As the sub-fidelity is a lower bound on the fidelity, that is \(E(\rho, \sigma) \leq F(\rho, \sigma)\), we can
   use `|toqito⟩` to observe that \(E(\rho, \sigma) \approx 0.599\leq F(\rho, \sigma \approx 0.774\).

   ```python exec="1" source="above"
   from toqito.states import basis
   from toqito.state_metrics import sub_fidelity

   e_0, e_1 = basis(2, 0), basis(2, 1)
   rho = 3 / 4 * e_0 @ e_0.conj().T + 1 / 4 * e_1 @ e_1.conj().T
   sigma = 1/8 * e_0 @ e_0.conj().T + 7/8 * e_1 @ e_1.conj().T

   print(sub_fidelity(rho, sigma))
   ```

   :raises ValueError: If matrices are not of equal dimension.

   :param rho: Density operator.
   :param sigma: Density operator.

   :returns: The sub-fidelity between `rho` and `sigma`.