:py:mod:`measurement_ops.measure`
=================================

.. py:module:: measurement_ops.measure

.. autoapi-nested-parse::

   Determine probability of obtaining measurement outcome.



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


Functions
~~~~~~~~~

.. autoapisummary::

   measurement_ops.measure.measure



.. py:function:: measure(measurement, state)

   Determine probability of obtaining a measurement outcome applied to state.

   A *measurement* is defined by a function

   .. math::
       \mu : \Sigma \rightarrow \text{Pos}(\mathcal{X}),

   for some choice of alphabet :math:`\Sigma` and a complex Euclidean space :math:`\mathcal{X}`
   that satisfies

   .. math::
       \sum_{a \in \Sigma} \mu(a) = \mathbb{I}_{\mathcal{X}}.

   Further information can be found here :cite:`WikiQuantMeas`.

   .. rubric:: Examples

   Consider the following state:

   .. math::
       u = \frac{1}{\sqrt{3}} e_0 + \sqrt{\frac{2}{3}} e_1

   where we define :math:`u u^* = \rho \in \text{D}(\mathcal{X})`.

   Define measurement operators

   .. math::
       P_0 = e_0 e_0^* \quad \text{and} \quad P_1 = e_1 e_1^*.

   >>> from toqito.states import basis
   >>> from toqito.measurement_ops import measure
   >>> import numpy as np
   >>> e_0, e_1 = basis(2, 0), basis(2, 1)
   >>>
   >>> u = 1/np.sqrt(3) * e_0 + np.sqrt(2/3) * e_1
   >>> rho = u * u.conj().T
   >>>
   >>> proj_0 = e_0 * e_0.conj().T
   >>> proj_1 = e_1 * e_1.conj().T

   Then the probability of obtaining outcome :math:`0` is given by

   .. math::
       \langle P_0, \rho \rangle = \frac{1}{3}.

   >>> measure(proj_0, rho)
   0.3333333333333334

   Similarly, the probability of obtaining outcome :math:`1` is given by

   .. math::
       \langle P_1, \rho \rangle = \frac{2}{3}.

   >>> measure(proj_1, rho)
   0.6666666666666666

   .. rubric:: References

   .. bibliography::
       :filter: docname in docnames

   :param measurement: The measurement to apply.
   :param state: The state to apply the measurement to.
   :return: Returns the probability of obtaining a given outcome after applying
            the variable :code:`measurement` to the variable :code:`state`.