Add `StateMeasurement.process_density_matrix` method

[Alex-Preciado]

Important Note

⚠️ This issue is part of an internal assignment and not meant for external contributors.

Context

Currently, measurements in PennyLane must define how to calculate the resulting number from "terminal" information, such as state, samples, or counts. For example, we have already implemented and required a StateMeasurement.process_state(state, wire_order) method. See the module documentation on measurements for more detailed information here.

But this method is specific to statevector-based simulators. To expand this framework to density-matrix based simulators like default.mixed, we need to add an additional method.

Implementation Details

This task is to add a new method to StateMeasurement:

class StateMeasurement(MeasurementProcess):

    def process_density_matrix(self, density_matrix, wire_order: Wires):
        raise NotImplementedError

And implement it for ProbabilityMP (bonus points for an implementation for StateMP, DensityMatrixMP)

Relevant functions include:

• qml.math.reduce_dm
• qml.math.reduce_statevector

Additional Requirements

A completed PR would:

• Include a docstring for the new StateMeasurement.process_density_matrix method
• Include a changelog entry at doc/releases/changelog-dev.md. Make sure to include your name in the list of contributors at the bottom!
• Add tests to tests/measurements/test_measurements.py, tests/measurements/test_state.py, and tests/measurements/test_probs.py
• Pass all existing tests including black and pylint checks