Expectation values
==================

See examples at :ref:`examples/mps/measurement:Expectation values`.

Overlaps and MPO expectation values
-----------------------------------

:ref:`Measurement<theory/mps/basics:Measurements>` of the overlap between two MPSs is calculated
by contracting a network formed by MPS and conjugate of another or the same MPS. Works also for two MPOs.

.. autofunction:: yastn.tn.mps.measure_overlap

While the above allows calculating the norm of MPS or MPO, a dedicated method extracts the norm through canonization.
This can be more precise for norms close to zero.

.. automethod:: yastn.tn.mps.MpsMpoOBC.norm

:ref:`Measurement<theory/mps/basics:Measurements>` of MPO's expectation value
is calculated by contracting a network formed by MPS, MPO, and a conjugate
of the same or different MPS. Works also for three MPOs.

.. autofunction:: yastn.tn.mps.measure_mpo

Shorthand notation
------------------

The above function can be also executed by

.. autofunction:: yastn.tn.mps.vdot

One- and two-point expectation values
-------------------------------------

.. autofunction:: yastn.tn.mps.measure_1site
.. autofunction:: yastn.tn.mps.measure_2site
.. autofunction:: yastn.tn.mps.measure_nsite

Schmidt values and entropy profile
----------------------------------

The Schmidt values are computed by performing bipartition of the MPS/MPO across
each of the bonds. This amounts to SVD decomposition with respect to a bond,
where all sites to the left are in :ref:`left-canonical form<theory/mps/basics:canonical form>` and all sites to the right
are in :ref:`right-canonical form<theory/mps/basics:canonical form>`.

.. automethod:: yastn.tn.mps.MpsMpoOBC.get_Schmidt_values

The SVD is used to calculate the entropy at each bond.
For :math:`\alpha \neq 1`, this gives the *Renyi entropy*,
:math:`S_\alpha(\rho)=\frac{1}{1-\alpha}\log_2 Tr(\rho^\alpha)`,
where :math:`\rho` indicates a reduced density matrix of the system with all sites to the right of the cut traced out.
For :math:`\alpha=1` we get *von Neumann entropy*, :math:`S_1(\rho)=-Tr(\rho \log_2 \rho)`.
The entropy is counted in bits, i.e., using base-2 logarithm.

.. automethod:: yastn.tn.mps.MpsMpoOBC.get_entropy

Sampling
--------

.. automethod:: yastn.tn.mps.sample