doc/source/guide/tomography.rst
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.. _tomography_guide:
.. currentmodule:: qinfer.tomography
Quantum Tomography
==================
Introduction
------------
Tomography is the most common quantum statistical problem being the subject
of both theoretical and practical studies. The :mod:`qinfer.tomography` module
has rich support for many of the common models of tomography including
standard distributions and heuristics, and also provides convenient plotting
tools.
If the quantum state of a system is :math:`\rho` and a measurement is obtained, then the probability of obtaining the outcome associated with effect :math:`E` is :math:`\operatorname{Pr}(E|\rho) = \operatorname{Tr}(\rho E)`, the Born rule. The tomography problem is the inverse problem and is succinctly stated as follows. Given an unknown state :math:`\rho`, and a set of count statistics :math:`\{n_k\}` from measurements, the corresponding operators of which are :math:`\{E_k\}`, determine :math:`\rho`.
In the context of Bayesian statistics, priors are distributions of quantum states and models define the Hilbert space dimension and the Born rule as a likelihood function (and perhaps additional complications associated with drift parameters and measurement errors).
The tomography module implements these details and has built-in support for common specifications.
The tomography module was developed concurrently with new results on Bayesian
priors in [GCC16]_. Please see the paper for more detailed discussions of these
more advanced topics.
QuTiP
~~~~~
Note that the Tomography module requires `QuTiP <http://qutip.org/>`_ which must be installed separately. Rather than reimplementing common operations on quantum states, we make use of QuTiP `Quantum objects <http://qutip.org/docs/3.1.0/guide/guide-basics.html>`_. For many simple use cases the QuTiP dependency is not exposed, but familiarity with `Quantum objects <http://qutip.org/docs/3.1.0/guide/guide-basics.html>`_ would be required to implement new distributions, models or heuristics.
Bases
-----
Bases define the map from abstract quantum theory to concrete representations. Once a basis is chosen, the tomography problem becomes a special case of a generic parameter estimation problem.
The tomography module is used in the same way as other but requires the specification of a basis via the :class:`TomographyBasis` class. QInfer comes with the following :ref:`tomography_bases`: the `Gell-Mann basis <https://en.wikipedia.org/wiki/Gell-Mann_matrices>`_, the `Pauli basis <https://en.wikipedia.org/wiki/Pauli_matrices>`_ and function to combine bases with the tensor product. Thus, the first step in using the tomography module is to define a basis. For example, here we define the 1-qubit Pauli basis:
>>> from qinfer.tomography import pauli_basis
>>> basis = pauli_basis(1)
>>> print(basis) #doctest: +ELLIPSIS
<TomographyBasis pauli_basis dims=[2] at ...>
User defined bases must be orthogonal Hermitian operators and have a 0'th component of :math:`I/\sqrt{d}`, where :math:`d` is the dimension of the quantum system and :math:`I` is the identity operator. This implies the remaining operators are traceless.
Built-in Distributions
----------------------
QInfer comes with several built-in distributions listed in :ref:`tomography_distributions`. Each of these is a subclass of :class:`DensityOperatorDistribution`. Distributions of quantum channels can also be subclassed as such with appeal to the Choi-Jamiolkowski isomorphism.
For density matrices, the :class:`GinibreDistribution` defines a prior over mixed quantum which allows for support also for states of fixed rank [OSZ10]_. For example, we can draw a sample from the this prior as follows:
>>> from qinfer.tomography import GinibreDistribution
>>> prior = GinibreDistribution(basis)
>>> print(prior.sample()) #doctest: +SKIP
[[ 0.70710678 -0.17816233 0.45195168 -0.08341437]]
Recall this is this representation of of a qubit in the Pauli basis defined above.
Quantum states are in general high dimensional objects which makes visualizing distributions of them challenging. The only 2-dimensional example is that of a rebit, which is usually defined as a qubit in the Pauli (or Bloch) representation with one of the Pauli expectations constrained to zero (usually :math:`\operatorname{Tr}(\rho \sigma_y)=0`).
Here we create a distribution of rebits accord to the Ginibre ensemble and use :func:`plot_rebit_prior` to depict this distribution through (by default) 2000 random samples. While discussing models below, we will see how to depict the particles of an :class:`~qinfer.SMCUpdater` directly.
.. plot::
basis = tomography.bases.pauli_basis(1)
prior = tomography.distributions.GinibreDistribution(basis)
tomography.plot_rebit_prior(prior, rebit_axes=[1, 3])
plt.show()
Using :class:`TomographyModel`
------------------------------
The core of the tomography module is the :class:`TomographyModel`. The key assumption in the current version is that of two-outcome measurements. This has the convenience of allowing experiments to be specified by a single vectorized positive operator:
>>> from qinfer.tomography import TomographyModel
>>> model = TomographyModel(basis)
>>> print(model.expparams_dtype) #doctest: +SKIP
[('meas', <type 'float'>, 4)]
Suppose we measure :math:`\sigma_z` on a random state. The measurement effects are :math:`\frac12 (I\pm \sigma_z)`. Since they sum to identity, we need only specify one of them. We can use :class:`TomographyModel` to calculate the Born rule probability of obtaining one of these outcomes as follows:
>>> expparams = np.zeros((1,), dtype=model.expparams_dtype)
>>> expparams['meas'][0, :] = basis.state_to_modelparams(np.sqrt(2)*(basis[0]+basis[1])/2)
>>> print(model.likelihood(0, prior.sample(), expparams)) # doctest: +SKIP
[[[ 0.62219803]]]
Built-in Heuristics
-------------------
In addition to analyzing given data sets, the tomography module is well suited for testing measurement strategies against standard heuristics. These built-in heuristics are listed at :ref:`tomography_heuristics`. For qubits, the most commonly used heuristic is the random sampling of Pauli basis measurements, which is implemented by :class:`RandomPauliHeuristic`.
>>> from qinfer.tomography import RandomPauliHeuristic
>>> from qinfer import SMCUpdater
>>> updater = SMCUpdater(model, 100, prior)
>>> heuristic = RandomPauliHeuristic(updater)
>>> print(model.simulate_experiment(prior.sample(), heuristic())) # doctest: +SKIP
0