Quantum Tomography

TomographyBasis

class qinfer.tomography.TomographyBasis(data, dims, labels=None, superrep=None, name=None)

Bases: object

A basis of Hermitian operators used for representing tomographic objects (states and channels) as vectors of real elements. By assumption, a tomographic basis is taken to have an initial (0th) element proportional to the identity, and all other elements are taken to be traceless. For example, the Pauli matrices form a tomographic basis for qubits.

Instances of TomographyBasis convert between representations of tomographic objects as real vectors of model parameters and QuTiP Qobj instances. The latter is convienent for working with other libraries, and for reasoning about fidelities and other metrics, while model parameter representations are useful for defining prior distributions and tomographic models.

Parameters:

• data (np.ndarray) – Dense array of shape (dim ** 2, dim, dim) containing all elements of the new tomographic basis. data[alpha, i, j] is the (i, j)-th element of the alpha-th matrix of the new basis.
• dims (list) – Dimensions specification used in converting to QuTiP representations. The product of all elements of dims must equal the dimension of axes 1 and 2 of data. For instance, [2, 3] specifies that the basis is over the tensor product of a qubit and a qutrit space.
• labels (str or list of str) – LaTeX-formatted labels for each basis element. If a single str, a subscript is added to each basis element’s label.
• superrep (str) – Superoperator representation to pass to QuTiP when reconstructing states.

data = None

Dense matrix... TODO: document indices!

dims = None

Dimensions of each index, used when converting to QuTiP Qobj instances.

labels = None

Labels for each basis element.

dim

Dimension of the Hilbert space on which elements of this basis act.

Type:int

name

Name to use when converting this basis to a string.

Type:str

flat()

Returns a NumPy array that represents this operator basis in a flattened manner, such that basis.flat()[i, j] is the \(j\text{th}\) element of the flattened \(i\text{th}\) basis operator.

state_to_modelparams(state)

Converts a QuTiP-represented state into a model parameter vector.

Parameters:state (qutip.Qobj) – State to be converted. Return type:np.ndarray Returns:The representation of the given state in this basis, as a vector of real parameters.

modelparams_to_state(modelparams)

Converts one or more vectors of model parameters into QuTiP-represented states.

Parameters:modelparams (np.ndarray) – Array of shape (basis.dim ** 2, ) or (n_states, basis.dim ** 2) containing states represented as model parameter vectors in this basis. Return type:Qobj or list of Qobj instances. Returns:The given states represented as Qobj instances.

covariance_mtx_to_superop(mtx)

Converts a covariance matrix to the corresponding superoperator, represented as a QuTiP Qobj with type="super".

Built-in bases

qinfer.tomography.gell_mann_basis(dim)

Returns a TomographyBasis on dim dimensions using the generalized Gell-Mann matrices.

This implementation is based on a MATLAB-language implementation provided by Carlos Riofrío, Seth Merkel and Andrew Silberfarb. Used with permission.

Parameters:dim (int) – Dimension of the individual matrices making up the returned basis. Return type:TomographyBasis Returns:A basis of dim * dim Gell-Mann matrices.

qinfer.tomography.pauli_basis(nq=1)

Returns a TomographyBasis for the Pauli basis on nq qubits.

Parameters:nq (int) – Number of qubits on which the returned basis is defined.

qinfer.tomography.tensor_product_basis(*bases)

Returns a TomographyBasis formed by the tensor product of two or more factor bases. Each basis element is the tensor product of basis elements from the underlying factors.

DensityOperatorDistribution

class qinfer.tomography.DensityOperatorDistribution(basis)

Bases: qinfer.distributions.SingleSampleMixin, qinfer.distributions.Distribution

Distribution over density operators parameterized in a given basis.

Parameters:basis (int or TomographyBasis) – Basis to use in representing sampled density operators. If an int, assumes a default (Gell-Mann) basis of that dimension.

n_rvs

Number of random variables represented by this distribution.

Type:int

dim

Dimension of the Hilbert space on which sampled density operators act.

Type:int

basis

Basis used to represent sampled density operators as model parameter vectors.

Specific Distributions

See also

Probability Distributions

class qinfer.tomography.TensorProductDistribution(factors)

Bases: qinfer.tomography.distributions.DensityOperatorDistribution

This class is implemented using QuTiP (v3.1.0 or later), and thus will not work unless QuTiP is installed.

Parameters:factors (list of DensityOperatorDistribution instances) – Distributions representing each factor of the tensor product used to generate samples.

class qinfer.tomography.GinibreDistribution(basis, rank=None)

Bases: qinfer.tomography.distributions.DensityOperatorDistribution

Distribution over all trace-1 positive semidefinite operators of a given rank. Generalizes the Hilbert-Schmidt (full-rank) and Haar (rank-1) distributions.

Parameters:

• basis (TomographyBasis) – Basis to use in generating samples.
• rank (int) – Rank of each sampled state. If None, defaults to full-rank.

class qinfer.tomography.GinibreReditDistribution(basis, rank=None)

Bases: qinfer.tomography.distributions.DensityOperatorDistribution

Distribution over all real-valued trace-1 positive semidefinite operators of a given rank. Generalizes the Hilbert-Schmidt (full-rank) and Haar (rank-1) distributions. Useful for plotting.

Parameters:

• basis (TomographyBasis) – Basis to use in generating samples.
• rank (int) – Rank of each sampled state. If None, defaults to full-rank.

class qinfer.tomography.BCSZChoiDistribution(basis, rank=None, enforce_tp=True)

Bases: qinfer.tomography.distributions.DensityOperatorDistribution

Samples Choi states for completely-positive (CP) or CP and trace-preserving (CPTP) maps, as generated by the BCSZ prior [BCSZ09]. The sampled states are normalized as states (trace 1).

class qinfer.tomography.GADFLIDistribution(fiducial_distribution, mean)

Bases: qinfer.tomography.distributions.DensityOperatorDistribution

Samples operators from the generalized amplitude damping prior for liklihood-based inference [GCC16], given a fiducial distribution and the desired mean for the prior.

Parameters:

• fiducial_distribution (DensityOperatorDistribution) – Distribution from which samples are initially drawn before transformation under generalized amplitude damping.
• mean (qutip.Qobj) – State which will be the mean of the GAD-transformed samples.

Models

class qinfer.tomography.TomographyModel(basis, allow_subnormalized=False)

Bases: qinfer.abstract_model.FiniteOutcomeModel

Model for tomographically learning a quantum state using two-outcome positive-operator valued measures (POVMs).

Parameters:

• basis (TomographyBasis) – Basis used in representing states as model parameter vectors.
• allow_subnormalized (bool) – If False, states \(\rho\) are constrained during resampling such that \(\Tr(\rho) = 1\).

dim

Dimension of the Hilbert space on which density operators learned by this model act.

Type:int

basis

Basis used in converting between Qobj and model parameter vector representations of states.

Type:TomographyBasis

n_modelparams

modelparam_names

is_n_outcomes_constant

expparams_dtype

n_outcomes(expparams)

are_models_valid(modelparams)

canonicalize(modelparams)

Truncates negative eigenvalues and from each state represented by a tensor of model parameter vectors, and renormalizes as appropriate.

Parameters:modelparams (np.ndarray) – Array of shape (n_states, dim**2) containing model parameter representations of each of n_states different states. Returns:The same model parameter tensor with all states truncated to be positive operators. If allow_subnormalized is False, all states are also renormalized to trace one.

trunc_neg_eigs(particle)

Given a state represented as a model parameter vector, returns a model parameter vector representing the same state with any negative eigenvalues set to zero.

Parameters:particle (np.ndarray) – Vector of length (dim ** 2, ) representing a state. Returns:The same state with any negative eigenvalues set to zero.

renormalize(modelparams)

Renormalizes one or more states represented as model parameter vectors, such that each state has trace 1.

Parameters:modelparams (np.ndarray) – Array of shape (n_states, dim ** 2) representing one or more states as model parameter vectors. Returns:The same state, normalized to trace one.

likelihood(outcomes, modelparams, expparams)

class qinfer.tomography.DiffusiveTomographyModel(basis, allow_subnormalized=False)

Bases: qinfer.tomography.models.TomographyModel

n_modelparams

expparams_dtype

modelparam_names

are_models_valid(modelparams)

canonicalize(modelparams)

likelihood(outcomes, modelparams, expparams)

update_timestep(modelparams, expparams)

Heuristics

Abstract Heuristics

class qinfer.tomography.StateTomographyHeuristic(updater, basis=None, other_fields=None)

Bases: qinfer.expdesign.Heuristic

class qinfer.tomography.ProcessTomographyHeuristic(updater, basis, other_fields=None)

Bases: qinfer.expdesign.Heuristic

class qinfer.tomography.BestOfKMetaheuristic(updater, base_heuristic, k=3, other_fields=None)

Bases: qinfer.expdesign.Heuristic

Draws \(k\) different state or tomography measurements, then selects the one that has the largest expected value under the action of the covariance superoperator for the current posterior.

Specific Heuristics

class qinfer.tomography.RandomStabilizerStateHeuristic(updater, basis=None, other_fields=None)

Bases: qinfer.tomography.expdesign.StateTomographyHeuristic

Randomly chooses rank-1 projectors onto a stabilizer state.

class qinfer.tomography.RandomPauliHeuristic(updater, basis=None, other_fields=None)

Bases: qinfer.tomography.expdesign.StateTomographyHeuristic

Randomly chooses a Pauli measurement. Defined for qubits only.

class qinfer.tomography.ProductHeuristic(updater, basis, prep_heuristic_class, meas_heuristic_class, other_fields=None)

Bases: qinfer.tomography.expdesign.ProcessTomographyHeuristic

Takes two heuristic classes, one for preparations and one for measurements, then returns a sample from each. The preparation heuristic is assumed to return only trace-1 Hermitian operators.

Plotting Functions

qinfer.tomography.plot_cov_ellipse(cov, pos, nstd=2, ax=None, **kwargs)

Plots an nstd sigma error ellipse based on the specified covariance matrix (cov). Additional keyword arguments are passed on to the ellipse patch artist.

Parameters:

• cov – The 2x2 covariance matrix to base the ellipse on.
• pos – The location of the center of the ellipse. Expects a 2-element sequence of [x0, y0].
• nstd – The radius of the ellipse in numbers of standard deviations. Defaults to 2 standard deviations.
• ax – The axis that the ellipse will be plotted on. Defaults to the current axis.

Returns:

A matplotlib ellipse artist.

qinfer.tomography.plot_rebit_prior(prior, rebit_axes=[1, 2], n_samples=2000, true_state=None, true_size=250, force_mean=None, legend=True, mean_color_index=2)

Plots rebit states drawn from a given prior.

Parameters:

• prior (qinfer.tomography.DensityOperatorDistribution) – Distribution over rebit states to plot.
• rebit_axes (list) – List containing indices for the \(x\) and \(z\) axes.
• n_samples (int) – Number of samples to draw from the prior.
• true_state (np.ndarray) – State to be plotted as a “true” state for comparison.

qinfer.tomography.plot_rebit_posterior(updater, prior=None, true_state=None, n_std=3, rebit_axes=[1, 2], true_size=250, legend=True, level=0.95, region_est_method='cov')

Plots posterior distributions over rebits, including covariance ellipsoids

Parameters:

• updater (qinfer.smc.SMCUpdater) – Posterior distribution over rebits.
• qinfer.tomography.DensityOperatorDistribution – Prior distribution over rebit states.
• true_state (np.ndarray) – Model parameters for “true” state to plot as comparison.
• n_std (float) – Number of standard deviations out from the mean at which to draw the covariance ellipse. Only used if region_est_method is 'cov'.
• level (float) – Credibility level to use for computing region estimators from convex hulls.
• rebit_axes (list) – List containing indices for the \(x\) and \(z\) axes.
• region_est_method (str) – Method to use to draw region estimation. Must be one of None, 'cov' or 'hull'.