ProductPOVM¶

class ProductPOVM(frames)¶

Bases: ProductFrame[MultiQubitPOVM], BasePOVM

Class to represent a set of product POVM operators.

A product POVM \(M\) is made of local POVMs \(M1, M2, ...\) acting on respective subsystems. Each global effect can be written as the tensor product of local effects, \(M_{k_1, k_2, ...} = M1_{k_1} \otimes M2_{k_2} \otimes \cdots\).

Below is an example of how to construct an instance of this class.

>>> from qiskit.quantum_info import Operator
>>> from povm_toolbox.quantum_info import SingleQubitPOVM, MultiQubitPOVM, ProductPOVM
>>> sqp = SingleQubitPOVM([Operator.from_label("0"), Operator.from_label("1")])
>>> mqp = MultiQubitPOVM(
...     [
...         Operator.from_label("00"),
...         Operator.from_label("01"),
...         Operator.from_label("10"),
...         Operator.from_label("11"),
...     ]
... )
>>> product = ProductPOVM({(0,): sqp, (1, 3): mqp, (2,): sqp})

Note

For most cases, you may find that ProductPOVM.from_list() works just fine and is easier to use.

Initialize from a mapping of local frames.

Parameters:

frames (dict[tuple[int, ...], T]) – a dictionary mapping from a tuple of subsystem indices to a local frame objects.

Raises:

ValueError – if any key in frames is not a tuple consisting of unique integers. In other words, every local frame must act on a distinct set of subsystem indices which do not overlap with each other.
ValueError – if any key in frames re-uses a previously used subsystem index. In other words, all local frames must act on mutually exclusive subsystem indices.
ValueError – if any key in frames does not specify the number of subsystem indices, which matches the number of systems acted upon by that local frame (MultiQubitFrame.num_subsystems()).

Inherited Attributes

dimension¶: The dimension of the Hilbert space on which the effects act.

informationally_complete¶: If the frame spans the entire Hilbert space.

num_operators¶: The number of effects of the frame.

num_outcomes¶: The number of outcomes of the POVM.

num_subsystems¶

The number of subsystems which the frame operators act on.

For qubits, this is always \(\log_2(\)dimension\()\).

shape¶: Give the shape of the product frame.

sub_systems¶: Give the number of operators per sub-system.

Methods

draw_bloch(*, title='', figure=None, axes=None, figsize=None, font_size=None, colorbar=False)¶

Plot a Bloch sphere for each single-qubit POVM forming the product POVM.

Parameters:

title (str) – a string that represents the plot title.
figure (Figure | None) – User supplied Matplotlib Figure instance for plotting Bloch sphere.
axes (Axes | list[Axes] | None) – User supplied Matplotlib axes to render the bloch sphere.
figsize (tuple[float, float] | None) – size of each individual Bloch sphere figure, in inches.
font_size (float | None) – Font size for the Bloch ball figures.
colorbar (bool) – If True, normalize the vectors on the Bloch sphere and add a colormap to keep track of the norm of the vectors. It can help to visualize the vector if they have a small norm.

Returns:

The resulting figure.

Raises:

NotImplementedError – if this product POVM contains a MultiQubitPOVM acting on more than a single qubit.

Return type:

Figure

Inherited Methods

analysis(hermitian_op, frame_op_idx=None)¶

Return the frame coefficients of hermitian_op.

This method implements the analysis operator \(A\) of the frame \(\{F_k\}_k\):

\[A: \mathcal{O} \mapsto \{ \mathrm{Tr}\left[F_k \mathcal{O} \right] \}_k,\]

where \(c_k = \mathrm{Tr}\left[F_k \mathcal{O} \right]\) are called the frame coefficients of the Hermitian operator \(\mathcal{O}\).

Parameters:

hermitian_op (SparsePauliOp | Operator) – a hermitian operator whose frame coefficients to compute.
frame_op_idx (tuple[int, ...] | set[tuple[int, ...]] | None) – label or set of labels indicating which coefficients are queried. If None, all coefficients are queried.

Returns:

Frame coefficients, specified by frame_op_idx, of the Hermitian operator hermitian_op. If a specific coefficient was queried, a float is returned. If a specific set of coefficients was queried, a dictionary mapping labels to coefficients is returned. If all coefficients were queried, an array with all coefficients is returned.

Raises:

TypeError – when the provided single or sequence of labels frame_op_idx does not have a valid type.
ValueError – when the dimension of the provided hermitian_op does not match the dimension of the frame operators.

Return type:

float | dict[tuple[int, …], float] | ndarray

classmethod from_list(frames)¶

Construct a ProductFrame from a list of MultiQubitFrame objects.

This is a convenience method to simplify the construction of a ProductFrame for the cases in which the local frame objects act on a sequential order of subsystems. In other words, this method converts the sequence of frames to a dictionary of frames in accordance with the input to ProductFrame.__init__() by using the positions along the sequence as subsystem indices.

Below are some examples:

>>> from qiskit.quantum_info import Operator
>>> from povm_toolbox.quantum_info import SingleQubitPOVM, MultiQubitPOVM, ProductPOVM

>>> sqp = SingleQubitPOVM([Operator.from_label("0"), Operator.from_label("1")])
>>> product = ProductPOVM.from_list([sqp, sqp])
>>> # is equivalent to
>>> product = ProductPOVM({(0,): sqp, (1,): sqp})

>>> mqp = MultiQubitPOVM(
...     [
...         Operator.from_label("00"),
...         Operator.from_label("01"),
...         Operator.from_label("10"),
...         Operator.from_label("11"),
...     ]
... )
>>> product = ProductPOVM.from_list([mqp, mqp])
>>> # is equivalent to
>>> product = ProductPOVM({(0, 1): mqp, (2, 3): mqp})

>>> product = ProductPOVM.from_list([sqp, sqp, mqp])
>>> # is equivalent to
>>> product = ProductPOVM({(0,): sqp, (1,): sqp, (2, 3): mqp})

>>> product = ProductPOVM.from_list([sqp, mqp, sqp])
>>> # is equivalent to
>>> product = ProductPOVM({(0,): sqp, (1, 2): mqp, (3,): sqp})

Parameters:: frames (Sequence[T]) – a sequence of MultiQubitFrame objects.
Returns:: A new ProductFrame instance.
Return type:: Self

get_operator(frame_op_idx)¶

Return a product frame operator in a product form.

Parameters:: frame_op_idx (tuple[int, ...]) – the label specifying the frame operator to get. The frame operator is labeled by a tuple of integers (one index per local frame).
Returns:: The product frame operator specified by frame_op_idx. The operator is returned in a product form. More specifically, is it a dictionary mapping the subsystems to the corresponding local frame operators forming the product frame operator.
Return type:: dict[tuple[int, …], Operator]

get_prob(rho, outcome_idx=None)¶

Return the outcome probabilities given a state, \(\rho\).

Each outcome \(k\) is associated with an effect \(M_k\) of the POVM. The probability of obtaining the outcome \(k\) when measuring a state rho is given by \(p_k = \text{Tr}\left[M_k \rho\right]\).

Note

In the frame theory formalism, the mapping \(A: \rho \mapsto \{\text{Tr}\left[M_k \rho\right]\}_k\) is referred to as the analysis operator, which is implemented by the analysis() method.

Parameters:

rho (SparsePauliOp | DensityMatrix | Statevector) – the state for which to compute the outcome probabilities.
outcome_idx (LabelT | set[LabelT] | None) – label or set of labels indicating which outcome probabilities are queried. If None, all outcome probabilities are queried.

Returns:

Probabilities of obtaining the outcome(s) specified by outcome_idx over the state rho. If a specific outcome was queried, a float is returned. If a specific set of outcomes was queried, a dictionary mapping outcomes to probabilities is returned. If all outcomes were queried, an array with all probabilities is returned.

Return type:

float | dict[LabelT, float] | ndarray