ProductDual

class ProductDual(frames: dict[tuple[int, ...], T])

Bases: ProductFrame[MultiQubitDual], BaseDual

Class to represent a set of product Dual operators.

A product Dual $D$ is made of local Duals $D1,D2,...$ acting on respective subsystems. Each global effect can be written as the tensor product of local effects, $\text{Missing open brace for subscript}$.

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 Dual.

num_subsystems

The number of subsystems which the frame operators act on.

For qubits, this is always ${\log }_2($dimension$)$.

shape

Give the number of operators per sub-system.

sub_systems

Give the number of operators per sub-system.

Methods

classmethod build_dual_from_frame(frame: BaseFrame, alphas: tuple[tuple[float, ...] | None, ...] | None = None) → ProductDual

Construct a dual frame to another (primal) frame.

Parameters:

• frame (BaseFrame) – The primal frame from which we will build the dual frame.

• alphas (tuple[tuple[float, ...] | None, ...] | None) – parameters of the frame super-operator used to build the dual frame. If None, the parameters are set as the traces of each operator in the primal frame.

Returns:

A dual frame to the supplied frame.

Return type:

ProductDual

is_dual_to(frame: BaseFrame) → bool

Check if self is a dual to another frame.

Parameters:

frame (BaseFrame) – the other frame to check duality against.

Returns:

Whether self is dual to frame.

Return type:

bool

Inherited Methods

analysis(hermitian_op: SparsePauliOp | Operator, frame_op_idx: tuple[int, ...] | set[tuple[int, ...]] | None = None) → float | dict[tuple[int, ...], float] | ndarray

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: Sequence[T]) → Self

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_omegas(observable: SparsePauliOp | Operator, outcome_idx: LabelT | set[LabelT] | None = None) → float | dict[LabelT, float] | ndarray

Return the decomposition weights of the provided observable.

Computes the ${\omega }_k$ in

$$\mathcal{O}=\sum _{k=1}^n{\omega }_k{M}_k$$

where $\mathcal{O}$ is the observable and $M_k$ are the effects of the POVM of which self is the dual. The closed form for computing ${\omega }_k$ is

$${\omega }_k=\text{Tr}\left[\mathcal{O}{D}_k\right]$$

where $D_k$ make of this dual frame (i.e. self).

Note

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

Parameters:

• observable (SparsePauliOp | Operator) – the observable for which to compute the decomposition weights.

• outcome_idx (LabelT | set[LabelT] | None) – label or set of labels indicating which decomposition weights are queried. If None, all weights are queried.

Returns:

Decomposition weight(s) associated to the effect(s) specified by outcome_idx. If a specific outcome was queried, a float is returned. If a specific set of outcomes was queried, a dictionary mapping outcome labels to weights is returned. If all outcomes were queried, an array with all weights is returned.

Return type:

float | dict[LabelT, float] | ndarray