Interfaces

The base interfaces for the quantum-informational frame, POVM, and Dual classes.

class BaseFrame

Abstract base class that contains all methods that any specific frame should implement.

A frame is a generalization of the notion of the basis of a vector space to sets that may not necessarily be linearly independent. Consider a Hilbert space of finite dimension $d$, then the set of Hermitian operators is an operator-valued vector space on the Hilbert space. Therefore, a set of Hermitian operators that spans the entire Hilbert space is said to be a frame.

If a set of operators does not span the entire Hilbert space, it can still be considered as a frame on the subspace it spans.

abstract property informationally_complete: bool

If the frame spans the entire Hilbert space.

abstract property dimension: int

The dimension of the Hilbert space on which the frame operators act.

property num_subsystems: int

The number of subsystems which the frame operators act on.

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

abstract property num_operators: int

The number of frame operators of the frame.

abstract analysis(hermitian_op: SparsePauliOp | Operator, frame_op_idx: LabelT | set[LabelT] | None = None) → float | dict[LabelT, 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 (LabelT | set[LabelT] | 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[LabelT, float] | ndarray

class BasePOVM

Abstract base class that contains all methods that any specific POVM should implement.

default_dual_class: type[BaseDual]

The default BaseDual associated with this POVM.

property num_outcomes: int

The number of outcomes of the POVM.

get_prob(rho: SparsePauliOp | DensityMatrix | Statevector, outcome_idx: LabelT | set[LabelT] | None = None) → float | dict[LabelT, float] | ndarray

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

class BaseDual

Abstract base class that contains all methods that any specific Dual should implement.

property num_outcomes: int

The number of outcomes of the Dual.

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

abstract 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

abstract classmethod build_dual_from_frame(frame: BaseFrame, alphas: tuple[Any] | None = None) → BaseDual

Construct a dual frame to another (primal) frame.

Parameters:

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

• alphas (tuple[Any] | 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:

BaseDual

class LabelT

Each operator in the frame is identified by a label.

This is the type of these labels. For instance, labels could be strings, integers, or it could be tuples of integers among other possibilities.

alias of TypeVar(‘LabelT’)