SingleQubitPOVM¶

class SingleQubitPOVM(list_operators: list[Operator])[source]¶

Bases: MultiQubitPOVM

A convenience class to represent a single-qubit MultiQubitPOVM instance.

Below is a simple example showing how you define a symmetric and informationally-complete POVM (SIC-POVM):

>>> import cmath
>>> import numpy as np
>>> from povm_toolbox.quantum_info import SingleQubitPOVM
>>> vecs = np.sqrt(1.0 / 2.0) * np.array(
...     [
...         [1, 0],
...         [np.sqrt(1.0 / 3.0), np.sqrt(2.0 / 3.0)],
...         [np.sqrt(1.0 / 3.0), np.sqrt(2.0 / 3.0) * cmath.exp(2.0j * np.pi / 3)],
...         [np.sqrt(1.0 / 3.0), np.sqrt(2.0 / 3.0) * cmath.exp(4.0j * np.pi / 3)],
...     ]
... )
>>> sic_povm = SingleQubitPOVM.from_vectors(vecs)
>>> print(sic_povm)
SingleQubitPOVM<4> at 0x...

Initialize from explicit operators.

Parameters:: list_operators (list[Operator]) – list that contains the explicit frame operators. The length of the list is the number of operators of the frame.
Raises:: ValueError – if the frame operators do not have a correct shape. They should all be hermitian and of the same dimension.

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\()\).

operators¶: Return the list of frame operators.

pauli_operators¶

Convert the internal frame operators to Pauli form.

Warning

The conversion to Pauli form can be computationally intensive.

Returns:: The frame operators in Pauli form. Each frame operator is returned as a dictionary mapping Pauli labels to coefficients.
Raises:: ValueError – when the frame operators could not be converted to Pauli form (e.g. when they are not N-qubit operators).

Methods

Plot the Bloch vector of each effect of the POVM.

>>> import cmath
>>> import numpy as np
>>> from povm_toolbox.quantum_info import SingleQubitPOVM
>>> vecs = np.sqrt(1.0 / 2.0) * np.array(
...     [
...         [1, 0],
...         [np.sqrt(1.0 / 3.0), np.sqrt(2.0 / 3.0)],
...         [np.sqrt(1.0 / 3.0), np.sqrt(2.0 / 3.0) * cmath.exp(2.0j * np.pi / 3)],
...         [np.sqrt(1.0 / 3.0), np.sqrt(2.0 / 3.0) * cmath.exp(4.0j * np.pi / 3)],
...     ]
... )
>>> sic_povm = SingleQubitPOVM.from_vectors(vecs)
>>> sic_povm.draw_bloch()
<Figure size 500x500 with 1 Axes>

(Source code, png, hires.png, pdf)

../_images/povm_toolbox-quantum_info-SingleQubitPOVM-1.png

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) – Figure size in inches. Has no effect if passing ax.
font_size (float | None) – Size of font used for Bloch sphere labels.
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.

Return type:

Figure

get_bloch_vectors() → ndarray[source]¶

Compute the Bloch vector of each effect of the POVM.

For a rank-1 POVM, each effect \(M_k\) can be written as

\[M_k = \gamma_k |\psi_k \rangle \langle \psi_k | = \gamma_k \frac{1}{2} \left( \mathbb{I} + \vec{a}_k \cdot \vec{\sigma} \right)\]

where \(\vec{\sigma}\) is the usual Pauli vector and \(||\vec{a}_k||^2=1\). We then define the Bloch vector of a rank-1 effect as \(\vec{r}_k = \gamma_k \vec{a}_k\), which uniquely defines the rank-1 effect.

Example:

>>> import cmath
>>> import numpy as np
>>> from povm_toolbox.quantum_info import SingleQubitPOVM
>>> vecs = np.sqrt(1.0 / 2.0) * np.array(
...     [
...         [1, 0],
...         [np.sqrt(1.0 / 3.0), np.sqrt(2.0 / 3.0)],
...         [np.sqrt(1.0 / 3.0), np.sqrt(2.0 / 3.0) * cmath.exp(2.0j * np.pi / 3)],
...         [np.sqrt(1.0 / 3.0), np.sqrt(2.0 / 3.0) * cmath.exp(4.0j * np.pi / 3)],
...     ]
... )
>>> sic_povm = SingleQubitPOVM.from_vectors(vecs)
>>> bloch_vectors = sic_povm.get_bloch_vectors()
>>> print(bloch_vectors)  
[[ 0.          0.          0.5       ]
 [ 0.47140452  0.         -0.16666667]
 [-0.23570226  0.40824829 -0.16666667]
 [-0.23570226 -0.40824829 -0.16666667]]

Returns:: The Bloch vector of all POVM effects.
Raises:: ValueError – if any effect of this POVM has a rank greater than 1.
Return type:: ndarray

Inherited Methods

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 (int | set[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[int, float] | ndarray

classmethod from_vectors(frame_vectors: ndarray) → Self¶

Initialize a frame from non-normalized bloch vectors.

The non-normalized Bloch vectors are given by \(|\tilde{\psi}_k \rangle = \sqrt{\gamma_k} |\psi_k \rangle\). The resulting frame operators are \(F_k = \gamma_k |\psi_k \rangle \langle \psi_k |\) where \(\gamma_k\) is the trace of the \(k\)’th frame operator.

Parameters:: frame_vectors (ndarray) – list of vectors \(|\tilde{\psi_k} \rangle\). The length of the list corresponds to the number of operators of the frame. Each vector is of shape \((\mathrm{dim},)\) where \(\mathrm{dim}\) is the dimension of the Hilbert space on which the frame acts.
Returns:: The frame corresponding to the vectors.
Return type:: Self

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