toqito.rand.random_povm

Generates a random POVM.

Module Contents

toqito.rand.random_povm.random_povm(dim, num_inputs, num_outputs, seed=None)

Generate random positive operator valued measurements (POVMs) [@WikiPOVM].

Randomness model

For each input we draw (n_{text{out}}) matrices from the real Ginibre ensemble, i.e., each entry is sampled independently from the standard normal distribution using numpy’s default_rng. We interpret these matrices as Kraus operators (A_{x,a}) and normalize them so that the measurement is complete. Concretely, for each input (x) we form

[

G_x = sum_a A_{x,a}^dagger A_{x,a}, qquad B_{x,a} = G_x^{-1/2} A_{x,a}, qquad M_{x,a} = B_{x,a}^dagger B_{x,a}.

]

The matrices (M_{x,a}) constitute a POVM satisfying (sum_a M_{x,a} = mathbb{I}). This procedure induces the (Hilbert–Schmidt) normalized Wishart measure on the POVM effects. Supplying seed reproduces the same sample sequence.

Examples

We can generate a set of dim-by-dim POVMs consisting of a specific dimension along with a given number of measurement inputs and measurement outputs. As an example, we can construct a random set of (2)-by-(2) POVMs of dimension with (2) inputs and (2) outputs.

```python exec=”1” source=”above” session=”povm_example” import numpy as np from toqito.rand import random_povm

dim, num_inputs, num_outputs = 2, 2, 2

povms = random_povm(dim, num_inputs, num_outputs)

print(povms) ```

We can verify that this constitutes a valid set of POVM elements as checking that these operators all sum to the identity operator.

`python exec="1" source="above" session="povm_example" print(np.round(povms[:, :, 0, 0] + povms[:, :, 0, 1])) `

It is also possible to add a seed for reproducibility.

```python exec=”1” source=”above” session=”povm_example” import numpy as np from toqito.rand import random_povm

dim, num_inputs, num_outputs = 2, 2, 2

povms = random_povm(dim, num_inputs, num_outputs, seed=42)

print(povms) ```

We can once again verify that this constitutes a valid set of POVM elements as checking that these operators all sum to the identity operator.

`python exec="1" source="above" session="povm_example" print(np.round(povms[:, :, 0, 0] + povms[:, :, 0, 1])) `

param dim:

The dimensions of the measurements.

param num_inputs:

The number of inputs for the measurement.

param num_outputs:

The number of outputs for the measurement.

param seed:

A seed used to instantiate numpy’s random number generator (Ginibre sampling).

returns:

A set of dim-by-dim POVMs of shape (dim, dim, num_inputs, num_outputs).

Parameters:

• dim (int)

• num_inputs (int)

• num_outputs (int)

• seed (int | None)

Return type:

numpy.ndarray