toqito.measurement_ops.measure

Apply measurement to a quantum state.

Module Contents

toqito.measurement_ops.measure.measure(state, measurement, tol=1e-10, state_update=False)[source]

Apply measurement to a quantum state.

The measurement can be provided as a single operator (POVM element or Kraus operator) or as a list of operators (assumed to be Kraus operators) describing a complete quantum measurement.

When a single operator is provided:

  • Returns the measurement outcome probability if state_update is False.

  • Returns a tuple (probability, post_state) if state_update is True.

When a list of operators is provided, the function verifies that they satisfy the completeness relation when state_update is True.

[

sum_i K_i^dagger K_i = mathbb{I},

]

when state_update is True. Then, for each operator (K_i), the outcome probability is computed as

[

p_i = mathrm{Tr}Bigl(K_i^dagger K_i, rhoBigr),

]

and, if (p_i > tol), the post‐measurement state is updated via

[

u = frac{1}{sqrt{3}} e_0 + sqrt{frac{2}{3}} e_1

]

where we define (u u^* = rho in text{D}(mathcal{X})).

Define measurement operators

[

P_0 = e_0 e_0^* quad text{and} quad P_1 = e_1 e_1^*.

]

```python exec=”1” source=”above” import numpy as np from toqito.states import basis from toqito.measurement_ops import measure

e_0, e_1 = basis(2, 0), basis(2, 1)

u = 1/np.sqrt(3) * e_0 + np.sqrt(2/3) * e_1 rho = u @ u.conj().T

proj_0 = e_0 @ e_0.conj().T proj_1 = e_1 @ e_1.conj().T print(measure(proj_0, rho)) ```

Then the probability of obtaining outcome (0) is given by

[

langle P_0, rho rangle = frac{1}{3}.

]

Similarly, the probability of obtaining outcome (1) is given by

[

langle P_1, rho rangle = frac{2}{3}.

]

```python exec=”1” source=”above” import numpy as np from toqito.measurement_ops.measure import measure

rho = np.array([[0.5, 0.5], [0.5, 0.5]]) K0 = np.array([[1, 0], [0, 0]]) K1 = np.array([[0, 0], [0, 1]])

# Returns list of probabilities. print(measure(rho, [K0, K1]))

# Returns list of (probability, post_state) tuples. print(measure(rho, [K0, K1], state_update=True)) ```

Raises:

ValueError – If a list of operators does not satisfy the completeness relation.

Parameters:

  • state (numpy.ndarray) – Quantum state as a density matrix shape (d, d) where d is the dimension of the Hilbert space.

  • measurement (numpy.ndarray | list[numpy.ndarray] | tuple[numpy.ndarray, Ellipsis]) – Either a single measurement operator (an np.ndarray) or a list/tuple of operators. When providing a

  • list

  • relation. (they are assumed to be Kraus operators satisfying the completeness)

  • tol (float) – Tolerance for numerical precision (default is 1e-10).

  • state_update (bool) – If True, also return the post-measurement state(s); otherwise, only the probability or

  • returned. (probabilities are)

Returns:

If a single operator is provided, returns a float (probability) or a tuple (probability, post_state) if state_update is True. If a list is provided, returns a list of probabilities or a list of tuples if state_update is True.

Return type:

float | tuple[float, numpy.ndarray] | list[float | tuple[float, numpy.ndarray]]