toqito.state_opt.state_distinguishability¶

Calculates the probability of optimally distinguishing quantum states.

Module Contents¶

toqito.state_opt.state_distinguishability.state_distinguishability(vectors, probs=None, strategy='min_error', solver='cvxopt', primal_dual='dual', **kwargs)[source]¶

Compute probability of state distinguishability [@Eldar_2003_SDPApproach].

The “quantum state distinguishability” problem involves a collection of (n) quantum states

[: rho = { rho_1, ldots, rho_n },

]

as well as a list of corresponding probabilities

[: p = { p_1, ldots, p_n }.

]

Alice chooses (i) with probability (p_i) and creates the state (rho_i). Bob wants to guess which state he was given from the collection of states.

For strategy = “min_error”, this is the default method that yields the minimal probability of error for Bob.

In that case, this function implements the following semidefinite program that provides the optimal probability with which Bob can conduct quantum state distinguishability.

[

begin{align*}: text{maximize:} quad & sum_{i=0}^n p_i langle M_i, rho_i rangle \ text{subject to:} quad & M_0 + ldots + M_n = mathbb{I},\

& M_0, ldots, M_n geq 0.

end{align*}

]

For strategy = “unambiguous”, Bob never provides an incorrect answer, although it is possible that his answer is inconclusive.

In that case, this function implements the following semidefinite program that provides the optimal probability with which Bob can conduct unambiguous quantum state distinguishability.

[

begin{align*}: text{maximize:} quad & mathbf{p} cdot mathbf{q} \ text{subject to:} quad & Gamma - Q geq 0,\

& mathbf{q} geq 0

end{align*}

]

[

begin{align*}: text{minimize:} quad & text{Tr}(Gamma Z) \ text{subject to:} quad & z_i + p_i + text{Tr}left(F_iZright)=0,\

& Z, z geq 0

end{align*}

]

where (mathbf{p}) is the vector whose (i)-th coordinate contains the probability that the state is prepared in state (left|psi_irightrangle), (Gamma) is the Gram matrix of (left|psi_1rightrangle,cdots,left|psi_nrightrangle) and (F_i) is (-|i\rangle\langle i|).

!!! Note: For unambiguous discrimination, this function supports both pure states (vectors) and mixed states (density matrices). For pure states, the states should be linearly independent. For mixed states, the Gram matrix is computed as Tr(ρᵢ ρⱼ). If the states cannot be unambiguously distinguished, the optimal probability will be low or zero.

Examples

Minimal-error state distinguishability for the Bell states (which are perfectly distinguishable).

```python exec=”1” source=”above” import numpy as np from toqito.states import bell from toqito.state_opt import state_distinguishability

states = [bell(0), bell(1), bell(2), bell(3)] probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]

res, _ = state_distinguishability(vectors=states, probs=probs, primal_dual=”dual”)

print(np.around(res, decimals=2)) ```

Note that if we are just interested in obtaining the optimal value, it is computationally less intensive to compute the dual problem over the primal problem. However, the primal problem does allow us to extract the explicit measurement operators which may be of interest to us.

```python exec=”1” source=”above” import numpy as np from toqito.states import bell from toqito.state_opt import state_distinguishability

states = [bell(0), bell(1), bell(2), bell(3)] probs = [1 / 4, 1 / 4, 1 / 4, 1 / 4]

res, measurements = state_distinguishability(vectors=states, probs=probs, primal_dual=”primal”)

print(np.around(measurements[0], decimals=5)) ```

Unambiguous state distinguishability for unbiased pure states.

```python exec=”1” source=”above” import numpy as np from toqito.state_opt import state_distinguishability

states = [np.array([[1.], [0.]]), np.array([[1.],[1.]]) / np.sqrt(2)] probs = [1 / 2, 1 / 2]

res, _ = state_distinguishability(vectors=states, probs=probs, primal_dual=”primal”, strategy=”unambiguous”)

print(np.around(res, decimals=2)) ```

Unambiguous state distinguishability for mixed states.

```python exec=”1” source=”above” import numpy as np from toqito.state_opt import state_distinguishability

# Two mixed states (Werner-like states) rho1 = 0.7 * np.array([[1., 0.], [0., 0.]]) + 0.3 * np.eye(2) / 2 rho2 = 0.7 * np.array([[0., 0.], [0., 1.]]) + 0.3 * np.eye(2) / 2 states = [rho1, rho2] probs = [1 / 2, 1 / 2]

res, _ = state_distinguishability(vectors=states, probs=probs, primal_dual=”primal”, strategy=”unambiguous”)

print(np.around(res, decimals=2)) ```

Parameters:

vectors (list[numpy.ndarray]) – A list of states provided as vectors (for pure states) or density matrices (for mixed states).
probs (list[float] | None) – Respective list of probabilities each state is selected. If no probabilities are provided, a uniform
assumed. (probability distribution is)
strategy (str) – Whether to perform unambiguous or minimal error discrimination task. Possible values are “min_error”
states. (and "unambiguous". Default option is strategy="min_error". Both strategies support pure and mixed)
solver (str) – Optimization option for picos solver. Default option is solver=”cvxopt”.
primal_dual (str) – Option for the optimization problem. Default option is “dual”.
kwargs – Additional arguments to pass to picos’ solve method.

Returns:

The optimal probability with which Bob can guess the state he was not given from states along with the optimal set of measurements.

Return type:

tuple[float, list[picos.HermitianVariable] | list[numpy.ndarray] | tuple[picos.SymmetricVariable]]