"""Calculates the probability of error of single state conclusive state exclusion."""
import numpy as np
import picos
from toqito.matrix_ops import calculate_vector_matrix_dimension, to_density_matrix
from toqito.matrix_props import has_same_dimension
[docs]
def state_exclusion(
vectors: list[np.ndarray],
probs: list[float] | None = None,
strategy: str = "min_error",
solver: str = "cvxopt",
primal_dual: str = "dual",
**kwargs,
) -> tuple[float, list[picos.HermitianVariable] | tuple[picos.HermitianVariable, picos.RealVariable]]:
r"""Compute probability of error of single state conclusive state exclusion.
The *quantum state exclusion* problem involves a collection of \(n\) quantum states
\[
\rho = \{ \rho_0, \ldots, \rho_n \},
\]
as well as a list of corresponding probabilities
\[
p = \{ p_0, \ldots, p_n \}.
\]
Alice chooses \(i\) with probability \(p_i\) and creates the state \(\rho_i\).
Bob wants to guess which state he was *not* given from the collection of states. State exclusion implies that
ability to discard at least one out of the "n" possible quantum states by applying a measurement.
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 exclusion.
\[
\begin{equation}
\begin{aligned}
\text{minimize:} \quad & \sum_{i=1}^n p_i \langle M_i, \rho_i \rangle \\
\text{subject to:} \quad & \sum_{i=1}^n M_i = \mathbb{I}_{\mathcal{X}}, \\
& M_0, \ldots, M_n \in \text{Pos}(\mathcal{X}).
\end{aligned}
\end{equation}
\]
\[
\begin{equation}
\begin{aligned}
\text{maximize:} \quad & \text{Tr}(Y) \\
\text{subject to:} \quad & Y \preceq p_1\rho_1, \\
& Y \preceq p_2\rho_2, \\
& \vdots \\
& Y \preceq p_n\rho_n, \\
& Y \in\text{Herm}(\mathcal{X}).
\end{aligned}
\end{equation}
\]
For `strategy = "unambiguous"`, Bob never provides an incorrect answer, although it is
possible that his answer is inconclusive. This function then yields the probability of an inconclusive outcome.
In that case, this function implements the following semidefinite program that provides the
optimal probability with which Bob can conduct unambiguous quantum state distinguishability.
\[
\begin{align*}
\text{minimize:} \quad & \text{Tr}\left(
\left(\sum_{i=1}^n p_i\rho_i\right)\left(\mathbb{I}-\sum_{i=1}^nM_i\right)
\right) \\
\text{subject to:} \quad & \sum_{i=1}^nM_i \preceq \mathbb{I},\\
& M_1, \ldots, M_n \succeq 0, \\
& \langle M_1, \rho_1 \rangle, \ldots, \langle M_n, \rho_n \rangle =0
\end{align*}
\]
\[
\begin{align*}
\text{maximize:} \quad & 1 - \text{Tr}(N) \\
\text{subject to:} \quad & a_1p_1\rho_1, \ldots, a_np_n\rho_n \succeq \sum_{i=1}^np_i\rho_i - N,\\
& N \succeq 0,\\
& a_1, \ldots, a_n \in\mathbb{R}
\end{align*}
\]
!!! Note
This function supports both pure states (vectors) and mixed states (density matrices).
It is known that it is always possible to perfectly exclude pure states that are linearly dependent.
Thus, calling this function on a set of states with this property will return 0.
The conclusive state exclusion SDP is written explicitly in [@Bandyopadhyay_2014_Conclusive]. The problem
of conclusive state exclusion was also thought about under a different guise in [@Pusey_2012_On].
Examples:
Consider the following two Bell states
\[
\begin{equation}
\begin{aligned}
u_0 &= \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right), \\
u_1 &= \frac{1}{\sqrt{2}} \left( |00 \rangle - |11 \rangle \right).
\end{aligned}
\end{equation}
\]
It is not possible to conclusively exclude either of the two states. We can see that the result of the
function in
`|toqito⟩` yields a value of \(0\) as the probability for this to occur.
```python exec="1" source="above"
import numpy as np
from toqito.states import bell
from toqito.state_opt import state_exclusion
vectors = [bell(0), bell(1)]
probs = [1/2, 1/2]
print(np.around(state_exclusion(vectors, probs)[0], decimals=2))
```
Unambiguous state exclusion for unbiased pure states.
```python exec="1" source="above"
import numpy as np
from toqito.state_opt import state_exclusion
states = [np.array([[1.], [0.]]), np.array([[1.],[1.]]) / np.sqrt(2)]
res, _ = state_exclusion(states, primal_dual="primal", strategy="unambiguous", abs_ipm_opt_tol=1e-7)
print(np.around(res, decimals=2))
```
State exclusion for mixed states.
```python exec="1" source="above"
import numpy as np
from toqito.state_opt import state_exclusion
# Two mixed 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]
res, _ = state_exclusion(states, primal_dual="dual")
print(np.around(res, decimals=2))
```
!!! Note
If you encounter a `ZeroDivisionError` or an `ArithmeticError` when using cvxopt as a solver (which is the
default), you might want to set the `abs_ipm_opt_tol` option to a lower value (the default being `1e-8`) or
to set the `cvxopt_kktsolver` option to `ldl`.
See https://gitlab.com/picos-api/picos/-/issues/341
Args:
vectors: A list of states provided as vectors (for pure states) or density matrices (for mixed states).
probs: Respective list of probabilities each state is selected. If no probabilities are provided, a uniform
probability distribution is assumed.
strategy: Whether to perform minimal error or unambiguous discrimination task. Possible values are "min_error"
and "unambiguous". Both strategies support pure and mixed states.
solver: Optimization option for `picos` solver. Default option is `solver_option="cvxopt"`.
primal_dual: Option for the optimization problem.
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.
"""
if not has_same_dimension(vectors):
raise ValueError("Vectors for state distinguishability must all have the same dimension.")
# Assumes a uniform probabilities distribution among the states if one is not explicitly provided.
n = len(vectors)
probs = [1 / n] * n if probs is None else probs
dim = calculate_vector_matrix_dimension(vectors[0])
if strategy == "min_error":
if primal_dual == "primal":
return _min_error_primal(vectors=vectors, dim=dim, probs=probs, solver=solver, **kwargs)
return _min_error_dual(vectors=vectors, dim=dim, probs=probs, solver=solver, **kwargs)
if primal_dual == "primal":
return _unambiguous_primal(vectors=vectors, dim=dim, probs=probs, solver=solver, **kwargs)
return _unambiguous_dual(vectors=vectors, dim=dim, probs=probs, solver=solver, **kwargs)
def _min_error_primal(
vectors: list[np.ndarray],
dim: int,
probs: list[float] | None = None,
solver: str = "cvxopt",
**kwargs,
) -> tuple[float, list[picos.HermitianVariable]]:
"""Find the primal problem for minimum-error quantum state exclusion SDP."""
n = len(vectors)
problem = picos.Problem()
measurements = [picos.HermitianVariable(f"M[{i}]", (dim, dim)) for i in range(n)]
problem.add_list_of_constraints([meas >> 0 for meas in measurements])
problem.add_constraint(picos.sum(measurements) == picos.I(dim))
dms = [to_density_matrix(vector) for vector in vectors]
problem.set_objective("min", np.real(picos.sum([(probs[i] * dms[i] | measurements[i]) for i in range(n)])))
solution = problem.solve(solver=solver, **kwargs)
return solution.value, measurements
def _min_error_dual(
vectors: list[np.ndarray],
dim: int,
probs: list[float] | None = None,
solver: str = "cvxopt",
**kwargs,
) -> tuple[float, list[picos.HermitianVariable]]:
"""Find the dual problem for minimum-error quantum state exclusion SDP."""
n = len(vectors)
problem = picos.Problem()
# Set up variables and constraints for SDP:
y_var = picos.HermitianVariable("Y", (dim, dim))
problem.add_list_of_constraints([y_var << probs[i] * to_density_matrix(vector) for i, vector in enumerate(vectors)])
# Objective function:
problem.set_objective("max", picos.trace(y_var))
solution = problem.solve(solver=solver, **kwargs)
measurements = [problem.get_constraint(k).dual for k in range(n)]
return solution.value, measurements
def _unambiguous_primal(
vectors: list[np.ndarray],
dim: int,
probs: list[float] | None = None,
solver: str = "cvxopt",
**kwargs,
) -> tuple[float, list[picos.HermitianVariable]]:
"""Solve the primal problem for unambiguous quantum state distinguishability SDP.
Implemented according to Equation (33) of [@Bandyopadhyay_2014_Conclusive].
"""
n = len(vectors)
problem = picos.Problem()
measurements = [picos.HermitianVariable(f"M[{i}]", (dim, dim)) for i in range(n)]
inconclusive_measurement = picos.I(dim) - picos.sum(measurements)
problem.add_list_of_constraints([meas >> 0 for meas in measurements])
problem.add_constraint(inconclusive_measurement >> 0)
unnormalized_dms = [p * to_density_matrix(vector) for (p, vector) in zip(probs, vectors)]
sums_of_unnormalized_dms = picos.sum(unnormalized_dms)
problem.add_list_of_constraints(m | rho == 0 for (m, rho) in zip(measurements, unnormalized_dms))
problem.set_objective("min", picos.trace(sums_of_unnormalized_dms * inconclusive_measurement))
solution = problem.solve(solver=solver, **kwargs)
return solution.value, measurements + [inconclusive_measurement]
def _unambiguous_dual(
vectors: list[np.ndarray],
dim: int,
probs: list[float] | None = None,
solver: str = "cvxopt",
**kwargs,
) -> tuple[float, tuple[picos.HermitianVariable, picos.RealVariable]]:
"""Solve the dual problem for unambiguous quantum state distinguishability SDP.
Implemented according to Equation (35) of [@Bandyopadhyay_2014_Conclusive].
"""
n = len(vectors)
problem = picos.Problem()
lagrangian_variable_big_n = picos.HermitianVariable("N", (dim, dim))
lagrangian_variables_a = picos.RealVariable("a", n)
problem.add_constraint(lagrangian_variable_big_n >> 0)
dms = [to_density_matrix(vector) for (p, vector) in zip(probs, vectors)]
unnormalized_dms = [proba * rho for (proba, rho) in zip(probs, dms)]
sum_of_unnormalized_dms = picos.sum(unnormalized_dms)
problem.add_list_of_constraints(
(lagrangian_variable_big_n + lagrangian_variables_a[i] * unnormalized_dms[i] >> sum_of_unnormalized_dms)
for i in range(n)
)
problem.set_objective("max", 1 - picos.trace(lagrangian_variable_big_n))
solution = problem.solve(solver=solver, primals=None, **kwargs)
return solution.value, (lagrangian_variable_big_n, lagrangian_variables_a)