toqito.state_opt.state_exclusion¶
Calculates the probability of error of single state conclusive state exclusion.
Module Contents¶
- toqito.state_opt.state_exclusion.state_exclusion(vectors, probs=None, strategy='min_error', solver='cvxopt', primal_dual='dual', **kwargs)[source]¶
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_1rho_1, \
& Y preceq p_2rho_2, \ & vdots \ & Y preceq p_nrho_n, \ & Y intext{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_irho_iright)left(mathbb{I}-sum_{i=1}^nM_iright) 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_1rho_1, ldots, a_np_nrho_n succeq sum_{i=1}^np_irho_i - N,\
& N succeq 0,\ & a_1, ldots, a_n inmathbb{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.
- 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 minimal error or unambiguous discrimination task. Possible values are “min_error”
states. (and "unambiguous". Both strategies support pure and mixed)
solver (str) – Optimization option for picos solver. Default option is solver_option=”cvxopt”.
primal_dual (str) – 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.
- Return type:
tuple[float, list[picos.HermitianVariable] | tuple[picos.HermitianVariable, picos.RealVariable]]