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.
\[
\begin{equation}
\begin{aligned}
\text{minimize:} \quad & \sum_{i=1}^n p_i \langle M_i, \rho_i \rangle \\
\text{subject to:} \quad & \sum_{i=1}^n M_i = \mathbb{I}_{\mathcal{X}}, \\
& M_0, \ldots, M_n \in \text{Pos}(\mathcal{X}).
\end{aligned}
\end{equation}
\]
\[
\begin{equation}
\begin{aligned}
\text{maximize:} \quad & \text{Tr}(Y) \\
\text{subject to:} \quad & Y \preceq p_1\rho_1, \\
& Y \preceq p_2\rho_2, \\
& \vdots \\
& Y \preceq p_n\rho_n, \\
& Y \in\text{Herm}(\mathcal{X}).
\end{aligned}
\end{equation}
\]
If measurement = "ppt", the measurement operators are also constrained to be positive under partial transpose on
the requested subsystem(s). In the primal minimum-error problem, this adds the constraints
\[
\Gamma_j(M_i) \succeq 0
\]
for every measurement operator \(M_i\) and for the subsystem(s) \(j\) listed in subsystems, where \(\Gamma_j\)
is the partial transpose map with subsystem dimensions dimensions. The corresponding dual uses the dual PPT cone,
\(\text{Pos} + \Gamma_j(\text{Pos})\), so the constraints become
\[
p_i\rho_i - Y \in \text{Pos} + \Gamma_j(\text{Pos}).
\]
If measurement = "locc", the measurement is restricted to one-way local operations and classical communication on
a bipartite system with subsystem dimensions dimensions = [dim_A, dim_B]. Alice measures her subsystem with a POVM
\(\{A_a\}\), sends the outcome to Bob, who applies a conditional POVM \(\{B^a_k\}\); the induced global operator for
guessing \(k\) is \(M_k = \sum_a A_a \otimes B^a_k\). Minimizing the error over \(\{A_a\}\) and \(\{B^a_k\}\) is
bilinear, so it is solved by a see-saw (alternating semidefinite programs), restarted from reps random POVMs with
the smallest value returned. The result is an upper bound on the optimal one-way LOCC exclusion error (and at least
the global error). Only strategy = "min_error" is supported for this measurement.
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).
Linear dependence does not imply perfect exclusion: the identical pair \(|0\rangle, |0\rangle\)
is linearly dependent yet cannot be excluded (Bob can never rule out \(|0\rangle\)). A
sufficient condition for perfect exclusion (a value of 0) is that the states span a strict
subspace of the full space, since then a measurement along a complementary direction excludes
with certainty.
The conclusive state exclusion SDP is written explicitly in 1. The problem
of conclusive state exclusion was also thought about under a different guise in 2.
Parameters:
-
vectors
(list[ndarray])
–
A list of states provided as vectors (for pure states) or density matrices (for mixed states).
-
probs
(list[float] | None, default:
None
)
–
Respective list of probabilities each state is selected. If no probabilities are provided, a uniform
probability distribution is assumed.
-
strategy
(str, default:
'min_error'
)
–
Whether to perform minimal error or unambiguous discrimination task. Possible values are "min_error"
and "unambiguous". Both strategies support pure and mixed states.
-
measurement
(str, default:
'positive'
)
–
The type of measurement to use. Possible values are "positive" (default) for standard positive
measurements, "ppt" for PPT (positive partial transpose) measurements, and "locc" for one-way LOCC
measurements (solved by a see-saw, see below).
-
subsystems
(list[int] | None, default:
None
)
–
A list of integers specifying which subsystems to transpose for PPT measurements. Required when
measurement="ppt".
-
dimensions
(list[int] | None, default:
None
)
–
A list of integers specifying the dimensions of each subsystem. Required when measurement="ppt",
and required as the two subsystem dimensions [dim_A, dim_B] when measurement="locc".
-
solver
(str, default:
'cvxopt'
)
–
Optimization option for picos solver. Default option is solver_option="cvxopt".
-
primal_dual
(str, default:
'dual'
)
–
Option for the optimization problem.
-
reps
(int, default:
5
)
–
Number of random see-saw restarts for measurement="locc". The smallest value found is returned.
-
tol
(float, default:
1e-07
)
–
Convergence tolerance on the see-saw error decrease for measurement="locc".
-
max_iters
(int, default:
100
)
–
Maximum number of see-saw iterations per restart for measurement="locc".
-
num_alice_outcomes
(int | None, default:
None
)
–
Number of outcomes available to Alice for measurement="locc". Defaults to the number of
states; allowing more outcomes can only lower the value.
-
seed
(int | None, default:
None
)
–
Base seed for the random initial POVMs when measurement="locc" (restart r uses seed + r).
-
kwargs
(Any, default:
{}
)
–
Additional arguments to pass to picos' solve method.
Returns:
-
float
–
The optimal probability with which Bob can guess the state he was not given from states along with the optimal
-
list[HermitianVariable] | list[ndarray] | tuple[HermitianVariable, RealVariable]
–
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.
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.
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.
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))
PPT-constrained minimum-error state exclusion.
import numpy as np
from toqito.state_opt import state_exclusion
from toqito.states import bell
states = [bell(0), bell(1), bell(2)]
res, _ = state_exclusion(
states,
measurement="ppt",
subsystems=[0],
dimensions=[2, 2],
primal_dual="primal",
cvxopt_kktsolver="ldl",
)
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
References
1 Bandyopadhyay, Somshubhro and Jain, Rahul and Oppenheim, Jonathan and Perry, Christopher. Conclusive exclusion of quantum states. Physical Review A. vol. 89(2). (2014). doi:10.1103/physreva.89.022336.
2 Pusey, Matthew and Barrett, Jonathan and Rudolph, Terry. On the reality of the quantum state. Nature Physics. vol. 8(6). (2012). doi:10.1038/nphys2309.
Source code in toqito/state_opt/state_exclusion.py
| def state_exclusion(
vectors: list[np.ndarray],
probs: list[float] | None = None,
strategy: str = "min_error",
measurement: str = "positive",
subsystems: list[int] | None = None,
dimensions: list[int] | None = None,
solver: str = "cvxopt",
primal_dual: str = "dual",
reps: int = 5,
tol: float = 1e-7,
max_iters: int = 100,
num_alice_outcomes: int | None = None,
seed: int | None = None,
**kwargs: Any,
) -> tuple[
float, list[picos.HermitianVariable] | list[np.ndarray] | 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.
\[
\begin{equation}
\begin{aligned}
\text{minimize:} \quad & \sum_{i=1}^n p_i \langle M_i, \rho_i \rangle \\
\text{subject to:} \quad & \sum_{i=1}^n M_i = \mathbb{I}_{\mathcal{X}}, \\
& M_0, \ldots, M_n \in \text{Pos}(\mathcal{X}).
\end{aligned}
\end{equation}
\]
\[
\begin{equation}
\begin{aligned}
\text{maximize:} \quad & \text{Tr}(Y) \\
\text{subject to:} \quad & Y \preceq p_1\rho_1, \\
& Y \preceq p_2\rho_2, \\
& \vdots \\
& Y \preceq p_n\rho_n, \\
& Y \in\text{Herm}(\mathcal{X}).
\end{aligned}
\end{equation}
\]
If `measurement = "ppt"`, the measurement operators are also constrained to be positive under partial transpose on
the requested subsystem(s). In the primal minimum-error problem, this adds the constraints
\[
\Gamma_j(M_i) \succeq 0
\]
for every measurement operator \(M_i\) and for the subsystem(s) \(j\) listed in `subsystems`, where \(\Gamma_j\)
is the partial transpose map with subsystem dimensions `dimensions`. The corresponding dual uses the dual PPT cone,
\(\text{Pos} + \Gamma_j(\text{Pos})\), so the constraints become
\[
p_i\rho_i - Y \in \text{Pos} + \Gamma_j(\text{Pos}).
\]
If `measurement = "locc"`, the measurement is restricted to one-way local operations and classical communication on
a bipartite system with subsystem dimensions `dimensions = [dim_A, dim_B]`. Alice measures her subsystem with a POVM
\(\{A_a\}\), sends the outcome to Bob, who applies a conditional POVM \(\{B^a_k\}\); the induced global operator for
guessing \(k\) is \(M_k = \sum_a A_a \otimes B^a_k\). Minimizing the error over \(\{A_a\}\) and \(\{B^a_k\}\) is
bilinear, so it is solved by a see-saw (alternating semidefinite programs), restarted from `reps` random POVMs with
the smallest value returned. The result is an upper bound on the optimal one-way LOCC exclusion error (and at least
the global error). Only `strategy = "min_error"` is supported for this measurement.
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).
Linear dependence does not imply perfect exclusion: the identical pair \(|0\rangle, |0\rangle\)
is linearly dependent yet cannot be excluded (Bob can never rule out \(|0\rangle\)). A
sufficient condition for perfect exclusion (a value of 0) is that the states span a strict
subspace of the full space, since then a measurement along a complementary direction excludes
with certainty.
The conclusive state exclusion SDP is written explicitly in [@bandyopadhyay2014conclusive]. The problem
of conclusive state exclusion was also thought about under a different guise in [@pusey2012reality].
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.
measurement: The type of measurement to use. Possible values are "positive" (default) for standard positive
measurements, "ppt" for PPT (positive partial transpose) measurements, and "locc" for one-way LOCC
measurements (solved by a see-saw, see below).
subsystems: A list of integers specifying which subsystems to transpose for PPT measurements. Required when
`measurement="ppt"`.
dimensions: A list of integers specifying the dimensions of each subsystem. Required when `measurement="ppt"`,
and required as the two subsystem dimensions `[dim_A, dim_B]` when `measurement="locc"`.
solver: Optimization option for `picos` solver. Default option is `solver_option="cvxopt"`.
primal_dual: Option for the optimization problem.
reps: Number of random see-saw restarts for `measurement="locc"`. The smallest value found is returned.
tol: Convergence tolerance on the see-saw error decrease for `measurement="locc"`.
max_iters: Maximum number of see-saw iterations per restart for `measurement="locc"`.
num_alice_outcomes: Number of outcomes available to Alice for `measurement="locc"`. Defaults to the number of
states; allowing more outcomes can only lower the value.
seed: Base seed for the random initial POVMs when `measurement="locc"` (restart `r` uses `seed + r`).
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.
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" result="text"
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" result="text"
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" result="text"
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))
```
PPT-constrained minimum-error state exclusion.
```python
import numpy as np
from toqito.state_opt import state_exclusion
from toqito.states import bell
states = [bell(0), bell(1), bell(2)]
res, _ = state_exclusion(
states,
measurement="ppt",
subsystems=[0],
dimensions=[2, 2],
primal_dual="primal",
cvxopt_kktsolver="ldl",
)
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
"""
if measurement not in {"positive", "ppt", "locc"}:
raise ValueError("Argument `measurement` must be 'positive', 'ppt', or 'locc'.")
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 len(probs) != n:
raise ValueError(f"The number of probabilities ({len(probs)}) must equal the number of states ({n}).")
# `probs` are weights, not necessarily a normalized distribution (e.g. antidistinguishability passes [1]*n), so we
# do not require them to sum to 1.
if any(p < 0 for p in probs):
raise ValueError("Probability vector must be nonnegative.")
if strategy not in ("min_error", "unambiguous"):
raise ValueError("strategy must be either 'min_error' or 'unambiguous'.")
if primal_dual not in {"primal", "dual"}:
raise ValueError("The primal_dual option must be either 'primal' or 'dual'.")
if measurement == "locc":
if strategy != "min_error":
raise ValueError("The 'locc' measurement supports only strategy='min_error'.")
if dimensions is None or len(dimensions) != 2:
raise ValueError("Argument `dimensions` must be `[dim_A, dim_B]` when measurement='locc'.")
return _locc_min_error(
vectors=vectors,
probs=probs,
dimensions=dimensions,
num_alice_outcomes=num_alice_outcomes,
reps=reps,
tol=tol,
max_iters=max_iters,
seed=seed,
)
if measurement == "ppt":
_validate_ppt_params(dim=dim, subsystems=subsystems, dimensions=dimensions)
if strategy == "min_error":
if primal_dual == "primal":
return _ppt_min_error_primal(
vectors=vectors,
dim=dim,
subsystems=subsystems,
dimensions=dimensions,
probs=probs,
solver=solver,
**kwargs,
)
return _ppt_min_error_dual(
vectors=vectors,
dim=dim,
subsystems=subsystems,
dimensions=dimensions,
probs=probs,
solver=solver,
**kwargs,
)
if primal_dual == "primal":
return _ppt_unambiguous_primal(
vectors=vectors,
dim=dim,
subsystems=subsystems,
dimensions=dimensions,
probs=probs,
solver=solver,
**kwargs,
)
return _ppt_unambiguous_dual(
vectors=vectors,
dim=dim,
subsystems=subsystems,
dimensions=dimensions,
probs=probs,
solver=solver,
**kwargs,
)
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)
|