"""Two-player extended nonlocal game."""
import itertools
from collections import defaultdict
import cvxpy
import numpy as np
from toqito.helper import update_odometer
from toqito.matrix_ops import tensor
from toqito.rand import random_unitary
from toqito.state_opt.npa_hierarchy import npa_constraints
[docs]
class ExtendedNonlocalGame:
r"""Create two-player extended nonlocal game object.
*Extended nonlocal games* are a superset of nonlocal games in which the
players share a tripartite state with the referee. In such games, the
winning conditions for Alice and Bob may depend on outcomes of measurements
made by the referee, on its part of the shared quantum state, in addition
to Alice and Bob's answers to the questions sent by the referee.
Extended nonlocal games were initially defined in [@Johnston_2016_Extended] and more
information on these games can be found in [@Russo_2017_Extended].
For a detailed walkthrough and several examples, including the BB84 and CHSH
games, please see the tutorial on
[Extended Nonlocal Games](../../../generated/gallery/extended_nonlocal_games/index.md).
"""
def __init__(self, prob_mat: np.ndarray, pred_mat: np.ndarray, reps: int = 1) -> None:
"""Construct extended nonlocal game object.
Args:
prob_mat: A matrix whose (x, y)-entry gives the probability that the referee will give Alice the value `x`
and Bob the value `y`.
pred_mat: A matrix representing the predictions for the game.
reps: Number of parallel repetitions to perform.
"""
if reps == 1:
self.prob_mat = prob_mat
self.pred_mat = pred_mat
self.reps = reps
else:
(
self.dim_x,
self.dim_y,
self.num_alice_out,
self.num_bob_out,
self.num_alice_in,
self.num_bob_in,
) = pred_mat.shape
self.prob_mat = tensor(prob_mat, reps)
pred_mat2 = np.zeros(
(
self.dim_x**reps,
self.dim_y**reps,
self.num_alice_out**reps,
self.num_bob_out**reps,
self.num_alice_in**reps,
self.num_bob_in**reps,
)
)
i_ind = np.zeros(reps, dtype=int)
j_ind = np.zeros(reps, dtype=int)
for i in range(self.num_alice_in**reps):
for j in range(self.num_bob_in**reps):
to_tensor = np.empty([reps, self.dim_x, self.dim_y, self.num_alice_out, self.num_bob_out])
for k in range(reps - 1, -1, -1):
to_tensor[k] = pred_mat[:, :, :, :, i_ind[k], j_ind[k]]
pred_mat2[:, :, :, :, i, j] = tensor(to_tensor)
j_ind = update_odometer(j_ind, self.num_bob_in * np.ones(reps))
i_ind = update_odometer(i_ind, self.num_alice_in * np.ones(reps))
self.pred_mat = pred_mat2
self.reps = reps
self.__get_game_dims()
def __get_game_dims(self):
"""Initialize game dimensions from the prediction matrix.
This private method checks whether the game dimensions have already been initialized by
inspecting the `_dims_initialized_by_get_game_dims` flag. If not, it extracts the dimensions
from the shape of 'self.pred_mat' and assigns the following instance attributes:
- referee_dim: The first dimension of self.pred_mat.
- num_alice_out: The third element of self.pred_mat.shape.
- num_bob_out: The fourth element.
- num_alice_in: The fifth element.
- num_bob_in: The sixth element.
After extracting these values, the flag '_dims_initialized_by_get_game_dims' is set to True
to prevent re-initialization on subsequent calls.
"""
if not hasattr(self, "_dims_initialized_by_get_game_dims") or not self._dims_initialized_by_get_game_dims:
(
self.referee_dim,
_,
self.num_alice_out,
self.num_bob_out,
self.num_alice_in,
self.num_bob_in,
) = self.pred_mat.shape
self._dims_initialized_by_get_game_dims = True
[docs]
def unentangled_value(self) -> float:
r"""Calculate the unentangled value of an extended nonlocal game.
The *unentangled value* of an extended nonlocal game is the supremum
value for Alice and Bob's winning probability in the game over all
unentangled strategies. Due to convexity and compactness, it is possible
to calculate the unentangled extended nonlocal game by:
\[
\omega(G) = \max_{f, g}
\lVert
\sum_{(x,y) \in \Sigma_A \times \Sigma_B} \pi(x,y)
V(f(x), g(y)|x, y)
\rVert
\]
where the maximum is over all functions \(f : \Sigma_A \rightarrow
\Gamma_A\) and \(g : \Sigma_B \rightarrow \Gamma_B\).
Returns:
The unentangled value of the extended nonlocal game.
"""
dim_x, dim_y, alice_out, bob_out, alice_in, bob_in = self.pred_mat.shape
max_unent_val = float("-inf")
for f_alice in itertools.product(range(alice_out), repeat=alice_in):
for g_bob in itertools.product(range(bob_out), repeat=bob_in):
p_win = np.zeros([dim_x, dim_y], dtype=complex)
for x_in in range(alice_in):
for y_in in range(bob_in):
p_win += self.prob_mat[x_in, y_in] * self.pred_mat[:, :, f_alice[x_in], g_bob[y_in], x_in, y_in]
unent_val = max(np.linalg.eigvalsh(p_win))
max_unent_val = max(max_unent_val, unent_val)
return float(max_unent_val)
[docs]
def nonsignaling_value(self) -> float:
r"""Calculate the non-signaling value of an extended nonlocal game.
The *non-signaling value* of an extended nonlocal game is the supremum
value of the winning probability of the game taken over all
non-signaling strategies for Alice and Bob.
A *non-signaling strategy* for an extended nonlocal game consists of a
function
\[
K : \Gamma_A \times \Gamma_B \times \Sigma_A \times \Sigma_B
\rightarrow \text{Pos}(\mathcal{R})
\]
such that
\[
\sum_{a \in \Gamma_A} K(a,b|x,y) = \rho_b^y
\quad \text{and} \quad
\sum_{b \in \Gamma_B} K(a,b|x,y) = \sigma_a^x,
\]
for all \(x \in \Sigma_A\) and \(y \in \Sigma_B\) where
\(\{\rho_b^y : y \in \Sigma_A, \ b \in \Gamma_B\}\) and
\(\{\sigma_a^x : x \in \Sigma_A, \ a \in \Gamma_B\}\) are
collections of operators satisfying
\[
\sum_{a \in \Gamma_A} \rho_b^y =
\tau =
\sum_{b \in \Gamma_B} \sigma_a^x,
\]
for every choice of \(x \in \Sigma_A\) and \(y \in \Sigma_B\)
where \(\tau \in \text{D}(\mathcal{R})\) is a density operator.
Returns:
The non-signaling value of the extended nonlocal game.
"""
dim_x, dim_y, alice_out, bob_out, alice_in, bob_in = self.pred_mat.shape
constraints = []
# The cvxpy package does not support optimizing over more than
# 2-dimensional objects. To overcome this, we use a dictionary to index
# between the questions and answers, while the cvxpy variables held at
# this positions are `dim_x`-by-`dim_y` cvxpy Variable objects.
# Define K(a,b|x,y) variable.
k_var = defaultdict(cvxpy.Variable)
for a_out in range(alice_out):
for b_out in range(bob_out):
for x_in in range(alice_in):
for y_in in range(bob_in):
k_var[a_out, b_out, x_in, y_in] = cvxpy.Variable((dim_x, dim_y), hermitian=True)
constraints.append(k_var[a_out, b_out, x_in, y_in] >> 0)
# Define \sigma_a^x variable.
sigma = defaultdict(cvxpy.Variable)
for a_out in range(alice_out):
for x_in in range(alice_in):
sigma[a_out, x_in] = cvxpy.Variable((dim_x, dim_y), hermitian=True)
# Define \rho_b^y variable.
rho = defaultdict(cvxpy.Variable)
for b_out in range(bob_out):
for y_in in range(bob_in):
rho[b_out, y_in] = cvxpy.Variable((dim_x, dim_y), hermitian=True)
# Define \tau density operator variable.
tau = cvxpy.Variable((dim_x, dim_y), hermitian=True)
p_win = 0
for a_out in range(alice_out):
for b_out in range(bob_out):
for x_in in range(alice_in):
for y_in in range(bob_in):
p_win += self.prob_mat[x_in, y_in] * cvxpy.trace(
self.pred_mat[:, :, a_out, b_out, x_in, y_in].conj().T @ k_var[a_out, b_out, x_in, y_in]
)
objective = cvxpy.Maximize(cvxpy.real(p_win))
# The following constraints enforce the so-called non-signaling
# constraints.
# Enforce that:
# \sum_{b \in \Gamma_B} K(a,b|x,y) = \sigma_a^x
for x_in in range(alice_in):
for y_in in range(bob_in):
for a_out in range(alice_out):
b_sum = 0
for b_out in range(bob_out):
b_sum += k_var[a_out, b_out, x_in, y_in]
constraints.append(b_sum == sigma[a_out, x_in])
# Enforce non-signaling constraints on Alice marginal:
# \sum_{a \in \Gamma_A} K(a,b|x,y) = \rho_b^y
for x_in in range(alice_in):
for y_in in range(bob_in):
for b_out in range(bob_out):
a_sum = 0
for a_out in range(alice_out):
a_sum += k_var[a_out, b_out, x_in, y_in]
constraints.append(a_sum == rho[b_out, y_in])
# Enforce non-signaling constraints on Bob marginal:
# \sum_{a \in \Gamma_A} \sigma_a^x = \tau
for x_in in range(alice_in):
sig_a_sum = 0
for a_out in range(alice_out):
sig_a_sum += sigma[a_out, x_in]
constraints.append(sig_a_sum == tau)
# Enforce that:
# \sum_{b \in \Gamma_B} \rho_b^y = \tau
for y_in in range(bob_in):
rho_b_sum = 0
for b_out in range(bob_out):
rho_b_sum += rho[b_out, y_in]
constraints.append(rho_b_sum == tau)
# Enforce that tau is a density operator.
constraints.append(cvxpy.trace(tau) == 1)
constraints.append(tau >> 0)
problem = cvxpy.Problem(objective, constraints)
ns_val = problem.solve()
return ns_val
[docs]
def quantum_value_lower_bound(
self,
iters: int = 20,
tol: float = 1e-8,
seed: int | None = None,
initial_bob_is_random: bool | dict = False,
solver: str = cvxpy.SCS,
solver_params: dict | None = None,
verbose: bool = False,
) -> float:
r"""Calculate lower bound on the quantum value of an extended nonlocal game.
Uses an iterative see-saw method involving two SDPs.
Args:
iters: Maximum number of see-saw iterations (Alice optimizes, Bob optimizes (default is 20).
tol: Tolerance for stopping see-saw iteration based on improvement (default is 1e-8).
seed: Optional seed for initializing random POVMs for reproducibility (default is None).
initial_bob_is_random: Optional
solver: Optional option for different solver (default is SCS).
solver_params: Optional parameters for solver (default is {"eps": 1e-8, "verbose": False}).
verbose: Optional printout for optimizer step (default is False).
Returns:
The best lower bound found on the quantum value.
"""
self.__get_game_dims()
if solver_params is None:
solver_params = {"eps_abs": tol, "eps_rel": tol, "max_iters": 50000, "verbose": False}
if seed is not None:
np.random.seed(seed)
# Get number of inputs and outputs for Bob's measurements.
_, _, _, num_outputs_bob, _, num_inputs_bob = self.pred_mat.shape
# Initialize Bob's POVMs (NumPy arrays) - Default to RANDOM
bob_povms_np = defaultdict(lambda: np.zeros((num_outputs_bob, num_outputs_bob), dtype=complex))
if isinstance(initial_bob_is_random, bool):
if initial_bob_is_random:
for y_ques in range(num_inputs_bob):
random_u_mat = random_unitary(num_outputs_bob)
for b_ans in range(num_outputs_bob):
ket = random_u_mat[:, b_ans].reshape(-1, 1)
bra = ket.conj().T
bob_povms_np[y_ques, b_ans] = ket @ bra
else:
for y_ques in range(num_inputs_bob):
bob_povms_np[y_ques, 0] = np.eye(num_outputs_bob)
for b_other_ans in range(1, num_outputs_bob):
bob_povms_np[y_ques, b_other_ans] = np.zeros((num_outputs_bob, num_outputs_bob))
elif isinstance(initial_bob_is_random, dict): # If you allow dict for custom POVMs
bob_povms_np = initial_bob_is_random
else:
raise TypeError(
f"Expected initial_bob_is_random to be bool or dict, " # Adjust if only bool supported
f"got {type(initial_bob_is_random).__name__} instead."
)
prev_win_val = -float("inf")
current_best_lower_bound = -float("inf")
if verbose:
init_bob_display = "dict" if isinstance(initial_bob_is_random, dict) else initial_bob_is_random
print(
f"Starting see-saw: max_steps={iters}, tol={tol}, seed={seed}, solver={solver}, "
+ f"random_init={init_bob_display}"
)
for step in range(iters):
opt_alice_rho_cvxpy_vars, problem_alice = self.__optimize_alice(bob_povms_np, solver, solver_params)
if (
opt_alice_rho_cvxpy_vars is None
or problem_alice.status not in [cvxpy.OPTIMAL, cvxpy.OPTIMAL_INACCURATE]
or problem_alice.value is None
):
if verbose:
print(
f"Warning: Alice optimization step failed (status: {problem_alice.status}) "
f"in see-saw step {step + 1}. Value: {problem_alice.value}"
)
return current_best_lower_bound if current_best_lower_bound > -float("inf") else 0.0
opt_bob_povm_cvxpy_vars, problem_bob = self.__optimize_bob(opt_alice_rho_cvxpy_vars, solver, solver_params)
if (
opt_bob_povm_cvxpy_vars is None
or problem_bob.status not in [cvxpy.OPTIMAL, cvxpy.OPTIMAL_INACCURATE]
or problem_bob.value is None
):
if verbose:
print(
f"Warning: Bob optimization step failed (status: {problem_bob.status}) "
f"in see-saw step {step + 1}. Value: {problem_bob.value}"
)
return current_best_lower_bound if current_best_lower_bound > -float("inf") else 0.0
current_win_val = problem_bob.value
current_best_lower_bound = max(current_best_lower_bound, current_win_val)
improvement = abs(current_win_val - prev_win_val)
if verbose:
print(
f"See-saw step {step + 1}/{iters}: Win prob = {current_win_val:.8f}, "
+ f"Improv = {improvement:.2e}, Best = {current_best_lower_bound:.8f}"
)
if (
improvement < tol and step > 0
): # step > 0 to ensure prev_win_val is not -inf for the first real improvement check
if verbose:
print(f"See-saw converged at step {step + 1} with value {current_best_lower_bound:.8f}")
break
prev_win_val = current_win_val
# If not the last iteration, update Bob's POVMs for the next step
if step < iters - 1:
for y_idx in range(self.num_bob_in):
for b_idx in range(self.num_bob_out):
povm_var = opt_bob_povm_cvxpy_vars.get((y_idx, b_idx)) # Use .get for safety
if povm_var is None or povm_var.value is None:
if verbose:
print(
f"Warning: Bob POVM var ({y_idx},{b_idx}) value is None in step {step + 1} "
f"during POVM update. Exiting see-saw early."
)
return current_best_lower_bound if current_best_lower_bound > -float("inf") else 0.0
bob_povms_np[y_idx, b_idx] = povm_var.value
else:
if verbose and iters > 0:
print(f"See-Saw reached max steps ({iters}) with value {current_best_lower_bound:.8f}")
return current_best_lower_bound if current_best_lower_bound > -float("inf") else 0.0
def __optimize_alice(
self, fixed_bob_povms_np: dict, solver: str = cvxpy.SCS, solver_params: dict | None = None
) -> tuple[dict | None, cvxpy.Problem]:
"""Fix Bob's measurements and optimize over Alice's measurements."""
# The cvxpy package does not support optimizing over 4-dimensional objects.
# To overcome this, we use a dictionary to index between the questions and
# answers, while the cvxpy variables held at this positions are
# `dim`-by-`dim` cvxpy variables.
self.__get_game_dims()
rho_xa_cvxpy_vars = defaultdict(cvxpy.Variable)
for x_ques in range(self.num_alice_in):
for a_ans in range(self.num_alice_out):
rho_xa_cvxpy_vars[x_ques, a_ans] = cvxpy.Variable(
(self.referee_dim * self.num_bob_out, self.referee_dim * self.num_bob_out),
hermitian=True,
name=f"rho_A_{x_ques}{a_ans}",
)
tau_A_cvxpy_var = cvxpy.Variable(
(self.referee_dim * self.num_bob_out, self.referee_dim * self.num_bob_out), hermitian=True, name="tau_A"
)
win_objective = cvxpy.Constant(0)
for x_q in range(self.num_alice_in):
for y_q in range(self.num_bob_in):
if self.prob_mat[x_q, y_q] == 0:
continue
for a_ans_alice in range(self.num_alice_out):
for b_ans_bob in range(self.num_bob_out):
# fixed_bob_povms_np is guaranteed to be np.ndarray here
v_xyab = self.pred_mat[:, :, a_ans_alice, b_ans_bob, x_q, y_q]
b_yb = fixed_bob_povms_np[y_q, b_ans_bob]
op_for_trace = np.kron(v_xyab, b_yb)
win_objective += self.prob_mat[x_q, y_q] * cvxpy.trace(
op_for_trace.conj().T @ rho_xa_cvxpy_vars[x_q, a_ans_alice]
)
objective = cvxpy.Maximize(cvxpy.real(win_objective))
constraints = []
for x_q_constr in range(self.num_alice_in):
rho_sum_a_constr = 0
for a_ans_constr in range(self.num_alice_out):
constraints.append(rho_xa_cvxpy_vars[x_q_constr, a_ans_constr] >> 0)
rho_sum_a_constr += rho_xa_cvxpy_vars[x_q_constr, a_ans_constr]
constraints.append(rho_sum_a_constr == tau_A_cvxpy_var)
constraints.append(cvxpy.trace(tau_A_cvxpy_var) == 1)
constraints.append(tau_A_cvxpy_var >> 0)
problem = cvxpy.Problem(objective, constraints)
problem.solve(solver=solver, **solver_params) # Use passed solver and params
return rho_xa_cvxpy_vars, problem
def __optimize_bob(
self, alice_rho_cvxpy_vars: dict | None, solver: str = cvxpy.SCS, solver_params: dict | None = None
) -> tuple[dict | None, cvxpy.Problem]:
"""Fix Alice's measurements and optimize over Bob's measurements."""
# Get number of inputs and outputs.
self.__get_game_dims()
# The cvxpy package does not support optimizing over 4-dimensional objects.
# To overcome this, we use a dictionary to index between the questions and
# answers, while the cvxpy variables held at this positions are
# `dim`-by-`dim` cvxpy variables.
bob_povm_cvxpy_vars = defaultdict(cvxpy.Variable)
for y_ques in range(self.num_bob_in):
for b_ans in range(self.num_bob_out):
bob_povm_cvxpy_vars[y_ques, b_ans] = cvxpy.Variable(
(self.num_bob_out, self.num_bob_out), hermitian=True, name=f"B_POVM_{y_ques}{b_ans}"
)
win_objective = cvxpy.Constant(0)
for x_q in range(self.num_alice_in):
for y_q in range(self.num_bob_in):
if self.prob_mat[x_q, y_q] == 0:
continue
for a_ans_alice in range(self.num_alice_out):
rho_xa_val = alice_rho_cvxpy_vars[x_q, a_ans_alice].value
for b_ans_bob in range(self.num_bob_out):
v_xyab = self.pred_mat[:, :, a_ans_alice, b_ans_bob, x_q, y_q]
win_objective += self.prob_mat[x_q, y_q] * cvxpy.trace(
cvxpy.kron(v_xyab, bob_povm_cvxpy_vars[y_q, b_ans_bob]) @ rho_xa_val
)
objective = cvxpy.Maximize(cvxpy.real(win_objective))
constraints = []
ident_bob_space = np.identity(self.num_bob_out)
for y_q_constr in range(self.num_bob_in):
sum_b_povm = 0
for b_ans_constr in range(self.num_bob_out):
constraints.append(bob_povm_cvxpy_vars[y_q_constr, b_ans_constr] >> 0)
sum_b_povm += bob_povm_cvxpy_vars[y_q_constr, b_ans_constr]
constraints.append(sum_b_povm == ident_bob_space)
problem = cvxpy.Problem(objective, constraints)
problem.solve(solver=solver, **solver_params) # Use passed solver and params
return bob_povm_cvxpy_vars, problem
[docs]
def commuting_measurement_value_upper_bound(self, k: int | str = 1, no_signaling: bool = True) -> float:
r"""Compute an upper bound on the commuting measurement value of an extended nonlocal game.
This function calculates an upper bound on the commuting measurement value by
using k-levels of the NPA hierarchy [@Navascues_2008_AConvergent]. The NPA hierarchy is a uniform
family of semidefinite programs that converges to the commuting measurement value of
any extended nonlocal game.
You can determine the level of the hierarchy by a positive integer or a string
of a form like '1+ab+aab', which indicates that an intermediate level of the hierarchy
should be used, where this example uses all products of one measurement, all products of
one Alice and one Bob measurement, and all products of two Alice and one Bob measurements.
Args:
k: The level of the NPA hierarchy to use (default=1).
no_signaling: Whether to enforce the no-signaling constraints (default=True).
Returns:
The upper bound on the commuting strategy value of an extended nonlocal game.
"""
dR, _, A_out, B_out, A_in, B_in = self.pred_mat.shape
K = {}
for x in range(A_in):
for y in range(B_in):
blocks = [[None for _ in range(B_out)] for _ in range(A_out)]
for a in range(A_out):
for b in range(B_out):
blocks[a][b] = cvxpy.Variable((dR, dR), hermitian=True, name=f"K({x},{y})_{a}{b}")
K[(x, y)] = cvxpy.bmat(blocks)
total_win = cvxpy.Constant(0)
for x in range(A_in):
for y in range(B_in):
for a in range(A_out):
for b in range(B_out):
P_ref = self.pred_mat[:, :, a, b, x, y]
blk = K[(x, y)][a * dR : (a + 1) * dR, b * dR : (b + 1) * dR]
total_win += self.prob_mat[x, y] * cvxpy.trace(P_ref.conj().T @ blk)
cons = npa_constraints(K, k, referee_dim=dR, no_signaling=no_signaling)
prob = cvxpy.Problem(cvxpy.Maximize(cvxpy.real(total_win)), cons)
cs_val = prob.solve(solver=cvxpy.SCS, eps=1e-8, max_iters=100_000, verbose=False)
return cs_val