CHSH Inequality¶

The CHSH scenario consists of two nonsignaling devices which share randomness in the classical case and entanglement in the quantum case.

The set of classical behaviors is bound by the CHSH inequality [Clauser1969]

where \(\langle A_xB_y \rangle = \text{Tr}[(A_x\otimes B_y )\rho^{AB}]\) is a bipartite correlator for dichotomic observables \(A_x\) and \(B_y\) with eigenvalues \(\pm 1\). Quantum entanglement yields a maximal CHSH score of of \(I_{CHSH}=\leq \beta^Q_{CHSH} = 2\sqrt{2}\) [Cirelson1980].

qnetvo.chsh_inequality_cost_fn(network_ansatz, parallel=False, **qnode_kwargs)[source]¶

Constructs a cost function for maximizing the score against the CHSH Bell inequality. This inequality is defined as

\[I_{CHSH} = \sum_{x,y\in\{0,1\}}(-1)^{x\cdot y}\langle A_x B_y \rangle\]

where \(\langle A_x B_y \rangle = \sum_{a,b\in\{-1,1\}} a\cdot b P(a,b|x,y)\) is the two-body correlator between dichotomic observables \(A_x\) and \(B_y\).

Parameters:

• network_ansatz (NetworkAnsatz) – A NetworkAnsatz class specifying the quantum network simulation.

• parallel (optional bool) – If True, remote qnode executions are made in parallel web requests.

• qnode_kwargs (optional dict) – Keyword arguments used only for pennylane.qnode construction.

Returns:

A cost function evaluated as cost(*network_settings) where the network_settings are obtained from the provided network_ansatz class.

qnetvo.parallel_chsh_grad_fn(network_ansatz, natural_grad=False, **qnode_kwargs)[source]¶

Constructs a parallelizeable gradient function grad_fn for the CHSH cost.

The parallelization is achieved through multithreading and intended to improve the efficiency of remote qnode execution.

The natural gradient

\[\nabla_{ng}I_{CHSH}(\vec{\theta}):= g^{-1}(\vec{\theta})\nabla I_{CHSH}(\vec{\theta})\]

scales the euclidean gradient \(\nabla\) by the pseudo-inverse of the Fubini-Study metric tensor \(g^{-1}(\vec{\theta})\). The natural gradient of the \(-I_{CHSH}(\vec{\theta})\) cost function is evaluated directly as

\[\begin{split}-\nabla_{ng}I_{CHSH}(\vec{\theta}) &= -\nabla_{ng}\sum_{x,y=0}^1 (-1)^{x \wedge y}\langle A_x B_y\rangle(\vec{\theta}_{x,y}) \\ &= -\sum_{x,y=0}^1 (-1)^{x \wedge y}\nabla_{ng}\langle A_x B_y \rangle(\vec{\theta_{x,y}}) \\ &= -\sum_{x,y=0}^1(-1)^{x \wedge y}g^{-1}(\vec{\theta}_{x,y})\nabla\langle A_x B_y \rangle(\vec{\theta}_{x,y})\end{split}\]

where \(\langle A_x B_y \rangle(\vec{\theta}_{x,y})\) is the expectation value of observables \(A_x\) and \(B_y\) parameterized by the settings \(\vec{\theta}_{x,y}\).

Parameters:

• network_ansatz (NetworkAnsatz) – The ansatz describing the network for which the CHSH inequality considered.

• natural_grad (optional Bool) – If True, then the natural gradient is evaluated. Default False.

• qnode_kwargs (optional dict) – A keyword argument passthrough to qnode construction.

Returns:

A parallelized (multithreaded) gradient function grad_fn(*network_settings).

Return type:

function

Warning

Parallel gradient computation is flaky on PennyLane v0.28+. Intermittent failures may occur.