I-3322 Inequality

Consider the bipartite non-signaling scenario where Alice and Bob each have three classical inputs and two classical outputs. This scenario is modeled as a prepare and measure network with one preparation node (entanglement source) and two measurement nodes (Alice and Bob). The local Hilbert space dimensions used by Alice and Bob depend on the particular ansatz and are specified by the number of wires in each measurement node.

The Bell inequalities bounding this scenario are computed in full by I. Pitowsky and K. Svozil [Pitowsky2001]. It was later shown by D. Collins and N. Gisin [Collins2004] that there is one novel Bell inequality for this scenario, the \(I_{3322}\) inequality. For inputs \(x,y\in \{0,1,2\}\) and outputs \(a,b\in \{0,1\}\), the \(I_{3322}\) Bell inequality is expressed as

\[\begin{split}I_{3322} := &-P_A(0|0) -2P_B(0|0) - P_B(0|1) + P(00|00) +P(00|01) + P(00|02) \\ &+ P(00|10) + P(00|11) - p(00|12) + P(00|20) - P(00|21)\end{split}\]

where \(P(a,b|x,y)\) are conditional joint probalities and \(P_A\) (\(P_B\)) are local probabilities for Alice (Bob). The classical upper bound is known to be \(I_{3322} \leq \beta_C = 0\) while the quantum upper bound for local qubit Hilbert spaces is \(I_{3322} \leq \beta_Q = 0.25\) and has been found to be bounded by \(\beta_Q \leq 0.250\: 875\: 38\) for local Hilbert spaces infinite dimension [Pal2010].

qnetvo.post_process_I_3322_joint_probs(probs_vec)

Applies post-processing to multi-qubit probabilities in order to coarse-grain them into the dichotomic parity observables required by the \(I_{3322}\) inequality.

An \(N\)-qubit circuit has \(2^N\) measurement outcomes. To construct the joint probabilitye \(P(00|xy)\) for binary outputs, the joint probabilities can be partitioned into two sets, \(\{Even\}\) and :math`{Odd}` which denote the set of Even and Odd parity bit strings. The \(2^N\) joint probabilities are expressed as \(P(\vec{a},\vec{b}|x,y)\) where \(\vec{a}\) and \(\vec{b}\) are each \(N\)-bit strings. Since the \(I_{3322}\) inequality only requires dichotomic probabilities \(P(00|xy)\), our post-processing only needs to calculate this value. To reduce the joint probabilities \(P(\vec{a},\vec{b}|x,y)\) to dichotomic probabilities \(P(00|x,y)\) we aggregate the probabilities of even parity bit strings with

\[P(00|xy) = \{Even\}_A\{Even\}_B = \sum_{\vec{a}\in \{Even\}} \sum_{\vec{b}\in\{Even\}} P(\vec{a},\vec{b}|x,y).\]

Parameters:

• n_qubits (int) – The number of wires measured to obtained the joint probabilites.

• probs_vec (list[float]) – A probability vector obtained by measuring all wires in the computational basis.

Returns:

The dichotomic probability \(P(00|xy)\).

qnetvo.I_3322_bell_inequality_cost_fn(network_ansatz, **qnode_kwargs)

Constructs a cost function that maximizes the score of the \(I_{3322}\) Bell inequality.

Parameters:

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

Returns:

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

References

[Pitowsky2001]

Pitowsky, Itamar, and Karl Svozil. “Optimal tests of quantum nonlocality.” Physical Review A 64.1 (2001): 014102.

[Collins2004]

Collins, Daniel, and Nicolas Gisin. “A relevant two qubit Bell inequality inequivalent to the CHSH inequality.” Journal of Physics A: Mathematical and General 37.5 (2004): 1775.

[Pal2010]

Pál, Károly F., and Tamás Vértesi. “Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems.” Physical Review A 82.2 (2010): 022116.