Optimal classical-communication-assisted local model of $n$-qubit Greenberger-Horne-Zeilinger correlations

We present a model, motivated by the criterion of reality put forward by Einstein, Podolsky, and Rosen and supplemented by classical communication, which correctly reproduces the quantum-mechanical predictions for measurements of all products of Pauli operators on an $n$-qubit GHZ state (or “cat state”). The $n-2\text{bits}$ employed by our model are shown to be optimal for the allowed set of measurements, demonstrating that the required communication overhead scales linearly with $n$. We formulate a connection between the generation of the local values utilized by our model and the stabilizer formalism, which leads us to conjecture that a generalization of this method will shed light on the content of the Gottesman-Knill theorem.

Figure 1

Quantum circuit that generates the three-qubit GHZ state.

Figure 2

Evolution of the LHV table during the creation of a three-qubit GHZ state using the circuit of Fig. 1. The initial table yields the correct quantum predictions for the state ∣000⟩. The rules for updating the table through Hadamard and C-NOT gates come from the operator transformations (2, 3). The Hadamard update rules are $X^F=Z^I$, $Y^F=-Y^I$, $Z^F=X^I$, where $I$ and $F$ denote the initial and final values of a table entry, before and after the application of the gate. The rules for updating through a C-NOT, with control $c$ and target $t$, are $X_c^F=X_c^I{X}_t^I$, $Y_c^F=Y_c^I{X}_t^I$, $Z_c^F=Z_c^I$, $X_t^F=X_t^I$, $Y_t^F=Z_c^I{Y}_t^I$, $Z_t^F=Z_c^I{Z}_t^I$. The update rules are local in that they only require changes to table entries corresponding to the qubits involved in a gate. The subscripts on the random variables in Table are now seen to represent the qubits to which these variables were initially associated. Correlations arising from the application of C-NOT gates then correspond to pairs of identical subscripts.