Modeling Pauli measurements on graph states with nearest-neighbor classical communication

We propose a communication-assisted local-hidden-variable model that yields the correct outcome for the measurement of any product of Pauli operators on an arbitrary graph state, i.e., that yields the correct global correlation among the individual measurements in the Pauli product. Within this model, communication is restricted to a single round of message passing between adjacent nodes of the graph. We show that any model sharing some general properties with our own is incapable, for at least some graph states, of reproducing the expected correlations among all subsets of the individual measurements. The ability to reproduce all such correlations is found to depend on both the communication distance and the symmetries of the communication protocol.

Figure 1

Example demonstrating that any communication-assisted LHV model with communication distance $d⩽1$ fails to reproduce some submeasurements for the ring with $n=12$ nodes. Five of the global measurements shown on the ring, $Y_1{\underset{̱}{X}}_2{Y}_3{\underset{̱}{X}}_4{Y}_5{\underset{̱}{X}}_6{Y}_7{\underset{̱}{X}}_8{Y}_9{\underset{̱}{X}}_{10}{Y}_{11}{\underset{̱}{X}}_{12}$, ${\underset{̱}{Y}}_1{\underset{̱}{X}}_2{\underset{̱}{Y}}_3{Y}_4{\underset{̱}{Y}}_5{\underset{̱}{X}}_6{\underset{̱}{Y}}_7{Y}_8{\underset{̱}{Y}}_9{\underset{̱}{X}}_{10}{\underset{̱}{Y}}_{11}{Y}_{12}$, ${\underset{̱}{Y}}_1{X}_2{\underset{̱}{Y}}_3{\underset{̱}{Y}}_4{Y}_5{\underset{̱}{X}}_6{Y}_7{\underset{̱}{X}}_8{Y}_9{\underset{̱}{X}}_{10}{Y}_{11}{\underset{̱}{Y}}_{12}$, $Y_1{\underset{̱}{X}}_2{Y}_3{\underset{̱}{Y}}_4{\underset{̱}{Y}}_5{X}_6{\underset{̱}{Y}}_7{\underset{̱}{Y}}_8{Y}_9{\underset{̱}{X}}_{10}{Y}_{11}{\underset{̱}{X}}_{12}$, $Y_1{\underset{̱}{X}}_2{Y}_3{\underset{̱}{X}}_4{Y}_5{\underset{̱}{X}}_6{Y}_7{\underset{̱}{Y}}_8{\underset{̱}{Y}}_9{X}_{10}{\underset{̱}{Y}}_{11}{\underset{̱}{Y}}_{12}$, contain submeasurements (underlined) useful for showing a contradiction. These submeasurements imply the follow-ing constraints on a nearest-neighbor communication model, $x_2{x}_4{x}_6{x}_8{x}_{10}{x}_{12}=1$, $y_1^Y{x}_2{y}_3^Y{y}_5^Y{x}_6{y}_7^Y{y}_9^Y{x}_{10}{y}_{11}^Y=-1$, $y_1^Y{y}_3^Y{y}_4{x}_6{x}_8{x}_{10}{y}_{12}=1$, $x_2{y}_4{y}_5^Y{y}_7^Y{y}_8{x}_{10}{x}_{12}=1$, $x_2{x}_4{x}_6{y}_8{y}_9^Y{y}_{11}^Y{y}_{12}=1$, which, when multiplied together, yield a contradiction.

Figure 2

Example demonstrating that any communication-assisted LHV model based on the hidden variables of Eq. (2) and assisted by a site-invariant communication protocol fails to reproduce some submeasurements. The global measurement $M=Y_1{Y}_2{Y}_3{Y}_4{Y}_5{Y}_6$ has a random outcome, but contains a submeasurement $\tilde{M}=Y_1{Y}_2{Y}_3{I}_4{Y}_5{I}_6$ such that $-\tilde{M}$ is an element of the stabilizer group. This means that an overall sign flip is required to correct the $+1$ prediction of the hidden variables for a measurement of $\tilde{M}$. The two qubits measuring $Y$ at nodes $1$ and $3$ are in symmetric situations, as are the qubits at nodes $2$ and $5$. Thus, under a site-invariant protocol, $1$ and $3$ must make the same sign-flipping decision, as must $2$ and $5$. For each pair, the sign-flipping decisions cancel one another, producing no overall sign flip and thus giving an incorrect result of $+1$ for the measurement of $\tilde{M}$.