Connection between measurement disturbance relation and multipartite quantum correlation

It is found that the measurement disturbance relation (MDR) determines the strength of quantum correlation and hence is one of the essential facets of the nature of quantum nonlocality. In reverse, the exact form of MDR may be ascertained through measuring the correlation function. To this aim, an optical experimental scheme is proposed. Moreover, by virtue of the correlation function, we find that the quantum entanglement, the quantum nonlocality, and the uncertainty principle can be explicitly correlated.

Figure 1

Illustration of different MDRs for the same kinds of quantum states with identical ensemble properties of $\Delta A,\Delta B,\langle C\rangle$. The allowed values of precision $\epsilon \left(A\right)$ and disturbance $\eta \left(B\right)$ fill the shaded areas, which correspond to (a) Heisenberg-type, (b) Ozawa's, and (c) Branciard's MDRs. The essential differences among those MDRs lie in the forbidden areas for $\epsilon \left(A\right)$ and $\eta \left(B\right)$ which are characterized by the minimal distances to the origin, $r_{\mathrm{He}},r_{\mathrm{Oz}}$, and $r_{\mathrm{B}1}$, with subscripts stand for the corresponding MDRs.

Figure 2

Schematic illustration of the measurement process of Theorem 1. A bipartite state $|{\psi }_{12}\rangle$ of particles 1 and 2 is prepared, and then the measurement process is carried out via an interaction with a third particle (particle 3) via $U_{13}$. After the interaction, the bipartite correlations are measured for the resulting tripartite state $|{\psi }_{123}\rangle =U_{13}|{\psi }_{12}\rangle |{\varphi }_3\rangle$ at detectors D1, D2, and D3.

Figure 3

The supremum for correlation functions predicted by different MDRs. (a) The supremum imposed by different MDRs on the sum of bipartite correlation functions $E(X_1,X_2)$ of particles 1 and 2 and $E(Z_2,Z_3)$ of particle 2 and 3. (b) Constraints of the sum of CHSH Bell operators for particles 1 and 2 and 2 and 3 imposed by the MDRs. Here the abbreviations He, Oz, Ha, We, B1, and B2 indicate the MDRs of Eqs. (4)–(9), respectively. The QM prediction (black line) contradicts the constraint of the Heisenberg-type MDR.

Figure 4

Experimental setup for the verification of MDRs. A pair of polarization-entangled photons (photons 1 and 2) is generated by means of spontaneous parametric downconversion (SPDC). A third photon passing through wave plates (WP), the meter system, interacts with photon 1 via a cnot operation.