Local Quantum Measurement and No-Signaling Imply Quantum Correlations

We show that, assuming that quantum mechanics holds locally, the finite speed of information is the principle that limits all possible correlations between distant parties to be quantum mechanical as well. Local quantum mechanics means that a Hilbert space is assigned to each party, and then all local positive-operator-valued measurements are (in principle) available; however, the joint system is not necessarily described by a Hilbert space. In particular, we do not assume the tensor product formalism between the joint systems. Our result shows that if any experiment would give nonlocal correlations beyond quantum mechanics, quantum theory would be invalidated even locally.

Figure 1

If the principle of no-signaling is obeyed and Alice and Bob are locally quantum, their nonlocal correlations can be obtained in quantum mechanics. Alice and Bob are locally quantum if their local systems can be described by a Hilbert space and they can choose to measure any local POVM $M_A=\{Q_a{\}}_a$ and $M_B=\{R_b{\}}_b$. A shared preparation between Alice and Bob corresponds to a function $\omega$ on the pair of POVM elements such that $p(a,b|M_A,M_B)=\omega (Q_a,R_b)$. This, with no-signaling, implies that the marginal distributions are given by Born’s rule, $p(a|M_A)=\mathrm{tr}(Q_a{\rho }_A)$ and $p(b|M_B)=\mathrm{tr}(R_b{\rho }_B)$, where ${\rho }_A$ and ${\rho }_B$ are quantum states (but the state of their joint system may not be quantum). We show that, in this setting, for any preparation $\omega$ there exist a joint quantum state ${\sigma }_{AB}$ and a relabeling of POVM measurements $\{{\tilde{M}}_A\}$ and $\{{\tilde{M}}_B\}$, such that $\omega (Q_a,R_b)=p(a,b|M_A,M_B)=\mathrm{tr}\mathbf{(}({\tilde{Q}}_a\otimes {\tilde{R}}_b){\sigma }_{AB}\mathbf{)}$, where ${\tilde{M}}_A=\{{\tilde{Q}}_a{\}}_a$ and ${\tilde{M}}_B=\{{\tilde{R}}_b{\}}_b$.