Entropic Tests of Multipartite Nonlocality and State-Independent Contextuality

We introduce a multipartite extension of an information-theoretic distance introduced by Zurek [Nature (London) 341, 119 (1989)]. We use it to develop a new tool for studying quantum correlations from an information-theoretic perspective. In particular, we derive entropic tests of multipartite nonlocality for three qubits and for an arbitrary even number of qubits as well as a test of state-independent contextuality. In addition, we rederive the tripartite Mermin inequality and a state-independent noncontextuality inequality by Cabello [Phys. Rev. Lett. 101, 210401 (2008)]. This suggests that the information-theoretic distance approach to multipartite nonlocality and state-independent contextuality can provide a more general treatment of nonclassical correlations than the orthodox approach based on correlation functions.

Figure 1

Bipartite correlations between two observers. Each local measurement is maximally random, i.e., $P(A=1)=P(A=-1)=1/2$, etc. Classical correlations (a) are transitive: if $A=B^{\prime }$, $B^{\prime }=A^{\prime }$, and $A^{\prime }=B$, then $A=B$. However, nonclassical correlations can be nontransitive: $A=-B$ [(b), left], or no correlations $H(AB)=H(A)+H(B)$ [(b), right]. Such extreme nonclassical bipartite correlations cannot be observed between quantum systems.

Figure 2

Tripartite correlations between three observers. Each local measurement is maximally random, i.e., $P(A=1)=P(A=-1)=1/2$, etc. Classical correlations (a) imply that if $A{B}^{\prime }{C}^{\prime }=A^{\prime }B{C}^{\prime }=A^{\prime }{B}^{\prime }C=1$, then $ABC=1$. However, for nonclassical tripartite correlations [(b), left] one can observe $ABC=-1$ (GHZ paradox). The above nonclassical correlations cannot be distinguished from the classical ones with entropic inequalities. No tripartite correlations between $A$, $B$, and $C$ can be observed [(b), right]. These can be detected via entropic inequalities. In contrast to the bipartite scenario, such extreme nonclassical tripartite correlations can be observed between quantum systems.

Figure 3

Peres-Mermin square. Classical correlations (a) imply $(Aa\alpha )(Bb\beta )(Cc\gamma )(ABC)(abc)(\alpha \beta \gamma )=1$. Nonclassical contextual correlations [(b), left] result in the product equal to $-1$. These nonclassical correlations cannot be distinguished from classical ones with entropic inequalities. This can be done by mixing them with the classical correlations [(b), right] since $H(\alpha ·\beta ·\gamma )=1$.