Simulating Maximal Quantum Entanglement without Communication

It is known that all causal correlations between two parties which output each 1 bit, $a$ and $b$, when receiving each 1 bit, $x$ and $y$, can be expressed as convex combinations of local correlations (i.e., correlations that can be simulated with local random variables) and nonlocal correlations of the form $a+b=xy\mathrm{mod}2$. We show that a single instance of the latter elementary nonlocal correlation suffices to simulate exactly all possible projective measurements that can be performed on a maximally entangled state of two qubits, with no communication needed at all. This elementary nonlocal correlation thus defines some unit of nonlocality, which we call a $nl$ bit.

Figure 1

Principle of $e$-bit simulation. The statistics of the output bits $A$ and $B$ should coincide with that predicted by quantum physics for the measurements defined by ${\overrightarrow{\nu }}_A$ and ${\overrightarrow{\nu }}_B$. The ${\lambda }_j$ denote random data that Alice and Bob can share beforehand, when they jointly agree on a strategy. The inputs ${\overrightarrow{\nu }}_A$ and ${\overrightarrow{\nu }}_B$ are given to Alice and to Bob, respectively, after they separate. Note that each party is oblivious of the other party’s input.

Figure 2

Scheme of the nonlocal PR machine, where $x$, $y$ and $a$, $b$ denote the input and output bits, respectively.

Figure 3

Scheme of a 3-party nonlocal machine.