Arbitrarily Many Independent Observers Can Share the Nonlocality of a Single Maximally Entangled Qubit Pair

Alice and Bob each have half of a pair of entangled qubits. Bob measures his half and then passes his qubit to a second Bob who measures again and so on. The goal is to maximize the number of Bobs that can have an expected violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality with the single Alice. This scenario was introduced in [R. Silva , Phys. Rev. Lett. 114, 250401 (2015)] where the authors mentioned evidence that when the Bobs act independently and with unbiased inputs then at most two of them can expect to violate the CHSH inequality with Alice. Here we show that, contrary to this evidence, arbitrarily many independent Bobs can have an expected CHSH violation with the single Alice. Our proof is constructive and our measurement strategies can be generalized to work with a larger class of two-qubit states that includes all pure entangled two-qubit states. Since violation of a Bell inequality is necessary for device-independent tasks, our work represents a step towards an eventual understanding of the limitations on how much device-independent randomness can be robustly generated from a single pair of qubits.

Figure 1

A schematic of the considered sequential CHSH scenario. All random variables $X,A,Y^{(i)}$, $B^{(i)}$ for $i=1,\dots ,n$ have only two outcomes. A quantum state ${\rho }_{A{B}^{(1)}}$ is initially shared between Alice and ${\mathrm{Bob}}^{(1)}$. After ${\mathrm{Bob}}^{(1)}$ has performed his randomly selected measurement and recorded the outcome he passes the qubit post-measurement state to ${\mathrm{Bob}}^{(2)}$ who repeats this process. Only the qubit post-measurement states are sent to the next Bob (the classical information regarding the measurement inputs and outputs are not conveyed). Each Bob also knows his position in the sequence.

Figure 2

The number of Bobs that can violate CHSH using our strategy as a function of $\theta$. This was computed numerically for several values of $ϵ$ by finding, for a fixed $\theta$, the maximum $k$ such that ${\gamma }_k(\theta )<1$. As the $\theta$ axis is plotted log scale, the nature of the plot indicates that the sequence $({\theta }_n)$ will likely decrease faster than exponentially in $n$.