Multiple Observers Can Share the Nonlocality of Half of an Entangled Pair by Using Optimal Weak Measurements

We investigate the trade-off between information gain and disturbance for von Neumann measurements on spin-$\frac{1}{2}$ particles, and derive the measurement pointer state that saturates this trade-off, which turns out to be highly unusual. We apply this result to the question of whether the nonlocality of a single particle from an entangled pair can be shared among multiple observers that act sequentially and independently of each other, and show that an arbitrarily long sequence of such observers can all violate the Clauser-Horne-Shimony-Holt–Bell inequality.

Figure 1

Square (solid line) and Gaussian (dashed line) pointers of equal width $\mathrm{\Delta }=1.5$ with (inset) the corresponding trade-off between the precision $G$ and quality factor $F$.

Figure 2

Plot of two optimal pointer distributions, $\{G=0.8,F=0.6\}$ (solid) and $\{G=0.2,F=0.98\}$ (dashed). Inset: Comparison of the optimal trade-off (dashed) to that attained by the Gaussian pointer (solid).

Figure 3

Bell scenario involving a single Alice and multiple Bobs, where the dashed lines indicate a spin-$\frac{1}{2}$ particle being transmitted, and the solid lines the inputs and outputs.

Figure 4

Plot of $I_{\mathrm{CHSH}}^{(1)}$ (solid) as a function of the precision $G$ of ${\text{Bob}}_1$, together with $I_{\mathrm{CHSH}}^{(2)}$ (dashed) for different pointer types, (from bottom) square, Gaussian and optimal.