Generalized Hardy’s Paradox

Here, we present the most general framework for $n$-particle Hardy’s paradoxes, which include Hardy’s original one and Cereceda’s extension as special cases. Remarkably, for any $n\ge 3$, we demonstrate that there always exist generalized paradoxes (with the success probability as high as $1/2^{n-1}$) that are stronger than the previous ones in showing the conflict of quantum mechanics with local realism. An experimental proposal to observe the stronger paradox is also presented for the case of three qubits. Furthermore, from these paradoxes we can construct the most general Hardy’s inequalities, which enable us to detect Bell’s nonlocality for more quantum states.

Figure 1

The success probability $P_n$ versus particle number $n$ $(n\ge 3)$. The blue points correspond to $P_n^G$ in the generalized Hardy’s paradox (with $|\alpha |=|\beta |=\text{even number}$), and the purple points correspond to $P_n^S$ in the standard Hardy’s paradox (with $|\alpha |=n$, $|\beta |=1$). The relation $P_n^G>P_n^S$ implies that there is always a stronger paradox in comparison to the standard Hardy’s paradox for $n\ge 3$. Especially, in the [3;2,2] scenario, we have $P_3^G=1/4$, which is twice $P_3^S=1/8$. Thus, it is feasible to observe the stronger quantum paradox in a three-qubit system.