Hardy-type paradoxes for an arbitrary symmetric bipartite Bell scenario

As with a Bell inequality, Hardy's paradox manifests a contradiction between the prediction given by quantum theory and local hidden-variable theories. In this work, we give two generalizations of Hardy's arguments for manifesting such a paradox to an arbitrary, but symmetric, Bell scenario involving two observers. Our constructions recover that of Meng et al. [Phys. Rev. A 98, 062103 (2018)] and that first discussed by Cabello [Phys. Rev. A 65, 032108 (2002)] as special cases. Among the two constructions, one can be naturally interpreted as a demonstration of the failure of the transitivity of implications (FTI). Moreover, one of their special cases is equivalent to a ladder-proof-type argument for Hardy's paradox. Through a suitably generalized notion of success probability called degree of success, we provide evidence showing that the FTI-based formulation exhibits a higher degree of success compared with all other existing proposals. Moreover, this advantage seems to persist even if we allow imperfections in realizing the zero-probability constraints in such paradoxes. Explicit quantum strategies realizing several of these proofs of nonlocality without inequalities are provided.

Figure 1

Schematic representation of (a) Hardy's argument and (b) Stapp's reformulation [16] for demonstrating the inadequacy of DLHVMs in reproducing quantum predictions. Here, $A_x$ and $B_y$ represent, respectively, the outcome (0 or 1) observed by Alice and Bob when she performs the $x\mathrm{th}$ measurement and he performs the $y\mathrm{th}$ measurement. In a DLHVM, the implications (black arrows) and the forbidden event (red) imply that the event $A_1=B_1=1$ is also forbidden, yet quantum theory defies this implication.

Figure 2

The maximal DS (which reduces to the success probability in the Hardy argument) in demonstrating a proof of nonlocality without inequality for given concurrence [29]. From top to bottom, we plot $q-r$ of Eq. (5) for our FTI argument (red, solid), $q-p$ of Eq. (2) for CLL's argument (green, dashed, see Ref. [23]), and $q$ of Eq. (1) for Hardy's argument (blue, dashed-dotted, see Ref. [14]). Note that the entangled state that gives the largest DS differs from one formulation to the other.

Figure 3

MGDS as a function of the deviation $\epsilon$ from the zero-probability constraints. The top curve (red, solid) shows a lower bound on the MGDS for our FTI argument (obtained by optimizing over two-qubit states and projective measurements); the middle curve (green, dashed) shows an upper bound on the MGDS for CLL's argument (see also Ref. [22]) obtained using level 3 of the SDP hierarchy described in Ref. [31]. The inset highlights the difference between the two MGDS (provided by FTI's MGDS less the CLL's), which is always positive within a numerical precision of ${10}^{-7}$. Note that two probabilities coincide when $\epsilon \approx 0.0730$.

Figure 4

Summary of the relationships between the various Hardy-type paradoxes discussed in this work. Our new constructions are printed in boldface.

Figure 5

Logical structure of the generalized CLL arguments in the $k$-input $d$-output Bell scenario. The generalization of Hardy's original argument to these cases is recovered by setting $p=0$.

Figure 6

Logical structure of our FTI arguments in the $k$-input $d$-output Bell scenario. Generalized Stapp's arguments as introduced in Ref. [26] are recovered by setting $r=0$.