Final answer:
1. In Hardy's version of the Bell tests, the three predictions are based on the correlation between measurement outcomes for entangled particles and the assumption of local realism.
Step-by-step explanation:
Hardy's version of the Bell tests aims to explore the violation of local realism, a concept that assumes physical properties exist independently of observation and that distant events are not influenced instantaneously.
The first prediction involves the existence of a correlation between measurements on entangled particles that exceeds a certain threshold, known as the Hardy correlation condition. Mathematically, this is expressed as P(A=1, B=1) + P(A=1, B=0) + P(A=0, B=1) > 0, where P represents the probability of a certain measurement outcome for particle A and B. The violation of this condition indicates a departure from local realism.
The second prediction involves a correlation between particle A and an auxiliary system C, given by P(A=1, C=1) + P(A=0, C=0) > 0. This correlation condition introduces an auxiliary system C and establishes a non-local connection between particles A and C, challenging the classical notion of local realism.
The third prediction extends the violation of local realism to include a correlation between particle B and the auxiliary system C, expressed as P(B=1, C=1) + P(B=0, C=0) > 0. Similar to the second prediction, this condition introduces a non-local connection between particle B and the auxiliary system C, providing further evidence against the validity of local realism. In summary, Hardy's version of the Bell tests and its three predictions offer a theoretical framework to test the limits of local realism and explore the intriguing phenomena of entanglement in quantum mechanics.