Hardy's paradox

Hardy's paradox is a thought experiment in quantum mechanics devised by Lucien Hardy in 1992-3 in which a particle and its antiparticle may interact without annihilating each other.

Experiments using the technique of weak measurement have studied an interaction of polarized photons and these have demonstrated that the phenomenon does occur. However, the consequence of these experiments is only that past events can be inferred after their occurrence as a probabilistic wave collapse. These weak measurements are considered to be an observation themselves, and therefore part of the causation of wave collapse, making the objective results only a probabilistic function rather than a fixed reality. However, a careful analysis of the experiment shows that Hardy's paradox only proves that a local hidden variable theory can not exist, as there can not be a theory that assumes that the system meets the states of reality regardless of the interaction with the measuring apparatus. This confirms that a quantum theory, to be consistent with the experiments, must be non-local (in the sense of Bell) and contextual.

The basic building block of Hardy’s thought experiment are two Mach–Zehnder interferometers for quantum particles and antiparticles. We will describe the case using electrons and positrons. Each interferometer consists of bent paths and two beam splitters (labeled BS¹ and BS² in the accompanying diagram) and is tuned so that when operating individually particles always exit to the same particle detector (the ones labeled "c" in the diagram – "c" is for "constructive interference" and "d" is for "destructive interference"). For example, for the right-hand side interferometer, when operating alone, entering electrons (labeled e⁻) become a quantum superposition of electrons taking the path v⁻ and electrons taking path w⁻ (in the diagram, the latter part of the w path is labeled u⁻), but these constructively interfere and thus always exit in arm c⁻:

Similarly, positrons (labeled e⁺) are always detected at c⁺. In the actual experiment the interferometers are arranged so that part of their paths overlap as shown in the diagram. If the amplitude for the particle in one arm, say w⁻, were to be obstructed by a second particle in w⁺ that collides with it, only the v amplitude would reach the second beam splitter, and would split into arms c⁺ and d⁺ with equal amplitude. The detection of a particle in d⁺ would thus indicate the presence of the obstructing particle, but without an annihilation taking place. For this reason, this scheme was named interaction-free measurement.

...
Wikipedia