Hardy's paradox is a thought experiment in quantum mechanics devised by Lucien Hardy in 1992–1993 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. This confirms that a hidden-variable quantum theory, to be consistent with the experiments, must be non-local (in the sense of Bell) and contextual.
Setup description and the results
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−:
| e − ⟩ → | v − ⟩ + i | w − ⟩ 2 → i | c − ⟩ . {\displaystyle \left|e^{-}\right\rangle \to {\frac {\left|v^{-}\right\rangle +i\left|w^{-}\right\rangle }{\sqrt {2}}}\to i\left|c^{-}\right\rangle .}
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 amplitudes. 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. If (classically speaking) both the electron and the positron take the w paths in their respective interferometers, they will annihilate to produce two gamma rays: | w + ⟩ | w − ⟩ → | γ ⟩ | γ ⟩ {\displaystyle \left|w^{+}\right\rangle \left|w^{-}\right\rangle \to \left|\gamma \right\rangle \left|\gamma \right\rangle } . There is a 1 in 4 chance of this happening. We can express the state of the system, before the final beam splitters, as
1 2 ( | v + ⟩ | v − ⟩ + i | v + ⟩ | w − ⟩ + i | w + ⟩ | v − ⟩ − | γ ⟩ | γ ⟩ ) . {\displaystyle {\frac {1}{2}}\left(\left|v^{+}\right\rangle \left|v^{-}\right\rangle +i\left|v^{+}\right\rangle \left|w^{-}\right\rangle +i\left|w^{+}\right\rangle \left|v^{-}\right\rangle -\left|\gamma \right\rangle \left|\gamma \right\rangle \right).}
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