Hardy's paradox
Encyclopedia
Hardy's paradox is a thought experiment
in quantum mechanics
devised by Lucien Hardy
in which a particle and its antiparticle
may interact without annihilating
each other. The paradox
arises in that this may only occur if the interaction is not observed and so it seemed that one might never be able to confirm this.
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 maintain only that past events can be inferred about after their occurrence as a probabilistic wave collapse. These weak measurements are considered by some 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.
would always exit in arm 
:
In the actual experiment the interferometers are arranged so that part of their arms overlap as shown in the diagram. If the amplitude for the particle in one arm, say
, were to be obstructed by a second particle in 
that collides with it, only the 
amplitude would reach the second beam splitter, and would split into arms 
and 
with equal amplitude. The detection of a particle in 
would thus indicate the presence of the obstructing particle without the latter being affected. For this reason, this scheme was named interaction-free measurement
.
If both the electron and the positron take arms w in their respective interferometers, they will annihilate with certainty to produce gamma radiation:
. Therefore the presence of either particle in its 
arm will affect the other’s interferometer output:
The situation can be analyzed in terms of two simultaneous interaction-free measurements: from the point of view of the interferometer on the left, a click at
implies the presence of the obstructing electron in 
. Similarly, for the interferometer on the right, a click at 
implies the presence of the positron in 
. Indeed, every time a click is recordered at 
the other particle is found in 
. If we assume the particles are independent (described by local hidden variables), we conclude that they can never emerge simultaneously in 
and 
. This would imply that they were in 
and 
, which cannot occur because of the annihilation process.
A paradox then arises because sometimes the particles do emerge simultaneously at
and 
(with probability p=1/16). Quantum mechanically, the 
term arises, in fact, from the nonmaximally entangled nature of the state just before the final beam splitters.
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
devised by Lucien Hardy
Lucien Hardy
Lucien Hardy is a theoretical physicist, known for his work on the foundation of quantum physics including Hardy's paradox, a thought experiment he devised in 1992....
in which a particle and its antiparticle
Antiparticle
Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.The...
may interact without annihilating
Annihilation
Annihilation is defined as "total destruction" or "complete obliteration" of an object; having its root in the Latin nihil . A literal translation is "to make into nothing"....
each other. The paradox
Paradox
Similar to Circular reasoning, A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition...
arises in that this may only occur if the interaction is not observed and so it seemed that one might never be able to confirm this.
Experiments using the technique of weak measurement
Weak measurement
Weak measurements are a type of quantum measurement, where the measured system is very weakly coupled to the measuring device. After the measurement the measuring device pointer is shifted by what is called the "weak value". So that a pointer initially pointing at zero before the measurement would...
have studied an interaction of polarized photons
Photon polarization
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. Individual photons are completely polarized...
and these have demonstrated that the phenomenon does occur. However, the consequence of these experiments maintain only that past events can be inferred about after their occurrence as a probabilistic wave collapse. These weak measurements are considered by some 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.
Setup description and the results
The basic building block of Hardy’s thought experiment are two Mach–Zehnder interferometers for quantum particles. Each interferometer is tuned so that when operating individually particles only ever exit to the same particle detector. For example, for the right hand side interferometer, when operating alone, particles entering in arm would always exit in arm :In the actual experiment the interferometers are arranged so that part of their arms overlap as shown in the diagram. If the amplitude for the particle in one arm, say
, were to be obstructed by a second particle in that collides with it, only the amplitude would reach the second beam splitter, and would split into arms and with equal amplitude. The detection of a particle in would thus indicate the presence of the obstructing particle without the latter being affected. For this reason, this scheme was named interaction-free measurementInteraction-free measurement
In physics, interaction-free measurement is a type of measurement in quantum mechanics that detects the position or state of an object without an interaction occurring between it and the measuring device...
.
If both the electron and the positron take arms w in their respective interferometers, they will annihilate with certainty to produce gamma radiation:
. Therefore the presence of either particle in its arm will affect the other’s interferometer output:The situation can be analyzed in terms of two simultaneous interaction-free measurements: from the point of view of the interferometer on the left, a click at
implies the presence of the obstructing electron in . Similarly, for the interferometer on the right, a click at implies the presence of the positron in . Indeed, every time a click is recordered at the other particle is found in . If we assume the particles are independent (described by local hidden variables), we conclude that they can never emerge simultaneously in and . This would imply that they were in and , which cannot occur because of the annihilation process.A paradox then arises because sometimes the particles do emerge simultaneously at
and (with probability p=1/16). Quantum mechanically, the term arises, in fact, from the nonmaximally entangled nature of the state just before the final beam splitters.