The first relation, derived by Daniel Greenberger and Allaine Yasin in 1988, is expressed as
It was later extended to, providing an equality for the case of pure quantum states by Gregg Jaeger, Abner Shimony, and Lev Vaidman in 1995.
This relation involves correctly guessing which of the two paths the particle would have taken, based on the initial preparation.
A year later Berthold-Georg Englert, in 1996, derived a related relation dealing with experimentally acquiring knowledge of the two paths using an apparatus, as opposed to predicting the path based on initial preparation.
The significance of the relations is that they express quantitatively the complementarity of wave and particle viewpoints in double-slit experiments.
The complementarity principle in quantum mechanics, formulated by Niels Bohr, says that the wave and particle aspects of quantum objects cannot be observed at the same time.
The wave–particle duality relations makes Bohr's statement more quantitative – an experiment can yield partial information about the wave and particle aspects of a photon simultaneously, but the more information a particular experiment gives about one, the less it will give about the other.
which expresses the degree of probability with which path of the particle can be correctly guessed, and the distinguishability
Fringes are visible over a wide range of distinguishability.
[5] This section reviews the mathematical formulation of the double-slit experiment.
The culmination of the development is a presentation of two numbers that characterizes the visibility of the interference fringes in the experiment, linked together as the Englert–Greenberger duality relation.
is the single hole wave function for an aperture centered on the origin.
In the far-field of the two pinholes the two waves interfere and produce fringes.
The intensity of the interference pattern at a point y in the focal plane is given by where
is the distance between the aperture screen and the far field analysis plane.
If a lens is used to observe the fringes in the rear focal plane, the angle is given by
denote the maximum and minimum intensity of the fringes respectively.
By the rules of constructive and destructive interference we have Equivalently, this can be written as And hence we get, for a single photon in a pure quantum state, the duality relation There are two extremal cases with a straightforward intuitive interpretation: In a single hole experiment, the fringe visibility is zero (as there are no fringes).
The above presentation was limited to a pure quantum state.
holds, following from the coherence properties of laser light.
The mathematical discussion presented above does not require quantum mechanics at its heart.
With slight modifications to account for the squaring of amplitudes, the derivation could be applied to, for example, sound waves or water waves in a ripple tank.
For the relation to be a precise formulation of Bohr complementarity, one must introduce wave–particle duality in the discussion.
This means one must consider both wave and particle behavior of light on an equal footing.
Wave–particle duality implies that one must A) use the unitary evolution of the wave before the observation and B) consider the particle aspect after the detection (this is called the Heisenberg–von Neumann collapse postulate).
Indeed, since one could only observe the photon in one point of space (a photon can not be absorbed twice) this implies that the meaning of the wave function is essentially statistical and cannot be confused with a classical wave (such as those that occur in air or water).
If both holes are open this implies that we don't know where we would have detected the photon in the aperture plane.
means that both holes are open and play a symmetric role.
and this means that a statistical accumulation of photons at (F) builds up an interference pattern with maximal visibility.
The above treatment formalizes wave particle duality for the double-slit experiment.