The wave–particle duality relation, also called the Englert–Greenberger–Yasin duality relation, or the Englert–Greenberger relation, relates the visibility, V {\displaystyle V} , of interference fringes with the definiteness, or distinguishability, D {\displaystyle D} , of the photons' paths in quantum optics. As an inequality:
D 2 + V 2 ≤ 1 {\displaystyle D^{2}+V^{2}\leq 1\,}
Although it is treated as a single relation, it actually involves two separate relations, which mathematically look very similar. The first relation, derived by Daniel Greenberger and Allaine Yasin in 1988, is expressed as P 2 + V 2 ≤ 1 {\displaystyle P^{2}+V^{2}\leq 1\,} . 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. Here P {\displaystyle P} can be called the predictability. 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. This relation is D 2 + V 2 ≤ 1 {\displaystyle D^{2}+V^{2}\leq 1\,} . Here D {\displaystyle D} is called the distinguishability. Since | P | ≤ | D | {\displaystyle \left|P\right|\leq \left|D\right|} , the former relation is a consequence of the Englert 1996 relation. The inequality | P | ≤ | D | {\displaystyle \left|P\right|\leq \left|D\right|} means that the which-way detector can only add to the predictability. 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. The predictability P {\displaystyle P} which expresses the degree of probability with which path of the particle can be correctly guessed, and the distinguishability D {\displaystyle D} which is the degree to which one can experimentally acquire information about the path of the particle, are measures of the particle information, while the visibility of the fringes V {\displaystyle V} is a measure of the wave information. The relations shows that they are inversely related, as one goes up, the other goes down. Fringes are visible over a wide range of distinguishability.
The mathematics of two-slit diffraction This section reviews the mathematical formulation of the double-slit experiment. The formulation is in terms of the diffraction and interference of waves. 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. The next section will discuss the orthodox quantum mechanical interpretation of the duality relation in terms of wave–particle duality. The wave function in the Young double-aperture experiment can be written as
Ψ Total ( x ) = Ψ A ( x ) + Ψ B ( x ) . {\displaystyle \Psi _{\text{Total}}(x)=\Psi _{A}(x)+\Psi _{B}(x).}
The function
Ψ A ( x ) = C A Ψ 0 ( x − x A ) {\displaystyle \Psi _{A}(x)=C_{A}\Psi _{0}(x-x_{A})}
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