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arXiv:1405.5721 · Progress of Theoretical and Experimental Physics

Three-slit interference: A duality relation: Fig. 1.

Mohd Asad Siddiqui, Tabish Qureshi

Published Aug 1, 2015• 28 citations
arXivHTMLPDF

Abstract

The issue of interference and which-way information is addressed in the context of three-slit interference experiments. A new path distinguishability |$\mathcal {D}_{Q}$| is introduced, based on unambiguous quantum state discrimination. An inequality connecting the interference visibility and path distinguishability, |$\mathcal {V} + {2\mathcal {D}_Q\over 3- \mathcal {D}_Q} \le 1$|⁠, is derived which puts a bound on how much fringe visibility and which-way information can be simultaneously obtained. It is argued that this bound is tight. For two-slit interference, we derive a new duality relation which reduces to Englert's duality relation and the Greenberger–Yasin duality relation, in different limits.

DOI
10.1093/ptep/ptv112

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The issue of interference and which-way information is addressed in the context of three-slit interference experiments. A new path distinguishability |DQ\mathcal {D}_{Q}DQ​| is introduced, based on unambiguous quantum state discrimination. An inequality connecting the interference visibility and path distinguishability, |V+2DQ3−DQ≤1\mathcal {V} + {2\mathcal {D}_Q\over 3- \mathcal {D}_Q} \le 1V+3−DQ​2DQ​​≤1|⁠, is derived which puts a bound on how much fringe visibility and which-way information can be simultaneously obtained. It is argued that this bound is tight. For two-slit interference, we derive a new duality relation which reduces to Englert's duality relation and the Greenberger–Yasin duality relation, in different limits.
It is well known that in a two-slit interference experiment, acquiring which-path information about the particle leads to a degrading of the interference. It is argued that path information has a meaning only when one can unambiguously tell which slit the particle went through. Using this idea, two duality relations are derived for the general case where the two paths may not be equally probable and the two slits may be of unequal widths. These duality relations, which are inequalities in general, saturate for all pure states. Earlier known results are recovered in suitable limit.
We derive a generalized wave-particle duality relation for arbitrary multipath quantum interference phenomena. Beyond the conventional notion of the wave nature of a quantum system, i.e., the interference fringe visibility, we introduce a quantifier as the normalized quantum coherence, recently defined in the framework of quantum information theory. To witness the particle nature, we quantify the path distinguishability or the which-path information based on unambiguous quantum state discrimination. Then, the Bohr complementarity principle for multipath quantum interference can be stated as a duality relation between the quantum coherence and the path distinguishability. For two-path interference, the quantum coherence is identical to the interference fringe visibility, and the relation reduces to the well-known complementarity relation. The duality relation continues to hold in the case where mixedness is introduced due to possible decoherence effects.
Abstract In an asymmetric multislit interference experiment, a quanton is more likely to pass through certain slits than some others. In such a situation one may be able to predict which slit a quanton is more likely to go through, even without using any path-detecting device. This allows one to talk of path predictability . It has been shown earlier that for a two-slit interference, the predictability and fringe visibility are constrained by the inequality  2 +  2 ≤ 1 . Generalizing this relation to the case of more than two slits is still an unsolved problem. A new definition for predictability for multi-slit interference is introduced. It is shown that this predictability and quantum coherence follow a duality relation  2 +  2 ≤ 1 , which saturates for all pure states. For the case of two slits, this relation reduces to the previously known one.
This thesis is a dissemination of the work I have done in the last four years, under supervision of Prof. Tabish Qureshi. The central theme is towards the understanding of entanglement as a physical resource and looking for its possible applications. This thesis contains six chapters.The Chapter 1 provides the background knowledge for the thesis work, which includes the mathematical formalism of quantum mechanics and the conceptual understanding of entanglement.The Chapter 2 unfolds the idea of fundamental concepts such as Bohr’s complementarity principle, quantum interference, which-path information, their trade-off relation, quantum eraser and also discuss how these issues play their role in building the foundations of quantum theory.The Chapter 3 is devoted to three slit interference experiments. The interference experiment is one of the most prominent concept of quantum theory. The two slit interference set-up is broadly studied and clearly understood, the synopsis for more than two slits are still not clear, recently a focussed research has been shifted towards three slit set-ups. We study the trade-off relation for three slit experiment in terms of visibility and the path distinguishability based on unambiguous quantum state discrimination (UQSD) method.The Chapter 4 discusses the wave-particle duality for multi-slit interference. Several attempts have been made in the past two decades to procure the trade-off relation for multi-slit experiments, yet a satisfactory answer was lacking. We obtain an analogous wave particle duality relation in terms of coherence and path-distinguishability based on UQSD. The role of coherence in building the interference pattern is largely admitted and their amount is readily stated by fringe visibility. Recently coherence has been understood as an information theoretical resource. We explore the measure of coherence as a wave quantifier rather than well known fringe visibility. We also analyze a recently proposed multi-path relation of Bagan et al. which claims to be in Englert’s form. We show that the proposed relation does not hold the essence of complementarity. We also derive a relation similar to Englert’s one to fill the exigency and to explicate its equivalence with our earlier proposed relation.The Chapter 5 discuss the scenario when the source is considered to be entangled. The entangled photon exhibits an interesting phenomenon known as ghost interference. The trade-off relation for two and three slit ghost experiment has been formulated, and also a doable quantum eraser has been proposed.The Chapter 6 aims to use the entanglement as a resource and develop the quantum information technologies, with Quantum Key Distribution (QKD). We propose a QKD protocol using entangled qubits which is shown to add an additional security over the well known BB84 protocol.
Bohr’s principle of complementarity has a long history and it is an important topic in quantum theory, among which the famous example is the duality relation. The relation between visibility C and distinguishability D, <?CDATA C2+D2≤1{C}^{2}+{D}^{2}\le 1C2+D2≤1?> C 2 + D 2 ≤ 1 , has long been recognized as the only representative of the duality relation. However, recent researches have shown that this inequality is not good enough because it is not tight for multipath interferometers. Meanwhile, a tight bound for the multipath interferometer has been put forward. Here we design and experimentally implement a three-path interferometer coupling with path indicator states. The wave property of photons is characterized by l 1-norm coherence measure, and the particle property is based on distinguishability of the indicator states. The new duality relation of the three-path interferometer is demonstrated in our experiment, which bounds the union of a right triangle and a part of elliptical area inside the quadrant of a unit circle. Data analysis confirms that the new bound is tight for photons in three-path interferometers.