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arXiv:0706.3902 · Physical Review A

Hierarchy of inequalities for quantitative duality

Jesús Martínez-Linares

Published May 23, 2007• 1 citation
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Abstract

We derive different relations quantifying duality in a generic two-way interferometer. These relations set different upper bounds to the visibility $\mathcal{V}$ of the fringes measured at the output port of the interferometer. A hierarchy of inequalities is presented which exhibits the influence of the availability to the experimenter of different sources of which-way information contributing to the total distinguishability $\mathcal{D}$ of the ways. For mixed states and unbalanced interferometers an inequality is derived, ${\mathcal{V}}^{2}+{\ensuremath{\Xi}}^{2}\ensuremath{\le}1$, which can be more stringent than the one associated with the distinguishability $({\mathcal{V}}^{2}+{\mathcal{D}}^{2}\ensuremath{\le}1)$.

DOI
10.1103/physreva.75.052112

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We derive different relations quantifying duality in a generic two-way interferometer. These relations set different upper bounds to the visibility V\mathcal{V}V of the fringes measured at the output port of the interferometer. A hierarchy of inequalities is presented which exhibits the influence of the availability to the experimenter of different sources of which-way information contributing to the total distinguishability D\mathcal{D}D of the ways. For mixed states and unbalanced interferometers an inequality is derived, V2+\ensuremathΞ2\ensuremath≤1{\mathcal{V}}^{2}+{\ensuremath{\Xi}}^{2}\ensuremath{\le}1V2+\ensuremathΞ2\ensuremath≤1, which can be more stringent than the one associated with the distinguishability (V2+D2\ensuremath≤1)({\mathcal{V}}^{2}+{\mathcal{D}}^{2}\ensuremath{\le}1)(V2+D2\ensuremath≤1).
We introduce a measure Q\mathcal{Q}Q of the ``quality'' of a quantum which-way detector, which characterizes its intrinsic ability to extract which-way information in an asymmetric two-way interferometer. The ``quality'' Q\mathcal{Q}Q allows one to separate the contribution to the distinguishability of the ways arising from the quantum properties of the detector from the contribution stemming from a priori which-way knowledge available to the experimenter, which can be quantified by a predictability parameter P.\mathcal{P}.P. We provide an inequality relating these two sources of which-way information to the value of the fringe visibility displayed by the interferometer. We show that this inequality is an expression of duality, allowing one to trace the loss of coherence to the two reservoirs of which-way information represented by Q\mathcal{Q}Q and P.\mathcal{P}.P. Finally, we illustrate the formalism with the use of a quantum logic gate: the symmetric quanton-detecton system (SQDS). The SQDS can be regarded as two qubits trying to acquire which-way information about each other. The SQDS will provide an illustrating example of the reciprocal effects induced by duality between system and which-way detector.
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.
We experimentally investigate the reduction of the fringe visibility in an atom interferometer due to the storage of which-way information. We focus on the case of incomplete which-way information and use the distinguishability DDD to quantify how much information is stored. For a given value of DDD, the fringe visibility VVV is limited by the duality relation ${D}^{2}{+V}^{2}\ensuremath{\le}1$. We have measured DDD and VVV independently. Combining the results, we find good agreement with the duality relation.
We propose for the experimental verification of an inequality concerning wave-particle duality by Englert [Phys. Rev. Lett. 77, 2154 (1996)] relating (or setting) an upper limit on distinguishability and visibility in a two-way interferometer. The inequality, quantifies the concept of wave-particle duality. The considered two-way interferometer is a Young's double-slit experiment involving two four-level atoms and is a slightly modified version of that of the recent experiment by Eichmann et al. [Phys. Rev. Lett. 70, 2359 (1993)]. The fringe visibility depends on the detected polarization direction of the scattered light and a read out of the internal state of one of the two atoms provides a partial which-way information.