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Chinese Physics Letters

An Inequality Concerning Quantitative Duality in an Asymmetry Two-Way Interferometer

Yang Han, Chun-Wang Wu, Wu Wei, Ping‐Xing Chen, Cheng-Zu Li

Published Mar 31, 2009• 4 citations
Publisher

Abstract

We reinvestigate the measures concerning the notion of duality of a quantum system (the ‘quanton’) in an asymmetry two-way interferometer. A new measure of the which-way information is introduced, which is based on the fidelity F instead of the trace distance Q of the final states of the which-way detector. An improved inequality is derived to be ν 2 ≤ (R 2 0 – P 2 )(1 − F) 2 , which is more stringent than the previous one of ν 2 ≤ (1 − P 2 )(1 − Q 2 ).

DOI
10.1088/0256-307x/26/4/040303

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We reinvestigate the measures concerning the notion of duality of a quantum system (the ‘quanton’) in an asymmetry two-way interferometer. A new measure of the which-way information is introduced, which is based on the fidelity F instead of the trace distance Q of the final states of the which-way detector. An improved inequality is derived to be ν 2 ≤ (R 2 0 – P 2 )(1 − F) 2 , which is more stringent than the previous one of ν 2 ≤ (1 − P 2 )(1 − Q 2 ).
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.
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, ${\mathcal{V}}^{2}+{\ensuremath{\Xi}}^{2}\ensuremath{\le}1$, which can be more stringent than the one associated with the distinguishability $({\mathcal{V}}^{2}+{\mathcal{D}}^{2}\ensuremath{\le}1)$.
Two entangled two-level system (or qubits) is the fundamental ‘brick’ in the construcion of quantum logic gates, which are the base of quantum networks. Quantum entanglement is also at the core of the duality principle relating fringe visibility and acquisition of which way information in a two-ways interferometer. We present here a quantum logic gate — the Symmetric Quanton-Detecton System, for which each qubit can play the role of quanton or which-way detector. Applying the results of Englert [Phys. Rev. Lett. 77 , 2154 (1996).] we derive a pair of coupled duality relations for the system. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
We propose quantitative measures of wave properties and particle properties for multibeam interferometers. We show that these quantities are connected by a few fundamental inequalities which express wave-particle duality. Our analysis includes which-way detection schemes and quantum erasure. The inequalities derived here are a generalization of similar limits which were studied previously in the special case of two-beam interferometers. In addition, we propose alternative measures that express the available information more efficiently, and we show that these measures fulfill similar inequalities.