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arXiv:quant-ph/0306057 · Physical Review A

“Quality” of a which-way detector

Jesús Martínez-Linares, David A. Harmin

Published Jun 23, 2004• 5 citations
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Abstract

We introduce a measure $\mathcal{Q}$ 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'' $\mathcal{Q}$ 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 $\mathcal{P}.$ 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 $\mathcal{Q}$ and $\mathcal{P}.$ 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.

DOI
10.1103/physreva.69.062109

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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.
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.
A formalism have been recently derived [J. Martinez-Linares and D. Harmin, quantum-ph/0306057] allowing one to separate different sources of which-way information contributing to the total distinguishability D of the ways in a two-way interferometer. Here we apply the formalism to a Quantum Optical Ramsey Interferometer where both sources, the a-priori predictability of the ways P and the quantum "Quality" Q of the which-way detector, stems from the same physical interaction. We show that the formalism is able to separate both sources of which-way information. Moreover, it is shown that Q succeeds in quantifying the amount of quantum which-way information stored in the which-way detector even in cases where D does not.
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 $({\mathcal{V}}^{2}+{\mathcal{D}}^{2}\ensuremath{\le}1)$.
A quantum system (quanton) traverses an interferometer with NNN equally probable paths and interacts with another quantum system (detector) that stores path information in a set of symmetric states. In this framework, we present a systematic quantitative study of wave-particle duality based on entropic trade-off relations between quantum coherence, characterized by the relative entropy of coherence of the quanton state, and which-path knowledge, quantified by the mutual information obtained through detector-state discrimination. Unlike prior upper-bound-based formulations, we obtain an exact quantification of which-path knowledge for the discrimination strategies considered here, encompassing previous strategies and extending beyond them. A general optimal measurement is implemented through a state-separation map governed by a continuous parameter ξ∈[0,1]ξ\in[0,1]ξ∈[0,1], followed by minimum-error (ME) discrimination of either the successful outputs only (standard approach) or both successful and failure outputs (concatenated approach). Our framework identifies a strategy-dependent hierarchy that generally emerges in the tightness of the resulting duality relations. At ξ=0ξ=0ξ=0, both approaches reduce to the ME measurement, which typically yields the tightest relation. For ξ>0ξ>0ξ>0, each approach yields a distinct relation that generally becomes less tight as ξξξ increases, with the concatenated strategy always yielding a relation at least as tight as its standard counterpart. We further characterize saturation of these relations by identifying the detector-state sets that lead to it and show that nontrivial saturation occurs only for interferometers with a nonprime number of paths. From these sets, we elucidate how the underlying structure of the quanton-detector correlations gives rise to this behavior and explains the lack of universal saturation in multipath duality relations for pure states.