Quantitative wave-particle duality in uniform multipath interferometers with symmetric which-path detector states

TL;DR

The paper addresses entropic wave-particle duality in a uniform $N$-path interferometer where path information is stored in symmetric detector states. It introduces an exact, two-step FRIO-based discrimination framework for which-path information and derives entropic duality bounds that tighten at $oldsymbol{\xi}=0$ and loosen with increasing $oldsymbol{\xi}$, with concatenated FRIO generally providing the tightest FRIO-bound and the discrete uncertainty principle identifying nontrivial saturation sets. Saturation occurs only when detector states span an $n$-dimensional subspace with $n|N$, implying nonprime $N$; these saturating sets correspond to maximally correlated, effectively projective detector measurements. The framework thus links exact which-path discrimination, coherence, and quanton-detector correlations, offering a path to experimental tests of multipath duality phenomena and insights into how detector-state structure governs saturation.

Abstract

A quantum system (quanton) traverses an interferometer with $N$ equally probable paths and interacts with another quantum system (detector) that stores path information in a set of symmetric states. In this interferometric framework, we present entropic wave-particle duality 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. By applying a general optimal discrimination measurement, which has a closed-form solution and encompasses other fundamental strategies as special cases, we provide an exact quantification of which-path knowledge in a variety of scenarios. This measurement is carried out in two steps. First, an optimal separation map with a prescribed separation level $ξ\in [0,1]$ probabilistically reduces the overlaps between the input detector states with maximum success rate, or increases them in case of failure. Then, a minimum-error (ME) measurement discriminates either only the successful outputs (standard approach) or both the successful and failure outputs (concatenated approach). We show that the duality relation is tighter at $ξ=0$, where both approaches reduce to the ME measurement. For $ξ>0$, each approach yields a distinct relation that becomes less tight as $ξ$ increases, with the concatenated one providing the tighter bound. Finally, by using the discrete uncertainty principle, we determine the sets of detector states that lead to saturation of the duality relation, showing that they span $n$-dimensional subspaces of the detector space, where $n$ divides $N$. As a result, nontrivial saturation occurs only for interferometers with a nonprime number of paths. From the identified saturating sets, we highlight how the quanton-detector correlations underlie this phenomenon.

Figures (5)

• Figure 1: Coherence vs which-path knowledge for uniform two-path interferometers with detector states discriminated via optimal FRIO measurement, using the separation parameter $\xi=0$ (solid line, ME measurement), $\xi=0.6$ (dashed line), and $\xi=1$ (dot-dashed line, optimal UD measurement). The shaded area is the region forbidden by the duality relation.
• Figure 2: Coherence vs which-path knowledge for uniform $N$-path interferometers with detector states discriminated via ME measurement. Orange dots: simulations with (a) $10^5$, (b) $3\times 10^5$, (c) $2.5\times 10^5$, and (d) $3\times 10^6$ randomly generated sets of $N$-dimensional detector states. Open black circles: simulations with all $\binom{N}{n}$ sets of uniform $n$-dimensional detector states; sets yielding the same coherence correspond to a fixed value of $n$, ranging from $n=1$ ($\mathcal{C}=1$) to $n=N$ ($\mathcal{C}=0$).
• Figure 3: Coherence vs which-path knowledge for uniform six-path interferometers with detector states discriminated via ME measurement. Orange dots: simulations with (a) $3\times 10^5$, (b) $10^5$, (c) $5\times 10^4$, and (d) $10^4$ randomly generated sets of $n$-dimensional detector states. Open black circles: simulations with all $\binom{N}{n'}$ sets of uniform $n'$-dimensional detector states; sets yielding the same coherence correspond to a fixed value of $n'$, ranging from $n'=1$ ($\mathcal{C}=1$) to $n'=n$ ($\mathcal{C}=0$).
• Figure 4: Coherence vs which-path knowledge for uniform $N$-path interferometers with $N$-dimensional detector states discriminated via standard FRIO (first and third rows; orange dots) and concatenated FRIO (second and fourth rows; green dots), using the separation parameter $\xi$ shown in the insets. The simulations were performed with (a) $10^5$, (b) $3\times 10^5$, (c) $10^5$, and (d) $10^6$ randomly generated sets of detector states. In each plot, the black solid line depicts the boundary of the corresponding region in the $\mathcal{C}$ vs $\mathcal{K}(\mathbf{\Pi}_\textsc{me})$ plots shown in Fig. \ref{['fig:ME']}.
• Figure 5: Coherence vs which-path knowledge for uniform $N$-path interferometers with uniform $n$-dimensional detector states discriminated via ME measurement. All $\binom{N}{n}$ sets of uniform detector states are generated, and those yielding the same coherence correspond to a fixed value of $n$, ranging from $n=1$ ($\mathcal{C}=1$) to $n=N$ ($\mathcal{C}=0$). The insets show the values of $n$ (beyond the trivial $n=1$ and $n=N$) for which the duality relation of Eq. (\ref{['eq:DualityME']}) is saturated.