Optimal universal bounds for waves with varied coherence based on supremum and infimum coherence spectra

TL;DR

This work develops a majorization-based framework to bound observables for waves with varied coherence, showing that the exact bounds for any measurement are attained by the maximal and minimal coherence spectra within the input set. By embedding coherence spectra in the complete lattice of $\Delta^{\downarrow}_n$, the authors define supremum and infimum spectra that yield universal outer and inner bounds, respectively, and prove their optimality for all Hermitian measurements. They provide practical algorithms to compute these spectra for finite and closed infinite sets, including inscribing/circumscribing polygon approximations with $\mathcal{O}(N^{-2})$ convergence, and classify the geometric locations of the extremal spectra (either as singular boundary points or outside the set). The results establish fundamental, measurement-independent constraints on how coherence variation constrains wave observables, applicable to arbitrary wave types and linear measurements.

Abstract

We establish a majorization-based theory for bounding observables of waves with varied coherence. For any measurement, exact bounds are attained by the maximal and minimal elements in the set of input coherence spectra. The set's supremum and infimum, which may lie outside the set, provide optimal universal bounds: any alternative spectrum yielding universal bounds produces weaker constraints. We present an algorithm to compute the supremum and infimum, and prove that they lie either at singular boundary points or strictly outside the set of coherence spectra.

Figures (12)

• Figure 1: Transport of waves with varied coherence. (a) Partially coherent waves $\{\rho\}$, subject to unitary control $U$, are injected into an $n$-mode diffusive waveguide (illustrated for the case $n=2$). Intensities $I_A$ and $I_B$ are measured in regions $A$ and $B$. (b--d) Results for $n=2$ with $\{\rho\} = \{\rho_a, \rho_b, \rho_c\}$. (b) Hasse diagram: edges indicate majorization, and higher positions correspond to lower entropy $H(\rho)$. (c,d) Intensity ranges satisfy $\{I\}_c \subseteq \{I\}_b \subseteq \{I\}_a$; light- and dark-shaded regions indicate the outer and inner bounds over $\{\rho\}$, attained by the maximum ($\rho_a$) and minimum ($\rho_c$) elements, respectively. (e--g) Results for $n=3$ with $\{\rho\} = \{\rho_d, \rho_e, \rho_f\}$. (e) Hasse diagram: the majorization order becomes partial. (f,g) Intensity ranges no longer nest. The supremum $\bm{\lambda}^{\downarrow}_{\sup}$ and infimum $\bm{\lambda}^{\downarrow}_{\inf}$, lying outside the input set, provide optimal universal outer and inner bounds for all measurements.
• Figure 2: Schematic of a compact set $\Lambda$ of coherence spectra. (a) Generic case: $\Lambda$ has sets of maximal and minimal elements $\Lambda_{\max}$ and $\Lambda_{\min}$. Outer bounds are attained by elements of $\Lambda_{\max}$, and inner bounds by elements of $\Lambda_{\min}$; the specific elements depend on the measurement $O$. The supremum $\bm{\lambda}^{\downarrow}_{\sup}$ and infimum $\bm{\lambda}^{\downarrow}_{\inf}$ lie outside $\Lambda$. (b) Special case: $\bm{\lambda}^{\downarrow}_{\sup}$ and $\bm{\lambda}^{\downarrow}_{\inf}$ coincide with the unique maximum and minimum of $\Lambda$.
• Figure 3: Supremum and infimum of a compact set $\Lambda \subseteq \Delta_n^\downarrow$. (a) Ternary plot of an elliptical disk $\Lambda$ (brown) in $\Delta^\downarrow_3$ (gray). Purple square and green triangle mark the supremum and infimum; red and orange stars mark the entropy minimum and maximum. (b) Achievable range $\{I_A\}$ for the $n=3$ waveguide in Fig. \ref{['fig:Fig1']}(a). The supremum and infimum bounds closely match the exact bounds (light and dark shades), whereas the entropy extrema do not. Results for $\{I_B\}$ are similar (Fig. \ref{['fig:Fig8']}). (c) Smooth (orange) and singular (blue) boundary points of a convex set. The supremum and infimum can lie only at singular points or outside $\Lambda$. (d) A hexagonal $\Lambda$; the supremum and infimum lie at vertices (singular points).
• Figure 4: Simulation setup for power delivery in diffusive waveguides. TM-polarized light at vacuum wavelength $\lambda_0 = 1.55\,\mu$m is injected from the left into silicon waveguides (refractive index $n_i = 3.48$, embedded in air) containing a $6\,\mu$m-long disordered region of randomly distributed air holes. (a) A $0.4\,\mu$m-wide waveguide supporting $n = 2$ eigenmodes. Target regions $A$ and $B$ (each $0.3\,\mu\text{m} \times 0.3\,\mu$m) are centered at depths of $5\,\mu$m and $5.3\,\mu$m, respectively. (b) A $0.6\,\mu$m-wide waveguide supporting $n = 3$ eigenmodes. Target regions $A$ and $B$ (each $0.48\,\mu\text{m} \times 0.48\,\mu$m) are centered at depths of $4.5\,\mu$m and $5\,\mu$m, respectively.
• Figure 5: Coherence spectra that attain the outer bounds (filled circles) and inner bounds (empty circles) for (a) $I_A$ and (b) $I_B$ measurements for the $n=3$ elliptical disk set in Fig. \ref{['fig:Fig3']}(a). These spectra are elements of $\Lambda_{\max}$ (purple) and $\Lambda_{\min}$ (green), confirming that maximal and minimal elements attain the exact outer and inner bounds.

Theorems & Definitions (10)

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