Investigating the Araki-Uhlmann relative entropy between two coherent states in relativistic Quantum Field Theory

TL;DR

Extends the numerical study of Araki-Uhlmann relative entropy to two coherent states localized in diamond regions on the right Rindler wedge in a massive scalar QFT in $(1+1)$ dimensions. The authors implement Weyl-operator-based coherent states, compute $S(f|g)$ from the relative modular operator using the Bisognano-Wichmann flow, and verify its positivity, region-size monotonicity, and mass dependence, while uncovering a linear increase with the spatial separation between the diamonds. They show that $S(f|g)$ decreases with the field mass $m$, tending to zero as $m\to\infty$, and that larger inter-diamond distances enhance distinguishability. This work provides a first quantitative characterization of the relative entropy between two coherent states in QFT and offers a framework for extending these techniques to other spacetimes and field theories.

Abstract

A numerical setup for investigating the Araki-Uhlmann relative entropy between two coherent states is presented for a scalar massive Quantum Field Theory in ($1+1$)-dimensional Minkowski spacetime. These states are constructed using smeared Weyl operators compactly supported in two diamond regions belonging to the right Rindler wedge. Using this setup, we verified the known properties of the relative entropy, namely: positivity, increase with the size of the spacetime regions considered, decrease with the increase of the mass parameter. A linear increase with respect to the spatial distance between the two diamond regions is also observed.

Figures (4)

• Figure 1: The test functions $f(\mathsf{x})$ and $g(\mathsf{x})$ (Eqs.\ref{['ft']} and \ref{['gt']}) are plotted as a function of $\mathsf{x} = (t,x)$. Here we used $r = 1$, $d = 3$, $c = 8$ and $\varepsilon = 10^{-3}$. Note the diamond-shape of the regions where $f$ and $g$ have support.
• Figure 2: The top plot shows the Araki-Uhlman relative entropy $S(f\vert g)$ (Eq.\ref{['ffA']}) between the coherent states $\ket{f}$ and $\ket{g}$ (Sect.\ref{['sec:Coherent states']}) constructed from the test functions $f(\mathsf{x})$ (Eq.\ref{['ft']}) and $g(\mathsf{x})$ (Eq.\ref{['gt']}) as a function of the mass $m$ of the scalar field $\hat{\varphi}$. The inset shows the causal diamonds $A$ and $B$, where the test functions $f$ and $g$ are, respectively, supported. The bottom plot shows the relative errors from the numerical evaluation of $S(f\vert g)$. The upper and lower black lines around the shaded region represent the error bands within which $S(f\vert g)$ lies. Being a quantification of the distinguishability between $\ket{f}$ and $\ket{g}$, the Araki-Uhlman relative entropy in the top-plot shows that these states become monotonically less distinguishable as the mass $m$ of $\hat{\varphi}$ increases.
• Figure 3: The top plot shows the Araki-Uhlman relative entropy $S(f\vert g)$ (Eq.\ref{['ffA']}) between the coherent states $\ket{f}$ and $\ket{g}$ (Sect.\ref{['sec:Coherent states']}) constructed from the test functions $f(\mathsf{x})$ (Eq.\ref{['ft']}) and $g(\mathsf{x})$ (Eq.\ref{['gt']}) as a function of $r$, the parameter that controls the size of the diamond-shaped regions $A$ and $B$ where each test function is supported. The bottom plot shows the relative errors from the numerical evaluation of $S(f\vert g)$ as function of $r$. The upper and lower black lines around the shaded region represent the error bands within which $S(f\vert g)$ lies. Here we see that $\ket{f}$ and $\ket{g}$ becomes increasingly distinguishable as the diamond size $r$ increases.
• Figure 4: The top plot shows the Araki-Uhlman relative entropy $S(f\vert g)$ (Eq.\ref{['ffA']}) between the coherent states $\ket{f}$ and $\ket{g}$ (Sect.\ref{['sec:Coherent states']}) constructed from the test functions $f(\mathsf{x})$ (Eq.\ref{['ft']}) and $g(\mathsf{x})$ (Eq.\ref{['gt']}) as a function of the coordinate separation $c$ between the centers of the diamond-shaped regions $A$ and $B$ where $f$ and $g$ are supported, respectively. The bottom plot shows the relative errors from the numerical evaluation of $S(f\vert g)$ as function of $c$. The large relative error as $c\to 0$ is expected since in that limit $S(f\vert g)\to 0$ as well and such value is difficult to obtain from a numerical integration performed with the Quasi-Monte Carlo method. The upper and lower black lines around the shaded region represent the error bands within which $S(f\vert g)$ lies. Here we see that $\ket{f}$ and $\ket{g}$ becomes increasingly distinguishable as the diamond separation $c$ increases, with a transition for a super-linear regime for $c \ll r$ to a linear regime when $c \gtrsim r$.