Entanglement Entropy of Dispersive Media from Thermodynamic Entropy in One Higher Dimension

A dispersive medium becomes entangled with zero-point fluctuations in the vacuum. We consider an arbitrary array of material bodies weakly interacting with a quantum field and compute the quantum mutual information between them. It is shown that the mutual information in $D$ dimensions can be mapped to classical thermodynamic entropy in $D+1$ dimensions. As a specific example, we compute the mutual information both analytically and numerically for a range of separation distances between two bodies in $D=2$ dimensions and find a logarithmic correction to the area law at short separations. A key advantage of our method is that it allows the strong subadditivity property to be easily verified.

Figure 1

Two dispersive media in $D$ dimensions coupled to a quantum field in vacuum. The mutual information between the two is related to the thermodynamic entropy of fluctuating fields in $D+1$ dimensions. The material bodies $A,B$ are reinterpreted as subregions in the plane $x_{D+1}=0$ in $(D+1)$-dimensional space, and impose a boundary condition on the fluctuating fields. This analogy finds a geometrical picture in terms of phantom chain polymers intersecting both regions.

Figure 2

QMI versus separation for circular (green circles) or square (red squares) material bodies in $D=2$ dimensions, as computed from Eqs. (7) and (8) using the numerical method outlined in the text with the bodies discretized into small triangles (upper insets). At large separations, the QMI obeys a power law, while at short separations the square-square data suggest a logarithmic dependence on $d$. The lower inset shows an enlarged view of the short-distance data for squares, together with a logarithmic fit ($a_0=0.050$).