Covariant Residual Entropy

TL;DR

The paper addresses the inadequacy of the previously proposed differential entropy as a general, covariant notion of residual entropy in AdS/CFT. It introduces two covariant, causality-based bulk constructs—the strip wedge and the rim wedge—to capture the idea of collective ignorance for boundary time strips and bulk holes, respectively. The authors establish an inclusion relation showing that the strip wedge is contained in the rim wedge and that the two coincide in tame, well-behaved scenarios, while caustics and non-static settings generally disrupt this equality. This covariant framework clarifies the relationship between boundary and bulk information measures in time-dependent holography and points toward a more robust understanding of holographic entropy beyond static AdS3 setups.

Abstract

A recently explored interesting quantity in AdS/CFT, dubbed 'residual entropy', characterizes the amount of collective ignorance associated with either boundary observers restricted to finite time duration, or bulk observers who lack access to a certain spacetime region. However, the previously-proposed expression for this quantity involving variation of boundary entanglement entropy (subsequently renamed to 'differential entropy') works only in a severely restrictive context. We explain the key limitations, arguing that in general, differential entropy does not correspond to residual entropy. Given that the concept of residual entropy as collective ignorance transcends these limitations, we identify two correspondingly robust, covariantly-defined constructs: a 'strip wedge' associated with boundary observers and a 'rim wedge' associated with bulk observers. These causal sets are well-defined in arbitrary time-dependent asymptotically AdS spacetimes in any number of dimensions. We discuss their relation, specifying a criterion for when these two constructs coincide, and prove an inclusion relation for a general case. We also speculate about the implications for residual entropy. Curiously, despite each construct admitting a well-defined finite quantity related to the areas of associated bulk surfaces, these quantities are not in one-to-one correspondence with the defining regions of unknown. This has nontrivial implications about holographic measures of quantum information.

Figures (6)

• Figure 1: Relation between bulk curve (bounding the inaccessible hole) and boundary time strip in residual entropy construction in AdS$_3$ for simplest type of scenario. Left: AdS$_3$ diagram, with bulk hole rim ${\cal C}$ [thick red curve], its null normals [lines color-coded by $\varphi_{ \infty}$] whose endpoints determine the extent the boundary time-strip ${\cal T}$ [shaded red] bounded by ${\Sigma}^\pm$ [thick black curves]. We also indicate boundary $t=0$ slice [thin gray curve], one boundary observer at $\vartheta=0$ [vertical black line] and his causal wedge; the latter is obtained from the construction of Hubeny:2012wa applied to the interval $I_0$ [thick purple curve] defining the boundary observer's causal domain. The causal information surface [thick blue curve] coincides with the extremal surface ${\mathfrak E}_0$ whose area defines the entanglement entropy $S(I_0)$. The axis are $(\rho \, \cos \varphi, \rho \, \sin \varphi, t)$ where $\rho = \tan^{-1} r$, so that radial null geodesics are inclined at 45 degrees. Right: The corresponding functions of $\varphi$ describing the temporal profile of the time strip boundary ${\Sigma}$ [black], the radial profile of the curve ${\cal C}$ [red], and the radial profile of the extremal surface ${\mathfrak E}_0$ [blue] which ends on $\partial I_0$. Note that ${\mathfrak E}_0$ is tangent to ${\cal C}$ at $\vartheta=0$ and strictly outside ${\cal H}$ everywhere else.
• Figure 2: Various boundary time strips [shaded region] and corresponding 'optimal' observers [solid, endpoints labeled by $p_\pm$] versus non-optimal observers [dashed]. For each observer [black line], we also show the corresponding causal development [bounded by the blue diagonal lines] and interval $I_0$ [purple line, for optimal observers labeled by endpoints $r_\pm$]. Left: For time-flip-symmetric uniform time strip, static observers are optimal. Middle: For a boosted time strip, correspondingly boosted observers are optimal. Right: For more complicated time strip, generically the intervals $I_\vartheta$ do not lie on the same spatial slice of the boundary. Moreover, there can be regions [shaded green] which are not traversed by longest-lived observers.
• Figure 3: Similar plot as Fig. \ref{['f:TS']} (except that the viewpoint is shifted and the causal wedge omitted for greater ease of visualization), exemplifying how smooth ${\cal C}$ can lead to kinky ${\cal T}$. Left: AdS$_3$ diagram showing the bulk curve ${\cal C}$ [red curve, lying at $t=0$], null normals [color-coded by starting $\varphi$], which end on $\{ p_\pm(\vartheta) \}$ [black and brown cuspy curve at $r=\infty$, the black parts specifying the future/past boundaries ${\Sigma}^\pm$ of the time strip ${\cal T}$], and a tangent extremal surface ${\mathfrak E}_0$ at $\vartheta =0$ [blue curve lying at $t=0$]. For orientation, boundary $t=0$ slice is also shown [thin grey curve at $r=\infty$]. Right: The corresponding functions of $\varphi$ describing the temporal profile of $t_{ \infty}$ [black] which includes ${\Sigma}$, the radial profile of ${\cal C}$ [red], and the radial profile of ${\mathfrak E}_0$ [blue]. Note that while ${\mathfrak E}_0$ is tangent to ${\cal C}$ at $\vartheta=0$, it passes through ${\cal H}$.
• Figure 4: Similar plot as Fig. \ref{['f:TS']}, exemplifying how smooth ${\cal T}$ can lead to kinky ${\cal C}$ (i.e. a converse of Fig. \ref{['f:TSsmC']}). Left: AdS$_3$ diagram with longer boundary time strip having same wiggliness of ${\Sigma}$ [black curves], which leads to cuspy intersection of the null normals with $t=0$ slice [red and brown curve; with only the red parts specifying ${\cal C}$ and therefore ${\cal H}$]. (Similar effect could also be achieved with shorter ${\cal T}$ but kinkier ${\Sigma}$.) Right: The corresponding temporal and radial profiles as in Fig. \ref{['f:TS']} and Fig. \ref{['f:TSsmC']}.
• Figure 5: Example of a 'tame' setup, wherein a smooth bulk rim ${\cal C}$ [thick red curve] yields smooth time strip ${\Sigma}^\pm$ [thick black curves] and vice-versa. In this case the strip wedge ${\cal W}_{{\Sigma}}$ and the rim wedge ${\cal W}_{{\cal C}}$ coincide, and the null generators [thin curves color-coded by $\varphi_{{\cal C}}$] are normal to both ${\cal C}$ and ${\Sigma}^\pm$. Note that both ${\cal C}$ and ${\Sigma}^\pm$ vary in $t$; no symmetries are present in either ${\cal C}$ or ${\Sigma}^\pm$. However, this 'genericity' is rather fragile: even innocuously small deformations to either starting point can easily destroy the smoothness. The greater the separation between ${\cal C}$ and ${\Sigma}^\pm$, the less wiggliness is admissible in order for this tameness to be maintained.