Additionally, the recently proposed timelike entanglement entropy, defined for timelike-separated regions in Lorentzian field theories, can also take complex values and is naturally interpreted as a pseudoentropy [8, 9, 26, 27, 28, 29, 30].

The compact Euclidean direction $t_{\mathrm{E}}\in [0,\pi ]$ serves as one of the angular coordinates on the global boundary ${\mathbb{S}}^d$. It arises from performing Wick rotation on the Lorentzian time $t$ in the AdS formulation.

The mixed nature of the bulk manifold is a consequence of the Hartle-Hawking geometry associated with the “no-boundary proposal” in dS/CFT holography [14, 16].

Throughout this Letter we use: (i) $\alpha$, $\beta$, $\gamma$, $\delta$ for indices in the conformal field theory, (ii) $\mu$, $\nu$, $\rho$, $\sigma$ as indices for the dual gravity theory, and (iii) $a$, $b$ for the codimension-two surface ${\mathrm{\Sigma }}_A$ and $i$, $j$ for normal directions to ${\mathrm{\Sigma }}_A$.

The piecewise construction is not generically valid in higher dimensions or for arbitrary subsystems; in particular, it can fail for strip-like regions even in AdS [9]. However, for spherical subregions, it provides a consistent representative of the relevant saddle in pure dS [9, 45]. We posit that this is still the case for small deformations thereof.

Notice that the extrinsic curvature along the $\theta$ direction vanishes identically, ${K^{\theta }}_{ab}=h_a^{\mu }{h}_b^{\theta }{\nabla }_{\mu }{n}_{\theta }^{\theta }=0$ since, for any value of $b$, $h_b^{\theta }=0$.

In the case of the undeformed sphere, the second junction (regularity) condition is unnecessary, since the final pseudoentropy is independent of the coefficient that would otherwise remain undetermined by this additional requirement [9].

The surface takes on a geometry reminiscent of a shuttlecock, with $\ell$ corresponding to the number of feathers in its skirt.

While, for generic entangling surfaces, cubic curvature invariants are affected by the splitting problem [58, 59, 60], which introduces ambiguities in the regularization of the conical singularity—and hence in the resulting holographic entropy functional, this issue does not arise at the perturbative level [55, 56, 61, 62, 63].