de Sitter space, extremal surfaces and "time-entanglement"

Abstract

We refine previous investigations on de Sitter space and extremal surfaces anchored at the future boundary $I^{+}$. Since such surfaces do not return, they require extra data or boundary conditions in the past (interior). In entirely Lorentzian de Sitter spacetime, this leads to future-past timelike surfaces stretching between $I^{\pm }$. Apart from an overall $-i$ factor (relative to spacelike surfaces in $AdS$) their areas are real and positive. With a no-boundary type boundary condition, the top half of these timelike surfaces joins with a spacelike part on the hemisphere giving a complex-valued area. Motivated by these, we describe two aspects of "time-entanglement" in simple toy models in quantum mechanics. One is based on a future-past thermofield double type state entangling timelike separated states, which leads to entirely positive structures. Another is based on the time evolution operator and reduced transition amplitudes, which leads to complex-valued entropy.

Latex, 16pgs, 3figs, v4: section added on time contours, rotations

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• Pseudo Entropy in dS/CFT and Time-like Entanglement Entropy
• Holography in de Sitter Space via Chern-Simons Gauge Theory
• de Sitter extremal surfaces
• CFT duals of three-dimensional de Sitter gravity
• Comments on Entanglement Entropy in the dS/CFT Correspondence
• Cosmologies, singularities and quantum extremal surfaces
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• Pseudo entropy under joining local quenches
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• Holographic Timelike Entanglement Entropy from Rindler Method
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• Horizon causality from holographic scattering in asymptotically dS${}_3$
• Non-extremal Island in de Sitter Gravity
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