Holographic Properties Of Kinematic Space And Its Tensor Network Representation

The Ryu-Takayanagi(RT)formula is a crucial concept in current theory of gauge-gravity duality and emergent phenomena of geometry.Recent reinterpretation of this formula in terms of a set of “bit threads” is an interesting effort in understanding holography.We investigate a quantum generalization of the “bit threads” based on tensor network,with particular interests in the multi-scale entanglement renormalization ansatz(MERA)living on the Kinematic space.We demonstrate that,in the large limit,isometries of the MERA can be regarded as “sources”(or “sinks”)of the information flow,which extensively modifies the original picture of the bit threads by introducing a new variable : density of the isometries.The large limit implies the classical gravity can be emerged from the information flow.In our picture,the emerged spacetime is nothing but the Kinematic space.Cosmology,on the other hand,plays a significant role in testing quantum gravity effects.How to apply ideas of holography and informational interpretation to a realistic universe is still missing.Fortunately,the Kinematic space is a de Sitter space,which is more related to our universe.Here we show some concepts in quantum information theory have their cosmological descriptions.Particularly,we show complexity of a MERA tensor network can be regarded as a Fisher information measure(FIM)of a de Sitter(dS)universe,followed by several observations:(i)the holographic entanglement entropy has a tensor-network description and admits an information-theoretical interpretation,(ii)on-shell action of dS spacetime has a same description of FIM,(iii)complexity/action(CA)duality holds for dS spacetime.Our result is also valid for gravity,whose FIM exhibits the same features of a recent proposed norm complexity.