Abstract
We consider the special case of Random Tensor Networks (RTN) endowed with gauge symmetry constraints on each tensor. We compute the R\`enyi entropy for such states and recover the Ryu-Takayanagi (RT) formula in the large bond regime. The result provides first of all an interesting new extension of the existing derivations of the RT formula for RTNs. Moreover, this extension of the RTN formalism brings it in direct relation with (tensorial) group field theories (and spin networks), and thus provides new tools for realizing the tensor network/geometry duality in the context of background independent quantum gravity, and for importing quantum gravity tools in tensor network research.
Highlights
Tensor networks [1] have developed into a powerful and ubiquitous formalism in quantum information and in the analysis of quantum many-body systems. It is first of all a very efficient way to capture the entanglement properties of such many-body systems, as well as quantum field theories, and it provides a general framework for describing quantum states characterized by area laws, which include the ground states of several interesting quantum many-body systems [2]
In group field theories (GFTs) [3] and random tensor models [4], as well as in loop quantum gravity (LQG) and spin foam models [5], pregeometric quantum degrees of freedom are encoded in random combinatorial network structures, labeled by algebraic data
They are encoded in spin networks, graphs labeled by irreps of SUð2Þ and endowed with a gauge symmetry at each node
Summary
Tensor networks [1] have developed into a powerful and ubiquitous formalism in quantum information and in the analysis of quantum many-body systems. In (tensorial) group field theories (GFTs) [3] and random tensor models [4], as well as in loop quantum gravity (LQG) and spin foam models [5], pregeometric quantum degrees of freedom are encoded in random combinatorial network structures, labeled by algebraic data They are encoded in spin networks, graphs labeled by irreps of SUð2Þ and endowed with a gauge symmetry at each node. [6], allows to map specific GFT partitions functions to tensor network states with a specific probability measure, providing a direct relation between the auxiliary tensor network bulk geometry and discrete gravity Along this direction, the present work aims at extending the formalism of RTN to incorporate one key feature of the random networks appearing in quantum gravity—the gauge symmetry constraint [16]—and deriving the RT formula in this extension.
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