As stated in the introduction of 1806.05007 (https://arxiv.org/pdf/1806.05007.pdf), a certain tensor network called the holographic code has a flat entanglement spectrum (ES), i.e. the reduced density matrix is always a maximally entangled state and this contradicts to the non-flat ES of vacuum in CFT. However, the entanglement entropy of such a CFT follows the Calabrese-Cardy formula or the Ryu-Takayanagi formula, which is realized in the above holographic codes. Why does this lead to a contradiction?

  • $\begingroup$ The question is solved. The ES is considering the all eigenvalues of the reduced density matrix. Thus even if the entanglement entropy (EE) matches (equivalently the RT formula is satisfied), it doesn’t mean the same ES. If all the eigenvalues of a state are equal, the state is maximally entangled and has a flat ES. In such a case, the Rényi entropies always equal to EE. But according to Calabrese and Cardy (eq. (27) in arxiv.org/abs/0905.4013), the Rényi entropies of the vacuum of 1+1 CFT do not equal to EE. Thus the ES is non-flat. $\endgroup$ – Amplituhedron Feb 7 at 8:36

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