Abstract
Holevo information is an upper bound for the accessible classical information of an ensemble of quantum states. In this work, we use Holevo information to investigate the ensemble theory interpretation of quantum gravity. We study the Holevo information in random tensor network states, where the random parameters are the random tensors at each vertex. Based on the results in random tensor network models, we propose a conjecture on the holographic bulk formula of the Holevo information in the gravity case. As concrete examples of holographic systems, we compute the Holevo information in the ensemble of thermal states and thermo-field double states in the Sachdev-Ye-Kitaev model. The results are consistent with our conjecture.
Highlights
Prepared state [19, 20]: these are global CFT states described by summation of two Euclidean semiclassical gravitational evolutions including the Hartle-Hawking geometry and a bra-ket wormhole geometry
For random tensor network states (RTN) with large bond dimension, we show that the Holevo information is determined by the difference between the generalized entropy with trivial entanglement wedge and that with the actual entanglement wedge (i.e. the Ryu-Takayanagi (RT) formula [29])
This assumption is true both in the random tensor network model, where the parameter J is related by the local random projections in the bulk, and in the SYK model and JT gravity where J is a random coupling in the Hamiltonian
Summary
To understand the Holevo information H in quantum gravity, we first consider the toy model of random tensor networks (RTN). Eq (2.7) and (2.8) tells us that the information we can obtain from a subsystem A about the random parameter J is equal to the entropy difference between the contribution of trivial entanglement wedge and the actual entanglement wedge To illustrate this result, we can consider two different cases, shown in figure 1(b) and (c). For simplicity we assume all EPR pairs have the same dimension D In this case, the averaged state ρA is maximally mixed, and S(ρA(J)) is given by the RT formula with quantum corrections, which leads to the Holevo information. A contains degrees of freedom from |ΨbS in addition to EPR pairs, leading to the entropy One example of this case is the RTN model for an evaporating black hole coupled with a non-gravitational bath [34].
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