for a given model, there is a single density matrix ρ
I
for the tensor product radiation
Hilbert space, constructed from the isometries defining the model, such that all of the
averaged entanglement invariants for the model may be expressed as linear combinations
of the entanglement invariants for this single “master” state.
In section 5, we evaluate these measures of entanglement more explicitly in the special
case of the maximally efficient models. In this class of models, we show that the entangle-
ment structure in the black hole radiation is universal: when averaged over initial states,
the results for all the entanglement measures described in section 3 do not depend on the
specific isometries I
n
defining the model, and are simply the results for a Haar-random
state of the multipart radiation system. For such systems, the entanglement entropy of a
subsystem is given by the well-known result of Page [15], and the subsystem R´enyi entropies
have been calculated previously in [16, 17]. We introduce new diagrammatic calculational
techniques that relate these quantities and the more general quantities of section 3 to
certain generating functions that appear in the theory of the symmetric group. These
techniques provide an alternative derivation of the previous results, a new simpler formula
for the R´enyi-entropies, and expressions for the more general multipartite entanglement
measures discussed in section 3.
In section 6, we return to the general case, and use the results of section 4 to investigate
which choices of isometries give models that best capture the “information free” property
of black hole horizons. Specifically, we determine constraints on the isometries that arise
from demanding that in each radiation step, the output state in the dimension D radiation
subsystem is as close as possible to the maximally mixed state of that subsystem. We find
that a sufficient condition is that the tensors I
n
mi
defining the model correspond to states
I
n
mi
|ni ⊗ |mi ⊗ |ii (1.2)
for which each subsystem is maximally mixed.
We end in section 7 with a discussion. Finally, some combinatoric details for the
derivations of section 5 are presented in an appendix. This paper is based on the UBC
undergraduate thesis [18].
Relation to earlier work. While many of the basic features of our models have ap-
peared before in the literature, we believe that our setup is more general than the existing
models, and that the connection to tensor networks, the investigation of the broader class
of entanglement measures discussed in section 3, and the calculational techniques for en-
tanglement measures in Haar-random states are novel, revealing an interesting connection
to enumerative combinatorics of the symmetric group.
2 Tensor network models of black hole evaporation
In this section, we introduce a general class of quantum mechanical models in which the
Hilbert space takes the form H
BH
⊗ H
rad
, and the evolution proceeds through a series of
discrete steps (corresponding to the release of radiation quanta) from an initial black hole
state |Ψ
BH
i ⊗ |0
rad
i to a final radiation state of the form |0
BH
i ⊗ |Ψ
rad
i.
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