for some state eρ
(n)
A
in the code subspace. We show that this is true if and only if χ
α
is maximally mixed. Second, when interpreted in the context of the AdS/CFT code, eρ
(n)
A
needs to manifest as the state eρ
A
with an inserted cosmic brane of exactly the right tension.
One of our primary focuses is to demonstrate that CFT states with geometric duals indeed
admit such an interpretation, as long as χ
α
is maximally mixed. Formulating this argument
requires that we carefully interpret eq. (1.4) in gravity. For example, we must understand
that each α-block corresponds to a particular geometry so that we can interpret some
α-blocks as geometries with cosmic branes. We must also understand that CFT states
with geometric duals have non-vanishing support p
α
on α-blocks corresponding to many
different classical geometries. This way, eρ
(n)
A
can have its support predominantly on a
different classical geometry than eρ
A
does. We provide these interpretations in section 3,
and we explicitly show how they manifest as a cosmic brane prescription within quantum
error-correction in section 4.
Also in section 4, we emphasize the fact that a maximally-mixed χ
α
for each α implies
both properties needed for a code to match the cosmic brane prescription. This leads us
to conclude that χ
α
is maximally-mixed in AdS/CFT. This has a number of interesting
implications, such as an improved definition of the area operator
L
A
= ⊕
α
log dim(χ
α
)1
a
α
¯a
α
. (1.13)
In section 5, we discuss the implications of these results for tensor network models of
AdS/CFT. While tensor networks tend to nicely satisfy the RT formula [14–16], historically
they have struggled to have a non-flat spectrum of Renyi entropies. Our results suggest
that there is a natural way to construct a holographic tensor network that not only has
the correct Renyi entropy spectrum, but also computes the Renyi entropies via a method
qualitatively similar to the cosmic brane prescription.
Finally, in section 6 we conclude with a discussion of implications, future directions
and related work. Note that this paper was released jointly with [17] where similar ideas
are discussed.
2 Operator-algebra quantum error correction
We start by reviewing the framework of operator-algebra quantum error correction (OQEC)
as discussed in [9]. Consider a finite dimensional “physical” Hilbert space H = H
A
⊗ H
¯
A
and a “logical” code subspace H
code
⊆ H.
2
In the context of holography, one can think of
H as the boundary Hilbert space and H
code
as the Hilbert space of the bulk effective field
theory (EFT).
Let L(H
code
) be the algebra of all operators acting on H
code
and M ⊆ L(H
code
) be a
subalgebra. In particular, we require that M be a von Neumann algebra, i.e. it is closed
under addition, multiplication, hermitian conjugation and contains all scalar multiples of
the identity operator.
2
More generally the physical Hilbert space need not factorize, e.g. if the boundary theory has gauge
constraints. We expect the qualitative features of our result to be unchanged in that case.
– 4 –