Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan.
Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Japan. *e-mail: firstname.lastname@example.org
he entanglement entropy is a unique quantity that character-
izes quantum entanglement between two subsystems A and B
for a given pure state. In the light of anti-de Sitter/conformal
field theory (AdS/CFT)
, the entanglement entropy has a simple
holographic counterpart given by the area of minimal surface
This gives a close relationship between spacetime geometry and
One of the most important properties of entanglement entropy,
called strong subadditivity, was derived geometrically using the
holographic entanglement entropy in ref.
. Moreover, a stronger
inequality called monogamy of mutual information was derived
and this gives an interesting characterization of quantum
states dual to a classical gravity background via the holography
(see also ref.
). Note that a classical gravity corresponds to large-N
strongly coupled quantum field theories. A large class of such entro
pic inequalities for holographic states has been found in ref.
On the other hand, for mixed states, many quantities that mea
sure quantum or classical correlations (including quantum entan-
glement) between two subsystems, called A and B below, are known
in quantum information theory
. We know essentially nothing
about their holographic interpretations. The one exception is the
mutual information I(A:B) = S(ρ
) + S(ρ
) − S(ρ
) (here AB ≡ A∪ B).
However, since this quantity is just a linear combination of entan-
glement entropy, we cannot regard it as a genuinely new quantity
from the viewpoint of either holographic or quantum information
theory. This motivates us to explore an independent quantity that
measures a correlation between two subsystems for a mixed state
and has a clear holographic interpretation.
If we have in mind holographic computations based on the AdS/
CFT correspondence, there is another interesting candidate that
measures correlation between two disjoint subsystems A and B.
Consider a static example of AdS/CFT whose boundary consists
of the subsystem A, B and the complement of AB at a fixed time.
The bulk region dual to a reduced density matrix ρ
is called the
(more precisely the restriction of entangle-
ment wedge on the canonical time slice), which we will write as
. The candidate that we would like to study here is the minimal
cross-section of the entanglement wedge, which separates the wedge
into two parts: one includes A and the other includes B. We write
this as E
) and call it the entanglement wedge cross-section.
This quantity measures a certain correlation between two subsys
tems. The main purpose of this article is to explore its properties
and interpretation in conformal field theories (CFTs) by employing
quantum information theoretic considerations.
Holographic entanglement entropy
Let us start with the holographic computation of entanglement
entropy. When the total Hilbert space H
is decomposed into
a direct product
, we define the reduced density
ρ ρ= Tr
, where ρ
is the total density matrix. The
entanglement entropy S(ρ
) for the subsystem A is defined by
ρρρ=−S() Tr log
We start with the definition of holographic entanglement entropy
in a general set-up, where we have a classical gravity dual. In most of
this article, we assume a static gravity background in AdS/CFT and
take a canonical time slice M, although a generalization to a time-
dependent background is straightforward. We set the total dimension
of the gravitational spacetime as d + 1 and then M is the d-dimen
sional manifold. The quantum state dual to the gravity lives on the
, which is, in general, a sum of disjoint manifolds
∂= …MN NN
We choose a subsystem A, which is also in general a sum of disjoint
d − 1-dimensional manifolds:
=… ⊂=…AA AAANin,(1, 2, ,)
We now introduce a d − 1-dimensional surface
Γ ⊂ M
, such that
= ∂ A with the condition that Γ
is homologous to A. Note that Γ
also, in general, consists of disjoint manifolds. There are infinitely
Entanglement of purification through
and Tadashi Takayanagi
The gauge/gravity correspondence discovered two decades ago has had a profound influence on how the basic laws in phys-
ics should be formulated. In spite of the predictive power of holographic approaches (for example, when they are applied to
strongly coupled condensed-matter physics problems), the fundamental reasons behind their success remain unclear. Recently,
the role of quantum entanglement has come to the fore. Here we explore a quantity that connects gravity and quantum informa-
tion in the light of the gauge/gravity correspondence. This is given by the minimal cross-section of the entanglement wedge
that connects two disjoint subsystems in a gravity dual. In particular, we focus on various inequalities that are satisfied by this
quantity. They suggest that it is a holographic counterpart of the quantity called entanglement of purification, which measures
a bipartite correlation in a given mixed state. We give a heuristic argument that supports this identification based on a tensor
network interpretation of holography. This predicts that the entanglement of purification satisfies the strong superadditivity
for holographic conformal field theories.
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NATURE PHYSICS | VOL 14 | JUNE 2018 | 573–577 | www.nature.com/naturephysics