Holographic Conjecture

"The holographic principle—dating back to ’t Hooft

1985 and Susskind 1995—goes even further, and suggests

that generally all information that is contained in a

volume of space can be represented by information that

resides on the boundary of that region. For an extensive

review, see Bousso 2002.the holographic principle Bousso, 2002

—the conjecture that

the information contained in a volume of space can

be represented by a theory which lives in the boundary

of that region—could be related to the area law

behavior of the entanglement entropy in microscopic

theories. ... 

Area laws also say something on

how quantum correlations are distributed in ground

states of local quantum many-body systems. Interactions

in quantum many-body systems are typically

local, which means that systems interact only over a

short distance with a finite number of neighbors. The

emergence of an area law then provides support for

the intuition that short ranged interactions require

that quantum correlations between a distinguished

region and its exterior are established via its boundary

surface. That a strict area law emerges is by no

means obvious from the decay of two-point correlators,

as we will see. Quantum phase transitions are

governed by quantum fluctuations at zero temperature,

so it is more than plausible to observe signatures

of criticality on the level of entanglement and

quantum correlations. This situation is now particularly

clear in one-dimensional 1D systems ...

It is hence not the decay behavior

of correlation functions as such that matters here,

but in fact the scaling of entanglement.

• Topological entanglement entropy: The topological

entanglement entropy is an indicator of topological

order a new kind of order in quantum many-body

systems that cannot be described by local order parameters

... Here a global feature is detected by

means of the scaling of geometric entropies.

...

In critical models the correlation length diverges and

the models become scale invariant and allow for a description

in terms of conformal field theories. According

to the universality hypothesis, the microscopic details

become irrelevant for a number of key properties. These

universal quantities then depend only on basic properties

such as the symmetry of the system, or the spatial

dimension. Models from the same universality class are

characterized by the same fixed-point Hamiltonian under

renormalization transformations, which is invariant

under general rotations. Conformal field theory then describes

such continuum models, which have the symmetry

of the conformal group including translations, rotations,

and scalings. The universality class is

characterized by the central charge c, a quantity that

roughly quantifies the “degrees of freedom of the

theory.” For free bosons c=1, whereas the Ising universality

class has c=1/2.

Once a model is known to be described by a conformal

field theory, powerful methods are available to compute

universal properties, and entanglement entropies

or even the full reduced spectra of subsystems. ...

On both sides of a

critical point in a system undergoing a quantum phase

transition, the quantum many-body system may have a

different kind of quantum order; but this order is not

necessarily one that is characterized by a local order parameter:

In systems of, say, two spatial dimensions, topological

order may occur. Topological order manifests

itself in a degeneracy of the ground-state manifold that

depends on the topology of the entire system and the

quasiparticle excitations then show an exotic type of

anyonic quasiparticle statistics. These are features that

make topologically ordered systems interesting for

quantum computation, when exactly this degeneracy can

be exploited in order to achieve a quantum memory robust

against local fluctuations. They even allow in theory

for robust instances of quantum computation, then referred

to as topological quantum computation"

"The fact that there is “little entanglement” in a system that satisfies an area law is at the core of the functioning of powerful numerical techniques such as the density-matrix renormalization group DMRG methods."

In this Colloquium, we presented the state of affairs in

the study of area laws for entanglement entropies. As

pointed out, this research field is presently enjoying

much attention for a number of reasons and motivations.

Yet, needless to say, there are numerous open

questions that are to be studied, of which we mention a

few to highlight further perspectives:

• Can one prove that gapped higher-dimensional general

local lattice models always satisfy an area law?

• In higher-dimensional systems, critical systems can

both satisfy and violate an area law. What are further

conditions to ensure that critical systems satisfy an

area law? What is the exact role of the Fermi surface

in the study of area laws in fermionic critical models?

• Can one compute scaling laws for the mutual information

for quasifree systems?

• For what 1D models beyond quasifree and conformal

settings can one find rigorous expressions for the

entanglement entropy?

• Under what precise conditions do quenched disordered

local models lead to having “less entanglement”?

• What are the further perspectives of using conformal

methods for systems with more than one spatial dimension?

• Can the link between the Bekenstein formula in the

AdS context and the scaling of geometric entropies

in conformal field theories be sharpened?

• To what extent is having a positive topological entropy

and encountering topological order one to

one?

• How can the relationship between satisfying an area

law and the efficient approximation of ground states

with PEPS be rigorously established?

• What efficiently describable states satisfy an area

law, such that one can still efficiently compute local

properties?

• Are there further instances for 1D systems satisfying

an area law that allow for certifiable approximations

in terms of matrix-product states?"

(from Area law Rev Mod Phys article cited below)