Quantum quenches and boundary entropy:
some new applications of relative entropy

Horacio Casini + Gonzalo Torroba, Ignacio Salazar, Raul Arias
Instituto Balseiro, Centro Atómico Bariloche

Relative entropy
Two states in the same
Hilbert space or algebra

K is called the modular Hamiltonian or entanglement Hamiltonian
Relative entropy measures distinguishability of two states (roughly inverse ofnumber of experiments needed to distinguish them)
It is positive and increasing with the size of the region: States become more
distinguishable for larger regions.

Preestablished relation between energy and entropy: Vacuum state in half spacedetermined by the energy density operator for all relativistic quantum field theories.

time

Bisognano Wichmann (1975). Unruh (1976):xThis is Bekenstein’s bound π E R> S .
In general K is expected to be non local (but localized inside V), and model dependent.
The useful feature of half space is that K is local, universaly given in terms of the stress tensor,
for any relativistic QFT.

Early time growth of entropy after a quench

Raul Arias, H.C

Unitary local change in the stateEvolution of entropy of a region in time after the quench

Calabrese, Cardy (2005)

Balasumramanian et al (2011)
Buchel, Myers, Niekerk (2013)
Abajo, Aparicio,Lopez (2010)
Liu, Suh

Holographic calculations

Liu, Suh (2014)

Different regimes:

Early growth:t<1/TUpper bound?

Linear Growth: 1/T< t <R

Saturation:t>RV<1 for relativistic theories
Relative entropy, Hartman, Afkhami-Jeddi
SSA, H.C, H. Liu, M. Mezei

Early time growth: Looks universal and related to relative entropy bound. Proportional
to area understood: surface phenomena. Can focus on half space. Why proportional
to time square and energy density?

The energy density is constant for t>0
Expectation value of modular Hamiltonian
and relative entropy are very large, increasing withVolume. State is very different from vacuum

For two states evolved with the sametime independent local Hamiltonian we dont careabout the choice of Cauchy surface, both DS and DK will be the same

Here however, here the state does change. But the change is an internal unitary that does notchange entropy
Changing the Cauchy surface for A(expressing the operators in A in a different surface)does not change the entropy because the different
states are related by a unitary in the algebra.

However, the state is not the same, and expectation valueof K changes because the stress tensor is not conserved

The minimum is for x=t

A universal bound (for all times) that explains the 𝑡 2 behavior.
Can this be improved? Most probably yes: we still have the freedom to choose a unitary in
the null interval segment. It is d times greater than the holographic result.

A possible way
How about going down the null line?
This is not a spatial surface anymore. And weCould be loosing relevant parts of the quench.
But the minimum of K, for the vertical surface,exactly reproduces the holographic result

Turn to a vertical (unitary) quench that keeps the state on the left
region constant, and eliminate half the horizontal quench. Entropy is the same.

Universal boundon entropy after
a global quench?

Boundary entropy. g-theorem

Gonzalo Torroba, Ignacio Salazar, H.C.
CFT in 1+1 with boundary conditions
or an impurity at the origin. Local
interaction at x=0 (local Hamiltoninandescription)

Thermal entropy. Boundary entropy Affleck, Ludwig (1991)
g-theorem Friedan, Konechny (2004)
At the fix points conformal boundary condition.

Classified Cardy (1989)

Entanglement entropy of an interval containing the origin
Calabrese, Cardy (2004)

But in general g(T) and g(R) not easily related to each otherA simple example: a free «Kondo» model

Momentum dependent
reflection coefficient

The model interpolates between theconformal boundary conditions

UVIRThermal boundary entropy

Entanglement entropy. BoundarycontributionLog(g) goes from log(2)in the UV to 0 in the IR

Mutual information
between impurity
and B. Increasing with B

It goes from 0 to2 log(2)Is there a g-theorem in terms of entanglement entropy?
Let us try with relative entropy between the state with and without the impurity:
little distinguishability?

Has the right sign:
Relative entropy isincreasing with size,g(R) would be decreasing
Stress tensor expectation values vanish in the bulk

Whatever this is, it is increasing linearly with R, ruining proof of monotonicity of g(R)Numerics shows relative entropy increases linearly.

Regularizing around the impurity (free fermion)

The subtracted (correlator-vacuum correlator) does not vanish in thelimit of small x around x=0 and gives a linearly increasing term in <K>

This relative entropy distinguishes too much!

Let us compare the states on a different Cauchy surfaces
Same idea as for the quench

The entropy is not modifyed because the evolution is unitaryinside the diamond (local Hamiltonian evolution for full system)

The modular Hamiltonian on the impurity region
vanishes as we move to the null horizon.
Less distinguishability because «high temperature»at the impurity position.

It is «Rindler like» near the boundaries in null coordinates, and forgets about the value of r
in the vecinity of the impurity
Eventually there might bea constant contact term… But<K> will not change with rRelative entropy is increasing
On the null Cauchy surface

In the model there is no constant contribution of the field <K> and log(g(0)) coincides with theimpurity entropy

Conclusions:
Relative entropy can be useful to bound entropy differences between two stateseven if the states are very distinguishable provided a unitary operator in the algebra
converts one of the states to one near to the other.
A universal bound on entropy grow after a global quench
Probably can be improved tosaturated by holographic calculations
Where this comes from geometrically? This is not simply DS=DH for spheres which
has been understood geometrically.
g-theorem for entanglement entropy. First of the list g, c, F, a, theorems to have a clear
QIT explanation: decrease in distinguishability as we eliminate operators of the algebra.
It is important in the proof relativistic CFT in 1+1. More dimensions, massive fields?

Mutual information between
Impurity and B. Increasing with B
It is relative entropy of state in AB
With state in A times state in B
Similar to previous, but does not use
the state without impurity