Quantum Entanglement and Gravity As an Entropic Force

Introduction Elliptic Minimal surfaces in AdS4 Ongoing research

Quantum Entanglement and Gravity as an Entropic Force

Georgios Pastras NCSR “Demokritos” - INPP

November 16 2017

Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force

Quantum Entanglement

Consider a composite quantum system in a state of the form

|Ψ⟩AB = |y1⟩A ⊗ |휓1⟩B + |y2⟩A ⊗ |휓2⟩B

Then,

I measurables of the two subsystems are correlated.

I there is no state description for each subsystem. This is the phenomenon of quantum entanglement. It has no classical analogue.

Entanglement Entropy

In general, subsystem A cannot be described by a state, but it is described by the density matrix

휌A = TrB휌AB.

In the absence of entanglement, 휌A corresponds to a pure state as there is a state description for subsystem A. The more entangled the overall system state, the “more mixed” is the state that 휌A describes, i.e. the more dispersed the spectrum of 휌A. Therefore, a natural measure of quantum entanglement is Shannon entropy deﬁned on the spectrum of 휌A,

SEE := −Tr휌A ln 휌A, commonly known as the entanglement entropy.

Holographic Dualities

Holographic dualities provide a broad framework allowing the description of gravitational theories with AdS asymptotics in d + 1 dimensions as emergent from strongly coupled conformal ﬁeld theories in d dimensions. As they connect the strongly coupled regime of one theory to the weakly coupled regime of the other, holographic dualities have found many applications in the description of strongly coupled condensed matter systems through gravitational dynamics, as well as the understanding of quantum gravity as emerging from the boundary CFT. A relatively recent development in the dictionary of the holographic duality is the Ryu-Takayanagi conjecture. 1 1S. Ryu, T. Takayanagi, PRL 96, 181602 (2006), arxiv:hep-th/0603001 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force

The RT conjecture

The latter connects the entanglement entropy for a region A deﬁned by the entangling surface 휕A in the boundary CFT to the area of an extremal co-dimension two open surface in the bulk gravitational dual theory with the same boundary 휕A. Area (Aextr) SA = . 4GN

r Aextr

휕A

Gravity vs Thermodynamics

I Black Hole Physics suggest Einstein equations could be effective thermodynamic relations for some underlying degrees of freedom. 2

I AdS/CFT correspondence could suggest that these underlying degrees of freedom are the boundary conformal ﬁeld theory degrees of freedom. 3

I RT conjecture points out that the relation between gravity and thermodynamics should not be attributed to thermal statistics, but rather to quantum statistics related to quantum entanglement physics. 2J. M. Bardeen, B. Carter, S. W. Hawking, Com. Math. Phys. 31, 161 (1973) 3E. P. Verlinde, JHEP 1104, 029 (2011), arXiv:1001.0785 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force

Gravity and Entanglement Thermodynamics

The density matrix 휌A is hermitian and positive semideﬁnite, thus, one may deﬁne the modular Hamiltonian as

−HA 휌A := e .

If we assume a variation in the pure state of the overall system, the variation of the entanglement entropy and the expectation value of the modular Hamiltonian are related as

훿SA = 훿 ⟨HA⟩ .

This is the direct analog of the ﬁrst law of thermodynamics for entanglement physics.

Gravity and Entanglement Thermodynamics

Variation of the state of the overall system is equivalent to the introduction of a metric perturbation in the bulk theory.

Using the holographic dictionary to calculate 훿SA and 훿 ⟨HA⟩ we ﬁnd that the ﬁrst law of entanglement thermodynamics is holographically realized if and only if the metric perturbation obeys the Einstein equations. 45

4N. Lashkari, M. B. McDermott, M. Van Raamsdonk, JHEP 1404, 195 (2014), arXiv:1308.3716 5I. Bakas, G. Pastras, Nucl. Phys. B 896, 440 (2015) arXiv:1503.00627 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Quantum Entanglement Elliptic Minimal surfaces in AdS4 Entanglement and Holography Ongoing research Gravity as an Entropic Force

Problems

Unfortunately the non-linearity of the equations that minimal surfaces in hyperbolic spaces obey allow their solution in very few cases, namely region A is either a sphere or an inﬁnite strip. This does not allow

I the complete proof of the equivalence of linearized Einstein equations to the 1st law of entanglement thermodynamics

I the study of the dependence of entanglement entropy on the geometric characteristics of the entangling surface

Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry

Elliptic Minimal surfaces in AdS4

In order to construct new minimal surfaces in AdS4 we use tools usually applied to 2d Non-linear Sigma Models.

This tool is the so called Pohlmeyer reduction which reduces the minimal surface equations to the equations of an integrable system of the family of the sine-Gordon equation.

For this system one can find many interesting solutions, however it is difficult to invert Pohlmeyer reduction to find the corresponding minimal surfaces due to its non-local nature

Elliptic Minimal surfaces in AdS4

We managed to invert Pohlmeyer reduction for a speciﬁc class of solutions of the reduced system (elliptic solutions) implementing an interesting relation between the corresponding minimal surfaces and the band structure of the analytically solvable n = 1 Lame periodic potential. 678

6I. Bakas and G. Pastras, JHEP 1607, 070 (2016), arXiv:1605.03920 7G. Pastras, arXiv:1612.03631 8G. Pastras, PoS CORFU 2016, 111 (2016), arXiv:1710.00545 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry Entangling Curve

The elliptic minimal surfaces correspond to entangling curves being the union of two logarithmic spirals, the one being the rotation of the other by 훿휙.

The family of elliptic minimal surfaces contain as special limits the helicoids in hyperbolic spaces,

the catenoids

and the cusps.

It is possible that more than one of the minimal surfaces correspond to the same entangling curve. This phenomenon is related to phase transitions (conﬁnement-deconﬁnement) of the boundary theory and the role of entanglement entropy as order parameter. In such cases there are three minimal surfaces with the same entagling curve

I one globally stable

I one locally stable, but globally unstable

I one locally unstable with one unstable mode

The expansion of holographic entanglement entropy with the energy cut-off L (radial cutoff) for an arbitrary smooth entangling curve assumes the form,

(︁ −1)︁ SEE = c1 (L0/L) + c0 + 풪 (L0/L) ,

where L0 is a characteristic length of the entangling curve. Cusps generate logarithmic terms,

(︁ −1)︁ SEE = c1 (L0/L) + a ln (L0/L) + c0 + 풪 (L0/L) .

The coefﬁcient a depends solely on the cusp angles.

This logarithmic term obeys some properties. 9 Obviously

a (Ω) = a (2휋 − Ω) .

At the limit Ω → 휋, the cusp disappears and thus,

a (휋) = 0.

The above two imply (︁ )︁ a (휋 + Ω) = 풪 Ω2 .

9P. Bueno and R. C. Myers, JHEP 1508, 068 (2015), arXiv:1505.07842 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry Logarithmic Terms from Spiral Spikes

In the case of elliptic minimal surfaces there are non-smooth points but an angle cannot be deﬁned. However we may explicitely ﬁnd that 10

√︂ 휔2 + 1 A (L) = 2휋Λ L 휔2 √︃ 2 √︃ 2 2 1 − 휔 2L 휔 + 1 (︁ −1)︁ −4Λ (휁 (휔1) + e2휔1) ln +풪 L . 2 2 2 e2−e3 3e2휔 Λ 휔 + (1 − 휔 ) 3e2

10G. Pastras, arXiv:1710.01948 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Construction Introduction The Entangling Curve Elliptic Minimal surfaces in AdS4 Interesting Limits Ongoing research Geometric Phase Transitions - Stability Dependence of Holographic EE on Geometry The Geometric Feature Determining Logarithms

The above implies that the logarithmic term is not a function of the angle taking values between 0 and 2휋, but of the ratio of separation of the neighbouring degrees of freedom,

A 휆 := lim sector within region A , (1) r →0 2 0 휋r0 taking values between 0 and 1. A smooth curve has always 휆 = 1/2. It is interesting to investigate whether for smooth surfaces the universal term can be derived in a similar manner as a function of the curvature, which determines the way in which 휆 tends to 1/2.

Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Elliptic Minimal surfaces in AdS4 Ongoing research

Ongoing research

I Study the dependence of EE on the entangling curve geometry and its dependence on the energy cutoff, as a geometric flow acting on the minimal surface I Extend the family of minimal surfaces using the “dressing technique” I Verify the equivalence between the first law of entanglement thermodynamics and linearized Einstein equations for elliptic minimal surfaces I Investigate whether the gravity can be related to quantum entanglement beyond holographic theories. For this purpose the study of entanglement in field theory is required, which is interesting on its own sake. 11 11D. Katsinis and G. Pastras, arXiv:1711.02618 Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Elliptic Minimal surfaces in AdS4 Ongoing research

Thank you!

Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force Introduction Elliptic Minimal surfaces in AdS4 Ongoing research

The research of G.P. is funded by the “Post-doctoral researchers support” action of the operational programme “human resources development, education and long life learning, 2014-2020”, with priority axes 6, 8 and 9, implemented by the Greek State Scholarship Foundation and co-funded by the European Social Fund - ESF and National Resources of Greece.

Georgios Pastras Quantum Entanglement and Gravity as an Entropic Force