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Fast scrambling in holographic Einstein-Podolsky-Rosen pair
HJE
Fast scrambling in holographic Einstein-Podolsky-Rosen pair
Keiju Murata 0
0 Department of Physics, Hiyoshi Campus, Keio University
We demonstrate that a holographic model of the Einstein-Podolsky-Rosen pair exhibits fast scrambling. Strongly entangled quark and antiquark in N = 4 super YangMills theory are considered. Their gravity dual is a fundamental string whose endpoints are uniformly accelerated in opposite direction. We slightly increase the acceleration of the endpoint and show that it quickly destroys the correlation between the quark and antiquark. The proper time scale of the destruction is τ∗ ∼ β ln S where β is the inverse Unruh temperature and S is the entropy of the accelerating quark. We also evaluate the Lyapunov exponent from correlation function as λL = 2π/β, which saturates the Lyapunov bound. Our results suggest that the fast scrambling or saturation of the Lyapunov bound do not directly imply the existence of an Einstein dual. When we slightly decrease the acceleration, the quark and antiquark are causally connected and an “one-way traversable wormhole” is created on the worldsheet. It causes the divergence of the correlation function between the quark and antiquark.
2D Gravity; AdS-CFT Correspondence; Black Holes
1 Introduction 2
Holographic EPR pair
Exact solution for time dependant open strings
Holographic EPR pair
Thermodynamical variables
3
4
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6
1
2.1
2.2
2.3
3.1
3.2
4.1
4.2
4.3
Holographic EPR pair with a shock
Perturbed holographic EPR string
Geodesic distance
Fast scrambling
Correlation function
Correlation in the laboratory frame
Decreasing the acceleration
Two shocks
Conclusion
Introduction
A local excitation of a quantum chaotic system spreads out over the entire system. This
delocalization of the quantum information is called “scrambling”. It has been believed
that the scrambling is related to the quantum chaos or the butterfly effect: initially similar
states evolves completely different state at the late time. The scrambling behaviour in
strongly coupled systems attracts much attention in the context of black hole physics or
AdS/CFT correspondence [1–3].
It has been conjectured that “Back holes are the fastest scramblers in nature” [4].
They demonstrated the delocalization of local information on a black hole horizon and
estimated its time scale as t∗ ∼ β ln S where β is the inverse Hawking temperature and S
is the Beckenstein-Hawking entropy. This time scale is much quicker than that for usual
quantum many body systems. In ref. [5], using the AdS/CFT correspondence, they
developed a formulation to quantify the scrambling in more concrete way. (See also refs. [
6–14
])
They found that the scrambling behaviour can be read out from the mutual information
or correlation function between two boundaries of an eternal AdS black hole. Especially,
the correlation function relates to the out-of-time-order correlator (OTOC). In ref. [15],
it has been proposed that the OTOC can be a measure of the quantum chaos and define
the “quantum Lyapunov exponent” from the OTOC. It has also been conjectured that the
– 1 –
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Poincaré horizon
A
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(a) String profile
(b) Spacetime structure
If the system exhibit chaos, the quantum state of the antiquark is scrambled by the
perturbation and, as the result, the subtle relation between quark and antiquark is destroyed.
We measure the strength of the entanglement by the correlation function between quark
and antiquark, which is equivalent to the OTOC. We will see that the proper time scale of
the decay of the correlation is given by τ∗ ∼ β ln S where β is the inverse Unruh
temperature and S is the thermal entropy of the quark and gluons surrounding it. The Lyapunov
– 2 –
exponent read from the OTOC saturates the bound as λL = 2π/β. The dual picture of
the EPR pair is the probe string in the fixed background. In that sense, the dual theory is
not Einstein gravity. Nevertheless, we can see the fast scrambling.
The organization of this paper is as follows. In section 2, we introduce the holographic
EPR pair as a solution in wide class of time dependant string solutions. We also see that
the spacetime structure of the string worldsheet is same as an eternal AdS black hole. In
section 3, we explicitly construct the string solution with a perturbation. Here, we consider
the tiny change of the acceleration as the perturbation. The geodesic distance between two
endpoints of the perturbed string along the worldsheet is computed. In section 4, we
compute the correlation function between the quark and antiquark. It decays quickly and
we find the fast scrambling result.
We also evaluate the Lyapunov exponent and find
that it saturates the bound in ref. [15]. The effect of the decreased acceleration is also
considered. It changes the causal structure of the worldsheet drastically and creates the
“one-way traversable wormhole”. We see the divergence of the correlation in th (...truncated)