Complexity of formation in holography
Received: November
Complexity of formation in holography
Shira Chapman 0 1 3
Hugo Marrochio 0 1 2 3
Robert C. Myers 0 1 3
Open Access 0 1
c The Authors. 0 1
0 University of Waterloo , Waterloo, ON N2L 3G1 , Canada
1 Waterloo , ON N2L 2Y5 , Canada
2 Department of Physics & Astronomy and Guelph-Waterloo Physics Institute
3 Perimeter Institute for Theoretical Physics
It was recently conjectured that the quantum complexity of a holographic boundary state can be computed by evaluating the gravitational action on a bulk region known as the Wheeler-DeWitt patch. We apply this complexity=action duality to evaluate the `complexity of formation' [1, 2], i.e. the additional complexity arising in preparing the entangled thermo eld double state with two copies of the boundary CFT compared to preparing the individual vacuum states of the two copies. We nd that for boundary dimensions d > 2, the di erence in the complexities grows linearly with the thermal entropy at high temperatures. For the special case d = 2, the complexity of formation is a constant, independent of the temperature. We compare these results to those found using the complexity=volume duality.
AdS-CFT Correspondence; Black Holes
1 Introduction
2 General framework 2.1 Evaluating the action
3 Complexity of formation Bulk contribution Surface contributions Joint contributions
d = 4
3.2 d = 3
Planar case for general d
4 Complexity of BTZ black holes
5 Comparison with complexity = volume 5.1 Planar geometry
5.2 Spherical and hyperbolic geometries
6 Discussion
A Fe erman-Graham near boundary expansions
A.1 Relating the cuto s
A.2 Cuto independence of the action
B Details for vacuum AdS actions
C Small hyperbolic black holes
C.1 d = 4
C.2 d = 3
C.3 Late-time growth of complexity
D Ambiguities in the action calculations
D.1 Rede nition of the function de ning the null hypersurface
D.2 Reparameterizations
D.3 Changing the normalization condition at the boundary
D.4 A comment on the cuto choice
E Insights from MERA
In recent years, it has become widely appreciated that quantum information theory is a
fruitful lens with which to examine the conundrums of quantum gravity. While most of
the ongoing research has focused on holographic entanglement entropy [3, 4], `quantum
complexity' (e.g., see [5{7]) is another concept from quantum information theory that has
recently found a place in this discussion. These ideas emerged from studies aimed at
understanding the growth of the Einstein-Rosen bridge for AdS black holes in terms of
quantum complexity in the dual boundary CFT [8{11].
Loosely speaking, the complexity C of a particular state j i is the minimum number of
quantum gates required to produce this state from a particular reference state j 0i.1 Now
in the context of the AdS/CFT correspondence, two proposals have been made to evaluate
the complexity of a boundary state: the rst is that the complexity should be dual to the
volume of the extremal codimension-one bulk hypersurface which meets the asymptotic
boundary on the time slice where the boundary state is de ned [10] | see section 5. The
second conjecture states [1, 2]
C =
where I is the gravitational action evaluated on a particular spacetime region in the bulk,
known as the `Wheeler-DeWitt (WDW) patch.' In particular, the WDW patch is the
region enclosed by past and future light sheets sent into the bulk spacetime from the time
slice on the boundary, e.g., see gure 1.
Both of these holographic conjectures satisfy a number of properties expected of
complexity, e.g., they continue to grow (linearly with time) after the boundary theory reaches
thermal equilibrium. However, the second conjecture has certain advantages. In
particrelating the bulk geometric quantity to the complexity | see eq. (5.1). However, the
posed, there was no rigorous method for evaluating the gravitational action on spacetime
regions with null boundaries. However, this problem was recently overcome with a careful
analysis of the boundary terms which must be added to the gravitational action for null
boundary surfaces and for joints where such null boundaries intersect with other boundary
surfaces [16].
The new boundary terms developed in [16] have opened up the possibility of
investiof the present paper. In particular, we use the new boundary terms to answer a question
posed in [1, 2]: what is the `complexity of formation' for a thermal state of temperature T ?
That is, the full geometry of an eternal AdS black hole can be interpreted as being dual to
the thermo eld double state [17],
1See [5, 11] for further details. A more re ned de nition is still required for application to continuum
quantum eld theories [2], perhaps using the geometric perspective of [12{14] | see also [15].
where we have two copies of the CFT, which are associated with the left (L) and right
(R) asymptotic boundaries | see the Penrose diagram in
gure 1. Of course, integrating
out either the left or right copy in the above state leaves us with th (...truncated)