# My Research

I work primarily on the AdS/CFT correspondence. Most of my work so far has been focused on the idea of holographic complexity. However, I’ve also spent some time thinking about extended black hole thermodynamics, the scrambling of quantum information, and the thermodynamics of finite causal diamonds. My interests for future work lie broadly at the intersection of quantum information and quantum gravity. I am also interested in thinking about the firewall paradox, bulk reconstruction in AdS/CFT, tensor networks, error-correcting codes, and more.

### Spacetime Thermodynamics

I began my work in AdS/CFT thinking about extended spacetime thermodynamics. The idea of extended thermodynamics is to consider the cosmological constant as a thermodynamic variable, in many ways analogous to the pressure term in the first law for a gas. By considering variations of the cosmological constant, one can find its conjugate ‘thermodynamic volume.’ In simple cases, such as for ordinary Schwarzchild black holes in 3+1 dimensions, this volume reproduces the ‘naive’ volume of the black hole interior, namely $\frac{4}{3} \pi r_h^3$, where $r_h$ is the horizon radius. Other black holes can have more complicated, less provocative expressions.

The project which led to my first published work started with the goal of better understanding the physical significance of the thermodynamic volume in extended thermodynamics. In other words, sure I can vary the cosmological constant as a thermodynamical variable, and sure this leads to a conjugate quantity, but what does that quantity represent physically? Within the AdS/CFT correspondence, one may further ask what this quantity means from the view of the dual theory on the boundary. Of course, from a stringy point of view, the cosmological constant is set by the number of coincident branes, and as such, the thermodynamic volume is simply a chemical potential on e.g., D3-branes in your stack (I believe this point of view is developed in the work of e.g., Clifford Johnson). Is this, however, all that can be said?

In the work of my collaborators and I, we ultimately noticed that the thermodynamic volume shows up naturally in the ‘complexity = action’ conjecture, when one computes the complexification rate for various black holes. This late-time asymptotic value of this complexification rate can be decomposed into geometric pieces, which for AdS-Schwarzschild black holes are associated to the horizon, to the singularity, and to the bulk respectively. The bulk portion, when computed, is proportional to the thermodynamic volume time the ‘pressure’ defined by the cosmological constant. This observation led us to conjecture that the thermodynamic volume is intimately connected with the rate of increase in the complexity of the boundary state.

More recently, I’ve spent some time thinking about ways to extend results about the thermodynamics of finite causal diamonds. In a recent paper by Jacobson and Visser, the authors derive a first law for maximally symmetric finite causal diamonds by considering the Iyer-Wald formalism for a conformal killing vector rather than an actual killing vector. Together with my advisor and other students, I’ve been discussing how these results might be generalized to less restrictive cases.

### Holographic complexity

Coming back to holographic complexity, I have, since the project mentioned above, thought more about the holographic complexity conjectures.

The idea of holographic complexity was first proposed by Leonard Susskind and collaborators, first as “complexity = volume” (see e.g. here and here) and later as “complexity = action” (e.g. here and here). To arrive at these conjectures, Susskind et al. start by noting that the size of the behind the horizon region of a two-sided black hole increases with time, even though in the AdS context, this should be dual to the thermofield double state. What property, they ask, could possibly be dual to this growth? The answer they come up with is the circuit complexity of the dual quantum state.

The circuit complexity of a quantum state is defined with respect to a reference state, a set of unitary gates, and a tolerance. It is defined as the minimum number of gates needed to build a quantum circuit which, when fed the reference state as input, outputs our state to within the tolerance. Susskind and his collaborators have presented various qualitative arguments suggestive of the idea that a version of this quantity may indeed be connected to the interior of black holes.

Given these conjectures, and others that have been made (one due to myself and collaborators), we would like to be able to determine whether any of these conjectures, or perhaps some refinement thereof, are correct. If they are correct, it would be nice to work out the details, such as what set of gates and what reference state are used to define the dual complexity on the boundary. In the absence of a rigorous derivation from first principles, the best one may hope to do is to work out the consequences of these conjectures in some concrete examples and check whether the results are consistent with what we know, both about AdS/CFT and about complexity.

It is in that spirit that my collaborators and I decided to study the complexity = action conjecture in black holes dual to $\mathcal{N} = 4$ Super Yang-Mills on a non-commutative manifold. In that work, we found that the complexification rate (i.e., the rate at which the purported complexity increases with boundary time) at late times increases with the Moyal scale, which characterizes the ‘size’ of the non-commutativity. We then develop a qualitative circuit argument that perhaps one should have expected such an increase for non-local systems, simply because the gates needed to build the time evolution operator are less likely to commute. The results of this study can be found here. In the future, we would like to test our qualitative argument by explicitly computing complexity (for some choice of gates set, reference, and tolerance) in a non-local theory, thereby completing the consistency check.

In addition to the work with complexity in a non-commutative theory, and the earlier work on extended thermodynamics, I have pursued a few other projects on the topic of holographic complexity. In a paper with collaborators at UT and Maryland, we revisit the ‘complexity = volume’ conjecture, studying its properties in more detail. In another paper with collaborators at UT, we study so-called ‘subregion complexity,’ the result of applying one of the conjectures above to the entanglement wedge of a subregion of the boundary, considered as its own spacetime with boundary. We attempted to do a consistency check on the idea that this quantity could be related to the so-called ‘purification complexity’ of a mixed quantum state. Finally, in work soon to be released on the arxiv, with collaborators at UT and Perimeter, I have studied how one might compute purification complexity for Gaussian mixed state in a regulated free scalar field theory.

### Other Work

In addition to the above, I have been working recently on understanding the relationships between various velocity scales involved in the scrambling of quantum information in holographic systems with a uniform insertion of energy and/or charge at a given point in time. Given an initially localized perturbation to the system, e.g., in which some local degrees of freedom are entangled with some reference system, the information speed measures the rate of growth of the smallest region from which one may recover the information content of the perturbation, e.g., its entanglement with the reference. The entanglement speed, on the other hand, is proportional to the rate of growth of the entanglement entropy of a fixed, large region, during the regime of linear growth after a quench.

Brian Swingle, who led the project, conjectured initially based on general quantum information arguments that these speeds are related by $V_I = \frac{V_E(f)}{1-f}$, which we have demonstrated for a large class of systems through a combination of analytic and numerical results in both holography and a spin chain system.

## Talks

Here is a list of my talks and presentations:

## Publications

#### E. Cáceres, J. Couch, S. Eccles and W. Fischler, Holographic Purification Complexity Phys. Rev. D 99, no. 8, 086016 (2019) doi:10.1103/PhysRevD.99.086016 [arxiv:1811.10650]

We study holographic subregion complexity, and its possible connection to purification complexity suggested recently by Agón et al. In particular, we study the conjecture that subregion complexity is the purification complexity by considering holographic purifications of a holographic mixed state. We argue that these include states with any amount of coarse-graining consistent with being a purification of the mixed state in question, corresponding holographically to different choices of the cutoff surface. We find that within the complexity = volume and complexity = spacetime volume conjectures, the subregion complexity is equal to the holographic purification complexity. For complexity = action, the subregion complexity seems to provide an upper bound on the holographic purification complexity, though we show cases where this bound is not saturated. One such example is provided by black holes with a large genus behind the horizon, which were studied by Fu et al. As such, one must conclude that these offending geometries are not holographic, that CA must be modified, or else that holographic subregion complexity in CA is not dual to the purification complexity of the corresponding reduced state.

#### J. Couch, S. Eccles, T. Jacobson and P. Nguyen, Holographic Complexity and Volume, JHEP 1811, 044 (2018) doi:10.1007/JHEP11(2018)044 [arxiv:1807.02186]

The previously proposed “Complexity=Volume” or CV-duality is probed and developed in several directions. We show that the apparent lack of universality for large and small black holes is removed if the volume is measured in units of the maximal time from the horizon to the “final slice” (times Planck area). This also works for spinning black holes. We make use of the conserved “volume current”, associated with a foliation of spacetime by maximal volume slices, whose flux measures their volume. This flux picture suggests that there is a transfer of the complexity from the UV to the IR in holographic CFTs, which is reminiscent of thermalization behavior deduced using holography. It also naturally gives a second law for the complexity when applied at a black hole horizon. We further establish a result supporting the conjecture that a boundary foliation determines a bulk maximal foliation without gaps, establish a global inequality on maximal volumes that can be used to deduce the monotonicity of the complexification rate on a boost-invariant background, and probe CV duality in the settings of multiple quenches, spinning black holes, and Rindler-AdS.

#### J. Couch, S. Eccles, W. Fischler and M. L. Xiao, Holographic complexity and non-commutative gauge theory, JHEP 1803, 108 (2018) doi:10.1007/JHEP03(2018)108 [arxiv:1710.07833]

We study the holographic complexity of noncommutative field theories. The four-dimensional N=4 noncommutative super Yang-Mills theory with Moyal algebra along two of the spatial directions has a well known holographic dual as a type IIB supergravity theory with a stack of D3 branes and non-trivial NS-NS B fields. We start from this example and find that the late time holographic complexity growth rate, based on the “complexity equals action” conjecture, experiences an enhancement when the non-commutativity is turned on. This enhancement saturates a new limit which is exactly 1/4 larger than the commutative value. We then attempt to give a quantum mechanics explanation of the enhancement. Finite time behavior of the complexity growth rate is also studied. Inspired by the non-trivial result, we move on to more general setup in string theory where we have a stack of Dp branes and also turn on the B field. Multiple noncommutative directions are considered in higher p cases.

#### J. Couch, W. Fischler and P. H. Nguyen, Noether charge, black hole volume, and complexity, JHEP 1703, 119 (2017) doi:10.1007/JHEP03(2017)119 [arxiv:1610.02038]

In this paper, we study the physical significance of the thermodynamic volumes of AdS black holes using the Noether charge formalism of Iyer and Wald. After applying this formalism to study the extended thermodynamics of a few examples, we discuss how the extended thermodynamics interacts with the recent complexity = action proposal of Brown et al. (CA-duality). We, in particular, discover that their proposal for the late time rate of change of complexity has a nice decomposition in terms of thermodynamic quantities reminiscent of the Smarr relation. This decomposition strongly suggests a geometric, and via CA-duality holographic, interpretation for the thermodynamic volume of an AdS black hole. We go on to discuss the role of thermodynamics in complexity = action for a number of black hole solutions, and then point out the possibility of an alternate proposal, which we dub “complexity = volume 2.0”. In this alternate proposal the complexity would be thought of as the spacetime volume of the Wheeler-DeWitt patch. Finally, we provide evidence that, in certain cases, our proposal for complexity is consistent with the Lloyd bound whereas CA-duality is not.-