Correlating Black Holes, The AdS/CFT Duality & The Second Law Of Quantum Complexity
Beyond Equilibrium: A Quantum Dance of Complexity
When it came to the pioneers of thermodynamics in the Victorian era, heat-death became common ground that the laws of thermodynamics were based upon.
You drop ice cubes into a glass of water, you create a state that is out of equilibrium. Then the ice melts, the liquid chills and it reaches a common temperature, or equilibrium( However, motion does not cease i.e. the water molecules continue to move randomly).
This concept of equilibrium, as the 19th century thermodynamics proponents discovered, could be applied to the universe as a whole.
They suggested that after the stars all burn out, what remains i.e.— dust, gas, stars and stellar corpses, radiation etc. will all come to a state of equilibrium.
“The universe from that time forward would be condemned to a state of eternal rest,” wrote Hermann von Helmholtz in 1854. Modern cosmology is based on this basic picture.
Which is why heat death shared a sort of romanticized fascination at the time. Which is perhaps because their little law described the universe.
When Black Holes Come Into The Picture
However, this equilibrium in a heat-dead universe has pushed modern theoretical physicists to question and probe further.
As recounted in Quanta Magazine:
“Lately physicists have been thinking that a supposedly heat-dead universe is a lot more interesting than it looks. Their story starts with a question about black holes — another riddle beyond the ones that get the most attention.”
“According to our standard understanding of black holes, they continue to change long after they should have come to equilibrium. An investigation into why has led researchers to reconsider how things in general evolve — including the universe itself.”
“No one much thought about this because it’s just sort of boring: It looks like equilibrium, and nothing happens,” said Brian Swingle, a physicist at Brandeis University. “But then along came black holes.”
“When an ice cube melts and attains equilibrium with the liquid, physicists usually say the evolution of the system has ended. But it hasn’t — there is life after heat death. Weird and wonderful things continue to happen at the quantum level.”
“If you really look into a quantum system, the particle distribution might have equilibrated, and the energy distribution might have equilibrated, but there’s still so much more going on beyond that,” said Xie Chen, a theoretical physicist at the California Institute of Technology.
“Chen, Swingle and others think that, if an equilibrated system looks boring and blah, we’re just not looking at it in the right way. The action has moved from quantities that we can see directly to highly delocalized ones that require new measures to track.”
Black holes created a sensation and the world’s greatest minds offered their hypotheses on how black holes function.
As explained on Quanta:
In the mid-20th century, black holes were mysterious because of their “singularity” at their core, a place where infalling matter becomes infinitely compacted, gravity intensifies without limit, and the known laws of physics break down.”
“In the 1970s Stephen Hawking realized that the perimeter or “horizon” of a black hole is equally weird, creating the much-discussed information paradox. Both puzzles continue to perplex theorists, and they drive the search for a unified theory of physics.”
“In 2014 Leonard Susskind of Stanford University identified yet another conundrum: the black hole’s interior volume. From the outside, a black hole looks like a big black ball. According to Einstein’s general theory of relativity, the ball grows when stuff falls in, but otherwise it just sits there.”
When it came to calculating the volume of a black hole’s interiors or its interior volume, you had to split up “space” and “time” from Einstein’s “spacetime” because the interiors are so “warped” so slicing became the only way to measure the black hole’s volume.
As Quanta explain:
The inside looks very different, however. The spherical volume formula that you learned in grade school doesn’t apply. The problem is that spatial volume is defined at one moment in time. To calculate it, you have to slice up the space-time continuum into “space” and “time,” and inside a black hole there is no unique way to do that.
Susskind argued that the most natural choice is a slicing process that maximizes the spatial volume at every moment; by the logic of relativity, it amounts to the shortest distance across the hole.”
“It’s a natural volume analogue of the shortest-line rule,” said Adam Brown, a physicist at Stanford. And because the interior space-time is so warped, the volume by this measure grows with time forever. “The slice on which I measure this volume gets deformed more and more,” said Luca Iliesiu, a physicist also at Stanford.
Basically, physicists theorized that a black hole should also observe the laws of thermodynamics, but it doesn’t.
“This growth is weird because the black hole should be governed by the same laws of thermodynamics as the glass of water. If the ice and liquid eventually reach equilibrium, so should the hole. It should stabilize, not grow forever.”
In essence what is described above is the paradox — Black holes continue to grow and evolve even after reaching equilibrium.
AdS/CFT Duality
This brings us to the AdS/CFT duality, which is a newer form of thinking that can explain the paradox, because of how it is theorized, which is explained in this short video below.
Using the AdS/CFT duality enabled Susskind to describe the black hole phenomenon, not exactly, but moreover as a system, which was as close to the real thing:
As Quanta explore:
“To formulate the paradox, Susskind applied a form of lateral thinking. The strategy, known as the AdS/CFT duality, conjectures that any situation in fundamental physics can be viewed in two mathematically equivalent ways, one with gravity, one without.
The black hole is a strongly gravitating system — there are none stronger. It is mathematically equivalent to a nongravitational but strongly quantum system. In technical terms, the black hole is equivalent to a thermal state of quantum fields — essentially, a hot plasma made up of nuclear particles.”
“A black hole doesn’t look anything like a hot plasma, nor does a plasma seem to have anything to do with a black hole. That is what makes the duality so powerful. It relates two things that should not be related.
If someone gave you such a plasma, you could measure its temperature, and that would be the temperature of the black hole. If you dropped material into the plasma, a ripple would reverberate through it, and that would be like the black hole swallowing an object.”
“The ripple gradually dissipates, and things return to equilibrium,” said Suvrat Raju, a theoretical physicist at the International Center for Theoretical Sciences in Bengaluru who has studied how AdS/CFT describes black holes.
“The duality swaps the weirdness of gravity for the complexities of quantum theory, which for Susskind was an improvement. It let him pose the question of how the black hole should or shouldn’t evolve.”
“A plasma reaches equilibrium quickly; its overall properties stop changing. But if it is mathematically equivalent to a black hole, whose interior volume continues to grow, something about the plasma must continue to evolve.”
“Casting about for what that property could be, he proposed something that seems, on the face of it, to have nothing at all to do with either plasmas or black holes — or indeed, any physical system at all.”
Circuit Complexity & Using It To Describe The Quantum Happenings
Coming to what is perhaps rather extraordinary, in the sense that, this theory was formulated mainly in the realms of computer science, but as theoretical physicists have found out; it describes quantum complexity.
That’s just like how Veneziano and Suzuki found the Euler Beta Function that seemed to describe string theory.
As Quanta Magazine suggest:
“The favorite measure, at the moment, is known as circuit complexity. The concept originated in computer science and has been appropriated — misappropriated, some have grumbled — to quantify the blossoming patterns in a quantum system.”
“The work is fascinating for the way it brings together multiple areas of science, not just black holes but also quantum chaos, topological phases of matter, cryptography, quantum computers, and the possibility of even more powerful machines.”
In particular, Susskind proposed a particular property known as circuit complexity.
As explained on Quanta:
“The word “circuit” has its origins in the “switching circuits” once used to route telephone calls. These circuits carry signals that are controlled by “gates,” which are electronic components that perform logical or arithmetic operations. A few basic types of gates can be strung together to implement more involved operations. All ordinary computers are built this way.
The inventors of quantum computers adopted the same framework. A quantum circuit acts on its basic units of information, qubits, using a standardized repertoire of gates. Some gates perform familiar operations such as addition, while others are quintessentially quantum. A “controlled NOT” gate, for example, can bind together two or more qubits into an indivisible whole, known as an entangled state.
Inside a quantum computer, qubits might be particles, ions or superconducting current loops.
But in general, their precise physical form doesn’t matter. Any system composed of discrete units can be recast as a circuit, even a system that looks nothing like a computer.
“The air molecules in a room are moving around and banging into each other, and we can think of any collision as a gate,” said Nicole Yunger Halpern, a quantum information theorist at the University of Maryland.
Though a technical concept, circuit complexity is not far from what we mean by “complexity” in daily life.
When we say a job is complex, we usually mean that it involves a large number of steps. In a quantum system, complexity is the number of elementary gates (or operations) needed to replicate a particular state.
Complexity according to this definition is an integer — the number of gates — but researchers have also explored using geometric concepts to define complexity as a continuous or real number.
However, here’s the part that correlates the ‘circuit complexity’ concept applying it to black holes:
Susskind applied this concept to the hot plasmas that, through the AdS/CFT duality, are equivalent to black holes. He suggested that, even after the plasma reaches a condition of thermal equilibrium, its quantum state does not stop evolving. It becomes ever more complex. The ripples that are reverberating through the plasma dissipate but do not entirely go away, and they are still there if you look at the plasma at the quantum level. Attempting to re-create another plasma with the same pattern of ripples would become increasingly laborious.”
Susskind’s findings theorized the following:
- The black hole is equivalent to a nuclear plasma
- the volume of the black hole is mathematically equivalent to the circuit complexity of the plasma
- and because the circuit complexity keeps growing, so must the volume
The Second Law Of Quantum Complexity
Out of this entire theoretical process, Susskind suggests a new law, called the second law of quantum complexity — a quantum analogue of the second law of thermodynamics.
As the abstract section of Adam R. Brown and Leonard Susskind’s paper explains:
We give arguments for the existence of a thermodynamics of quantum complexity that includes a “second law of complexity.” To guide us, we derive a correspondence between the computational (circuit) complexity of a quantum system of K qubits, and the positional entropy of a related classical system with 2K degrees of freedom.
We also argue that the kinetic entropy of the classical system is equivalent to the Kolmogorov complexity of the quantum Hamiltonian. We observe that the expected pattern of growth of the complexity of the quantum system parallels the growth of entropy of the classical system.
We argue that the property of having less-than-maximal complexity (uncomplexity) is a resource that can be expended to perform directed quantum computation. Although this paper is not primarily about black holes, we find a surprising interpretation of the uncomplexity resource as the accessible volume of spacetime behind a black hole horizon.
And further as Quanta suggest:
“Not letting the lack of a rigorous proof stop them, Susskind and Brown suggested in 2018 that the steady growth of complexity qualifies as a new law of nature, the second law of thermodynamics holds that closed systems increase in entropy until they reach thermal equilibrium, the state of maximal entropy.”
“According to Susskind and Brown, the same happens with complexity. A system increases in complexity for eons after it reaches thermal equilibrium. But it does eventually plateau, reaching “complexity equilibrium.” At that point, a quantum system has explored every possible state it is capable of and will finally lose any sense of progress.”
“The eventual plateauing of circuit complexity led Susskind to revisit his original motivation for considering circuit complexity — namely, the growth of black hole interiors.
General relativity predicts that they grow forever, but the fun has to end sometime. That means general relativity itself must eventually fail. Theorists already had plenty of reasons to suspect that black holes ultimately need to be described by a quantum theory of gravity, but the cessation of volume growth is a new one.”
In 2021 Iliesiu, Márk Mezei of Oxford University, and Gábor Sárosi of CERN studied what that means for black holes. They used a standard quantum physics method known as the path integral, which has the nice feature of being agnostic to whatever the full quantum theory of gravity is, be it string theory or one of its competitors.
As Luca V. Iliesiu, Márk Mezei, Gábor Sárosi’s paper on The volume of the black hole interior at late times explains:
Understanding the fate of semi-classical black hole solutions at very late times is one of the most important open questions in quantum gravity. In this paper, we provide a path integral definition of the volume of the black hole interior and study it at arbitrarily late times for black holes in various models of two-dimensional gravity.
Because of a novel universal cancellation between the contributions of the semi-classical black hole spectrum and some of its non-perturbative corrections, we find that, after a linear growth at early times, the length of the interior saturates at a time, and towards a value, that is exponentially large in the entropy of the black hole.
This provides a non-perturbative confirmation of the complexity equals volume proposal since complexity is also expected to plateau at the same value and at the same time.
Closing Thoughts (Simplified)
This might seem a bit overwhelming when it comes to understanding the premise and concepts in this article. However, I will try to summarize and break it down is a simple terms as possible.
Coming to summarizing the concepts and this article: Could the complexity used in quantum computing explain the workings of black holes and in a larger perspective the system that is the universe?
In summary, this article dissects and explores the connection between black holes, the AdS/CFT duality, and the second law of quantum complexity.
I started by mentioning the concept of heat-death and equilibrium in thermodynamics and how it applies to the universe as a whole. And then introduced black holes and their role in challenging the idea of equilibrium.
Physicists have discovered that black holes continue to evolve and change even after reaching equilibrium, contrary to the expectations based on thermodynamics.
The AdS/CFT duality is introduced as a theoretical framework that relates black holes to quantum systems, specifically hot plasmas.
Through this duality, circuit complexity, a concept from computer science, is applied to describe the evolving quantum state of the plasma and its relation to the volume of a black hole.
This leads to the proposal of the second law of quantum complexity, which suggests that quantum systems continue to increase in complexity even after reaching thermal equilibrium.
I conclude by discussing the implications of this law for black holes and the eventual plateauing of complexity.
It suggests that the cessation of volume growth in black holes indicates the need for a quantum theory of gravity.
Overall, to conclude, the article explores the potential of complexity theory to explain the behavior of black holes and the universe as a whole.
Note: This article is based on this article on Quanta Magazine.
