The AdS/CFT duality has been reliably blowing people’s minds for over 25 years. It’s hard to visualize how such different systems as a black hole and a hot plasma could be equivalent. Now it seems that the difficulty is not just a failure of the human imagination, but a feature of the mathematics.
As a result of all this, computer scientists have come around to Susskind’s view that circuit complexity is a perfectly legitimate physical quantity. They hadn’t liked it because it was hard, verging on impossible, to measure or compute. But if the black-hole-to-plasma translation is hard, then any quantity that is equivalent to black hole volume is going to be hard to compute. The difficulty of calculating circuit complexity is not a strike against it. To the contrary, it is precisely what you would expect. When the translation is hard, a measurable physics quantity on one side is necessarily “unfeelable” on the other. “The unfeelability is simply a reflection of the extreme difficulty of making the dictionary from one to the other,” Susskind said. “I think physicists have not really realized the implications of this.”
Susskind is also gratified that his sharpest critics have become his closest allies. “I was amused,” he said. “I watched them go through their contortions. They’re very good scientists. And in the end the conclusion was no, complexity is the only possible thing that it could be.”
The Pace of Space
A second potential problem with Susskind’s conjecture is that the circuit complexity of a hot plasma might not grow at the right rate. It seems intuitive, even trivial, that circuit complexity would grow with time. With each passing moment, more happens to the hot plasma. So it stands to reason that ever more operations would be required to reproduce its present state.
The trouble, though, is that circuit complexity is being pressed into a task for which it was not originally intended. The operations occurring in the hot plasma are uncontrolled random interactions, not the predictable logic operations of a computer algorithm. So theorists can’t be sure what will happen. The plasma might undergo a million interactions, creating an increasingly complex quantum state, and then the next interaction might abruptly leave it in a simple state — one that could have been created using a mere 1,000 interactions. It wouldn’t matter that the plasma had undergone a million and one interactions; the complexity is defined by the number of interactions it needs to undergo to reach the end point.
It would be like setting out to explore your neighborhood, turning left at some intersections and right at others, and eventually arriving at a hole-in-the-wall restaurant you’d never seen before. Your sense of accomplishment would turn to chagrin when you realized it was just across the street from your house. The distance from your house to the restaurant depends on their relative positions, not on how much walking you’ve done.
Susskind’s original argument for why this shouldn’t happen — why the complexity should grow in a continuous linear trend — was that the space of possibilities is, to quote Douglas Adams, vastly, hugely, mind-bogglingly big. Susskind thought it very unlikely that the system would stumble into a simpler state. But turning this intuition into a solid argument has been tough.
In one of several approaches that theorists have taken, Fernando Brandão, a quantum computing scientist at Caltech, and his co-authors studied what happens when a system undergoes one random interaction after another. It enters states that are uniformly spread through the space of possibilities, forming a set known as a design. It turns out that a chaotic system will naturally create a sequence of designs that approximate a truly random distribution with increasing refinement. Because randomness is maximal complexity, getting closer to randomness means that the system grows ever more complex, and at nearly the same rate at which the black hole interior grows bigger.
But Brandão’s approach and others make some debatable simplifications, and not all match the black hole perfectly, so a full proof remains on theorists’ to-do list.
A New Second Law
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 quantum complexity — a quantum analogue of the second law of thermodynamics. 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. The theorists found that quantum effects accumulate like barnacles on a ship’s hull and eventually arrest the growth of the interior. At that point, the black hole’s interior geometry changes. This is an additional milestone in black hole evolution, with no obvious relation to events that theorists already knew about, such as the final evaporation and disappearance of the object.
The Five Stages of Quantum Systems
So far, all of this concerns black holes. But the black holes really just reveal a more general principle about matter. Gradually emerging from all this work is a picture of the full life cycle of quantum systems — the chaotic ones, which means most of them, including the universe as a whole. According to this picture, they go through five distinct stages.
The first is initialization. The system starts simply: just a bunch of particles or other building blocks, acting independently.
Then comes thermalization. The particles bounce around and collide with one another, eventually reaching thermal equilibrium. Their shenanigans also begin to link the particles through quantum entanglement. In a process that Susskind calls “scrambling,” information is disseminated through the system until it no longer resides in localized places, much as a butterfly flapping its wings in Brazil can affect weather over the whole globe. “Operators that are initially local have spread over the entire system in a butterfly-effect-like way,” said Nick Hunter-Jones, a theoretical physicist at the University of Texas, Austin.
Next is complexification. Here, the system is in thermal equilibrium but has not stopped evolving. It keeps getting more complex, but in a way that is almost invisible to standard measures such as entropy. Theorists rely instead on circuit complexity, which expresses the increasingly intricate linkages among entangled particles. “Complexity is really like a microscope into the entanglement structure of the system,” Hunter-Jones said. This stage lasts exponentially longer than thermalization.
Then the system reaches complexity equilibrium, where complexity hits a ceiling. Although the system continues to change, it can no longer be said to evolve — it has no sense of directedness, but wanders among equal states of maximal complexity.
The last stage is called recurrence, where the system stumbles back into its original simple condition. For this to happen by accident is highly improbable. But eternity is a long time, so it ultimately does happen, after a period of time that is not merely exponential, but an exponential of an exponential. The whole process then repeats.
In short, quantum systems that reach thermal equilibrium are like the happy couples in romantic comedies. The film typically ends when the couple gets married, as if that were the end of one’s love life. In reality, it’s just the start.