# How Is a Black Hole Like a Glass of Ice Water?

### Both tend toward thermal equilibrium—and, beyond that, to an ever-increasing quantum complexity.

Welcome to the inaugural post of my Substack blog! I’ll be sharing ideas and epiphanies about physics and other sciences—things that for whatever reason never make it into magazine articles that I write, but that I stilll find fascinating.

One thing that makes fundamental physics so much fun is the lateral thinking, the connections that researchers make between seemingly unrelated phenomena. Black holes, for example, seem about as far from everyday life as it is possible to imagine, yet physicists keep finding that the principles governing black holes apply to everything else, too. This past June I wrote for *Quanta* magazine about one such discovery, which concerns the tendency of all things to approach thermal equilibrium.

Everything from the air in a room to ice water in a glass will eventually reach thermal equilibrium unless some outside influence acts to keep it out of equilibrium. Physicists think of equilibrium as a condition of general stasis. Molecules continue to bounce around vigorously, but the material as a whole reaches a common temperature and ceases to change in any lasting way.

But it turns out this is true only at the classical level. At the quantum level, the material continues to change. Thermal equilibrium is just a waypoint at which those changes become too subtle to detect using localized observations. In particular, the molecules or other building blocks become ever more quantum-entangled.

Entanglement is a peculiarly quantum type of harmony among molecules, particles. or other building blocks of matter. Through it, a group of particles can gain collective properties above and beyond those of the individuals; in fact, the particles can lose their own identities altogether. Quantum physicists have had since 1935 to get used to entanglement, and at the rate they’re going it may take them until 2135. “Entanglement is something that physics people find hard to think about,” says Xie Chen, a theorist at Caltech. “It’s something that’s not easily measured.” It’s not easy because all we see are particles’ behavior—where they are, how fast they move, what magnetic effects they exert, and so on. Only from the correlations among those properties can we tell their fates are linked. In short, our knowledge of entanglement is indirect and requires extra effort to quantify.

One way that theorists quantify the degree of entanglement is the concept of entanglement entropy, which is a quantum counterpart to ordinary thermal entropy. Theorists imagine cleaving a material system in two and considering each piece in isolation. Because this procedure deliberately ignores entanglement, it makes each piece appear to behave erratically. The ensuing uncertainty becomes a measure of how much entanglement there is in the combined system. As useful as this measure has been, it is very crude. It captures the raw amount of entanglement, but little of the form it takes. Particles might be entangled with their neighbors or with distant ones, and entanglement entropy can’t tell the difference.

So theorists have reached for new quantities, and their favorite at the moment is known as circuit complexity. The concept originates in computer science. It quantifies the complexity of a task in terms of how many steps it takes to perform it, each step typically being a logic or arithmetic operation such as addition. Physicists co-opted the concept to help explain black holes and later found it goes way beyond those cosmic sinkholes.“Complexity is really like a microscope into the entanglement structure of the system,” says Nick Hunter-Jones, a theoretical physicist at Stanford University.

Short-range entanglement is simple: Just a single interaction between two neighboring particles will entangle them. Long-range entanglement is complex: It takes a lot of such interactions to forge a link between far-flung particles. “The range of entanglement generated will depend on how big a circuit you apply,” Chen says. In July I caught up with her at the Max Planck Institute for Quantum Optics in Garching, in the suburbs of Munich. She described how she has gone beyond the short vs. long dichotomy to consider intermediate cases. A so-called sequential circuit acts on each particle or other element a fixed number of times (as for short-range entanglement), but the total number of interactions grows indefinitely (as for long-range entanglement).

Circuit complexity continues to increase long after a system has reached equilibrium, indicating that the entanglement among the particles is spreading out. “Long-range entanglement is key to interesting post-thermalization physics, where distant regions of your system continue to become more entangled even after local quantities become equilibrated,” Hunter-Jones says. In fact, as particles become bound together into larger collective structures, the particles themselves become increasingly irrelevant; instead, the collective structures become the fundamental unit of the system. “Everything is so entangled that even thinking about that box of gas of particles in a room, as this collection of classical things, doesn’t even make sense,” Hunter-Jones says. Eventually, even complexity will max out and the system will reach a quantum analogue of thermal equilibrium. “The system has reached the limits of how nonlocally it can entangle itself,” he says.

Circuit complexity shows that, if you take any system and wait long enough, it will become ineluctably quantum. Quantum mechanics is usually described as a theory of atoms and particles—small things. “If I only look at the properties of a single atom, then quantum mechanics becomes important there,” says Stanford theorist Luca Iliesiu. But it is also a theory of long spans of time, he says: “There is another parameter that does not necessarily have to be small—rather, it can be very large—which is the time.”

In another feat of lateral thinking, Chen and her colleagues have also applied circuit complexity to classify phases of matter. A classical phase such as a gas or liquid is basically unentangled. Superconductors and superfluids have short-range entanglement, giving them low complexity. Topological phases, which have been the subject of so much physics research in recent decades and are notoriously hard to visualize, are highly entangled and thus highly complex. Circuits can also describe transitions from one phase to another.

Chen says circuit complexity is the first way of classifying topological phases that does not try to force-fit them into older categories, but takes them on their own terms. “We have to come up with fully quantum ways to think about quantum systems,” she says. “Quantum circuit complexity is one such effort.”