## Introduction

Brian Swingle was a graduate student studying the physics of matter at the Massachusetts Institute of Technology when he decided to take a few classes in string theory to round out his education — “because, why not?” he recalled — although he initially paid little heed to the concepts he encountered in those classes. But as he delved deeper, he began to see unexpected similarities between his own work, in which he used so-called tensor networks to predict the properties of exotic materials, and string theory’s approach to black-hole physics and quantum gravity. “I realized there was something profound going on,” he said.

Tensors crop up all over physics — they’re simply mathematical objects that can represent multiple numbers at the same time. For example, a velocity vector is a simple tensor: It captures values for both the speed and the direction of motion. More complicated tensors, linked together into networks, can be used to simplify calculations for complex systems made of many different interacting parts — including the intricate interactions of the vast numbers of subatomic particles that make up matter.

Swingle is one of a growing number of physicists who see the value in adapting tensor networks to cosmology. Among other benefits, it could help resolve an ongoing debate about the nature of space-time itself. According to John Preskill, the Richard P. Feynman professor of theoretical physics at the California Institute of Technology in Pasadena, many physicists have suspected a deep connection between quantum entanglement — the “spooky action at a distance” that so vexed Albert Einstein — and space-time geometry at the smallest scales since the physicist John Wheeler first described the latter as a bubbly, frothy foam six decades ago. “If you probe geometry at scales comparable to the Planck scale” — the shortest possible distance — “it looks less and less like space-time,” said Preskill. “It’s not really geometry anymore. It’s something else, an emergent thing [that arises] from something more fundamental.”

Physicists continue to wrestle with the knotty problem of what this more fundamental picture might be, but they strongly suspect that it is related to quantum information. “When we talk about information being encoded, [we mean that] we can split a system into parts, and there is some correlation among the parts so I can learn something about one part by observing another part,” said Preskill. This is the essence of entanglement.

It is common to speak of a “fabric” of space-time, a metaphor that evokes the concept of weaving individual threads together to form a smooth, continuous whole. That thread is fundamentally quantum. “Entanglement is the fabric of space-time,” said Swingle, who is now a researcher at Stanford University. “It’s the thread that binds the system together, that makes the collective properties different from the individual properties. But to really see the interesting collective behavior, you need to understand how that entanglement is distributed.”

Tensor networks provide a mathematical tool capable of doing just that. In this view, space-time arises out of a series of interlinked nodes in a complex network, with individual morsels of quantum information fitted together like Legos. Entanglement is the glue that holds the network together. If we want to understand space-time, we must first think geometrically about entanglement, since that is how information is encoded between the immense number of interacting nodes in the system.

## Many Bodies, One Network

It is no easy feat to model a complex quantum system; even doing so for a classical system with more than two interacting parts poses a challenge. When Isaac Newton published his *Principia* in 1687, one of the many topics he examined became known as the “three-body problem.” It is a relatively simple matter to calculate the movement of two objects, such as the Earth and the sun, taking into account the effects of their mutual gravitational attraction. However, adding a third body, like the moon, turns a relatively straightforward problem with an exact solution into one that is inherently chaotic, where long-term predictions require powerful computers to simulate an approximation of the system’s evolution. In general, the more objects in the system, the more difficult the calculation, and that difficulty increases linearly, or nearly so — at least in classical physics.

Now imagine a quantum system with many billions of atoms, all of which interact with each other according to complicated quantum equations. At that scale, the difficulty appears to increase exponentially with the number of particles in the system, so a brute-force approach to calculation just won’t work.

Consider a lump of gold. It is comprised of many billions of atoms, all of which interact with one another. From those interactions emerge the various classical properties of the metal, such as color, strength or conductivity. “Atoms are tiny little quantum mechanical things, and you put atoms together and new and wonderful things happen,” said Swingle. But at this scale, the rules of quantum mechanics apply. Physicists need to precisely calculate the wave function of that lump of gold, which describes the state of the system. And that wave function is a many-headed hydra of exponential complexity.

Even if your lump of gold has just 100 atoms, each with a quantum “spin” that can be either up or down, the total number of possible states totals 2^{100}, or a million trillion trillion. With every added atom the problem grows exponentially worse. (And worse still if you care to describe anything in addition to the atomic spins, which any realistic model would.) “If you take the entire visible universe and fill it up with our best storage material, the best hard drive money can buy, you could only store the state of about 300 spins,” said Swingle. “So this information is there, but it’s not all physical. No one has ever measured all these numbers.”

Tensor networks enable physicists to compress all the information contained within the wave function and focus on just those properties physicists can measure in experiments: how much a given material bends light, for example, or how much it absorbs sound, or how well it conducts electricity. A tensor is a “black box” of sorts that takes in one collection of numbers and spits out a different one. So it is possible to plug in a simple wave function — such as that of many non-interacting electrons, each in its lowest-energy state — and run tensors upon the system over and over, until the process produces a wave function for a large, complicated system, like the billions of interacting atoms in a lump of gold. The result is a straightforward diagram that represents this complicated lump of gold, an innovation much like the development of Feynman diagrams in the mid-20th century, which simplified how physicists represent particle interactions. A tensor network has a geometry, just like space-time.

The key to achieving this simplification is a principle called “locality.” Any given electron only interacts with its nearest neighboring electrons. Entangling each of many electrons with its neighbors produces a series of “nodes” in the network. Those nodes are the tensors, and entanglement links them together. All those interconnected nodes make up the network. A complex calculation thus becomes easier to visualize. Sometimes it even reduces to a much simpler counting problem.

There are many different types of tensor networks, but among the most useful is the one known by the acronym MERA (multiscale entanglement renormalization ansatz). Here’s how it works in principle: Imagine a one-dimensional line of electrons. Replace the eight individual electrons — designated A, B, C, D, E, F, G and H — with fundamental units of quantum information (qubits), and entangle them with their nearest neighbors to form links. A entangles with B, C entangles with D, E entangles with F, and G entangles with H. This produces a higher level in the network. Now entangle AB with CD, and EF with GH, to get the next level in the network. Finally, ABCD entangles with EFGH to form the highest layer. “In a way, we could say that one uses entanglement to build up the many-body wave function,” Román Orús, a physicist at Johannes Gutenberg University in Germany, observed in a paper last year.

Why are some physicists so excited about the potential for tensor networks — especially MERA — to illuminate a path to quantum gravity? Because the networks demonstrate how a single geometric structure can emerge from complicated interactions between many objects. And Swingle (among others) hopes to make use of this emergent geometry by showing how it can explain the mechanism by which a smooth, continuous space-time can emerge from discrete bits of quantum information.

## Space-Time’s Boundaries

Condensed-matter physicists inadvertently found an emergent extra dimension when they developed tensor networks: the technique yields a two-dimensional system out of one dimension. Meanwhile, gravity theorists were subtracting a dimension — going from three to two — with the development of what’s known as the holographic principle. The two concepts might connect to form a more sophisticated understanding of space-time.

In the 1970s, a physicist named Jacob Bekenstein showed that the information about a black hole’s interior is encoded in its two-dimensional surface area (the “boundary”) rather than within its three-dimensional volume (the “bulk”). Twenty years later, Leonard Susskind and Gerard ’t Hooft extended this notion to the entire universe, likening it to a hologram: Our three-dimensional universe in all its glory emerges from a two-dimensional “source code.” In 1997, Juan Maldacena found a concrete example of holography in action, demonstrating that a toy model describing a flat space without gravity is equivalent to a description of a saddle-shaped space with gravity. This connection is what physicists call a “duality.”

Mark Van Raamsdonk, a string theorist at the University of British Columbia in Vancouver, likens the holographic concept to a two-dimensional computer chip that contains the code for creating the three-dimensional virtual world of a video game. We live within that 3-D game space. In one sense, our space is illusory, an ephemeral image projected into thin air. But as Van Raamsdonk emphasizes, “There’s still an actual physical thing in your computer that stores all the information.”

The idea has gained broad acceptance among theoretical physicists, but they still grapple with the problem of precisely how a lower dimension would store information about the geometry of space-time. The sticking point is that our metaphorical memory chip has to be a kind of quantum computer, where the traditional zeros and ones used to encode information are replaced with qubits capable of being zeros, ones and everything in between simultaneously. Those qubits must be connected via entanglement — whereby the state of one qubit is determined by the state of its neighbor — before any realistic 3-D world can be encoded.

Similarly, entanglement seems to be fundamental to the existence of space-time. This was the conclusion reached by a pair of postdocs in 2006: Shinsei Ryu (now at the University of Illinois, Urbana-Champaign) and Tadashi Takayanagi (now at Kyoto University), who shared the 2015 New Horizons in Physics prize for this work. “The idea was that the way that [the geometry of] space-time is encoded has a lot to do with how the different parts of this memory chip are entangled with each other,” Van Raamsdonk explained.

Inspired by their work, as well as by a subsequent paper of Maldacena’s, in 2010 Van Raamsdonk proposed a thought experiment to demonstrate the critical role of entanglement in the formation of space-time, pondering what would happen if one cut the memory chip in two and then removed the entanglement between qubits in opposite halves. He found that space-time begins to tear itself apart, in much the same way that stretching a wad of gum by both ends yields a pinched-looking point in the center as the two halves move farther apart. Continuing to split that memory chip into smaller and smaller pieces unravels space-time until only tiny individual fragments remain that have no connection to one another. “If you take away the entanglement, your space-time just falls apart,” said Van Raamsdonk. Similarly, “if you wanted to build up a space-time, you’d want to start entangling [qubits] together in particular ways.”

Combine those insights with Swingle’s work connecting the entangled structure of space-time and the holographic principle to tensor networks, and another crucial piece of the puzzle snaps into place. Curved space-times emerge quite naturally from entanglement in tensor networks via holography. “Space-time is a geometrical representation of this quantum information,” said Van Raamsdonk.

And what does that geometry look like? In the case of Maldacena’s saddle-shaped space-time, it looks like one of M.C. Escher’s *Circle Limit* figures from the late 1950s and early 1960s. Escher had long been interested in order and symmetry, incorporating those mathematical concepts into his art ever since 1936 when he visited the Alhambra in Spain, where he found inspiration in the repeating tiling patterns typical of Moorish architecture, known as tessellation.

His *Circle Limit* woodcuts are illustrations of hyperbolic geometries: negatively curved spaces represented in two dimensions as a distorted disk, much the way flattening a globe into a two-dimensional map of the Earth distorts the continents. For instance, *Circle Limit IV (Heaven and Hell)* features many repeating figures of angels and demons. In a true hyperbolic space, all the figures would be the same size, but in Escher’s two-dimensional representation, those near the edge appear smaller and more pinched than the figures in the center. A diagram of a tensor network also bears a striking resemblance to the *Circle Limit* series, a visual manifestation of the deep connection Swingle noticed when he took that fateful string theory class.

To date, tensor analysis has been limited to models of space-time, like Maldacena’s, that don’t describe the universe we inhabit — a non-saddle-shaped universe whose expansion is accelerating. Physicists can only translate between dual models in a few special cases. Ideally, they would like to have a universal dictionary. And they would like to be able to derive that dictionary directly, rather than make close approximations. “We’re in a funny situation with these dualities, because everyone seems to agree that it’s important, but nobody knows how to derive them,” said Preskill. “Maybe the tensor-network approach will make it possible to go further. I think it would be a sign of progress if we can say — even with just a toy model — ‘Aha! Here is the derivation of the dictionary!’ That would be a strong hint that we are onto something.”

Over the past year, Swingle and Van Raamsdonk have collaborated to move their respective work in this area beyond a static picture of space-time to explore its dynamics: how space-time changes over time, and how it curves in response to these changes. Thus far, they have managed to derive Einstein’s equations, specifically the equivalence principle — evidence that the dynamics of space-time, as well as its geometry, emerge from entangled qubits. It is a promising start.

“‘What is space-time?’ sounds like a completely philosophical question,” Van Raamsdonk said. “To actually have some answer to that, one that is concrete and allows you to calculate space-time, is kind of amazing.”

*Part three of this series, featuring an interactive presentation that illustrates the relationship between entanglement, tensor networks and space-time, will appear on Thursday, April 30.*

*This article was reprinted on Wired.com.*