Hannes Hummel for Quanta Magazine

Tensor networks could connect space-time froth to quantum information.

Next in the series
Interactive: What Is Space?

Chapter 2: Network Tapestry

How Quantum Pairs Stitch Space-Time

New tools may reveal how quantum information builds the structure of space.

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.”

Natalie Wolchover/Quanta Magazine

Video: Brian Swingle explains what quantum entanglement and gravity have to do with each other.

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 2100, 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.”

Olena Shmahalo/Quanta Magazine

Mark Van Raamsdonk imagines entanglement creating space-time gradually: Along the outside of the figure, individual particles (dots) become entangled with each other. These entangled pairs then become entangled with other pairs. As more particles become entangled, the three-dimensional structure of space-time emerges.

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.

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  • At this time, the speculation of the existence of forces beyond our perception (e.g. the dark force acting on dark matter), yields an interesting question: Are the 2D geometries that describe quantum entanglement also able to explain, or predict, the existence of these forces? If the entire “code” of the universe is in 2D form (sounds a bit like human DNA!), how many true dimensions does the universe exist in?

  • Seems like a circular argument: locality assumes some type of spacial relationship between particles, and from that locality we get space?

  • This is not the first time that people propose to put together tensor networks and quanta gravity. There is a whole programa going on employing tensor network renormalization group techniques to understand better the continuum limit of spinfoam models for quantum gravity:

  • Ok! After researching a bit, I see that the proposal of Brian Swingle is not new at all.
    Very interesting that different approaches to quantum gravity try to benefit from tensor networks!

  • If Brian Swingle is attributing “universality” to the connection between entanglement and gravity (in the above video); for clarity, do we know if he’s referring to the same universality that’s covered in these recent Quanta articles (as first coined by Leo Kadanoff)?
    If so, it seems we’re well on the eponymous way to covering the ToE bases from the quantum to the relativistic, via the classical scale – including consciousness itself, just as Roger Penrose predicted over 20 years ago.

  • I would like to mention a precursor to this tensor network-entanglement stuff. Since the mid-nineties we developed in many papers what we called the Structurally Dynamic Cellular Network Approach to Quantum Space-Time Physics. A recent review is arXiv:1501.00391. In arXiv:gr-qc/0110077 (J.Math.Phys. 44(2003)5588) we constructed a Geometric Renormalization Group dealing with the various hierarchies of entanglement up to continuous space-time. In arXiv:0910.4017 we introduced socalled Wormhole Spaces as the common cause of BH Entropy-Area Law, the Holographic Principle and Quantum Entanglement.

  • Sounds a bit like Reginald Cahill’s concept of random gebits (http://sprott.physics.wisc.edu/pickover/pc/random_reality.html).

  • Jon makes an excellent point. Consider this paragraph from the article:

    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.

    The key problem in applying notions like this from condensed matter physics to understanding the basis of spacetime geometry is finding a way to avoid assuming a geometric background in terms of which notions like locality are defined.
    A way forward is suggested by reconsidering how cellular automata are described. In the normal mode of presentation a grid of cells is given, viewed as laid out in a two-dimensional background space. (Each cell is allowed to have a discrete state, which is often but not necessarily binary.) The notion of neighboring cells can then be defined (usually implicitly) by picking out cells associated with neighboring points in that background space.
    One can then move to dynamics and discuss various possible transition rules defined on the cells in the grid, and consider whether the transition rules act on neighboring cells or not.

    An alternative viewpoint is to simply view the cells as a collection, with each cell assigned a unique but otherwise arbitrary label. Transition rules are then stated using those labels. One can *define* whether or not two cells are neighbors in terms of their mutual participation in a transition rule. A given cell may participate in more than one rule, so the cells in the collection are in general tied together into a network of interlocking transition rules. The geometrical structure of the collection can be considered to arise from the structure of the network, which was in turn dictated by the transition rules, i.e., by a dynamics imposed on the collection.

    That is the key point; the geometrical structure arises from the structure of the dynamics, initially defined in terms of opaque labels, without reference to a pre-defined geometrical structure.

    This raises at least two large questions:
    1. What principle(s) dictate the dynamics? If we arrange the dynamics to produce the geometrical structure we expect spacetime to have then we have arguably described a mere simulation; we haven’t really learned much about the origins of spacetime structure.
    2. The framework described above is discrete. How do we get the apparent continuity of spacetime from it? In particular, how do we formulate and precisely preserve local Lorentz invariance, which is so central to the spacetime structure we know, and has been checked to very high accuracy?

    Forms of these questions have already arisen and received a great deal attention in various approaches to quantum gravity. I expect they are implicit in the work that Swingle and his collaborators are doing. By the way, there is some similarity to earlier ideas of Seth Lloyd, who is also at MIT. (See his 2007 book ‘Programming the Universe’, http://www.amazon.com/Programming-Universe-Quantum-Computer-Scientist/dp/1400033861.)

  • If Hawking’s virtual micro black holes exist, they’re continually forming an evaporating and likely creating entangled pairs of virtual Hawking radiation, therefore forming virtual Einstein-Rosen bridges (EPR=ER). This means that the distinction been the virtual and non-virtual (rest-mass & entanglement) can be a closed 2pshere surface with zero length non interior (non-local).

    It doesn’t seem very hard to see how the flat fundamentals under QFT can be holographic (and super-deterministic btw). There is an experiment proposed by Jacob Bekenstein to indirectly observe virtual micro black holes. My guess is that this requires a space-based vacuum experiment of some cost & energy. But it is doable.

  • There is a serious causality problem with the idea the space emerges from qbits. Qbits are states of physical entities, which themselves must exist in space. So space cannot emerge from qbits since qbits must emerge as states within space.

  • D. L. Burnstein: Your comment is very insightful. Van Raamsdonk is quoted in the text as saying: “Space-time is a geometrical representation of … quantum information.” The causal chain appears to go from “real” Space to particles in that Space, to properties of those particles, to geometrical representations of those properties. Logical consistency appears to require that we define clearly the difference between the ontological Space with matter and the abstract space with information about the properties of matter. To capture the first one, we need a “map of the Universe”. To capture the second, we need a “map of energy distribution in the Universe”.

  • I’d like to make note that MERA was created by Guifre Vidal who is currently at Perimeter Institute (he was at University of Queensland when it was created some years ago). Indeed the whole idea of MERA was to represent highly entangled quantum many-body states in condensed matter lattices.

  • Please delete my comment if it doesn’t comply with your expectations. I love science and have thought deeply about physics which I set out to study when I was young but felt unworthy . My sense of the architecture of the universe has not changed the more I read. What I read here agrees with much of my thoughts. I believe there are two functions in the universe, pattern and motion. Motion is spin and counter – gyrating spins entangle and create gravity and that gravity that links the energy of spins forms matter. Matter is the entangled form of energy which is spin or vortex. Pattern or geometry is the extension by contraction or extension of spins creating or releasing gravity throughout space and time. I have not expressed this as correctly as I have done in previous writings but I have not read all the related material above so My simplistic understanding may be my own.I have many writings on this as I believe naming particles is a never ending game. The theries above seem like a much more workable approach. Forgive me for saying holographic offers good simulation patterns. Thank you for your patience if anyone reads this a.

  • Could a black hole be an empty shell with no singularity at the center? From an outsider’s perspective, any matter falling into a black hole would never reach beyond the Event Horizon due to Time Dilation.

    Could this hypothesis be tested? If all of the mass of the black hole were located at the event horizon instead of the center, it should affect nearby objects differently. From a large distance away, the gravitational field would still approximate a point source. But light passing close to a black hole, particularly a supermassive one, should bend differently.

    Have measurements ruled this out?

  • Note to web designer: Please add an EDIT option to the comments page.

    That said, while spacetime is normally thought to be frothy or quantized only at the Plank scale, I wonder if perhaps a black hole is a macroscale example of this. I refer back to my previous comment about a black hole being hollow (devoid of spacetime) with all of its mass and properties located along the event horizon , i.e. the holographic description of a black hole.

  • Answer for LizR : Quantum entanglement occurs when pairs of particles interact in ways that the quantum state of each particle cannot be described independently

  • @jon: by “locality” in this context, is it really meant “spatial locality”? i doubt…

  • I was so impressed by the clarity of this excellent article that I signed up for the Quanta newsletter.

    No background in physics but very interested in quantum research and its strange implications for our view of reality.

  • Good article on quantum entanglement, but I can sense that the physisists working on this have yet to completely wrap their minds aorund the fundamental mechanism that entanglement is. Interesting way of looking at things. Love your web site…good to see people pressing on with something that’s vexing a lot of us. Thanks.

  • Interesting that 3D emerges gradually as more particles are entangled, indeed our universe is fractal in it's nature as I have suspected.

  • Any one ever seen a Kabbalah tree of life and the similarities between the illustration heading this article.

  • I'm curious as to why the author credited Susskind with advancing the notion of holography in the universe. To my understanding, it was David Bohm who pioneered holographic notions and I think it's shameful to see him ignored by the physics community. IMO. Great article otherwise.

  • Hierarchical matrix math! But how to pick adjacencies & entanglements, when we don't know what "entanglement" actually means?

  • If the nature of reality is consciousness then, there are no perfect symmetries, there is no pure randomness. We are in the gray region between truth and chaos. These extremes can only be ideals, not reality. Process only occurs in the gray region, time does not exist at the extremes. If we think of coin tosses, with truth, the coin is either heads or tails as a frozen expression of meaning, and with chaos the coin is always both, it never stops spinning, and contains no meaning. The dynamic tension between the two is where time comes from. Either spin is imparted to truth or the perfect randomness of the perpetual spin symmetry is broken. At each extreme is a different form of symmetry, one is a symmetry in the relationship of meaning and the other is a symmetry of potential.
    Truth as a static structure vs a dynamic system. To simplify, think of a stack of copy paper with one word on each page. In time, we see each page one at a time, outside of time all of the words, on all of the pages combine to make a single word. This single word is truth, it is the entire story, told in an instant of time. The fractal version of this story has another feature. As each page is presented to us, our intent creates a slightly new meaning that branches out, changing the story, an effect that turns the stack into a tree like structure.
    The direction of time's arrow is the breaking of the symmetry of the potential of the boundary condition. In other words, if I toss a coin and it has perfect symmetry of potential it will land heads half the time and tails half the time. The symmetry of the potential is broken if the coin tosses are not 50/50. In a perfectly random system, after a sufficient number of tosses, the symmetry for all even number tosses would always be 50/50. Coin tosses are a lot like squaring the circle. You get closer and closer to the true value but you never reach it, like an infinite recursive iteration.

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