Spacetime and Information

There's a pattern in twenty-first-century theoretical physics: every time someone pokes at the foundations of spacetime, they find information theory underneath. Entanglement turns out to hold geometry together. Quantum error-correcting codes turn out to describe the fabric of space. Complexity classes in computer science turn out to constrain what's physically possible. The emerging picture is that spacetime isn't fundamental — it's a structure that emerges from patterns of quantum information, the way temperature emerges from molecular motion.

Spacetime as a Code

In 2014, three young physicists — Ahmed Almheiri, Xi Dong, and Daniel Harlow — made a startling connection. Working in anti-de Sitter (AdS) space, a theoretical toy universe that works like a hologram (the interior geometry is a projection from quantum particles on the boundary), they found that the way the holographic projection works is structurally identical to a quantum error-correcting code.¹

In quantum error correction, you protect fragile information by encoding it in patterns of entanglement across many physical qubits. A single logical qubit gets spread among three, five, or more physical qubits, so that damage to any one physical qubit can be detected and corrected without ever directly measuring the encoded information. The key property of a good code: you can reconstruct the logical information from slightly more than half the physical qubits, even if the rest are corrupted.¹

This is exactly how holographic spacetime works. Any point in the interior of AdS space can be reconstructed from slightly more than half the boundary. Different subsets of boundary qubits reconstruct overlapping "entanglement wedges" in the bulk — just as a logical qubit in a quantum computer can be recovered from many different subsets of physical qubits.¹

John Preskill's way of putting it is memorable: "We're not walking on eggshells to make sure we don't make the geometry fall apart." Spacetime has a built-in robustness, and quantum error correction explains why. The geometry of space is robust for the same reason that error-corrected quantum computations are robust — the information that defines the geometry is spread across entanglement patterns in a way that tolerates local disturbances.¹

In 2015, the HaPPY code (Harlow, Pastawski, Preskill, Yoshida) made this even more concrete, constructing a holographic code out of five-sided tiles — "little Tinkertoys" tessellating hyperbolic space like the fish in an Escher print. Each tile represents a point of bulk spacetime, and the whole structure captures multiple properties of AdS geometry.¹

Black Holes as the Breakdown of Correctability

The language of error correction gives a clean definition of what a black hole is: the breakdown of correctability. When so many boundary qubits are corrupted (or equivalently, when the bulk distortion becomes too extreme) that you can no longer recover the interior geometry, you've got a black hole. As Patrick Hayden puts it, it's "a sink for your ignorance."¹

This reframes the black hole information paradox. In the holographic picture, information that falls into a black hole is always encoded on the boundary — it can never truly be lost. But reconstructing the interior requires access to roughly three-quarters of the boundary qubits, not just slightly more than half. Why that fraction? Still an open question. But the framework predicts it, and it hints that black hole interiors are more deeply entangled with their surroundings than ordinary spacetime.¹

Almheiri speculates that quantum error correction is what prevents the formation of "firewalls" — hypothetical energy barriers at black hole horizons that his own earlier work helped conceive. The error-correcting structure of spacetime maintains smoothness at the horizon by protecting information even as it crosses the boundary. If true, the resolution of one of physics' sharpest paradoxes would come down to the same principle that makes quantum computers fault-tolerant.¹

Entanglement as the Fabric

The deepest implication: "It's really entanglement which is holding the space together," as Preskill says. "If you want to weave space-time together out of little pieces, you have to entangle them in the right way. And the right way is to build a quantum error-correcting code."¹

This connects to the Information And Computation story about Landauer's principle. If erasing a bit costs kT ln 2 joules, and spacetime geometry is encoded in patterns of quantum information, then there's a thermodynamic cost to the geometry of space itself. The Bekenstein-Hawking entropy of a black hole — proportional to horizon area, not volume — is precisely the entropy you'd expect from a holographic code where all the information lives on the boundary.

The MIP* = RE Bombshell

Meanwhile, from a completely different direction, computer scientists proved something equally disorienting about the structure of quantum information. In 2020, Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen posted a 165-page proof that MIP* = RE — and it shattered a fifty-year-old conjecture in pure mathematics.²

The translation for non-specialists: MIP* is the class of computational problems that can be verified by interrogating two provers who can't communicate but share quantum entanglement. RE is the class of problems solvable by a machine that can solve the halting problem. Showing these are equal means that games with entangled quantum players are extraordinarily more powerful than anyone expected — so powerful that figuring out the optimal strategy for winning such games is, in general, impossible.²

The proof worked by combining quantum entanglement with the PCP theorem (a cornerstone of complexity theory) in a recursive way. Classical verifiers run out of random bits when you try to iterate the verification process. But entangled players can use CHSH games — the same games that prove quantum mechanics violates Bell's theorem — to generate certified fresh random bits at each stage. This lets the verification recur indefinitely.²

The collateral damage was the Connes embedding conjecture, a major open problem in operator algebras since 1976. The conjecture asked whether infinite-dimensional von Neumann factors can always be approximated by finite-dimensional matrices. MIP* = RE implies the answer is no — these mathematical objects are irreducibly infinite. A problem born in the theory of quantum mechanics, solved using ideas from computational complexity, refuting a conjecture in pure mathematics. Three blind men finally comparing notes about the elephant.²

For physics, the most provocative implication is that MIP* = RE proves the existence — in principle — of an experiment that could verify whether nature is truly infinite-dimensional. The proof doesn't say how to build the experiment, but it establishes that the question isn't metaphysical; it's empirically accessible.²

Relativity as Operational Physics

The informational turn in physics didn't start this century. It goes back to Einstein's 1905 paper on special relativity, which was — under the surface — an argument about operational definitions and measurement. But as a pedagogical essay on special relativity makes painfully clear, most presentations of the subject obscure the crucial physical insight.³

The standard story: Einstein postulated that the speed of light is the same in all inertial frames, and everything else follows. Students walk away thinking this is some arbitrary convention about how to synchronize clocks. It isn't. The physical content is that light-based coordinate systems and inertia-based coordinate systems are the same — and this is an empirical fact, not a definition.³

Newton's laws already define a unique coordinate system for each inertial frame: the one in which mechanical inertia is isotropic (the same in all directions). Two identical particles exerting equal and opposite forces travel equal distances in equal times — that's Newton's third law, and it implicitly defines simultaneity. Einstein's contribution was showing that the coordinate systems defined by the isotropy of light speed are identical to these inertia-based coordinates. This isn't tautological. It would be entirely possible for light to obey one set of symmetries and mechanical inertia another. That they agree is a deep fact about the universe.³

Eddington took this operational perspective further in his 1923 Mathematical Theory of Relativity. Physical quantities, he argued, are not features of reality that we passively observe — they are "manufactured articles," results of specific operations of comparison. Distance is what you get when you lay rulers end to end. Time is what you get when you follow a specific astronomical protocol. Parallax and cubic parallax are equally real results of operations; our preference for distance over cubic parallax reflects inherited theoretical prejudice, not ontological priority.⁴

This sounds like pure philosophy, but it's the same operational stance that the quantum reconstruction program takes a century later — physics describes the structure of measurement results, not an observer-independent reality. From Eddington's manufactured physical quantities to QBism's quantum probabilities as personal beliefs to AdS/CFT's holographic boundary-to-bulk reconstruction, the thread is consistent: what we call "the world" is a structure in our measurements, and the laws of physics are constraints on that structure.

The Convergence

These threads are converging. Quantum foundations researchers rebuild quantum mechanics from axioms about information. Quantum gravity researchers find that spacetime is an error-correcting code built from entanglement. Complexity theorists show that entanglement makes verification problems as hard as the halting problem. Relativists from Einstein onward have insisted that physics is about operational measurements, not metaphysical substances.

The picture that's forming — tentatively, and with enormous gaps — is that the fundamental layer of reality is informational. Spacetime, particles, fields, and forces might all be emergent structures in a network of quantum correlations. Whether this informational picture will survive contact with a full theory of quantum gravity, or collapse under its own weight like so many beautiful frameworks before it, remains to be seen. But the convergence across such different subfields — each approaching the elephant from their own direction — is hard to dismiss as coincidence.