The corner of Hilbert space that is ever visited

Monogamy is stated between three parties: the more entanglement two of them share, the less either can share with the third, with an exact trade. That is a constraint on a small system and it looks like a curiosity about protocols — useful for moving a link between parties who never met and for arguing that an eavesdropper can be detected, and otherwise a detail.

Read across a boundary in a system of many particles, the same constraint becomes something else. Each particle in a region can be strongly entangled with only a few others; a particle deep inside the region has its allowance used up by its neighbours, who are also inside; so only the particles near the boundary have anything left over to spend on the outside. Locality is doing the work: the interactions that bind a solid’s electrons are between neighbours, and a Hamiltonian with only near-neighbour terms is what makes the ground state’s correlations short-ranged in the first place.

The entanglement between a region and everything else is therefore proportional to the area of the boundary rather than to the volume of the region — and that is not the behaviour of a typical state at all.

Three states, three answers #

The figure is computed rather than taken on trust, and the three cases are worth separating because each is a different fact.

A random state. Pick a state of the whole chain at random, in the only natural sense — uniformly on the unit sphere of Hilbert space. Its entanglement with any block of $\ell$ sites is $\ell\ln 2$ to within an exponentially small correction: the block is very nearly maximally mixed, and knowing everything about the rest tells nothing about it. That is a volume law and it is what the overwhelming majority of states do.

A gapped ground state. Take a chain with a gap between its ground state and the first excited one, diagonalise it, and compute the block’s entanglement. It does not grow. From a block of eight sites to a block of forty it changes by less than a twentieth of a per cent, because the boundary of a one-dimensional block is two points however long the block is, and an area law in one dimension is a constant.

A gapless ground state. Close the gap and the entanglement grows again, but only as the logarithm of the block’s length — $\tfrac{1}{3}\ln\ell$ for this chain, and the figure measures 0.333 against the exact third. That logarithm is the signature of a critical point, and its coefficient is a number with a name.

The contrast between the first and the other two is the substance. A ground state is an extraordinarily atypical state of its own Hilbert space, and the property that makes it atypical is one number: how much a region knows about its surroundings.

Why the coefficient of the logarithm is worth having #

The third goes as $\tfrac{c}{3}\ln\ell$, where $c$ is the central charge of the conformal field theory that describes the critical point — a number that classifies which universality class a continuous transition belongs to. Free fermions have $c = 1$; the Ising transition has $c = \tfrac{1}{2}$.

That gives an unusual measurement. A universality class is normally identified from critical exponents, which means measuring how some correlation function decays over a range of distances and fitting a power. The entanglement gives the same classification from the coefficient of a logarithm, computed from the ground state alone, with no correlation function and no fitting over a range.

It is also a quantity that is not an expectation value of anything. The block entropy cannot be written as the average of an operator; it is a property of a reduced density matrix, and there is no experiment that measures it by averaging repeated readings of an instrument — which makes it unlike everything a measurement ordinarily returns. That makes it a strange thing to have become a standard diagnostic, and the reason it did is that it is easy to compute and sharply different between phases.

What it means for a correlation #

There is a consequence for ordinary correlation functions that is worth having, because it connects the entanglement to quantities an experiment measures.

A state with a bounded entanglement across every cut has correlations that decay exponentially with distance. That is a theorem in one dimension and it is the reason a gap and a correlation length are the same statement: a system with an energy gap $\Delta$ has correlations falling over a length of order the velocity divided by the gap, and a system with no gap has correlations falling as a power.

So the three regimes the hero figure draws correspond to three regimes of ordinary physics. A gapped phase has a constant entanglement and exponential correlations; a critical point has a logarithm and power-law correlations; and a state with volume-law entanglement has no reason for its correlations to be anything in particular. Entanglement scaling and correlation decay are the same classification seen through two instruments, and the first is computable where the second has to be measured over a range.

What the area law buys #

A state of $N$ spins is a vector with $2^N$ components. At forty spins that is a million million numbers and a machine is full; at three hundred it is more numbers than there are atoms in the observable universe, and the difficulty is not one that better hardware addresses.

A state with bounded entanglement across every cut is a different object. It can be written as a product of small matrices, one per site, with the matrix size $\chi$ set by how much entanglement there is — and the number of parameters is then about $2N\chi^2$, which is linear in the length. A chain of a thousand costs a thousand times a chain of one.

That representation is the matrix-product state, and the algorithm built on it — the density-matrix renormalisation group — is the most successful numerical method in many-body physics. It works on chains of hundreds of sites to a dozen significant figures where exact diagonalisation stops at twenty.

And the reason it works is entirely the area law. The method is not cleverer than exact diagonalisation; it restricts itself to a manifold of states, and it succeeds because the state being looked for happens to lie in that manifold. On a state with volume-law entanglement it fails completely, and does so no matter how much computer is applied.

How small the corner is #

The quantitative statement is hard to write down because the numbers involved have no names.

Hilbert space for $N$ spins has $2^N$ dimensions. The set of matrix-product states with a fixed bond dimension has of order $N\chi^2$ parameters. The ratio of the two is an exponential of an exponential, and at three hundred spins the logarithm of the first is already thirty times the logarithm of the second.

Every ground state of every local Hamiltonian lies in the small set. So does every state a quantum computer running for a reasonable time can reach, by a similar argument about how fast entanglement can spread. So does every state that has ever been prepared in any laboratory.

The overwhelming majority of Hilbert space is states that nothing prepares, nothing evolves into in any reasonable time, and nothing has any use for. That is a strange thing to be true of a space the theory treats as entirely uniform, and it is the practical content of the area law: quantum mechanics offers an enormous space of possibilities and physics uses a vanishing corner of it.

Where the law fails, and what that means #

An area law is not a theorem in general, and the cases where it fails are informative.

It is proved in one dimension for gapped systems — Hastings’ theorem, 2007 — and the proof is genuinely hard for a statement that sounds obvious.

It is not proved in two or three dimensions and is believed on numerical evidence and physical argument. There are no known counterexamples among gapped local Hamiltonians and no proof.

It fails at criticality, where the logarithm appears, and it fails much more thoroughly for excited states: a typical state at finite energy density has a volume law, which is why simulating a system’s thermal behaviour is far harder than finding its ground state — and is the same statement as a system’s approach to equilibrium being a property of its interactions rather than of its size — and why a quantum quench — a system suddenly disturbed and then left — becomes intractable after a time proportional to how far it has been simulated.

And it fails for systems with long-range interactions, where a particle’s allowance is not spent on neighbours because it has no neighbours in the relevant sense. That is why a trapped-ion chain with Coulomb couplings behaves differently from a lattice of nearest neighbours, and why the numerical methods need modifying for it.

The pattern across all four is worth stating. The area law is a consequence of interactions being local and the state being the lowest one — and where either fails, so does it, which is a reason to regard it as a fact about which states nature settles into rather than as a fact about Hilbert space.

Two dimensions, where it matters most and is proved least #

Everything above is drawn in one dimension because that is where exact computation is available, and the interesting materials are not one-dimensional.

In two dimensions a region of linear size $L$ has an area $L^2$ and a boundary of length $L$, so an area law says the entanglement grows as $L$ rather than as $L^2$. That is still an enormous quantity — a region of ten sites on a side has ten units rather than a hundred — and the matrix-product representation, which is built for a line, does not straightforwardly extend.

The generalisation exists and is called a projected entangled-pair state, which places a tensor at every lattice site with bonds to each neighbour. It has the right entanglement scaling by construction. What it does not have is an efficient way of computing anything from it: contracting the network to get an expectation value is, in general, a computationally hard problem in a precise sense, where the one-dimensional contraction is a sequence of small matrix multiplications.

So the area law in two dimensions is believed, is almost certainly true, and does not deliver the same benefit. That gap — between a structural property of the states and an algorithm that exploits it — is where most of the effort in the subject currently is, and it is the reason simulating a two-dimensional magnet remains hard while a one-dimensional chain is routine.

There is one further reason two dimensions matter. The correction to the area law in two dimensions is not always zero: a topologically ordered state has an entanglement of $\alpha L - \gamma$, with $\gamma$ a constant that is a property of the phase and not of the region. Measuring that constant identifies a phase which has no local order parameter at all — which is a classification by something other than a symmetry breaking, and is one of the few places where an entanglement is the only available diagnostic.

The one place the law is the whole subject #

There is a system for which the area law is not a useful property of the state but the central fact about it, and this subject has met it from the other end.

The entropy of a black hole is proportional to the area of its horizon rather than to the volume it encloses — one quarter of the area in units of the Planck area, for a hole of any size. That is a statement of exactly the shape of the ones above, arrived at from thermodynamics, with no many-body Hamiltonian anywhere in the argument.

Whether the two are the same statement is a live question and the reason so much attention is paid to entanglement in this subject at all. The quantum fields outside a horizon are entangled with the fields inside; tracing out the inside gives a reduced state with an entropy; and that entropy is divergent and has to be cut off, at which point it comes out proportional to the area with a coefficient depending on the cutoff. So the horizon’s entropy has an entanglement interpretation, and if it is the right one then the Planck area appearing in the formula is the area of one entangled degree of freedom.

That is the most-cited connection in theoretical physics and it is not established. The counting is ambiguous, it depends on how many species of field there are, and the coefficient has to be fixed by hand. What can be said is that two arguments with nothing in common — monogamy in a lattice and thermodynamics at a horizon — produce the same scaling with the same shape, and that the coincidence has been taken seriously for forty years without being resolved.

Free fermions, one measure, and circuits of bounded depth #

The chain computed here is of free fermions. Both ground states are obtained by diagonalising a single-particle problem and building a correlation matrix, which is exact for a model with no interactions between the particles and is not available for one that has them. The scaling behaviour is believed to be the same, on strong numerical evidence, and the exact computation is only available in the free case.

A random state means uniformly distributed on the unit sphere, which is a definite mathematical object and is not obviously what “typical” should mean physically. Other measures — states reachable in a given time, states of a given energy — give different typical entanglements, and the volume law survives all of them.

Entanglement entropy is one of many measures, and it is the one used because it is computable. Other measures — the negativity, the mutual information, the entanglement spectrum — carry different information, and in particular the entropy of a block is blind to where in the block the entanglement is.

The statement about quantum computers is a statement about circuits of bounded depth. A circuit of local gates spreads entanglement at a bounded rate — a limit with the shape of the speed at which a state can stop being itself — so a state reached in polynomial time has entanglement bounded by the depth rather than by the volume — which is a different argument from the ground-state one and reaches a similar conclusion. It does not say that every reachable state is a matrix product, only that none of them is typical.

And the parameter counting ignores precision. A matrix-product state with a given bond dimension represents a state approximately, and the error falls with $\chi$ at a rate that depends on the state. The statement “linear in N” is a statement about a fixed accuracy, and near a critical point the $\chi$ needed for a fixed accuracy grows with the length.

One number standing in for a whole spectrum #

The block figure draws an entropy against a length and cannot show what is entangled with what. A block’s entropy is a single number summarising a reduced density matrix with exponentially many eigenvalues, and the spectrum of those eigenvalues carries far more — it is, for a topologically ordered state, a fingerprint of the state’s topological character that no single number contains. Drawing the summary is what makes the comparison possible and it throws away the structure.

The parameter-count figure draws two lines on one plot and hides that they are not the same kind of quantity. The exact count is a dimension of a vector space; the matrix-product count is a number of coordinates on a curved manifold embedded in it. The manifold is not a subspace — sums of matrix-product states are not matrix-product states of the same bond dimension — and the geometry of that embedding is most of what makes the algorithms delicate.

And the corner figure cannot draw its own subject. A ratio whose logarithm is exponential cannot be plotted, described by analogy or written down; the figure shows the logarithms of both quantities and asks the reader to remember that they are logarithms. That is the honest limit of any picture of this comparison.

Still open: whether the law explains the world or describes an accident #

The area law’s consequences are practical and the reason for it is not fully settled, and the two facts sit oddly together.

The physical argument is monogamy plus locality plus the state being the lowest one, and it is convincing without being a proof outside one dimension. What is not clear is how much of the structure of physics that argument accounts for. Every state anything prepares obeys the law; every state a computer can simulate obeys it; the states that do not obey it are the ones nothing reaches. There is a reading on which that is a deep statement about why the world is describable at all, and a reading on which it is a tautology about what “reachable” means.

There is a second and sharper question underneath. If physical states occupy an exponentially small corner of Hilbert space, in what sense are the rest of the states there? A theory whose state space is overwhelmingly composed of objects that nothing can prepare, nothing evolves into, and no measurement distinguishes is a theory with a great deal of unused structure — and the history of physics is not encouraging about unused structure. Whether that is a hint about a better formulation or merely an observation about a convenient one is a question with no experimental handle at all.

The habit worth carrying away is the one the three curves make together. When something is easy to compute, ask what is special about the case rather than about the method. The numerical methods of many-body physics are often described as clever, and they are; what makes them work is that ground states are atypical in one specific and measurable way, and the measurement of that atypicality — an entropy against a length — is a more useful thing to know about a system than any amount of knowledge about the algorithm.