Where the interference goes

A measurement produces an answer that was not there before, and the standard account has the state collapsing when the measurement happens. This essay is about what can be said without that step — which turns out to be most of what is observed, and not all of it.

Coherence is a property of a pair #

Write a particle going through two slits as $|1\rangle + |2\rangle$, and let something in the environment — an air molecule, a photon — scatter off it, ending in a state that depends on which way the particle went. The joint state is

$$|1\rangle|e_1\rangle + |2\rangle|e_2\rangle$$

and it is still a perfectly coherent superposition. Nothing has collapsed and nothing is random.

What has changed is what a measurement on the particle alone can see. The interference term in the pattern is multiplied by the overlap $\langle e_1|e_2\rangle$, so if the environment’s two states are distinguishable the fringes are gone — even though the total state is as coherent as ever.

The coherence has not been destroyed. It has been relocated, into a correlation between the particle and the environment, where no measurement on the particle can reach it.

Arrivals accumulate into a pattern one at a time, and the pattern exists because there is no record anywhere of which slit each particle used. Make such a record — anywhere, in anything, in a stray photon that will never be caught — and the pattern goes. Whether the record is ever consulted makes no difference at all, which is the sentence the rest of this essay is about.

The exact trade #

The relation between what is visible and what is recorded is not a slogan but an equation.

Let the visibility $V$ be the fringe contrast and the distinguishability $D$ be the best any measurement on the environment could do at telling the two paths apart. Then for a pure joint state

$$V^2 + D^2 = 1$$

exactly, and the figure above verifies it to a part in $10^{12}$ from two calculations with nothing in common.

That equation says something stronger than the usual complementarity slogan. It is not that measuring disturbs; it is that the interference is reduced by exactly the amount of which-path information that exists, in the environment, in principle, whether or not anybody reads it.

The converse matters as much. An interaction that is violent but leaves no distinguishing record does not reduce the visibility at all. A particle can be kicked hard by something that ends in the same state whichever path it took, and the fringes survive — which rules out disturbance as the mechanism.

How fast it happens #

The rate is what makes decoherence an explanation of the world rather than a curiosity, and it is set by how well the environment distinguishes positions.

Each scattering event carries away some which-path information, and the coherence between two positions separated by $\Delta x$ decays at a rate $\Lambda(\Delta x)^2$, where $\Lambda$ depends on what is doing the scattering.

The dust grain in air is the number that settles the ordinary world. There is no mystery about why a chair is not in two places: any object of that size interacts with so much that a superposition of two positions is correlated with the surroundings in a time far shorter than anything that could be measured.

And the molecule column is why matter-wave interferometry works. Molecules of a few hundred atoms have been interfered — the record is around 25,000 atomic mass units — and the experimental art is entirely the art of pushing the decoherence time above the flight time — one arrival at a time, with nothing in the apparatus knowing which way each went: high vacuum, cold sources, and short paths.

The reason large objects do not show interference is usually given as the shortness of their wavelength, and that is a separate and much weaker obstacle. A wavelength can be made measurable by going slowly and building a long apparatus, and experiments have done exactly that for molecules of tens of thousands of atomic mass units. What defeats the next step up is not the wavelength: it is that the coherence would be gone before the fringes had time to form.

The density matrix, and why it is the right object #

The bookkeeping deserves a paragraph, because it is where “the coherence has moved” becomes a calculation rather than a phrase.

A system correlated with something else has no state of its own in the usual sense. What it has is a reduced density matrix, obtained by summing over the possibilities of everything it is correlated with — and for the two-path case that matrix is

$$\rho = \begin{pmatrix} \tfrac{1}{2} & \tfrac{1}{2}\langle e_2|e_1\rangle \\ \tfrac{1}{2}\langle e_1|e_2\rangle & \tfrac{1}{2}\end{pmatrix}$$

The diagonal entries are the probabilities of the two paths and they do not move. The off-diagonal ones carry the interference, and they are multiplied by the environment states’ overlap — so decoherence is, in one sentence, the decay of the off-diagonal elements in the basis the environment monitors.

What a density matrix adds to a list of levels is the phases between them. The diagonal entries say how much of each state there is; the off-diagonal ones say whether those states can interfere. Decoherence removes the second while leaving the first untouched — which is why probabilities are conserved by it, and why the object it acts on has to be the matrix rather than the list.

Two consequences follow immediately. The probabilities are unchanged, so decoherence never alters what fraction of particles land where in a which-path experiment — only whether the fringes appear. And the reduced matrix is a mixture rather than a superposition, so a system that has decohered looks exactly as though it had collapsed, to anybody with access only to it.

That is the sense in which the theory explains what it explains: not that collapse happens, but that the difference between a collapsed state and a decohered one is unobservable from inside the system.

Which states survive #

Decoherence does something else that is easy to miss and is arguably its most important output: it selects a preferred basis.

Quantum mechanics has no intrinsic preference between describing a state in one basis or another. But the environment does. The interaction with the surroundings is nearly always a function of position — photons scatter off where a thing is, air molecules bounce off where it is — so it is superpositions of different positions that decay, and states of definite position that survive.

Those surviving states are called pointer states, and they are the ones that the environment monitors without disturbing. That is why the classical world has definite positions rather than definite momenta or definite superpositions of positions: not because position is fundamental, but because the interaction happens to be local.

A pointer state is a localised packet, and it is robust for a reason that can be stated in one line: the environment’s interaction commutes with position. A photon scattering off a well-localised object leaves it well localised, so nothing about the monitoring degrades that shape — which is why localised states are what a monitored system settles into, rather than the energy eigenstates a textbook usually starts from.

For a system where the coupling is to something else — a superconducting circuit coupled through its flux, say — the preferred basis is something else, and engineering it is what building a qubit consists of.

What the argument does not do #

This is where care is required, because decoherence is routinely oversold and the overselling has a long history.

It does not produce a single outcome. After decoherence the state is $|1\rangle|e_1\rangle + |2\rangle|e_2\rangle$, which is a superposition of two branches that no longer interfere. Both branches are still there. Nothing in the dynamics has removed one, and the step from “two non-interfering branches” to “this one happened” is exactly the measurement problem.

What it explains is why the branches stop talking to each other. That is a real and quantitative achievement: it says why interference is unobservable for large systems, how fast, and in which basis. It converts the measurement problem from “why is a superposition never seen” — which is answered — into “why is there one outcome” — which is not.

Decoherence works by creating correlations between a system and its surroundings — correlations of exactly the kind no shared instruction list can reproduce. So the coherence that has left the system is still present in the world, in precisely the sense that an entangled pair’s correlations are present while neither half of the pair shows anything unusual on its own. Nothing has been destroyed; it has been moved somewhere no local measurement can reach.

And it is not irreversible in principle. The correlations are still there, and a sufficiently complete manipulation of the environment would bring the interference back. That is done routinely on small systems: a spin echo reverses the dephasing of a spin ensemble, and quantum error correction is an industry built on undoing decoherence before it becomes irrecoverable. What makes it effectively irreversible for a dust grain is the number of degrees of freedom involved, which is the same kind of practical irreversibility as the arrow of time in thermodynamics and rests on the same counting.

Why that move is irreversible in practice is a counting argument. Recovering the coherence would mean controlling every degree of freedom it went into, and the number of those grows so fast with the size of the system that recovery stops being a question of skill and becomes one of arithmetic — the same arithmetic that makes a shuffled deck stay shuffled.

The echo, and what it cannot undo #

The claim that decoherence is reversible in principle is worth demonstrating rather than asserting, and there is a routine laboratory procedure that does it — for one particular kind of coherence loss, which is what makes it instructive.

Take a large number of identical spins in a magnetic field. They should all precess at the same rate, and they do not: the field is never perfectly uniform, so a spin here sees a slightly different field from a spin there, and after some time the spins have fanned out in phase and the total signal has died away.

That looks exactly like decoherence and it is not. Nothing has been lost: each spin is still precessing perfectly coherently, and what has vanished is only the sum. Apply a pulse that flips every spin through half a turn, and every accumulated phase is reversed — the fast spins now find themselves behind and the slow ones ahead — so after an equal interval they all come back into step and the signal reappears.

That is a spin echo, it is a standard technique, and it is the cleanest available demonstration that a signal disappearing is not the same as information being destroyed.

What the echo cannot recover is the part where the environment genuinely changed. If a spin’s local field fluctuates during the interval rather than merely differing from its neighbour’s, the flip does not reverse it, because the phase it accumulated in the first half was not the phase it will accumulate in the second. That part is gone.

So there are two timescales and the distinction is the whole point. One is how fast the signal disappears, which includes the recoverable fanning-out; the other is how fast the echo’s height itself decays as the interval is lengthened, which measures only what cannot be undone. The second is often orders of magnitude longer than the first, and it is the number that matters for anything one wants to build.

The lesson generalises past spins. A measurement that a system has stopped showing coherence is not a measurement that its coherence has been destroyed, and telling the two apart requires trying to bring it back.

Correcting an error without learning it #

The other half of the practical response to decoherence is stranger than the echo and is the reason anybody believes a large quantum computation is possible at all.

The obvious approach is to check whether an error has occurred, and the obvious approach is fatal: any measurement that reveals the state of a quantum system destroys the superposition being protected. Looking to see whether the information survived is the thing that would destroy it.

The escape is to measure something else. Encode one logical bit of quantum information across several physical systems, in such a way that there exist joint measurements — of whether two of them agree, say, rather than of what either of them is — whose answers say which error occurred without saying anything about what is encoded. Those answers are called syndromes, they are ordinary classical bits, and reading them collapses nothing that matters.

Given the syndrome, the error is known, and applying its inverse restores the state. Done fast enough and often enough, the encoded information survives indefinitely in a system whose individual parts are decohering the whole time.

There is a threshold in it, and the threshold is what makes it an engineering programme rather than a proposal. If the physical error rate is below some critical value, adding more layers of encoding makes the logical error rate fall faster than the overhead rises, so an arbitrarily long computation is possible with a finite — if large — amount of hardware. Above the threshold, correction adds errors faster than it removes them and no amount of hardware helps.

That the threshold exists is a theorem; where it is depends on the code and the noise model, and the values for the codes now being built are in the region of a per cent. The whole of the past two decades of superconducting and trapped-ion work is a campaign to get physical error rates below that number and keep them there while the number of devices grows.

Which is a curious relationship to have with a physical process. Decoherence is not being prevented — it cannot be — and it is not being reversed in the echo’s sense either. It is being detected and undone continuously, by an apparatus that never finds out what it is protecting.

Where it is measured #

The theory is quantitative and has been tested against experiment in several systems, which is what distinguishes it from an interpretation.

Molecular interferometry. Fullerenes and larger molecules can be sent through a grating and their interference watched as decoherence is turned up deliberately — by admitting gas at a controlled pressure, or by heating the molecules until they radiate thermal photons. The fringe visibility falls exactly as the theory predicts, over several orders of magnitude in pressure, with no free parameters.

Cavity experiments. A superposition of two coherent states of a microwave field — a “cat” of a few photons — has its decoherence watched directly as the field leaks from the cavity, and the decay rate scales with the separation of the two states as predicted.

And superconducting circuits, where coherence times have been pushed from nanoseconds to milliseconds over twenty-five years by systematically removing the couplings this theory identifies. That programme is the most economically consequential application of decoherence theory, and it works.

The size of the largest thing put in two places #

It is worth recording the state of the art, because the number is the practical measure of everything above.

Molecule interferometry has reached masses around 25,000 atomic mass units — molecules of a few hundred atoms, with de Broglie wavelengths of a few picometres and interferometers metres long. The limiting factors are exactly the ones this theory names: residual gas pressure, thermal emission from the molecules themselves if they are warm, and the difficulty of making a coherent source.

Mechanical oscillators have been cooled to their ground state and put into superpositions of two vibrational states, at masses of nanograms — far heavier than any molecule, though the superposed separation is far smaller, and it is the product that decoherence cares about.

Every experiment that puts something in two places at once is an analyser and a re-analyser in some form: sort the states, let the branches travel apart, bring them back and sort again. What limits the size of the thing being sorted is not the sorting. It is how long the two branches can be kept from becoming correlated with anything else, which is a question about vacuum, temperature and vibration rather than about optics.

Neither number is anywhere near a dust grain, and the gap is not one of ingenuity: closing it means beating a decoherence rate that scales as the square of the separation and rises steeply with size. The proposals for going further — levitated nanoparticles in ultra-high vacuum, cooled to millikelvin, with the superposition held for milliseconds — are all attempts on the same quantity, and their difficulty is a direct reading of the second figure in this essay.

What the pictures cannot show #

The Λ values are quoted, not computed here. The localisation rates for air, sunlight, thermal photons and the microwave background come from the literature; what this essay computes from them is the coherence time for a stated separation. Deriving them requires a scattering calculation for each environment.

The model is a two-state superposition. Real systems have continuous position, and the density matrix’s decay is a function of separation rather than a single rate. The (Δx)² law drawn on holds only for separations small compared with the scattering wavelength; for larger separations the rate saturates.

Nothing here is about the interpretation of quantum mechanics. Everything computed above is standard unitary quantum mechanics with no additional assumptions, and it is common to every interpretation. Which interpretation one prefers changes what one says about the leftover branch and changes no number on any figure.

And the environment is assumed to be uncorrelated with itself. A structured environment — one with memory, or with its own coherences — can produce non-monotonic decay, and information can flow back into the system temporarily. That is the subject of non-Markovian decoherence and it is not the simple exponential drawn here.

Why the word “classical” can now be defined #

The last thing decoherence supplies is a definition of a word that was previously used and not explained.

A system behaves classically when three things hold: its state is one of the pointer states the environment selects; the coherences between those states have decayed; and the correlations that carry them away are spread over so many degrees of freedom that no feasible operation recovers them. All three are quantitative, and all three are consequences of the coupling rather than of a boundary between two kinds of physics.

That removes the need for a line drawn somewhere between the small and the large. Where the quantum picture hands back the old one had one answer in terms of large quantum numbers; this is a second, and it is the one that applies to a dust grain, which has no large quantum number to speak of and is classical for a different reason entirely.

A quantum system’s behaviour approaches a classical one as its quantum numbers grow, which is the correspondence principle and is where most accounts stop. Decoherence supplies a second and independent route to the same limit: a system with small quantum numbers also behaves classically, once its coherences are held by an environment rather than by itself. Two mechanisms, one destination, and only the second explains why a dust grain has a position.

The residue is the part worth being honest about. Decoherence explains why the world looks classical, and does not explain why any particular thing happens. Those are different questions, and the first is now answered quantitatively while the second is exactly where it was in 1935.

The ladder from here #

Later rungs on this anchor: the master equation for the reduced density matrix, and the Lindblad form its evolution takes; einselection and the derivation of pointer states from the interaction Hamiltonian; quantum Darwinism, which asks why many observers agree about a classical fact and answers it by counting redundant copies in the environment; and the measurement problem proper, where the interpretations differ.

The neighbouring ladders are the answer that was not there before, which is the collapse this essay works without, the measurement that never touched it, which shows that information rather than disturbance is what costs the interference, and one arrival at a time, where the coherence being discussed is directly visible.