The fastest a state can stop being itself

The questions that can be asked together replaced the famous inequality with the condition behind it: two quantities can have definite values at once exactly when their operators commute, and when they do not, the product of their spreads has a floor fixed by the commutator. Position and momentum are the standard pair. Energy and time look like another, and the relation $\Delta E\,\Delta t \ge \hbar/2$ is written in the same form and quoted as often.

It cannot be the same kind of statement. Sharpness has to be paid for noted the reason: a system does not have a time the way it has a momentum, because time is the parameter the state evolves in, not a quantity measured on it. Pauli showed in 1933 that no operator can stand for it — an operator conjugate to the energy would shift energies by any amount, so the energy would have no lowest value and no atom would be stable. With no time operator there is no commutator, and the inequality written with $\Delta t$ has to mean something else.

What it can mean is a statement about how fast a state changes. That statement has two sharp forms, both theorems, and both are testable by the most elementary calculation quantum mechanics has.

Three theorems wearing one inequality #

The relation written $\Delta E\,\Delta t \gtrsim \hbar$ is quoted for at least three different facts, and they need separating before any of them can be tested.

The first is not about quantum mechanics at all. A pulse lasting a time $\Delta t$ contains a spread of frequencies of about $1/\Delta t$, because a short burst of any wave is a sum of many frequencies — the reason a short note has no definite pitch. Multiplying the frequency by $\hbar$ turns it into an energy, and the statement becomes one about the energy content of a short-lived wave. It bounds the duration of a signal by its bandwidth.

The second is about decay. A state that decays with lifetime $\tau$ has an energy distribution of width $\hbar/\tau$, and the width of a line is its lifetime in another coordinate. It relates a rate at which probability leaks away to the shape of an energy distribution, and it is exact only for the exponential decay that holds only in the middle.

The third is the one this essay is about, and it is the only one that is a universal inequality about any state under any Hamiltonian: a bound on how quickly a state can become distinguishable from what it was. The other two are relatives of it rather than consequences, each bounding something narrower. None of the three says that energy conservation can be suspended for a short time.

A floor under the survival #

A state that is not an energy eigenstate changes, and the natural measure of how much is the survival probability $|\langle\psi(0)|\psi(t)\rangle|^2$: the chance that a measurement asking “is it still the initial state?” answers yes. If the state is a superposition of energies $E_k$ with probabilities $p_k$, the overlap is a sum of phases, $\sum_k p_k\, e^{-iE_k t/\hbar}$. At $t = 0$ every phase is zero and the survival is one; as the phases fan out, it falls.

The exponential that is only true in the middle found how it starts to fall: quadratically, as $1 - (\Delta E\, t/\hbar)^2$, with the energy spread setting the curvature. In 1945 Mandelstam and Tamm proved what happens after the start. The survival can never fall faster than

$$|\langle\psi(0)|\psi(t)\rangle|^2 \ge \cos^2\!\left(\frac{\Delta E\, t}{\hbar}\right) \quad\text{for } 0 \le \frac{\Delta E\, t}{\hbar} \le \frac{\pi}{2},$$

so a state cannot reach an orthogonal state — one a measurement could distinguish from the start with certainty — in less than $\pi\hbar/2\Delta E$.

The geometry behind the inequality is short. The angle between the evolving state and its starting point, $\arccos|\langle\psi(0)|\psi(t)\rangle|$, is a distance in the space of states, and the rate at which a state can move through that space is its energy spread divided by $\hbar$. A state cannot cover an angle of $\pi/2$ — the distance to an orthogonal state — faster than that speed allows. The energy spread is a speed, and the orthogonal state is a distance.

Spending a spread well and badly #

Every curve in the figure has the same energy spread, and they do very different things with it.

The two equally weighted levels are the fastest possible state. Their survival is $\cos^2(\Delta E\,t/\hbar)$ exactly — they run along the edge of the forbidden region and arrive at an orthogonal state at the limit. This is not a special curiosity: a spin one half whose axis turns through 180 degrees about a perpendicular field, and a qubit flipped from 0 to 1 by a resonant pulse, are both this state, and both reach the orthogonal state in exactly the time their energy spread permits. A resonant π pulse is the fastest flip that its drive strength allows, which is why the speed of a quantum gate is fixed by how hard the qubit is driven.

The same two levels driven off resonance by as much as the drive have the same energy spread, and waste it. The state precesses about a tilted axis, never reaches the opposite pole, and its survival bottoms out at one half. Three equally spaced levels with equal weights do reach an orthogonal state, but only at 1.0887 times the limit, because three phases cannot all oppose each other at once. A coherent state of an oscillator, spread over many levels with Poisson weights, never becomes orthogonal at all; with an average of one quantum its survival bottoms out at 0.0183 half a period in, when the packet is at the far side of its swing, and then climbs back.

The inequality therefore bounds how fast a state may change, not how fast it does. A large energy spread is a permission. Whether it is used depends on how the spread is arranged, and the two-level arrangement is the only one that uses all of it.

The other speed limit #

Mandelstam and Tamm’s bound uses the spread of the energy. In 1998 Margolus and Levitin found a second bound that uses its mean: a state whose average energy above the ground state is $\langle E - E_0\rangle$ cannot reach an orthogonal state in less than $\pi\hbar/2\langle E - E_0\rangle$. The two are independent — a state can have a large spread and a small mean above the ground, or the reverse — so both apply at once, and the one that binds is whichever gives the longer time.

For sets of equally weighted levels, the spread bound is the tight one. The true time falls as $2\pi/N$; the spread grows almost in proportion to $N$, so its bound falls almost as fast and stays within 15.5 per cent. The mean energy above the ground also grows with $N$, but only as fast as half the span of the levels, so its bound ends up at half the true time. The mean bound wins instead for states whose energy sits mostly near the ground with a thin tail far above it, where the spread is inflated by the tail while the mean is not. Levitin and Toffoli showed in 2009 that the larger of the two bounds is attained, and only by the two-level state with equal weights — the state that sits at 1.0000 on both curves.

The mean bound has a consequence the spread bound does not. A state’s energy spread can be made large without supplying much energy, but its mean energy above the ground state is energy that must actually be provided. So Margolus and Levitin’s theorem says that changing a system into something distinguishable costs energy per unit of speed, not merely spread per unit of speed — and it holds for any system, whatever it is made of.

Six hundred states and no exception #

A bound checked on hand-picked examples invites the suspicion that the examples were picked to obey it. The inequality in its general form covers any amount of change, not only orthogonality: the angle $\arccos|\langle\psi(0)|\psi(t)\rangle|$ can never exceed $\Delta E\,t/\hbar$, so a survival of one half, an angle of $\pi/4$, cannot be reached before $(\pi/4)\hbar/\Delta E$.

None of the six hundred crosses the diagonal, and most of them sit close to it: 429 of the 599 that reach a survival of one half get there within five per cent of the earliest time allowed. The bound is not a generous overestimate that typical states ignore. For losing half of the initial state, the energy spread nearly always sets the pace, and it is only for losing all of it — reaching exact orthogonality — that the arrangement of the phases usually costs a large factor.

Moving and spreading #

A free particle’s wave packet is the most familiar evolving state, and it changes in two ways at once: its centre moves and its width grows.

The moving packet runs almost along the limit. Almost all of its energy spread is the spread of its velocity: a packet containing a range of wavenumbers around a large mean has energies spread in proportion to that mean, and the same spread of velocities is what carries it bodily away from where it started. When it has moved by one width, its angle from the initial state is 97.7 per cent of the largest the theorem allows. The packet at rest has an energy spread of a different origin — its wavenumbers run both ways around zero — and that spread goes into widening the packet rather than moving it. It stays much more like its initial state for much longer than its energy spread would permit.

The comparison says what the speed limit is about. Translation is the most efficient kind of change, because displacing a state by its own width makes it nearly distinguishable while using only the energy spread it needs; broadening is inefficient, because a packet that has doubled in width still overlaps most of what it was. How close a process comes to the limit is a measure of how directly its energy goes into making the state different.

What a joule buys per second #

Margolus and Levitin’s bound turns into a statement about computing that has been quoted ever since, because an elementary logical step is, at the least, a change of a physical state into one distinguishable from it.

The rate is proportional to the energy available: 6.04 × 10³³ orthogonal steps per second for each joule above the ground state. A two-level system sits exactly on the line — a caesium clock’s pair of levels, in an equal superposition, passes through an orthogonal state twice per cycle, 18.4 billion times a second. Seth Lloyd used the same number in 2000 to bound an idealised computer of one kilogram, all of whose rest energy is spent on computing: at most $5.43\times10^{50}$ elementary operations a second, about $10^{40}$ times what a present processor of that mass does.

The figure is often misread as a cost per operation. It is a cost per operation per second. The energy needed to flip a bit quickly is set by this bound; the energy that must be dissipated to erase one is set by a quite different argument, Landauer’s, about entropy; and a computation carried out slowly enough can use arbitrarily little energy for each step. The speed limit prices time, not work.

A clock is a state that changes #

The bound has a reading that makes it concrete. A clock is a physical system that passes through a sequence of states that can be told apart — the positions of a hand, the phases of an oscillation — and a tick is one such passage. The number of distinguishable states a clock passes through per second is its resolution, and the speed limits cap it: a clock whose state has energy spread $\Delta E$ cannot tick faster than once per $\pi\hbar/2\Delta E$, and one whose mean energy above the ground is $\langle E - E_0\rangle$ cannot tick faster than once per $\pi\hbar/2\langle E - E_0\rangle$.

The second form carries a point that is easy to miss. It counts energy above the ground state, not total energy. An oscillator sitting in its ground state has zero-point energy that cannot be removed, and that energy buys no change at all: the ground state is stationary, and a clock made of it never ticks. Only energy in excitations, energy that could in principle be extracted, can drive a state through distinguishable configurations.

Distinguishable is doing real work in that sentence. Orthogonal states are exactly the ones that can be told apart with certainty, and so they are the only states whose record can be copied without disturbing them. A clock’s successive readings are meant to be read and written down, so a tick is a passage between orthogonal states, and the speed limit on reaching an orthogonal state is a speed limit on keeping time.

Where the limits stop applying #

The Hamiltonian was held fixed. When it changes in time — a pulse switched on and off, a field swept — the bounds still hold with the energy spread replaced by its average over the evolution, a form Anandan and Aharonov gave in 1990. The limit is then on the path the state takes through the space of states, and a cleverly shaped pulse can follow a shorter path than a constant one.

The states were pure. A system in contact with an environment is described by a mixed state, and its change is measured by a different distance between states. Speed limits exist for that case too, but several inequivalent versions have been derived, and which is tightest depends on the kind of environment.

Orthogonal is a mathematical statement. A state orthogonal to its initial state could be distinguished from it with certainty by the right measurement, which asks the question a measurement answers only in one basis. A measurement in the wrong basis sees nothing, and nothing in the bound says the right one is available.

And the energy spread must be finite. The exponential decay of an idealised unstable state corresponds to an energy distribution with infinite spread, for which the Mandelstam–Tamm bound says nothing. A real decaying state has a finite spread, a quadratic start, and a bound that holds but is so loose that the width of its line, which fixes its lifetime, is the more useful number.

What the figures leave out #

Every figure measures time in units of $\hbar/\Delta E$, which makes all the curves comparable and hides the times themselves. For a qubit driven at a few megahertz the limit is tens of nanoseconds; for an electron in an atom, attoseconds. The units are chosen so that the theorem looks the same everywhere, which it is.

The random-state figure uses six levels because six are enough for most random superpositions to reach a survival of one half within a period; with fewer the phases recur too soon, and with many more the scatter plot would look the same. It is a test of an inequality, not a survey of typical quantum dynamics.

Still open: a speed limit for things that are not closed #

Mandelstam and Tamm’s argument needs a closed system with a Hamiltonian. Nearly every system worth controlling — a qubit in a processor, a molecule in a solvent — is open, and its evolution is not a rotation in the space of states but a contraction towards some mixture. Speed limits have been derived for open systems from several directions, using different measures of distance between mixed states, and they disagree about whether an environment can speed up the change of the system it acts on. Some bounds say the environment can make a state leave itself faster than its own energy spread would allow; others are constructed so that it cannot.

Which of these bounds can actually be attained, which measure of distinguishability they should be stated in, and whether the resulting limits are genuinely quantum or have classical counterparts of the same form for probability distributions — as calculations since 2018 suggest — has not been settled, and experiments on driven superconducting qubits have begun to test some of them against one another.

The habit worth carrying away is to ask what an inequality with a $\Delta t$ in it is actually about. Time is not measured on a state; it is what a state changes along, and the honest energy–time relation is a limit on how far a state can travel through the space of states in a given time, with the energy spread as its speed.