The value a gentle measurement returns

The answer that was not there before found that measuring a spin gives one of two answers, and that the answer is created by the measurement rather than read from the spin: send the survivors of one analyser through another at right angles and they split evenly, as though the first answer had been forgotten. The state that cannot be copied closed off the obvious escape, making spare copies to measure in different ways, and listed what remained, including the trade between how much a measurement learns and how much it disturbs, and weak measurement, “where a little is learned and little is disturbed, repeatedly”.

This essay takes that up. A measurement can be made arbitrarily gentle by coupling the pointer only slightly to the thing measured. Each run then learns almost nothing, and disturbs almost nothing. Averaged over many runs, the gentle measurement recovers the ordinary average. The surprise comes when the runs are sorted by what happens to the system afterwards. The average pointer reading in a selected subset can lie far outside the range of values the measured quantity can have. That weak value is real, measurable and useful, and it is also one of the most argued-over objects in quantum mechanics, because it seems to say something about a system between two measurements that ordinary quantum mechanics says nothing about.

A measurement too gentle to split #

A measurement has two parts: the system, and a pointer that ends up in a different place depending on the system’s value. A Stern–Gerlach magnet is the classic case — the spin is the system, the atom’s position is the pointer — and if the pointer’s shift is large compared with its initial spread, the two outcomes land in separate places and each run reads one value.

The figure shows both regimes. With the coupling three times the pointer’s spread, the dashed curve, the pointer splits into two clean spots at $\pm3$. With the coupling reduced to a tenth of the spread, the two possible positions overlap almost completely and the pointer is a single blob barely displaced. For a spin prepared at an angle close to the $x$ direction, the average of the spin’s $z$ component is small, and the blob moves by only 0.020 spreads, the average times the coupling. A single run tells almost nothing about which value the spin had, and it disturbs the spin almost not at all, because the pointer has barely become correlated with it. Averaged over many runs, the shift still recovers the ordinary expectation value. Nothing yet is unusual.

Now add a second step. After the weak coupling, measure the spin strongly in a different direction — along $x$ — and keep only the runs where it points along $x$. With the preparation chosen close to the opposite direction, only 0.99 per cent of runs pass. In those runs the pointer is centred at 0.802 spreads, eight times the coupling. Neither value of the spin could move the pointer by more than one coupling. The kept runs have moved it by eight.

The value that comes out #

Aharonov, Albert and Vaidman worked out the general rule in 1988. For a system prepared in a state $|i\rangle$, coupled weakly to a pointer through an observable $A$, and then post-selected in a state $|f\rangle$, the kept pointer is shifted by the coupling times the weak value

$$A_w = \frac{\langle f|A|i\rangle}{\langle f|i\rangle}.$$

When the initial and final states are the same, this is the ordinary expectation value. When they differ it can be anything: large, negative, or complex, since it is a ratio of amplitudes and not an average of values. The title of their paper announced that the result of a measurement of a spin-½ particle can turn out to be 100.

The figure shows the price. The weak value is large when the denominator $\langle f|i\rangle$ is small, which means choosing a final state nearly orthogonal to the initial one, and then almost every run fails the post-selection. For the spin here the kept fraction is exactly $1/(1 + A_w^2)$: half the runs for a weak value of one, one per cent for ten, one in ten thousand for a hundred. The large values come from rare events, and the rarer the events, the larger the values.

The arithmetic behind it is the one that makes a wiggle faster than any wave in it possible. That essay noted that superoscillation — a function oscillating locally faster than any of its Fourier components — was discovered by people thinking about exactly this measurement. The kept pointer is a superposition of two displaced Gaussians with amplitudes of opposite sign and nearly equal size, $\cos\theta\,\varphi(x - g) + \sin\theta\,\varphi(x + g)$. The two nearly cancel everywhere, and what survives the cancellation is a small Gaussian centred far from either of them. No new physics enters: two overlapping, almost cancelling waves leave a residue that can sit anywhere.

A value that needs the touch to stay light #

Because the weak value comes from near-cancellation, it survives only while the two displaced pointers overlap strongly.

The figure computes the kept pointer’s shift, divided by the coupling, as the coupling is strengthened. At a thousandth of the pointer’s spread the shift is exactly the weak value, 10 or 3. As the coupling grows the shift falls away, beginning once the coupling times the weak value approaches the pointer’s spread — which is why the first figure’s shift was 8 rather than 10 — and at strong coupling it comes back inside the range $\pm1$ where every ordinary measurement of a spin lies. At a coupling of three spreads, the kept runs show 0.20 and 0.61.

That dependence is the first thing to hold on to when interpreting a weak value. It is not a number the spin carries between preparation and post-selection, waiting to be read. It is the limit of what a particular procedure returns as its coupling is made gentler, and a stronger version of the same procedure returns something else. The procedure is also not one measurement but an ensemble statistic: the weak value is an average over many kept runs, each of which individually reads almost nothing.

Where the information goes #

A pointer shifted ten times further looks like an amplifier, and weak values have been used as one. The question is what the amplification buys.

The figure measures it with the Fisher information: how much each run tells about the size of a small coupling $g$, in units of one over the pointer’s spread squared. The most any run can carry, with this spin and this pointer, is one. If every pointer is read and the spin is ignored, the information depends on the spin’s average, which goes to zero as the preparation is turned towards the large-weak-value settings: 0.039 at a weak value of ten. If instead only the kept runs are counted, their share of the total is $A_w^2/(1+A_w^2)$ of the ceiling — half for a weak value of one, 0.990 for ten, 0.9999 for a hundred.

So post-selection does something striking and something modest at once. It concentrates almost all the information about $g$ that the whole experiment could have carried into the one per cent, or the one in ten thousand, of runs it keeps. It creates none. The amplified shift is exactly paid for by the runs thrown away, and the best possible estimate of $g$ from the kept runs is no better than the best possible estimate from all runs.

The practical advantage lies in the noise. A real pointer is read by a real detector, and detectors have noise that is not quantum — electronic noise, drifts, vibrations, a detector that saturates at high light levels. Those do not care whether a run was post-selected. Concentrating the signal into a few runs with large shifts can lift it above noise that would swamp a small shift spread over many runs, and it lets a detector handle a weak beam instead of a strong one. That is the regime in which weak-value amplification has earned its keep. In 2008 Hosten and Kwiat measured a displacement of a light beam of about an ångström, the spin Hall effect of light, with weak-value amplification of about ten thousand. Dixon, Starling, Jordan and Howell measured beam deflections of a few hundred femtoradians the following year. Neither beat a quantum limit, which weak values cannot do; both beat technical noise in ways ordinary measurement could not.

Three boxes and a negative presence #

Weak values become strangest when applied to questions that sound like counting.

Aharonov and Vaidman’s three-box arrangement prepares a particle in an equal superposition of three boxes, A, B and C, and later keeps only runs in which it is found in the superposition of A plus B minus C, which happens in 11.1 per cent of runs. The figure computes weak measurements of whether the particle is in each box, done one box at a time with the pointer shifted by $g$ for a yes. The kept pointers move by 1.00 times $g$ for box A, 1.00 for B, and $-0.99$ for C. The weak values are 1, 1 and $-1$.

The three add to one, as the probabilities of being in some box must, because the weak values of projectors that add to the identity must themselves add to one. But the one for box C is negative. Read literally, the particle is certainly in box A, certainly in box B, and in box C minus once. That is not a probability of anything, and it is the clearest sign that weak values follow the algebra of the operators rather than the rules of counting. The same kind of object appears in the probability that goes below zero, where the Wigner function, a quantum state’s closest analogue of a distribution over position and momentum, takes negative values that no measurement ever sees directly. Weak values are negative in the same sense: as the coefficients of an expansion, not as the frequencies of an event.

A disturbance made small, and the coherence it leaves #

Weak measurement sits between two situations earlier essays treated as opposites. At one extreme, a strong measurement records which value the system had and destroys any superposition of the two, the process where the interference goes described as coherence moving from the system into the pointer. At the other, the measurement that never touched it learned about an object without any interaction at all, by an interference arrangement in which the object’s mere possible presence changed the outcome.

A weak coupling does a little of the first. The pointer becomes slightly correlated with the spin, and the spin’s superposition is slightly degraded — by an amount that shrinks as the square of the coupling, and the information gained per run shrinks as the same square. That is the information–disturbance trade in its simplest form. Neither vanishes faster than the other, and a weak measurement is a strong one diluted, not a different kind of process. What makes it useful is that the disturbance is small enough for the post-selection afterwards to see the state almost as it was prepared. The near-cancellation that produces the weak value depends on the spin’s superposition surviving the coupling nearly intact, so a strong coupling, which destroys the superposition, destroys the effect.

The first experiment was optical rather than atomic. In 1991 Ritchie, Story and Hulet used a birefringent crystal to shift a laser beam’s position very slightly depending on its polarisation — the polarisation playing the spin, the beam’s transverse position the pointer — and post-selected with a nearly crossed polariser. The beam’s centre moved by many times the crystal’s own shift, as predicted. Almost every weak-value experiment since has used light for the same reason: polarisation and beam position are an ideal system and pointer, prepared and read with high precision and coupled by a piece of glass.

The atomic version, with a real Stern–Gerlach magnet, is harder, and for electrons impossible in the form drawn here: the experiment that defines spin and cannot be done on it explained why a magnetic field cannot separate a free electron’s spin states, because the Lorentz force on the charge blurs the splitting. A weak measurement needs no clean splitting — it deliberately avoids one — but it still needs a pointer whose shift is controlled, and for charged particles the charge’s own response to the field swamps it.

Post-selection also appears in the tests of Bell’s inequalities, where the correlation no instructions produce required that no run be discarded in a way that could depend on the settings. There, selecting a subset of runs is a loophole to be closed. Here it is the method. The same operation that can fake a violation of Bell’s inequalities, if done carelessly, produces the weak value when done deliberately, and that is one reason weak values need careful reading.

What a weak value is, and is not #

The disagreements about weak values are real, and they are about interpretation rather than prediction. Everyone agrees on what a weak measurement followed by post-selection will show, because it is standard quantum mechanics applied to a pointer. The disputes concern what, if anything, the weak value says about the system between preparation and post-selection.

One reading, Aharonov’s, takes the pair of states — the prepared one running forwards and the post-selected one running backwards — as a complete description of the system in the interval, with weak values as its properties there. On that reading the three-box particle really does have a negative presence in box C, in the weak sense. Another reading treats a weak value as a conditioned average with no special status, and notes that the same numbers can be produced by classical post-selected statistics with appropriately designed disturbances. That makes anomalous weak values evidence of something quantum only when classical explanations are carefully excluded. A third reading, more technical, points out that the imaginary part of a weak value measures the disturbance the measurement causes, which ties weak values firmly to the measurement process rather than to the system alone.

What is not in dispute is the constraint the fourth figure drew. Weak values cannot beat the quantum limits on how much can be learned. They can reorganise where the information sits, and that reorganisation is useful. They also cannot be used to send signals faster than light or to copy a state, because post-selection is a filter on outcomes that have already happened, and the state that cannot be copied is not threatened by a procedure that throws most of its runs away.

Where the picture stops #

The figures use the simplest possible system and pointer, and three things lie beyond them.

Complex weak values. With general initial and final states the weak value is complex. Its real part shifts the pointer’s position, and its imaginary part shifts the pointer’s momentum instead. Measurements of both parts together have been used to reconstruct a photon’s transverse wavefunction directly, by Lundeen and colleagues in 2011, with the wavefunction appearing as a weak value measured point by point.

Sequential weak measurements. Measuring two observables weakly one after the other, and correlating the pointers, gives joint weak values of quantities that cannot be measured together strongly, such as a particle’s position and its momentum. These have been used to map average trajectories of photons in a two-slit experiment. Whether those trajectories mean anything beyond the statistics that defined them is the same interpretive question in a sharper form.

Real pointers are noisy. The information figure assumes a perfect pointer with Gaussian quantum spread. Real experiments have pointers with their own imperfections, and the analysis of when weak-value amplification helps against which kinds of noise has become a technical literature of its own, with the answer depending on whether the noise is correlated in time and whether the detector saturates.

Still open: whether anomalous weak values prove anything quantum #

The anomalous weak value — one outside the range of the observable’s eigenvalues — has been proposed as a signature of genuinely quantum behaviour, a kind of contextuality that no classical model could reproduce. Pusey showed in 2014 that anomalous weak values, under certain assumptions about how the measurement disturbs the system, do witness contextuality. Other analyses have constructed classical models that reproduce the statistics when those assumptions are relaxed. Whether an anomalous weak value, measured in a real experiment with its imperfections, can by itself certify that something non-classical has happened, and under exactly which assumptions, is debated in current work on the foundations of quantum measurement.

The habit worth carrying away is to ask what a statistic is conditioned on before reading it as a property. A weak value is the average pointer reading among runs selected by a later outcome, and the selection can place that average anywhere, because it is a ratio of amplitudes that nearly cancel. It is exactly as real as any other conditioned average and no more. It can make a small effect visible above technical noise, it cannot beat the quantum limit on information, and it cannot tell what a quantum system was doing when nobody was looking closely enough to disturb it.