The third question pairwise bounds miss

The sharpness a variance cannot see replaced Heisenberg’s product of spreads with a statement about missing information: the Shannon entropies of the outcomes of two incompatible measurements add up to at least a fixed amount, a bit for a qubit’s x and z. That essay ended with a question it could not answer — what the tightest relation is for more than two questions — and noted that it matters for quantum cryptography, where how much an eavesdropper can know depends on it. The simplest case is a single spin-½ asked about all three directions in space, and even there the answer is not what adding up pairs would suggest.

Three questions to a spin #

A spin-½ can be asked whether it points up or down along any axis. Take the three perpendicular axes x, y and z. These questions are mutually unbiased: a spin that answers one of them with certainty answers each of the other two with a fifty–fifty coin toss. The questions that can be asked together found that no two of the three spin components commute, so no state can give definite answers to any two of them. Mutually unbiased is the strongest version of that: complete knowledge of one answer implies complete ignorance of the others.

Any pure state of the spin is a direction in space, the Bloch vector, and the probability of “up” along each axis is $(1 + r_i)/2$, with $r_i$ the vector’s component along that axis. The uncertainty in each answer, in bits, is the binary entropy of that probability, and the total for three questions is the sum.

A spin pointing along z has no uncertainty about z and one bit about each of x and y: two bits in all. A spin pointing along any axis does the same. Tilting it away from the axes spreads the certainty more evenly and raises the total, to a maximum when the spin points equally towards all three, at 54.7° from each axis, where each answer is up with probability 0.789 and the total is 2.232 bits. Every pure state lies in that narrow band. Two bits is the floor, and it is reached.

That floor is not what simpler arguments give. The two-question relation, the Maassen–Uffink bound, says each pair of these questions carries at least one bit between them. There are three pairs; add the three inequalities and each entropy appears twice, so twice the total is at least three bits, and the total at least 1.5. That is true but weak.

The questions as an experiment #

The three questions are not abstractions; they are three orientations of a Stern–Gerlach magnet, the apparatus that splits a beam of spin-½ atoms into two according to the spin’s component along the magnet’s axis. The answer that was not there before followed the sequence of such magnets that first showed how a measurement along one axis destroys the answer along another: atoms selected as up along z, sent through a magnet along x, divide fifty–fifty, and those selected as up along x, sent through another magnet along z, divide fifty–fifty again, though every one of them had been up along z two magnets earlier.

The entropy sum is the bookkeeping for that experiment. Prepare a beam in any state, send one third of it through a magnet along each of the three axes, and record the fractions. From each fraction comes an entropy, and the three add to at least two bits. A beam that comes out entirely on one side of one magnet is, by that very fact, split evenly by the other two. A beam that is lopsided on all three — say 79 per cent up on each — can be made, but it cannot be made more lopsided than that on all three at once, and its total uncertainty is then larger than the minimum, not smaller. There is no preparation of atoms that the three magnets together find predictable.

That makes the floor a practical limit on what an experimenter can know about a source of spins. However the source is built, measuring a third of its output along each axis will find at least two bits of unpredictability per atom across the three, and an experimenter who finds less has made an error. The floor is a check on the measuring apparatus as much as a statement about spins.

A bound that is true and not tight #

There is also a general bound for any complete set of mutually unbiased questions. In a system with $d$ distinguishable states, at most $d + 1$ questions can be mutually unbiased, and Ivanović and Sánchez-Ruiz showed in the early 1990s that their entropies together satisfy

$$\sum_{i=1}^{d+1} H_i \ge (d + 1)\log_2\frac{d+1}{2}.$$

For a qubit, $d = 2$, three questions, the bound is $3\log_2 1.5 = 1.755$ bits.

So there are two valid lower bounds, at 1.5 and 1.755, and a true floor at 2. The gap is the information that general arguments throw away. A pairwise bound treats each pair as if the third question did not exist, and the general bound is derived from a property — the total “collision probability” of the outcomes — that holds for any dimension but does not capture everything about two. Sánchez-Ruiz proved in 1995 that the true minimum for a qubit is 2, using the specific geometry of the Bloch sphere: the three probabilities are tied together by the requirement that the Bloch vector have length at most one. That constraint links all three questions at once, and no argument working pair by pair can reproduce it.

The measure that cannot see it #

Variances, the measure of spread in Heisenberg’s original relation, fail more completely.

The variance of the answer to each question, counted as ±1, is $1 - r_i^2$. Add the three and the result is $3 - |r|^2$, which for a pure state, with $|r| = 1$, is exactly 2, always. A spin along z, certain about one question, and a spin tilted equally towards all three, uncertain about every question, have identical total variance. A relation written in variances for these three questions is not a constraint at all, merely an identity. The Robertson relation for any two of them, $\Delta S_x \Delta S_y \ge \tfrac{\hbar}{2}|\langle S_z\rangle|$, is weaker still, since its right side vanishes whenever the third component’s average does.

That is the strongest form of the point the previous essay made. A variance measures how far outcomes lie from their average, which for a yes–no question is a poor measure of how unpredictable it is. The entropy measures unpredictability directly, and for three questions it is the only one of the two that says anything.

Why entropy suits a yes-or-no question #

The failure of variances is not special to spins. Sharpness has to be paid for derived Heisenberg’s relation from Fourier analysis, where position and momentum are continuous and their spreads are natural measures of uncertainty. For a quantity with only two outcomes, spread is a strange measure. A question answered “up” with probability 0.99 and one answered with probability 0.5 differ in their variances by a factor of 25, from 0.04 to 1, but in their information by a factor of 12, from 0.08 bits to 1. Variance rewards outcomes for being far from their average; entropy rewards them for being hard to guess. Only the second is what “uncertain” means.

The angle the uncertainty principle cannot be written for met the same problem for an angle, where variance depends on where the circle of values is cut and the relation with angular momentum cannot be written as a product of spreads at all. The entropic form needed no repair there, and here it needs none for three questions. Entropy treats every outcome as a label rather than a number, so it does not care whether the outcomes are ±1, angles or colours, and that is why the theory of uncertainty for many questions is now written entirely in its terms.

There is one more advantage that matters for the next section. Entropy has a natural conditional form: the uncertainty about one thing given knowledge of another. Variance has a conditional form too, but it does not combine with quantum correlations in the clean way entropy does, and the uncertainty relation with a quantum memory has no satisfactory variance version.

A partner who can answer #

Entropic relations have a feature that variance relations lack: they can be extended to include what an observer already knows from somewhere else. Suppose the spin is one of an entangled pair, and the same question is put to its partner as to the spin. The relevant uncertainty is then the conditional entropy, what remains unknown about the spin’s answer once the partner’s answer is known.

For a perfectly entangled Bell pair, the partner’s answer to any of the three questions predicts the spin’s answer with certainty — the same, or opposite, depending on the Bell state. The total conditional uncertainty is zero. That does not contradict the uncertainty relation, which is about a single system’s state; it shows that the relation’s floor is a property of what one system can know about itself, and that a second system entangled with it changes the accounting. Mario Berta and colleagues gave the general form in 2010: the bound is lowered by an amount set by how entangled the system is with the memory.

That is the property quantum cryptography exploits from the other side. In protocols in which two parties check their measurements against each other, any eavesdropper is a would-be partner. What two have they cannot give a third found that entanglement is monogamous: if the two legitimate parties are strongly entangled, a third cannot be. The entropic relation with memory turns that into a number. The more the two parties’ results agree on all their questions, the less any eavesdropper can know, and the six-state protocol, which uses all three questions on a spin instead of two, tolerates more noise because the three-question floor is higher than any two-question floor.

Randomness with a guarantee #

The two-bit floor has a use that inverts its usual reading. If a spin is prepared along z and then measured along x and y, the answers to those two questions are each a perfect coin toss, and the uncertainty relation guarantees that no hidden description of the spin could have predicted them better: if the z answer is certain, the other two together must carry two bits. That is randomness certified by physics rather than by the hope that a pseudo-random algorithm is unpredictable. Commercial quantum random number generators work on variants of this principle, usually with photons and their polarisation in place of spins.

The guarantee is only as good as the assumption that the device really prepares and measures what it claims to, and an untrusted device might be built to look random while following a hidden list. The correlation no instructions produce described the Bell correlations that no list of predetermined answers can imitate, and they allow randomness to be certified without trusting the devices at all: if two separated boxes produce correlations violating a Bell inequality, their outputs cannot have been written down in advance. The entangled pair a list of answers can imitate found where that guarantee fails — states entangled but still describable by lists — and the boundary between the two regimes is set by entropic relations of exactly the kind drawn here.

A bound reached in three dimensions #

The question of how tight the general bound is does not have one answer.

For a system with three states, such as a spin-1 or an atom with three levels, there are four mutually unbiased questions, and the general bound is $4\log_2 2 = 4$ bits. A numerical search over all pure states finds a minimum of exactly 4, so in three dimensions the bound is tight, met by particular states, while in two it misses by a quarter of a bit. Whether the bound is tight depends on the arithmetic of the dimension, on how the complete sets of unbiased questions are built, and in general it is known only case by case. For dimensions that are not powers of primes, it is not even known how many mutually unbiased questions exist — whether six-level systems have a complete set of seven is a famous unsolved problem.

What the picture cannot show #

The figures concern one spin, three particular questions and pure states, with a single entangled-pair model, a Bell state mixed with white noise, for the memory figure. Mixed states have higher entropies, filling the range from 2 to 3 bits for a qubit, and the floor for them is the same 2 for the purest and higher for the rest. Questions that are not mutually unbiased — measurements along axes not at right angles — have lower bounds that depend on the angles between them, and the general problem of the tightest relation for arbitrary sets of measurements is harder than the unbiased case shown.

The conditional-uncertainty figure uses a particular, classical way of using the partner: measuring it in the same basis and reading the answer. A partner kept as a quantum memory, measured later in whatever way is best, can do as well or better, and the bound with memory is stated in terms of quantum conditional entropies. For the Bell-plus-noise state the figure’s numbers are the classical ones, which are what an experiment comparing measurement records gives.

Measured entropies carry a statistical trap. An entropy estimated from a finite number of outcomes is biased low, because a short run of a fair coin usually comes out a little lopsided, and the bias is about a half divided by the number of samples times the natural logarithm of two for each binary question. An experimenter checking the two-bit floor with a few hundred atoms per magnet can therefore find a total slightly below 2 without anything being wrong, and has to correct for the bias before concluding that a state beats the bound. The numerical search for the three-level system is a search, not a proof: it finds states at the bound to the precision of the computation, which shows the bound is reached; it does not say which states reach it or why, which an analytic argument would have to supply.

Still open: the tightest relation for many questions #

For two questions the tightest entropic relation is known in great generality. For three or more it is known in special cases — the qubit’s three, some sets of unbiased questions in prime dimensions — and otherwise only bounds exist, some known to be loose. The problem matters practically, because the security of quantum key distribution with many measurement settings, and the certification of randomness from quantum devices, rely on bounds of exactly this kind, and a looser bound than necessary means less key or less certified randomness from the same data. Finding tight relations efficiently, for arbitrary sets of measurements and for systems entangled with memories, is an active area of quantum information theory.

For a spin-½ asked about x, y and z, the three answers carry at least 2 bits of uncertainty in total, reached by any eigenstate of one question, with every pure state between 2 and 2.232; adding pairwise bounds gives only 1.5 and the general bound for unbiased questions 1.755, the variance sum is exactly 2 for every pure state and so says nothing, and an entangled partner removes the floor entirely, below 2 bits above a visibility of 0.65 — while in three dimensions the general bound is met exactly. The habit worth carrying is to distrust a bound assembled from its parts, because the constraints that tie several questions together belong to none of them alone.