Quantum contextuality

A quantum measurement returns one result from a fixed list, and quantum mechanics gives only the odds of each. The simplest picture of why is that the result was already there and the measurement only looked: the particle carries a card of answers, one for every question that could be put to it, and the odds come from not knowing which card. Physicists call the entries on such a card hidden variables.

Some measurements can be made together, because the order in which they are made makes no difference, and one measurement can be grouped with different partners. A card of the simplest kind holds one answer per measurement, the same whatever it is measured with. In 1966 John Bell put his finger on this assumption in the arguments against such cards: “It was tacitly assumed that measurement of an observable must yield the same value independently of what other measurements may be made simultaneously.” Nothing forces it, he went on: “The result of an observation may reasonably depend not only on the state of the system (including hidden variables) but also on the complete disposition of the apparatus.” The partners a measurement is made with are now called its context, and a card with one answer per measurement, whatever the context, is called noncontextual.

Simon Kochen and Ernst Specker showed in 1967 that quantum mechanics rules such a card out; Specker had raised the question in 1960. For a particle of spin one, the squares of the spin along three mutually perpendicular directions can be measured together, and exactly one of the three always comes out zero. Kochen and Specker chose 117 directions in ordinary space and proved that they cannot be marked in advance with “zero” or “not zero” so as to agree with quantum mechanics on every perpendicular set among them: exactly one “zero” in every perpendicular three, and never two in a perpendicular pair. Bell’s paper of 1966 reached the same conclusion by a different route, through a theorem of Andrew Gleason’s from 1957. The result is the Kochen–Specker theorem.

The 117 directions are hard to hold in the head. In 1990 Asher Peres found a short argument of the same kind for two particles of spin one half in one particular state, using six measurements. The same year David Mermin, building on it, set nine measurements in a three-by-three square, so that the argument holds for every state. That square, now named for both, is the subject of this page.

Take two systems that each have two levels, such as two ions: four levels in all, which is what this square needs. (For a single system with two levels a card of answers that agrees with quantum mechanics exists, as Bell showed in 1966; the theorem needs three levels or more, and this square at least four, which two properties of one particle can also give.) On each system two basic measurements with results +1 and −1, and products of them, are enough. Z, the sign, gives +1 on the first level and −1 on the second. X, the flip, exchanges the two levels; its +1 goes with their sum and its −1 with their difference (the textbook’s superpositions). A subscript says which system: Z₁ is the sign of the first, X₂ the flip of the second. Z₁X₂ is their product: in the textbook’s account, made together with Z₁ and X₂, its result is the product of theirs. The ninth, Y₁Y₂, uses Y, the textbook’s third basic measurement on one system, which combines the sign and the flip; Y₁Y₂ equals Z₁X₂ times X₁Z₂, the other two in its row.

Call a row or a column a line. The three measurements of any line can be made together, or one after another in any order. In quantum mechanics, make the three in a row and multiply their results, and the product is always +1. The first two columns also always give +1. The third column always gives −1. These are not averages. The product of the three measurements in a row or column, multiplied out as matrices (Figure 2), is exactly +1 or −1 times the one, the matrix that leaves every state as it is, so in the textbook’s account every single run obeys it; a real apparatus falls a little short (section 03).

Where does the −1 come from? On one system the sign and the flip do not commute: sign then flip is minus flip then sign. Call that a clash; Y clashes with both the sign and the flip, and a measurement that leaves a system alone clashes with nothing there. Two of the nine commute when they clash on both systems, because the two minus signs cancel, or on neither. That is why each row and each column can be made together. Multiply out the third column and one sign has to pass one flip on the same system with nothing to cancel it, which leaves a single −1.

Now suppose each measurement had its answer written in advance: nine signs, one per cell, the same whichever row or column it is measured in. Multiply the three row products, and every sign is used once: you get the product of all nine. Multiply the three column products, and again every sign is used once. Both must give the same number. Quantum mechanics demands +1 × +1 × +1 = +1 from the rows and +1 × +1 × −1 = −1 from the columns. No table of nine signs gives both. The table of nine plus signs misses one, the third column; changing one sign flips one row product and one column product together, so every table misses an odd number of the six, at least one.

From a count to a test

A real apparatus is imperfect, so a laboratory test has to be a sum of averages. Adán Cabello gave it in 2008: average the product over many runs for each row and each column, add the three rows and the first two columns, and subtract the third column.

test = R₁ + R₂ + R₃ + C₁ + C₂ − C₃R and C are the average products of the rows and the columns. Mixing tables at random, without regard to which line will be read, cannot beat the best table.

In quantum mechanics the nine measurements score 6 on every state, because every row and column product is fixed. A table scores 1 for each product it meets and −1 for each it misses, so each miss costs 2, and of the 2⁹ = 512 tables of nine signs the best score exactly 4.

In 2009 Gerhard Kirchmair and colleagues ran the square in Innsbruck on a pair of calcium ions held in a trap, two energy levels of each ion serving as its two levels. Their nine measurements are the nine above, in the same places, and each was made the same way in whichever row or column it stood. A laser reads an ion’s level by whether the ion glows. For a flip, the ion’s levels were first turned so that the flip became a level; for a product such as X₁X₂, a gate acting on both ions turned the product into the level of one ion. That ion was read while the other’s state was hidden from the light, and the turns were then undone. Each measurement gave one sign (for a product, one sign for the pair and none for either ion alone) and left the ions in the part that matched it, ready for the next. This matters for the third column: its three products can be read together, but Z₁ and X₁ cannot, so the products cannot be had by reading each ion and multiplying. For each row and each column they prepared the pair afresh and made the three measurements one after another, many times over.

For one entangled state of the two ions, the singlet, in which the two ions’ results for the same measurement always come out opposite, the test came to 5.46 ± 0.04: more than 36 times its quoted uncertainty (one standard deviation) above the 4 that any table of answers allows. A score of 5.46 means that about one read in 22 missed its line; answers written in advance must miss at least one read in six. Quantum mechanics gives 6 for every state, so the team repeated the run on nine other states, from entangled to unentangled to almost completely mixed (close to random). All ten passed 4, with values from 5.23 to 5.46. Every run gave a result: the ions’ glow is read almost without error, so, unlike the photon tests of the time, no run was lost and nothing had to be assumed about lost runs (the detection loophole). Imperfect operations pull the measured value below 6; no improvement in the apparatus can lift a table of answers above 4.

In the experiment the three measurements of a line, made one after another, were not perfectly compatible, so a skeptic can ask whether each measurement disturbed answers written in advance before the next one read them (the compatibility loophole). The team answered part of that: for four measurements of a square of the same kind (with Y and Z swapped), on one state prepared for it, and with one further assumption (that an extra measurement never lessens the disturbance of an answer written in advance), their data passed the limit of a test built to allow for such disturbance. The test could in principle be extended to the whole square, they wrote, but the experimental techniques would have to improve to pass its limit there. For the singlet the team also found the limit of 4 passed whatever the order of the three measurements in a line. Later trapped-ion experiments ran other tests of contextuality, not this square: one studied how compatible and repeatable its measurements were (Leupold et al. 2018), and one closed the detection, sharpness and compatibility loopholes together (Wang et al. 2022).

The same square was tested in 2009 with single photons, each carrying two levels in its path and two in its polarization, and in a form for one state with neutrons; in 2010 with nuclear spins. How best to allow for measurements that are not perfectly compatible is still being worked out (Budroni et al. 2022).

The record is everything that has happened, written down posting by posting. One cell of the record has three on/off axes, one for each dimension of space, so 2 × 2 × 2 = 8 positions (the paper takes these from earlier work); its register is what the cell holds at them. The square uses two of the three axes: 2 × 2 = 4 configurations, written 00, 01, 10 and 11, the same kind of four-level system the two ions carry between them (that the two ions’ registers act as two blocks of one larger set of axes is composition, owed). A state gives each configuration a number, its value. Between postings the register holds these values.

An act is the least a two-valued measurement can be: it has two results, +1 and −1, and doing it twice gives the same result both times. Each result picks out a part of the state: picture the four values as the coordinates of an arrow (real values are enough here: all nine matrices are real); the act splits that arrow into two arrows at right angles that add up to it, one for each result. The result’s weight, the textbook’s probability once the total is set to one, is the squared length of its arrow (for parts made of whole configurations the paper proves this, on the conditions of section 05; for other parts it takes it from the textbook). Some acts are plain: the sign of an axis splits the state along whole configurations, so its result is already fixed in each configuration. Others move the state: the flip of an axis splits it into the sum and the difference of the axis’s two settings, so no configuration holds its result.

Which acts go together is a theorem. Two acts can be read together exactly when they commute (von Neumann, 1932, and Lüders, 1951; the paper proves it again in the record’s terms as the theorem “Acts can be read together exactly when they commute”, Theorem 24). If they commute, every state divides cleanly into the four parts of their joint results, and reading one first leaves the other’s weights exactly as they were. If they do not, there is a state on which reading one first shifts the other’s weights, by an exact amount the paper gives, never more than one half (“What reading first costs”, Corollary 23). There is no third case. In the square the three acts of each row commute in pairs, and so do those of each column, so each row and each column can be read together (for three at once, the proof in the record’s terms is in work not yet published; see section 05).

Reading Z₁ with Z₂ (row 1) and reading Z₁ with X₂ (column 1) are two different reads, each writing its own results. A table of nine signs would be a set of entries waiting to be looked at, and section 02 counted that no such table matches the reads.

Take moving away. Make each of the nine a plain read: a sign for each configuration of the two axes, moving nothing, as Z₁ and Z₂ already are. Plain reads all commute, so every row and every column can still be read together. But now each configuration carries all nine answers in advance: it is a table of nine signs. A state weighs the four configurations, so its score on the test is a weighted average of four tables’ scores, and none of them passes 4. With plain reads the order of multiplying never matters, so all nine multiplied row by row equal all nine multiplied column by column, and rows of +1, +1, +1 cannot sit with columns of +1, +1, −1. The square needs acts that move, and on one axis the sign and the flip do not commute (section 02).

What the square rests on

• Proved in the paper: the rule for reading two acts together (the theorem “Acts can be read together exactly when they commute”, Theorem 24). For the three acts of a row or column, the paper notes, the proof in the record’s terms is in work not yet published.
• The textbook’s own computations, which the paper recomputes and its theorems do not produce: the six products, the 6 for the nine acts and the at most 4 for any table of signs.
• Proved from premises: for parts made of whole configurations, a result’s weight is the squared length of its part (the theorem “The read is the square”, Theorem 16; the Born rule page shows why a square). It rests on five premises, two principal and three mild; the first principal premise is the position read, which is owed.
• Taken from the textbook: the same squared weights for other parts, such as the sum and the difference a flip picks out; and that the state after a result is the part it picks out (deriving that belongs to attachment).
• Owed: attachment, the position read and composition, as the three lines name them.

Isn’t this just the Kochen–Specker theorem?

Yes. The count is the Peres–Mermin square of 1990, a short proof of the Kochen–Specker theorem for four levels; the test and its limit of 4 are Cabello’s of 2008; and the rule for which measurements go together is von Neumann’s and Lüders’ (1932, 1951). The paper adds no new number: it proves that rule again in the record’s terms for two acts and places the nine measurements on two axes of the record’s register.

Is any of this different from ordinary quantum mechanics?

No number is: the six products, the 6 and the 4 are the textbook’s, so these experiments cannot tell the record’s account from the textbook’s. What differs is what each takes as given. The textbook takes its rules of measurement as given. The record’s account says what a measurement is, a read that writes a posting, and the paper proves in the record’s terms when two acts can be read together; the account still rests on five premises (one of them the owed position read) and two of the textbook’s rules of measurement, and owes three statements, all listed in section 05.

Does this rule out hidden variables?

Only those that give each measurement one answer, whatever it is measured with. Accounts in which an answer depends on the whole arrangement, as Bell allowed in 1966 (Bohm’s account of 1952 is one), are untouched by the count.

How is this related to Bell’s inequality?

Both rule out answers written in advance, under different conditions. Bell’s test needs two distant stations and an entangled pair, and the answers it rules out are those each station holds by itself; the square needs no distance and works on every state, but the three measurements in each line must be compatible. Mermin’s 1990 paper gave the square beside a three-particle version tied to Bell’s theorem, and played as a game between two distant players, the “magic square” game, the square becomes a Bell test (Cabello 2001; Aravind 2002; Cleve, Høyer, Toner and Watrous 2004).

What is not proved?

Attachment, the position read and composition, the three owed statements named under “The paradox in three lines”. Beside them the account rests on the other four of the five premises (the first is the position read) and on two of the textbook’s rules, and the paper proves the rule for reading together for two acts; for three at once the proof in the record’s terms is in work not yet published. Section 05 lists them all.

What this page does not claim

• A proof of the Kochen–Specker theorem in general, or any account on the record of other tests of contextuality, such as the five-measurement cycle on three levels (Klyachko, Can, Binicioğlu and Shumovsky, 2008).
• That Recognition Science gives a different number for any test of contextuality, or that these experiments test the record against the textbook.
• That the paper derives contextuality from the record. The square is a count, and its numbers are the textbook’s computation; what the record adds is what a read is: a posting that writes its result.

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