Pauli matrices

In plain terms

The simplest quantum system has just two levels, like the spin of an electron, which when measured along any axis comes out either up or down. Yet it has to answer infinitely many possible questions, since spin can be measured along any direction in three-dimensional space, so you can ask what the minimal toolkit for describing all of them is.

It's three matrices, one for each of the three perpendicular axes of space. In quantum mechanics a measurable quantity is represented by a matrix, and every measurable quantity of a two-level system is a weighted mix of these three and the identity matrix, the matrix that changes nothing, so a spin measurement along any tilted direction is represented by the corresponding mix of the three. Four building blocks answer every question.

Their crucial property is what they refuse to do, which is commute, that is, give the same result regardless of order. Applying the matrix for one axis and then the matrix for another gives a different result from applying them the other way round, by a computable amount. That failure to commute is the uncertainty principle for spin, the impossibility of definite values along two axes at once, and not merely an analogy to it. The whole two-level world, strangeness included, fits in these three matrices.

In CSD

These are the workhorses wherever CSD's machine-checked corpus touches spin. The detector settings in the Bell and GHZ experiments are built from them, the correlations of the singlet state (the two-particle state whose spins are perfectly opposed along every axis) are computed through them, the Tsirelson bound, the ceiling on how far quantum mechanics can exceed Bell's limit, is proved in terms of them, and the Bloch geometry, the picture of two-level states as points on a sphere, is theirs.

The corpus claims nothing about the matrices themselves, as they're two centuries of linear algebra, but their concreteness does real work. The programme's empirical twins, its reproductions of flagship experiments, are exact finite computations rather than schematic arguments precisely because the measurement settings are explicit matrices, and several results that would be awkward in the abstract become arithmetic in this basis.

Mathematically

Three traceless Hermitian involutions, pairwise anticommuting, which together with the identity form a basis of the two-by-two Hermitian matrices.

Exponentials of i times their real linear combinations generate SU(2), the double cover of the rotation group, which is why a spin-half state needs two full turns to return to itself, sign included.

The name

Wolfgang Pauli (1900-1958) was born in Vienna, with Ernst Mach as his godfather, and announced himself at nineteen with a review of general relativity that Einstein praised in print. He took his doctorate under Sommerfeld at Munich, found the exclusion principle at twenty-four, held the ETH Zurich chair from 1928, spent the war at Princeton, and received the Nobel Prize in 1945.

He was the century's most feared critic, and his phrase not even wrong remains the epitaph for untestable theorising, and he was also its most collegial, since half the era's results were sharpened in savage letters from Zurich. The matrices come from his 1927 theory of spin. His decades-long correspondence with Carl Jung about physics and the psyche remains the strangest document left by any major physicist.

Related entries

• Bloch sphereThe states of the simplest quantum system form an ordinary ball, which is a lovely accident of low dimension and the single most misleading picture in quantum mechanics.
• Hermitian operatorA single symmetry condition on an operator guarantees that every number it can produce in a measurement is real, and it delivers the whole machinery of outcomes and eigenstates at no extra cost.
• Gottesman-Knill theoremThe Gottesman-Knill theorem marks the boundary of quantum advantage, as circuits built from the Clifford gates can be simulated classically, because in the right picture they're nothing more than linear algebra on bits.
• Shor codeShor's nine-qubit code was the first proof that quantum information can be protected against any error on any single qubit without ever being read, and the corpus verifies that protection as explicit constructed operations, including the subtle case where two different errors share one repair.
• Steane codeSeven qubits protect one, as the Steane code was the first to correct any single- qubit error by using the same classical Hamming code twice, once for bit flips and once for phase flips.