Kronecker product

In plain terms

Two quantum systems sit side by side, and you want to know what describes the pair. Classically you'd simply list both descriptions, one after the other, and the size of the joint description would be the sum of the two sizes. Quantum mechanics demands more, as every state of the first system can be paired with every state of the second, and every superposition of such pairings, every way of combining them with weights, is also a state of the pair, so the number of independent directions the pair can point in, its dimension, is the product of the two individual dimensions and not their sum.

The Kronecker product is this fact written out in matrix form. It's a recipe that takes a square grid of numbers describing one system and a square grid describing the other, and multiplies them into a much larger grid describing both, with one row and one column for every pairing of a row from the first with a row from the second. It's mechanical to compute and explosive in its consequences.

That multiplication instead of addition is the origin of quantum computing's power and of the difficulty of simulating quantum systems alike, as thirty qubits, thirty two-level systems, need about a billion dimensions to describe. The exponential lives here.

In CSD

Throughout CSD's machine-checked corpus the Kronecker product is how composition, the joining of systems into a larger one, is realised, and in one important place it's also how composition is earned rather than assumed. The results on composite systems don't start from the tensor-product structure of quantum mechanics, as they derive it. Starting from two collections of operations, one for each part, such that operations on different parts can be done in either order and the two collections together generate every operation on the whole, the corpus proves that the whole must have the structure of a tensor product, with dimension equal to the product of the parts' dimensions, and it proves this for arbitrary ways of embedding the parts in the whole. The programme's necessity audit, its internal review of its results, rates that among the strongest in the development.

Elsewhere it's infrastructure. The partial trace that discards an environment, the reduced states that describe one part of a larger system, and the marginals of thermal states are all Kronecker bookkeeping, consumed daily.

Mathematically

The Kronecker product is the matrix form of the operator tensor product, with the mixed-product identity, (A tensor B)(C tensor D) = (AC) tensor (BD), doing the functorial work.

The partial-trace machinery consumes it, and the composite reconstruction earns it rather than assuming it, as commuting generating subalgebras force tensor structure and multiplicative dimension.

The name

Leopold Kronecker (1823-1891) was born in Liegnitz, Silesia, and was privately tutored by Kummer, the start of a lifelong alliance. He then did something almost no mathematician has done, as he stopped at his doctorate, spent a decade running the family's banking and estate business, grew rich, and returned to mathematics at thirty with no need of a salary. He lectured at Berlin for twenty years before accepting a chair at all.

He's remembered for his constructivist conviction, summed up in his remark that God made the integers and all else is the work of man, and for prosecuting that conviction against Cantor's infinities with a persistence that shaded into persecution and darkened Cantor's life. The product bearing his name is, ironically, the least controversial thing attached to it.

Related entries

• Partial traceMathlib has no partial trace for matrices, so the corpus built one, the staple operation of quantum information, with its working properties proved and staged for contribution upstream.
• Hilbert spaceHilbert space is the stage on which quantum mechanics is usually performed, and CSD's claim is that it's scenery, the shadow cast by the physical object and not the object itself.
• Bell state is not a joinThe arena CSD assigns to two systems together is strictly larger than the two arenas simply paired, and the Bell state is a point of the excess, so entanglement is located exactly.
• Field arenaTake finitely many field modes, cap the number of quanta each can hold, and the whole toolbox of quantum field theory, propagators, light cones, Wick's theorem and thermal states, reappears as exact theorems on one finite arena.
• Stinespring dilationEvery noisy quantum process is a reversible process on a larger system with part of that system ignored, and in CSD the larger system is the constraint surface itself, so every channel the programme uses is the environment's marginal of a volume- preserving flow.