Choi's theorem

In plain terms

You can ask which transformations of a quantum state are physically allowed, and the abstract answer is the completely positive, trace-preserving maps. Trace-preserving means total probability is conserved, and completely positive means probabilities stay nonnegative not only for the system itself but for the system together with any bystander system it might be entangled with. That last clause quantifies over every possible bystander, which isn't something you can check by computation.

Choi's insight of 1975 collapses the quantifier. Encode the whole map into a single matrix by feeding it one half of a maximally entangled pair, a pair of systems whose joint state is as correlated as quantum mechanics allows. The map is legitimate precisely when that one matrix is positive semidefinite, meaning that it has no negative eigenvalues, which is the matrix analogue of a number being nonnegative, so an infinite family of conditions becomes one eigenvalue check.

The corpus proves both directions. Every channel written in Kraus form, as a family of simpler operators applied and summed, has a positive Choi matrix, and every positive semidefinite matrix on the combined system arises from some Kraus family, and the equivalence is the theorem.

In CSD

Choi's theorem is the workhorse of the corpus's channel layer. The positivity witnesses at the generator level in the open-system results, the channel constructions of the coarse-graining tier, and the arguments about the jump part in the proof that the Lindblad flow is completely positive all route through Kraus decompositions certified by it.

It sits on the operational stratum, standard quantum information formalised because the programme's claims lean on it, and the construction is fully explicit, as the converse direction actually builds the Kraus family from the spectral decomposition of the Choi matrix.

Mathematically

The theorem choi_iff_posSemidef states that a matrix on the composite index is the Choi matrix of some Kraus family if and only if it's positive semidefinite.

The forward direction is the rank-one decomposition of the Choi matrix into outer products of vectorised Kraus operators, and the converse reconstructs Kraus operators from a PSD matrix's spectral square roots, de-vectorised along the index pairing.

The name

Man-Duen Choi, born in 1945 in Hong Kong, spent his career at the University of Toronto working in operator algebras. The 1975 paper, Completely positive linear maps on complex matrices, in Linear Algebra and its Applications, is a model of brevity, and its central object was independently studied by Andrzej Jamiolkowski, which is why the encoding is often called the Choi-Jamiolkowski isomorphism.

Related entries

• Quantum channelThe most general thing that can happen to a quantum state is captured in CSD as a concrete structure, a family of operators with one completeness law, and every certificate of its physicality is proved rather than assumed.
• 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.
• Lindblad equationThe master equation of open quantum systems is formalised end to end, as the evolution generated by every Lindblad generator is proved to be completely positive, with the bystander-system argument carried out rather than gestured at.
• Loewner orderThere's a workable meaning of bigger for operators, and it brings a strange discovery, as taking square roots respects it while taking squares can break it.
• Kronecker productTwo quantum systems become one by multiplying their descriptions rather than adding them, which is why quantum resources multiply where classical ones merely add.