Lindblad equation

In plain terms

Real systems leak. They lose the delicate coordination between the components of their state, they decay, and they decohere, because none of them is perfectly isolated. The equation that governs the density matrix of an open quantum system, the description of a system whose state is known only statistically, is the Lindblad equation, and its defining virtue is that it should generate legitimate physics at every time, so probabilities stay probabilities, even when the system is entangled with a bystander the equation never mentions.

That virtue is called complete positivity, and it's precisely what's delicate to prove, because the evolution is the exponential of a generator with competing parts. The corpus proves it from first principles. Split the generator into a drift part, whose evolution is a change of frame, and a jump part, whose exponential is a series with only positive terms, reassemble the two with the Lie-Trotter product formula, proved for the purpose, and then show that the evolution of the generator enlarged by a bystander is literally the original evolution acting alongside the bystander left untouched, so positivity of the enlarged evolution is complete positivity in so many words.

It's a staple of every open-systems textbook, with a proof checked by a proof assistant.

In CSD

The open-system tier is where the programme prices imperfect isolation. The dephasing and amplitude-damping semigroups it exhibits, families of evolutions indexed by elapsed time that describe loss of coherence and loss of energy, are Lindblad flows, and the general theorem certifies that every such flow, for any Hamiltonian and any set of jump operators, the terms describing the leaks, is a legitimate quantum channel at every nonnegative time.

The supporting product formula was deliberately proved in a generality suitable for contribution to Mathlib. An earlier label in the corpus classed exponentiated positivity as out of reach, and that label was retired when the assessment was re- checked and the route was found to lie entirely within the corpus.

Mathematically

The theorem lindbladSemigroup_completelyPositive states that for Hermitian H, jump family L, t >= 0 and every finite ancilla, the identity-tensor amplification of the flow maps positive semidefinite matrices to positive semidefinite matrices.

The route is as follows. The drift flow is conjugation by a matrix exponential, the jump exponential is a positive Kraus series with positivity closed under limits, the general Lie-Trotter formula assembles the flow, and the blockwise identification of the amplified generator's flow with id-tensor-Phi makes the amplified statement literally complete positivity.

The name

Goran Lindblad (1940-2022) was a Swedish mathematical physicist at the Royal Institute of Technology in Stockholm. His 1976 paper on generators of quantum dynamical semigroups gave the general form of the equation, and Gorini, Kossakowski and Sudarshan published the finite-dimensional classification in the same year, which is why careful writers say GKSL. By the standards of a result used daily across quantum optics and quantum information, its authors are remarkably rarely read.

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.
• Lie-Trotter product formulaAlternate many tiny steps of two operations that don't commute and the result converges to the exponential of their sum, with an error that shrinks like one over the number of steps, and CSD proves this in textbook generality, in a form ready for the Mathlib library, because its open-systems tier needed it and nothing on the shelf existed.
• De-isolationNothing collapses, as a measured system stops being isolated, and measurement is what de-isolation dynamics looks like from inside.
• Choi's theoremA transformation of quantum states is physically allowed exactly when one matrix built from it passes a single positivity test, and CSD proves both directions, so whether a map is physical becomes a decidable question of linear algebra.
• Schrodinger equationIn CSD the Schrodinger equation isn't postulated, as it's what any continuous motion that preserves the measurable relations between quantum states is forced to look like.