Unbounded Lindblad generators
On a specified infinite-dimensional operator space and topology, characterize when an unbounded generator or quadratic form defines a unique conservative completely positive semigroup, with explicit operator domains and boundary conditions.
Problem statement
On a specified infinite-dimensional operator space and topology, characterize when an unbounded generator or quadratic form defines a unique conservative completely positive semigroup, with explicit operator domains and boundary conditions.
Lean 4
No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.
Background
A time-homogeneous quantum Markov semigroup has a complete GKLS generator characterization in finite dimensions and in the appropriate bounded-generator setting. With unbounded Hamiltonians and noise operators, domains, closures and possible probability escape become essential: a formal expression need not define a conservative evolution.
What is known: The bounded-generator case is fully solved (Lindblad; GKS, 1976). For unbounded generators, the theory of quantum dynamical semigroups (Davies; Chebotarev–Fagnola) gives sufficient conditions for the minimal semigroup to be conservative (no probability leak), often via quantum analogues of non-explosion criteria for Markov processes and Lyapunov-function methods. Form-generator approaches and results on when the minimal solution is unique are available in important cases (quantum Ornstein–Uhlenbeck, quadratic bosonic generators).
Siemon–Holevo–Werner construct nonstandard unbounded generators and examples where first-order trace preservation does not imply finite-time conservativity. Thus classification of all generators cannot assume a naive unbounded GKLS representation from the outset.
Separate three targets: recognition of a proposed expression on a stated core, conservativity and uniqueness of the associated minimal evolution, and representation of more general generators. Seek necessary-and-sufficient, checkable conditions for specified operator classes beyond existing sufficient criteria. Related: O2, O3.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Gorini, Kossakowski & Sudarshan, 'Completely positive dynamical semigroups of N-level systems', J. Math. Phys. 17, 821 (1976) link
- [2]Lindblad, 'On the generators of quantum dynamical semigroups', CMP 48 (1976) link
- [3]Chebotarev & Fagnola, 'Sufficient conditions for conservativity of minimal quantum dynamical semigroups', J. Funct. Anal. 153 (1998) link
- [4]Siemon, Holevo & Werner, 'Unbounded generators of dynamical semigroups', Open Systems & Information Dynamics 24 (2017) link
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
Machine attempts
0 recordedAttempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.
No machine attempts recorded.
Sign in to run an attempt. New accounts start with enough credits for ten.
Discussion
0 notesNotes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.
No notes yet.
Sign in to add a note.