Quantum dynamics is usually written relative to a smooth external time parameter, while actual control schedules often contain waits, pulses, gates, resets, and event-like interventions. This paper develops the first unitary part of a sigma-time quantum dynamics framework. We define a bounded-Hamiltonian quantum evolution over a finite-horizon bounded-variation sigma clock with a continuous clock component, flat clock intervals, and a locally finite set of atom times carrying unitary gates. Under these hypotheses the sigma-chronological product exists, is unitary, composes as an evolution family, conserves wave-function norm, is strongly continuous away from atom times, and obeys a sharp atom/gate insertion law. We also prove a fixed-clock perturbation estimate and a bounded isolated-pulse corridor in which shrinking finite-width pulses converge in operator norm to declared atomic gates. The result is deliberately scoped: it does not claim an unbounded Schroedinger-operator theorem, a measurement theorem, an open-system theorem, or a Zeno limit theorem. A finite qubit benchmark with two bounded waits and one atomic gate is included as a reproducibility appendix and verifies unitary defect, trace preservation, Choi positivity, and output-state validity to machine precision
Quantum dynamics is usually written relative to a smooth external time parameter, while actual control schedules often contain waits, pulses, gates, resets, and event-like interventions. This paper develops the first unitary part of a sigma-time quantum dynamics framework. We define a bounded-Hamiltonian quantum evolution over a finite-horizon bounded-variation sigma clock with a continuous clock component, flat clock intervals, and a locally finite set of atom times carrying unitary gates. Under these hypotheses the sigma-chronological product exists, is unitary, composes as an evolution family, conserves wave-function norm, is strongly continuous away from atom times, and obeys a sharp atom/gate insertion law. We also prove a fixed-clock perturbation estimate and a bounded isolated-pulse corridor in which shrinking finite-width pulses converge in operator norm to declared atomic gates. The result is deliberately scoped: it does not claim an unbounded Schroedinger-operator theorem, a measurement theorem, an open-system theorem, or a Zeno limit theorem. A finite qubit benchmark with two bounded waits and one atomic gate is included as a reproducibility appendix and verifies unitary defect, trace preservation, Choi positivity, and output-state validity to machine precision
This paper develops the measurement and reset layer of sigma-time quantum dynamics. The first paper treats unitary BV-clock evolutions with gates and pulses; the second treats bounded open-system channel products with CPTP atom maps and random-clock averages. Here we add finite-outcome measurement and reset events. A measurement event is modeled as a quantum instrument placed at a locally finite atom of a BV sigma clock. Selective outcomes are completely positive trace-nonincreasing branch maps; their traces are joint outcome probabilities. Positive-probability branches normalize to density matrices, while the nonselective sum over all outcomes is a completely positive trace-preserving channel. Reset atoms and measurement-thenreset chains are included by finite Kraus certificates. We also give a finite adaptive scaffolding theorem: when the next event instrument is chosen from a finite recorded outcome history, the same branch and nonselective closure follow by induction on the event tree. The paper includes event ledger schemas, examples, refusal walls, and a reproducible qubit benchmark for projective measurement, weak POVM measurement, reset, measurement-reset, and a two-stage adaptive event tree. Zeno accumulation, continuous outcomes, unbounded measurement domains, and hardware fidelity are explicitly outside the Scope of the results.
I present a short approach to discretizing continuous unitary evolution. It gives a correspondence in both directions between continuous unitary evolutions and products of unitaries in finite dimensions, and I use it to reduce discrete adiabatic theorems to continuous ones. At fine steps, holding truncated samples of a regulated Hamiltonian on a suitable finite partition gives an approximating product. Conversely, rounding the clock identifies a product of T unit-step samples with a continuous evolution, up to two half-step phases. I bound its operator-norm distance from the original evolution by O(1/T) through cancellation within each time step. This assumes bounded first and second derivatives, bounded spectral width, and eigenvalue differences bounded away from nonzero multiples of 2π, uniformly in T. For a unitary walk with these derivative bounds, a fixed common free spectral arc supplies a Hermitian logarithm path satisfying the comparison hypotheses. Continuous simulation guarantees then transfer to the phase-corrected product, and continuous adiabatic bounds to the product itself, with an explicit additive O(1/T) error and each adiabatic theorem's hypotheses retained. For positive definite quantum linear systems of condition number κ, this comparison and continuous adiabatic theory give a rescaled qubitized walk with O(κ/δ) steps for leakage δ, matching the scaling previously obtained through a discrete adiabatic theorem. These products need no discrete adiabatic theorem.
This paper develops the open-system companion to the unitary BV-clock quantum dynamics paper. We consider density matrices evolving over a finite-horizon sigma clock with bounded Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) continuous generators and locally finite completely positive trace-preserving (CPTP) atom maps. Under bounded measurable coefficients with integrable generator norm, the sigma-chronological product exists as a CPTP channel evolution, composes through admissible cuts, preserves trace, positivity, and complete positivity, and inserts atom maps in clock order. We also prove that averaging over a probability distribution of admissible sigma clocks preserves CPTP structure by convexity. This random-clock result is deliberately not stated as a semigroup or divisibility theorem: mixtures of admissible channels need not inherit Markovian divisibility without additional hypotheses. A finite open-qubit benchmark checks a GKSL flow with amplitude damping, excitation, dephasing, reset/depolarizing/bit-flip atoms, and random timing averages using trace, Hermiticity, state eigenvalue, Choi eigenvalue, and trace-preservation ledgers.
Unrefereed research candidate. A quantum system evolves under a fixed gate, step by step, and is occasionally reset; the waiting times between resets are independent and may have an infinite mean. Wald, Yao, Platini, Hooley and Carollo (Quantum 9, 1742, 2025, doi:10.22331/q-2025-05-13-1742) inferred from estimates and numerics how the averaged state behaves for the hazard r_n = gamma/(n+1)^alpha. This paper proves it: for alpha < 1 the state converges to a survival-weighted mixture, for alpha = 1 to the dephased state, and for alpha > 1 it approaches a rotating orbit and converges iff the reset state commutes with the gate. For complete resets the behaviour is classified for every renewal clock (finite mean, infinite mean, periodic, defective); a clock makes every finite-dimensional unitary reset system converge iff it is proper and aperiodic. This part is a short consequence of classical renewal theory. The claimed contributions are: the exact infinite-dimensional trace-norm defect 1 - m_pp (continuous spectral mass) for proper infinite-mean clocks, with no trace-norm convergent subsequence unless m_pp = 1; a laboratory-time reduction of imperfect resets (general jump channels, possibly noncommuting) to complete resetting at an effective state under a nonresonance condition (W), which is characterised, decidable for finitely supported clocks and open and dense, but is sufficient, not necessary, and not a classification; and uniform two-term asymptotics, endpoint and near-resonance crossovers for the critical clock gamma/(n+1), with practical time scales (dephasing can need more than 10^9 steps; numerical estimates). The archive contains the manuscript (PDF and LaTeX), a single deterministic verifier with its JSON report, semantic negative controls and SHA-256 sums. Assurance: written proofs; exact finite checks and labelled numerical checks; producer-side clean replay; internal AI-assisted review, AI-assisted adversarial audits before and after revision (OpenAI gpt-6-astra via Codex) and an AI-assisted verification re-review. This is not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.
In this chapter, we finally introduce time evolution. The development is carried out for finite, enumerable systems (i.e., systems whose states live in a finite dimensional, inner product space) which, of course, allows a straight forward extension to the continuum and infinite dimensional state spaces. The necessary mathematical background of unitary operators, and their properties relevant to quantum mechanics are first introduced. In particular, the fact that preservation of norm under the action of an operator implies unitarity, and that every unitary operator can be expressed as the exponential of i timesa Hermitian operator are introduced. Time evolution is then introduced in its integral form through the time evolution operator, and the Hamiltonian operator is introduced in this context. This approach, however, necessitates the restriction to conservative systems. Next, the differential form of the law of time evolution, i.e., Schrodinger equation, is derived from the time evolution operator, and this is postulated to be the general law of time evolution. We then demonstrate how, for conservative systems, the solution of the Schrodinger equation essentially reduces to solving the eigenvalue problem of the Hamiltonian. Constants of motion and Heisenberg picture have been relegated to the problem section.
The physical--observation dual-axis (PODA) structure treats physical and observation configurations as distinct arguments of joint dynamics. We study the analytic conditions under which this structure admits rigorous path integrals and finite complex measures for sequential quantum histories. On compact smooth joint geometry, a specified covariant Hamiltonian has a Feynman--Kac representation. A uniform complex-time product estimate permits simultaneous removal of Abel damping and time slicing at arbitrary relative speeds. For fixed finite bounded-readout protocols, class operators converge strongly and the trace-class branch-pair operators converge in trace norm. Finite instruments with complete Kraus families also yield trace-norm convergence of their record states. Energy-form regularity gives quantitative error bounds. A completed cylinder-test space carries a consistent, strongly positive history functional. The stronger finite-measure question is controlled by covariant transport: calibrated rank-one readouts on a finite-dimensional fibre satisfy the sharp estimate ∥μγ∥TV≤e2cγ, where cγ is the off-diagonal covariant variation. In the specified common-action class, conserved energy and smooth, globally labelled rank-one readouts imply this regularity when the kinetic geometry is complete and the effective potential is bounded below, or when the chosen energy confines the trajectory to a compact set. In infinite dimension, finitely many readout sectors allow unbounded internal energies and bounded transitions. The construction extends to countable outcomes when the operator of block norms is bounded on ℓ2; nonatomic readouts admit a double-Dyson construction when an absolute integral kernel is bounded on L2. In these Dyson constructions, pure preparations with rank-one discrete readouts or scalar continuous fibres have exact variation ∥eTBv∥22. Whole-sector and spatial-domain cutoffs converge in variation, whereas nonatomic readout quantization converges weakly and remains separated in variation norm. Mixed curvature survives in the covariant constructions, and invariant quantum embeddings recover physical-sector histories. Unrestricted basis switching, a bounded-Hamiltonian countable example and free-particle position histories establish the limits of finite-measure existence; exact physical recovery transfers the last obstruction to the joint theory. Analytic two-axis models exhibit the resulting dynamics and verify the distinct convergence statements. A proposed two-mode trapped-ion realization links the magnetic oscillator's transport phase to independent spectral and driven-response tests. Exact finite-time Gaussian evolution shows that nearly complete wave-packet return can coexist with a substantial phase correction. The protocol therefore infers the coupling from finite-time interference and predicts the other observables without refitting, with calibration and uncertainty requirements explicit. These are model predictions, not measured data. Each construction is tied to its stated geometric, dynamical and readout hypotheses.