Using tools from classical signal processing, we show how to determine the dimensionality of a quantum system as well as the effective size of the environment's memory from observable dynamics in a model-independent way. We discuss the dependence on the number of conserved quantities, the relation to ergodicity and prove a converse showing that a Hilbert space of dimension D+2 is sufficient to describe every bounded sequence of measurements originating from any D-dimensional linear equations of motion. This is in sharp contrast to classical stochastic processes which are subject to more severe restrictions: a simple spectral analysis shows that the gap between the required dimensionality of a quantum and a classical description of an observed evolution can be arbitrary large.
Nonlinear dynamics generate non-Gaussian states and operations, but they also make continuously monitored quantum systems difficult to track. A quantum filter performs this task by compressing a noisy measurement record into a set of evolving variables that predict future observables. For Gaussian dynamics, this compression is exact and finite. In nonlinear systems, conventional moment equations generally expand without bound, but the failure of one representation does not prove that every finite description is impossible. Here we formulate quantum filtering as a realization problem for general quantum input--output maps. We introduce the linearized observable Hankel operator
KO, which maps changes in the past measurement record to changes in a future conditional observable. Any finite-dimensional smooth filter forces
KO to have finite rank, so infinite rank provides a coordinate-independent proof that no exact finite-dimensional filter exists. Applied to single-mode polynomial bosonic systems under continuous quadrature monitoring, this yields a sharp boundary: Gaussian dynamics and the measurement-aligned Conditional Momentum Moment class admit finite realizations, whereas dynamics outside these classes produce infinitely many response directions and admit no finite-dimensional smooth exact filter. Exact impossibility, however, need not imply large practical complexity. The singular values of
KO quantify how many response directions matter at a chosen local accuracy, defining the effective memory of an observable. For the Kerr and Duffing oscillators, these singular values decay rapidly despite infinite exact rank, showing that infinite exact memory can coexist with low finite-accuracy complexity. The framework therefore separates exact realizability from effective complexity in nonlinear quantum filtering.
We develop a constructive procedure for the explicit extraction of the physical spectrum in systems described by structural operators with non-abelian internal degrees of freedom, within the context of Spectral Phase Mechanics. The formulation is based on representing the dynamics through the resolvent of self-adjoint operators, allowing the evolution to be interpreted as a spectral filtering process. Starting from a structural operator with
SU(3) symmetry, we show that the diagonalization of the internal sector induces a decomposition into independent tridiagonal effective operators. This reduction naturally leads to the factorization of the characteristic polynomial into lower-degree factors, whose coefficients are explicitly determined by the structural parameters of the model. The resolvent analysis associated with this decomposition establishes an intrinsic criterion for spectral selection: the physical relevance of the modes is determined by the pole structure of the operator, without the need for externally imposing additional constraints. We demonstrate, within this class of operators and under the established hypotheses, that the reduction of the observable spectrum emerges from the spectral structure of the system itself. The obtained results provide an explicit mechanism for extracting the physical spectrum, in which the interaction between algebraic factorization and spectral filtering determines the effectively observable modes. This construction reinforces the interpretation of dynamics as spectral filtering of the action phase, offering a natural extension of Spectral Phase Mechanics to systems with non-abelian internal structure.
Open quantum systems can rapidly lose most microscopic information, leaving only a few degrees of freedom to govern their long-time dynamics. Classical Markov state models (MSMs) describe such metastable dynamics as transitions among a few representative phases and are widely used to reduce complex-system dynamics in condensed matter and chemical physics. In quantum systems, however, phase labels alone are insufficient when coherence persists between metastable states. Even when the slow modes are known, their spectral projection need not produce valid quantum states. We construct quantum Markov state models (QMSMs) that describe the surviving information and its evolution on a small physical state space containing classical sectors and quantum matrix blocks. We quantify metastability by assuming that the evolution channel
C changes little when applied a second time, with sufficiently small defect
η=∥C2−C∥⋄, and that the number of slow degrees of freedom is bounded independently of the full system size. Under these assumptions, we construct compression and reconstruction channels whose composition recovers every reduced state exactly, while the reverse composition gives an exactly idempotent channel approximating
C, answering Kitaev's exact-rounding question [Kit25]. Our constructions give an optimal diamond-norm bound of
O(η1/3) on all input states, as well as an improved bound of
O(η1/2) on metastable states prepared by
C, with constants depending only on the slow dimension. The reduced transition channel can be iterated to predict the microscopic dynamics with controlled error. We illustrate the QMSM through a weakly driven dissipative spin chain supporting either a metastable logical qubit or long-lived classical phases.
Open quantum systems lose energy and phase coherence through different dissipative processes, but these processes can produce overlapping dynamical signatures. The Liouvillian spectrum summarizes how such a system relaxes, yet it is not obvious how much of that spectrum is needed to distinguish the underlying dissipation rates. We study this question for amplitude damping and dephasing in a six-qubit Lindblad model whose spectrum can be derived analytically. We retain only the slowest non-steady spectral modes and ask how many are required before each dissipative rate becomes recoverable. We show that population modes contain no dephasing information, which creates a lower bound of D = 2^n retained modes for uniform dephasing identifiability in the relevant rate regime. The measured recovery threshold reaches this bound at n = 4,5,6, while n = 3 remains above it. At n = 6, least squares achieves a mean joint absolute error of order 10^-9, compared with 4.355 x 10^-4 for four tabular learning methods. Robustness tests show that this advantage weakens when the spectra are perturbed and when a transverse field breaks the commuting structure. These results show that the amount and structure of retained spectral information can determine whether dissipative parameters are recoverable, independently of the estimator used. The present conclusions apply to noise-free simulator spectra rather than measurement-derived spectra.
We present a quantum eigenstate filtering algorithm based on quantum signal processing (QSP) and minimax polynomials. The algorithm allows us to efficiently prepare a target eigenstate of a given Hamiltonian, if we have access to an initial state with non-trivial overlap with the target eigenstate and have a reasonable lower bound for the spectral gap. We apply this algorithm to the quantum linear system problem (QLSP), and present two algorithms based on quantum adiabatic computing (AQC) and quantum Zeno effect respectively. Both algorithms prepare the final solution as a pure state, and achieves the near optimal Oe(dκlog(1/∊)) query complexity for a d-sparse matrix, where κ is the condition number, and ∊ is the desired precision. Neither algorithm uses phase estimation or amplitude amplification.