Interval Quantum Mechanics (IQM)

• IQM is a finite-precision reformulation of quantum mechanics that replaces exact point states with quantum parcels, defined as weak* open convex sets of density matrices characterized by expectation intervals.
• The framework leverages parcel dynamics and fuzzy POVMs to model measurement, using geometric contraction as an indicator of information gain and thermalization.
• IQM also encompasses interval-valued probabilities and multi-interval formulations, providing new perspectives on contextuality, thermal equilibration, and the foundations of quantum theory.

Interval Quantum Mechanics (IQM) denotes a family of interval-based reformulations of quantum theory in which at least one exact object of standard quantum mechanics is replaced by an interval-like structure. In the most explicit recent usage, IQM replaces point states by quantum parcels: weak$^*$ open convex sets of density matrices defined by finitely many expectation intervals, intended to represent exactly the epistemic content of finite-precision measurements. In adjacent strands of the literature, intervals appear instead as interval-valued probabilities, as local domains for factorization-algebraic observables, or as components of multi-interval boundary-value problems. The unifying theme is the rejection of idealized pointwise descriptions in favor of structures adapted to finite resolution, locality on intervals, or singular interface data, but the mathematical content of “interval” differs substantially across these programs (Edalat, 19 May 2026, Tai et al., 2017, Chiaffrino et al., 2024, Pitelli et al., 2023).

1. Quantum parcels as the basic state concept

In the finite-precision formulation of IQM, the standard state assignment $\rho \in \mathcal D(\mathcal H)$ is treated as an idealization requiring infinite precision. The physically meaningful state is instead a quantum parcel, typically written as

$$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$

where $H_1,\dots,H_m$ are bounded Hermitian observables and $(a_j,b_j)$ are open intervals determined by finite-resolution measurements. More generally, a parcel is taken to be a non-empty weak$^*$ open convex subset of $\mathcal D(\mathcal H)$ with compact closure. On this view, the parcel is not a coarse approximation to a hidden exact state; it is the exact mathematical representation of the set of microscopic states compatible with the available macroscopic data (Edalat, 19 May 2026).

The weak$^*$ topology is central because it is the coarsest topology for which all maps $\rho \mapsto \operatorname{Tr}(\rho B)$ are continuous for bounded $B$. IQM therefore makes finite collections of expectation-value constraints primary, rather than deriving them from an underlying point state. For a parcel $\rho \in \mathcal D(\mathcal H)$0 and Hermitian $\rho \in \mathcal D(\mathcal H)$1, the associated expectation interval

$\rho \in \mathcal D(\mathcal H)$2

replaces the single expectation value of standard quantum mechanics. Because $\rho \in \mathcal D(\mathcal H)$3 is convex and the expectation functional is linear, this interval is open and every intermediate value is realized by some $\rho \in \mathcal D(\mathcal H)$4 (Edalat, 19 May 2026).

The same framework introduces a geometric measure of informativeness. Using the Hilbert–Schmidt inner product on the affine trace-one hyperplane, a parcel has a volume $\rho \in \mathcal D(\mathcal H)$5, and for basic parcels cut out by orthonormal observables with widths $\rho \in \mathcal D(\mathcal H)$6, one has

$\rho \in \mathcal D(\mathcal H)$7

Smaller parcels therefore correspond to more informative state assignments. This geometric reading is one of the distinctive features of parcel-based IQM: information is identified with contraction of admissible state volume rather than with entropy of a point state (Edalat, 19 May 2026).

2. Parcel dynamics, fuzzy measurement, and geometric information

IQM lifts unitary dynamics from point states to parcels by transport of sets: $\rho \in \mathcal D(\mathcal H)$8 This induced evolution is stated to be reversible, order-preserving, and volume-preserving. In particular, $\rho \in \mathcal D(\mathcal H)$9, so closed-system evolution becomes a deterministic flow on parcel space rather than a trajectory in state space (Edalat, 19 May 2026).

Measurement is modeled not by sharp projection but by a fuzzy POVM. Given projectors $$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$0, the corresponding effects are

$$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$1

with Kraus operators $$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$2. Conditional on outcome $$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$3, the parcel update is

$$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$4

The updated set remains weak$$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$5 open, and for sufficiently sharp measurements the update is volume-contracting. For qubits the contraction can be made explicit in Bloch coordinates, where the Jacobian of the update map is

$$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$6

This geometric contraction is interpreted as information gain (Edalat, 19 May 2026).

To represent both positive and negative information, the framework introduces a double parcel $$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$7, where $$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$8 is the possible set and $$O=\{\rho\in\mathcal{D}(\mathcal{H}) : a_j < \operatorname{Tr}(\rho H_j) < b_j,\; j=1,\dots,m\},$$9 the impossible set, with $H_1,\dots,H_m$0. The corresponding geometric information is

$H_1,\dots,H_m$1

Under unitary evolution $H_1,\dots,H_m$2 is constant, while under fuzzy measurement it increases because $H_1,\dots,H_m$3 contracts and $H_1,\dots,H_m$4 expands. The paper presents this as a resolution of the “von Neumann entropy paradox,” arguing that operational information gain should be tracked geometrically rather than by the entropy of an exact density matrix (Edalat, 19 May 2026).

Within this interpretive scheme, several standard foundational puzzles are reformulated. Wave-particle duality becomes a smooth trade-off controlled by the fuzziness parameter $H_1,\dots,H_m$5; Schrödinger’s cat is not taken to be literally in a superposition but rather associated with a parcel compatible with macroscopically distinct alternatives; and the apparent nonlocality of entanglement is described as a purely epistemic geometric update of the parcel rather than as a physical superluminal influence. These claims belong to the interpretive layer of the proposal, but they follow directly from the parcel-and-update ontology adopted there (Edalat, 19 May 2026).

3. Thermalization and ergodicity in parcel IQM

A further development combines IQM with Reimann’s spectral typicality theorem to obtain a parcel-level notion of thermalization. Here one works in a finite-dimensional energy shell

$H_1,\dots,H_m$6

with microcanonical state $H_1,\dots,H_m$7. For a state $H_1,\dots,H_m$8, the relevant control parameter is the effective dimension

$H_1,\dots,H_m$9

and for a parcel $(a_j,b_j)$0 one sets

$(a_j,b_j)$1

The crucial hypothesis is a uniform lower bound $(a_j,b_j)$2 with $(a_j,b_j)$3, ensuring that every state in the parcel lies in the thermalizing regime (Edalat, 30 May 2026).

Under this condition, parcel thermalization is formulated uniformly over the whole set of admissible states. For any bounded observable $(a_j,b_j)$4 and any $(a_j,b_j)$5, there exists a set $(a_j,b_j)$6 of good times such that for all $(a_j,b_j)$7,

$(a_j,b_j)$8

Thus, for most late times, the expectation interval of any bounded observable becomes narrow and concentrated near the microcanonical value. The asymptotic bound depends on the target precision, the observable norm, the covering number $(a_j,b_j)$9, and the minimal effective dimension $^*$0, but not on the detailed internal shape of the parcel (Edalat, 30 May 2026).

The framework also distinguishes sharply between conserved and nonconserved observables. If $^*$1, then

$^*$2

for all $^*$3, so constants of motion remain exact interval invariants. For nonconserved observables, the interval typically contracts toward equilibrium. The physical interpretation is therefore not that a single exact state relaxes to equilibrium, but that an epistemic set of compatible microscopic states becomes observationally indistinguishable from the microcanonical ensemble for most late times (Edalat, 30 May 2026).

The same paper extends the analysis to a double parcel $^*$4 separated by a conserved quantity $^*$5 satisfying $^*$6. If initially

$^*$7

then the separation remains exact for all times. At the same time, both parcels thermalize for bounded observables with $^*$8, and for $^*$9 sufficiently close to $\mathcal D(\mathcal H)$0 the updated double parcel after fuzzy measurement remains valid. The paper further states that the geometric information

$\mathcal D(\mathcal H)$1

is constant under unitary evolution but strictly increases under fuzzy measurement (Edalat, 30 May 2026).

4. Interval-valued probabilities and finite-precision foundational theorems

A different interval-based program replaces exact probabilities, rather than states, by Quantum Interval-Valued Probability Measures (QIVPMs). A QIVPM is a map

$\mathcal D(\mathcal H)$2

from projectors to intervals satisfying $\mathcal D(\mathcal H)$3, $\mathcal D(\mathcal H)$4, $\mathcal D(\mathcal H)$5, and for commuting projectors $\mathcal D(\mathcal H)$6,

$\mathcal D(\mathcal H)$7

Here interval arithmetic weakens exact additivity to an inclusion relation, thereby encoding finite precision and experimental imperfections directly at the level of probability assignments (Tai et al., 2017).

This framework introduces the core of a QIVPM on a subspace $\mathcal D(\mathcal H)$8,

$\mathcal D(\mathcal H)$9

namely the set of ordinary density matrices consistent with the interval assignment. For commuting families, the core is non-empty; in the fully quantum case it may be empty. Expectation values of observables become intervals obtained by minimizing and maximizing over the core. This is a distinct route to finite-precision quantum theory: the basic state may remain point-valued, but the admissible probabilities and expectations do not (Tai et al., 2017).

The most prominent application concerns contextuality. A QIVPM is called $^*$0-deterministic if every projector receives an interval contained in either $^*$1 or $^*$2. The paper proves that in dimension $^*$3 there is no $^*$4-deterministic QIVPM for $^*$5, and that the bound is sharp in the sense that a $^*$6-deterministic example exists. This yields a quantitative finite-precision version of the Kochen–Specker theorem and provides a middle position in the Meyer–Mermin debate: finite precision does not automatically trivialize contextuality, but sufficiently coarse imprecision can (Tai et al., 2017).

The same article also gives an interval-valued analogue of Gleason-type uniqueness. In the singleton case one recovers ordinary Born-rule uniqueness. For genuinely blurred measures, the consistent states form a neighborhood rather than a single point: if a QIVPM arises by applying an interval map to the Born probabilities of a state $^*$7, then any consistent $^*$8 satisfies

$^*$9

where $\rho \mapsto \operatorname{Tr}(\rho B)$0 is the maximum interval width. This identifies the zero-width limit as the regime in which standard quantum probability is recovered exactly (Tai et al., 2017).

5. Interval-local and multi-interval formulations

Not all interval-based quantum mechanics is finite-precision IQM. In a homological and local-to-global direction, quantum mechanics can be formulated on intervals via factorization algebras. For each open interval $\rho \mapsto \operatorname{Tr}(\rho B)$1, one assigns a chain complex of quantum observables

$\rho \mapsto \operatorname{Tr}(\rho B)$2

with $\rho \mapsto \operatorname{Tr}(\rho B)$3, and for disjoint intervals $\rho \mapsto \operatorname{Tr}(\rho B)$4 the factorization product is the wedge product

$\rho \mapsto \operatorname{Tr}(\rho B)$5

For the harmonic oscillator and the spin-$\rho \mapsto \operatorname{Tr}(\rho B)$6 system, the resulting off-shell factorization algebra is proved quasi-isomorphic to the familiar on-shell quantum-mechanical factorization algebra. A major technical extension is the inclusion of half-open and closed intervals, where half-open intervals encode bra or ket states and closed intervals encode amplitudes (Chiaffrino et al., 2024).

A related but geometrically different interval construction appears in supersymmetric reductions from disks or cigars. There, a 2-deformed B-model on a disk $\rho \mapsto \operatorname{Tr}(\rho B)$7 reduces by cigar compactification to supersymmetric quantum mechanics on an interval $\rho \mapsto \operatorname{Tr}(\rho B)$8, with one boundary inherited from $\rho \mapsto \operatorname{Tr}(\rho B)$9 and the other from the tip of the cigar. Shrinking the interval then localizes the theory to constant maps and produces a zero-dimensional sigma model over a middle-dimensional cycle $B$0. In the gauged case, the same procedure yields gauged quantum mechanics on $B$1 and then a zero-dimensional gauged sigma model with complexified gauge group $B$2. In this literature, the interval is primarily a localization domain mediating a disk-to-point reduction, not an interval-valued state or probability object (Ishtiaque et al., 2020).

A third use of intervals arises in singular one-dimensional geometry. For a sharply bent quantum wire, the singular corner splits the domain into two intervals,

$B$3

and the physics is encoded by self-adjoint transfer conditions at the junction rather than by a universal singular potential. The matching can be written in the canonical form

$B$4

leading to energy-dependent reflection and transmission amplitudes. The paper emphasizes that the effective parameters depend on the regularization of the bend, so the interval decomposition supplies a controlled low-energy model of a singular junction rather than a unique sharp-corner Hamiltonian (Pitelli et al., 2023).

6. Analytical constraints, limits, and neighboring “interval” notions

A common misconception is that an interval-based reformulation naturally supports piecewise-discontinuous or abruptly switching quantum evolution. A theorem on time evolution points in the opposite direction for standard Hamiltonian dynamics: for amplitudes

$B$5

if $B$6 on a nonzero interval, then $B$7 for all times. The proof uses the holomorphic continuation $B$8 into the upper half-plane under the relevant boundedness assumptions, together with contour deformation and the identity theorem. The resulting conclusion is that nontrivial amplitudes cannot switch on or off across finite time intervals; they must vary continuously on every interval. This places a strong analytical constraint on any IQM interpretation that might be read as endorsing genuinely discontinuous time evolution (Rajabpour, 2018).

Parcel-based IQM is explicitly designed so that standard quantum mechanics reappears as an ideal infinite-precision limit. Given a point state $B$9, one considers nested parcels $\rho \in \mathcal D(\mathcal H)$00 defined by tighter and tighter expectation constraints. For bounded observables $\rho \in \mathcal D(\mathcal H)$01, the corresponding expectation intervals satisfy

$\rho \in \mathcal D(\mathcal H)$02

and in infinite dimension the intersection of the nested parcels is $\rho \in \mathcal D(\mathcal H)$03 when constructed from a separating family of observables. Thus IQM does not propose new empirical predictions at perfect precision; it reinterprets ordinary quantum theory as the unattainable limiting case of exact information (Edalat, 19 May 2026).

The literature also contains interval notions that are adjacent to IQM but not identical with it. In “quantum language” or measurement theory, confidence intervals are reconstructed from an observable $\rho \in \mathcal D(\mathcal H)$04, an estimator $\rho \in \mathcal D(\mathcal H)$05, a quantity $\rho \in \mathcal D(\mathcal H)$06, and a semi-distance $\rho \in \mathcal D(\mathcal H)$07. The resulting object

$\rho \in \mathcal D(\mathcal H)$08

is a confidence domain in parameter space. Here the interval belongs to inference about quantities compatible with data, not to the physical quantum state itself. This use of interval language is therefore methodologically related to finite-precision IQM, but it is not an interval-state reformulation of quantum mechanics in the parcel sense (Ishikawa et al., 2013).

Taken together, these results show that “Interval Quantum Mechanics” is best understood as a heterogeneous research area rather than a single formalism. Its most developed current form is the parcel-based finite-precision program, but interval-valued probability, factorization-algebraic localization on intervals, and multi-interval boundary mechanics all contribute distinct mathematical realizations of the broader interval idea.