A Qualitative Modal Representation of Quantum Register Transformations

We introduce two modal natural deduction systems that are suitable to represent and reason about transformations of quantum registers in an abstract, qualitative, way. Quantum registers represent quantum systems, and can be viewed as the structure of quantum data for quantum operations. Our systems provide a modal framework for reasoning about operations on quantum registers (unitary transformations and measurements), in terms of possible worlds (as abstractions of quantum registers) and accessibility relations between these worlds. We give a Kripke–style semantics that formally describes quantum register transformations and prove the soundness and completeness of our systems with respect to this semantics.

💡 Research Summary

The paper introduces two labelled modal natural deduction systems—MSQR and its variant MSpQR—to provide a qualitative logical framework for reasoning about transformations of quantum registers. Quantum registers, abstracted as “worlds,” evolve through two fundamental operations: unitary transformations and measurements. The authors model these operations using binary accessibility relations U (unitary) and M (total measurement) between worlds, and introduce modal operators ◻₁ (written as ) and ◻₂ (written as ) to capture statements that hold after any unitary transformation or after any total measurement, respectively.

The language consists of propositional symbols, the usual connectives, and the two modal operators. Formulas are labelled with world identifiers (e.g., x : A), and relational formulas (e.g., x U y, x M y) express the accessibility relations. The deduction system combines standard natural‑deduction rules for implication, falsum, and reductio ad absurdum with specialised modal rules (introduction and elimination for each modal operator) and relational rules that enforce the intended properties of U and M. U is axiomatized as an equivalence relation via reflexivity, symmetry, and transitivity rules, reflecting the invertibility and composability of unitary operations. M is given a seriality rule (every world has an M‑successor), an idempotence rule (a world measured twice yields the same world), and a “shift‑reflexivity” rule, capturing the idea that a total measurement produces a classical (idempotent) register.

A Kripke‑style semantics is defined: a frame F = ⟨W, U, M⟩ with a non‑empty set of worlds W, an equivalence relation U ⊆ W × W, and a relation M ⊆ W × W satisfying (i) M ⊆ U, (ii) ∀v∃w (v M w) (total measurability), (iii) ∀v,w (v M w → w M w) (measurements lead to classical worlds), and (iv) ∀v,w (v M v ∧ v M w → v = w) (idempotence). An interpretation function V maps each world to the set of propositional formulas true there. Truth conditions for the modal operators follow the usual universal quantification over the corresponding accessibility relation.

The authors prove soundness (every derivable sequent is semantically valid) and completeness (every semantically valid sequent is derivable) for MSQR with respect to this semantics. They then extend the framework to handle non‑total (partial) measurements, which are common in quantum computing (e.g., measuring a single qubit of a multi‑qubit register). In MSpQR, the M relation is replaced by a more general P relation, and a new modal operator ◻₃ (written as ) denotes “true after any measurement (not necessarily total).” Corresponding rules are adjusted to reflect that measurements need not be idempotent, and the semantics is adapted accordingly. Completeness and soundness are also established for MSpQR.

Throughout the paper, illustrative derivations demonstrate how familiar quantum‑theoretic facts become provable modal statements: the identity unitary transformation (x :  A → A), invertibility of unitary operations (x :  A →  A), composability of measurements (x :  A →  A), and the equivalence between a formula and its double measurement (x : (A ↔  A)). The authors argue that this qualitative modal approach offers a clean, abstract way to reason about quantum register dynamics without delving into the underlying Hilbert‑space mathematics.

In conclusion, the paper provides a novel logical apparatus that captures the essential dynamical aspects of quantum registers via modal logic, establishes rigorous meta‑theoretical properties, and opens avenues for further extensions (e.g., incorporating entanglement, error correction, or automated proof tools) to support formal verification of quantum algorithms and protocols.