Lean-QuantumInfo: Verified Quantum Protocols

Updated 13 October 2025

Lean-QuantumInfo Library is a formalized infrastructure built on Lean that encodes quantum states, protocols, and resource theories using dependent types.
It employs rigorous type-theoretic foundations to verify quantum processes like teleportation and key distribution, ensuring mathematical correctness.
The library supports collaborative formalization by addressing subtleties in hypothesis testing and resource theories, aiding in error detection in quantum proofs.

The Lean-QuantumInfo Library is a formalized software infrastructure built on the Lean interactive theorem prover for quantum information theory, quantum resource theories, and quantum hypothesis testing. Its foundation comprises rigorous mathematical models encoded as dependent types, supported by an extensive base of verified theorems, definitions, and formalized protocols, and is designed for a collaborative program aimed at the complete formalization of quantum theory.

1. Foundational Principles and Type-Theoretic Infrastructure

The Lean-QuantumInfo Library encodes quantum states and protocols using dependent types and constructive type theory, following methodologies developed for Coq-based quantum information frameworks (Boender et al., 2015). In Lean, quantum bits (qubits) are represented as normalized $2^n \times 1$ complex matrices accompanied by proofs of unitarity and invariance (e.g., normalization constraints):

Qubit Representation: For $n$ qubits, a state is a dependent type constrained to be a unit vector in $2^n$ dimensions.
Quantum Gates: Unitary $2^n \times 2^n$ matrix-valued operators, each accompanied by a formal proof of unitarity.
Measurement Primitives: Formalized as functions computing outcome probabilities using matrix traces and post-measurement states through explicit normalization and side proofs ensuring denominators are nonzero.
Tensor Operations: The tensor product is implemented via well-defined index manipulations ensuring the output remains within admissible bounds, using algebraic proofs for closure.

All manipulations and transformations (addition, multiplication, contraction, normalization) are subject to Lean’s strict typing and proof requirements, ensuring not only the computational correctness but also the mathematical soundness of every protocol step. Model components are closely matched to their Coq analogues, such as the construction of maximally entangled states and explicit gate composition using tensor algebra.

2. Formalization of Quantum Protocols and Theorems

The library supports the explicit, rigorous modeling of quantum protocols such as quantum teleportation, quantum key distribution, bit commitment, and blind quantum computing, closely mirroring the stepwise methodology given in (Boender et al., 2015). For example:

Teleportation Protocol: The unknown state $\varphi$ and an EPR pair $\psi$ are composed as $(\varphi \otimes \psi)$ ; a sequence of gates—Hadamard, CNOT, and identities—are tensor-composed and applied as per the protocol. Measurement is simulated through a function returning probability distributions and collapsed states, and classical bits (type Bit with constructors z and o) drive conditional correction steps (applying $I$ , $X$ , $Z$ , $Y$ gates based on outcome).
Verification of Protocols: Each protocol step is accompanied by a Lean formal proof of correctness; for instance, the “teleportation” theorem establishes the equivalence of Bob’s output state with the original input qubit, up to normalization and choice of classical message.

The inclusion of auxiliary constructs (e.g., rigorous handling of division in state collapse, dependent proofs of normalization after measurement, explicit encoding of outcome types) ensures that protocol modeling is robust to mathematical and computational subtleties present in practical implementations.

3. Quantum Resource Theories and the Generalized Quantum Stein’s Lemma

A substantial portion of recent development focuses on the formalization of quantum resource theories and the Generalized Quantum Stein’s Lemma (GQSL), as per (Meiburg et al., 9 Oct 2025). Key elements include:

Resource Theory Definition: The “FreeStateTheory” data structure assigns, to each Hilbert space, a set of free states that is compact, convex, and tensor-product closed, with at least one full-rank state. The tensor-product operation is not assumed to be strictly canonical, and explicit indexing and associators are managed within a unital (monoidal) framework, embedding the theory into symmetric monoidal categories.
GQSL Formalization: Lean encodes mixed quantum states (MState as Hermitian PSD matrices of unit trace), probability types (Prob in $[0,1]$ ), and operational quantities:

$\lim_{n \to \infty} \left[ -\frac{1}{n} \log \beta_\epsilon(\rho^{\otimes n} \| \mathcal{S}_n) \right] = \lim_{n \to \infty} \left[ \frac{1}{n}\min_{\sigma \in \mathcal{S}_n} D(\rho^{\otimes n} \| \sigma) \right]$

where $\beta_\epsilon$ is the minimal type-II error over all $\epsilon$ -type-I-verified POVMs and $D(\cdot\|\cdot)$ is quantum relative entropy.

The formalization process identified and rectified subtle mathematical imprecisions, especially in the manipulation of extended real numbers (e.g., avoiding subtraction of infinities, requiring explicit proof of finiteness before operations). Modular proofs encompass continuity of relative entropy, data processing inequalities, and pinching theorems, each constructed as reusable lemma modules.

The Lean-QuantumInfo Library distinguishes itself from other frameworks such as QuantumInformation.jl (Gawron et al., 2018), QuAlg (Dahlberg, 2020), and Qualtran (Harrigan et al., 6 Sep 2024) through its focus on formal proof and verification:

Library	Key Methodology	Domain Coverage
Lean-QuantumInfo	Formal verification	Resource theory, QIT
QuantumInformation.jl	Numerical analysis	Channels, states, entropies
QuAlg	Symbolic algebra	Infinite-dimensional, QOptics
Qualtran	Algorithmic analysis	Simulation, resource estimates

Binary distinction exists between Lean-QuantumInfo’s type-theoretic rigor and the numerical / symbolic nature of related Julia or Python packages. While QuantumInformation.jl and QuAlg emphasize computational and symbolic accessibility (for tasks like simulation or symbolic integration), Lean-QuantumInfo ensures mathematical certainty in protocol correctness and resource-theoretic claims. Qualtran provides cost modeling and resource tabulation rather than theorem-proving, but in principle could be integrated with Lean-QuantumInfo for a hybrid approach to verified resource analysis.

5. Practical Applications and Verification Workflow

The library is deployed as a tool for both mathematical research and protocol testing. Its practical applications include:

Verification of quantum communication and cryptographic protocols such as QKD, teleportation, and error correction.
Testing and debugging circuit designs through interactive Lean proofs and tactics.
Development of resource theory frameworks suitable for analyzing operational interpretations of relative entropy and hypothesis testing in quantum settings.
Formal assurance: By formalizing protocols, bugs and subtle errors in implementation or theoretical assumptions can be detected early, mitigating risks in experimental or commercial quantum technologies.

Specific case studies described include the modeling and verification of quantum teleportation, with explicit gate and measurement operations, as well as foundational work relevant for quantum bit commitment, blind computation, and error correction (Boender et al., 2015, Meiburg et al., 9 Oct 2025).

6. Infrastructure, Collaboration, and Future Directions

The Lean-QuantumInfo Library encompasses a substantial codebase—with over 15,000 lines, 250 definitions, and 1,000+ theorems—serving as infrastructure for community-wide formalization efforts:

Modularity: All definitions, lemmas, and proofs are packaged as reusable components, facilitating extension, proof automation, and integration with broader Lean projects such as mathlib.
Collaborative Program: The library enables a large-scale, distributed effort targeting the formalization of quantum theory, resource theories, and advanced theorems. Potential generalizations (non-unital resource theories, general second laws, formal treatment of data processing inequalities) are anticipated as seamless extensions once core foundations are established.
Readability and Simplification: The design emphasizes clarity and structure, aiming to make advanced quantum constructs accessible for further formalization, teaching, and cross-domain verification.
Integration of Automated Techniques: While not the current focus, future work may incorporate model checking, equivalence checking, or more sophisticated tactics for circuit and protocol verification.

A plausible implication is that, as the mathematical and type-theoretic infrastructure matures, the Lean-QuantumInfo Library could become a standard repository for verified quantum information results, with interoperability across subfields of quantum science modeled after successful mathematics libraries such as mathlib (Meiburg et al., 9 Oct 2025).

7. Mathematical Rigor and Correction of Subtle Errors

The Lean-QuantumInfo project has demonstrated that computer-assisted formalization enforces a rigorous standard, compelling the authors to clarify and correct subtle points often overlooked in traditional mathematical presentations. Key aspects include:

Handling Extended Real Values: Expressions using quantum relative entropy (ranging in $[0, \infty]$ ) are managed without making unjustified subtractions; sums are preferred, and explicit proofs of finiteness precede operations.
Foundations for Quantum Resource Theory: The refined definition, including explicit indexing, associators, and monoidal structure, provides a transparent blueprint for further generalization and ensures compatibility with categorical and algebraic formulations.
Identification of Imprecision: The formalization forced discovery—not present in original literature—of necessary conditions such as lower semicontinuity and the existence of minimizers for quantum relative entropy.
Blueprint for Future Theorems: The modular infrastructure and methodologies serve as templates for subsequent formalizations, including the data processing inequality, composable protocols, and quantum channel characterizations.

This suggests that Lean-QuantumInfo is positioned not only as a verification tool but as a mechanism to clarify and potentially correct entrenched technical arguments in quantum information theory (Meiburg et al., 9 Oct 2025).

The Lean-QuantumInfo Library provides a robust, formally verified foundation for quantum information theory in Lean, encompassing a rigorous encoding of states, channels, measurements, resource theories, and hypothesis-testing theorems. Its infrastructure supports detailed protocol modeling, theorem formalization, correction of subtle errors, and lays the groundwork for a collaborative program aiming at complete formalization and verification of quantum theory.