Lean-QIT provides a machine-readable foundation for formal QIT and a compositional knowledge substrate for emerging AI-assisted formalization, automated proof search, and agentic reasoning in quantum information and computation.
Abstract
Quantum information theory (QIT) characterizes the capabilities and fundamental limits of quantum information processing, underpinning quantum communication, computation, and error correction. Formalizing its coding theorems requires connecting finite-block protocols, analytic inequalities, and asymptotic limits within a unified machine-checked framework. Existing developments, however, lack a reusable operational layer that defines codes, error criteria, achievable rates, and capacities independently of their information-theoretic characterizations. In this work, we present LeanQIT, a Lean 4 library for finite-dimensional QIT. It provides composable, kernel-checked interfaces for quantum states and channels, source and channel codes, finite-block performance criteria, hypothesis testing, one-shot quantities, and asymptotic rate constructions. Using this infrastructure, we formalize Schumacher's quantum source-coding theorem, the Holevo--Schumacher--Westmoreland classical-capacity theorem, and the entanglement-assisted classical-capacity theorem together with its strong converse. By separating operational definitions from analytic characterizations and exposing reusable achievability, converse, and asymptotic components, Lean-QIT provides a machine-readable foundation for formal QIT and a compositional knowledge substrate for emerging AI-assisted formalization, automated proof search, and agentic reasoning in quantum information and computation.
This work establishes the first rigorous finite-$n bounds on quantum resource testing and hence quantum resource manipulation, thus strengthening the GQSL and providing explicit estimates on the number of copies needed to achieve a prescribed performance.
Dmitry Grinko, Ludovico Lami· arXiv.org· 0 citations
This framework provides foundational tools for exploring quantum information regardless of whether the physical world using finite- or infinite-dimensional spaces, and shows that the operational interpretation of entropic quantities through these achievable rates constitutes a universal principle beyond finite dimensions.
Quantum locally recoverable codes (qLRCs) allow a single qudit erasure to be corrected by accessing only a small number of other qudits. Standard CSS and Hermitian constructions, however, impose dual-containing or self-orthogonal constraints on the underlying classical codes, thereby restricting the well-structured classical LRCs (cLRCs) that can be used to construct qLRCs. To relax these constraints, we introduce entanglement-assisted quantum locally recoverable codes (EAQLRCs) by assuming that halves of the pre-shared maximally entangled pairs are noiseless. We characterize sufficient support conditions on extended stabilizers under which entanglement-assisted stabilizer codes have locality $r$ and derive a CSS-like construction from two classical codes without imposing the ordinary dual-containing condition. We further establish an upper bound on locality and a Singleton-like bound for arbitrary CSS-like EAQLRCs, and characterize the pure codes attaining equality in the latter bound. These results yield a general framework for constructing optimal pure EAQLRCs from pairs of cLRCs. Applying this framework to $\ell$-intersection pairs of MDS codes and block parity-check matrices, we obtain two families of optimal pure CSS-like EAQLRCs with flexible parameters and nontrivial localities. To the best of our knowledge, these represent the first explicit families of EAQLRCs.
Yang Li, San Ling, Zhenliang Lu et al.· arXiv.org· 2 citations· ⚡1
Quantum computational advantage is generally attributed to coherent interference and other non-classical resources, yet their respective roles remain difficult to disentangle in experimental platforms where multipartite entanglement is inherently present. High-dimensional quantum systems provide an attractive route for investigating these resources while simultaneously reducing hardware overhead for quantum information processing. Here we realize a programmable four-dimensional optical qudit encoded in a single trapped $^{138}\mathrm{Ba}^{+}$ ion and demonstrate universal coherent control through phase-programmable optical rotations. Using this platform, we implement an entanglement-free realization of Grover's quantum search algorithm, achieving target-state identification probabilities of up to $94.5\pm2.0\%$. Within the same processor, we further demonstrate state-dependent quantum contextuality through a Clauser--Horne--Shimony--Holt (CHSH)-type noncontextuality inequality, obtaining a maximum violation of $S = 2.816 \pm 0.082$, in close agreement with the Tsirelson bound. By integrating programmable quantum computation and contextuality measurements within a single multilevel trapped-ion platform, our work establishes a versatile architecture for investigating the relationship between coherent interference and contextuality in quantum information processing and provides a scalable route toward high-dimensional quantum technologies.
T. Dutta, Jasper Phua Sing Cheng, Alex Jin et al.· 0 citations
This paper studies entanglement-assisted quantum locally recoverable codes (EA-qLRCs) built via a CSS-like stabilizer construction from pairs of classical locally recoverable codes (cLRCs), without requiring dual-containment. We define such codes through local recovery channels, give a sufficient stabilizer criterion for the construction, and derive Singleton-, Griesmer-, Plotkin-, and sphere-packing-like converse bounds on the parameters of the resulting pure CSS-like EA-qLRCs, along with a Cadambe--Mazumdar-like bound that, as in the classical case, lacks a closed form, plus a comparison of their relative tightness across finite-length and asymptotic regimes. We give necessary and sufficient conditions for a pure CSS-like EA-qLRC to attain the Singleton-like bound with equality; for the single-code case $\mathcal{C}_1=\mathcal{C}_2=\mathcal{C}$, this reduces to a simple condition on the hull dimension $s=\dim(\mathcal{C}\cap\mathcal{C}^\perp)$, which also fixes the entanglement count via $c=n-k-s$. We present CSS-like EA-qLRC constructions from classical LRC families---Tamo--Barg and cyclic codes---and characterize when these attain the Singleton-like bound, showing the cyclic families yield optimal codes while the Tamo--Barg construction, though valid, attains the bound only in the degenerate regime $k \le r$, where locality is vacuous. We complement these constructions with two Gilbert--Varshamov-like achievability bounds, via a classical parity-check augmentation and a sharper concatenated-code construction, and show both hold unconditionally for field size $q>3$ via a monomial-equivalence argument. Finally, we unify all bounds---converse and achievability alike---under a common maximally entangled regime, giving a single comparison of the achievable and forbidden rate--distance--locality region for CSS-like EA-qLRCs.
A theoretical framework in Pauli-Liouville space is developed that provides a unified analytical treatment of the echo state property (ESP), nonlinear expressive power, and quantum resources and shows that the ESP is natively guaranteed by the Liouvillian spectral gap, decoupling it from quantum magic.
Wei Xia, S. Cao, Xingze Qiu et al.· 3 citations
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