quantum information theory

This work addresses the efficient computation of the measured relative entropy—a quantity with operational significance in quantum hypothesis testing and recent relevance to hybrid quantum-classical hardware. We establish, for the first time, exact semidefinite programming (SDP) characterizations of the measured relative entropy for both quantum states and quantum channels, unifying optimal value computation with explicit construction of optimal measurement strategies. Methodologically, we combine SDP representations of weighted geometric means of operators, convex optimization formulations of logarithmic operator connections, and variational principles—yielding polynomial-time exact computability. Our framework delivers not only tight theoretical bounds but also directly implementable optimal measurements, thereby significantly enhancing both the design efficiency and experimental feasibility of quantum hypothesis testing protocols.

Quantifying the minimal entanglement cost required to prepare quantum states and processes is a fundamental problem in quantum information theory. This work addresses the long-standing challenge of characterizing entanglement cost under positive-partial-transpose (PPT) operations. We introduce a computable measure—the logarithmic k-negativity—and establish, for the first time, a faithful lower bound on the PPT entanglement cost for all non-PPT states. We further prove the asymptotic irreversibility of full-rank entangled states under PPT operations and extend our results to point-to-point and bipartite quantum channels. Leveraging semidefinite programming optimization and asymptotic information-theoretic techniques, our bound strictly improves upon all previously known computable bounds: it is efficiently computable, applies broadly, and yields nontrivial values for a wide range of states and channels. These results deepen our understanding of the ultimate limitations and structural nature of entanglement manipulation.

This work addresses the lack of efficient numerical methods for computing the quantum channel relative entropy—a fundamental metric in quantum channel discrimination and resource theories. Methodologically, we introduce the first scalable, error-controllable computational framework: we discretize and linearize the integral representation of quantum state relative entropy, thereby reformulating the channel relative entropy maximization problem as a sequence of semidefinite programs (SDPs); tight upper and lower bounds are constructed to enable arbitrarily precise sandwich estimation. Crucially, our approach overcomes prior limitations restricted to minimization settings, enabling rigorous optimization over input states for the first time. Experiments demonstrate high accuracy and low computational complexity, significantly enhancing the tractability and practical utility of channel relative entropy in real-world discrimination tasks and resource quantification.

This work addresses the equivalence of success probabilities and strong converse exponents between entanglement-assisted (EA) and no-signaling-assisted (NS) quantum channel coding. We propose the first theoretically guaranteed rounding algorithm that converts optimal NS solutions into feasible EA implementations: achieving a $(1-e^{-1})$-approximation for measurement channels, and a dimension-dependent yet tight approximation for general quantum channels. Crucially, we establish, for the first time, the exact equality of success probabilities under NS assistance and the semidefinite programming relaxation—termed the “meta-converse”—thereby resolving their operational equivalence. Our technical toolkit includes position-based decoding, quantum decoupling, matrix Chernoff bounds, and input flattening. The results precisely characterize the strong converse exponent for EA channel coding, yielding new analytical tools and fundamental benchmarks for quantum error correction and quantum information theory.

This work addresses the existence problem of binary quantum codes. We propose the first complete hierarchy of semidefinite programs (SDPs), constructed via state polynomial optimization and pseudo-Clifford algebras, applicable to arbitrary quantum codes—not restricted to stabilizer codes. Exploiting symmetry via the Terwilliger algebra and group representation theory, we reduce the SDP size to $O(n^4)$. We derive, for the first time, quantum analogues of the Lovász theta bound and the Delsarte bound, unifying and generalizing their classical counterparts. Our method yields a new proof of the nonexistence of the $(7,1,4)_2$ code and establishes, for the first time, the nonexistence of both the $(8,9,3)_2$ and $(10,5,4)_2$ codes—significantly advancing the existence map for small-parameter binary quantum codes.

This study addresses the urgent challenge posed by quantum computing to classical cryptographic systems and advocates for a systematic transition to post-quantum cryptography (PQC). It comprehensively reviews the principal PQC approaches—including lattice-based, code-based, hash-based, multivariate, and isogeny-based schemes—and provides an in-depth analysis of the threats from Shor’s algorithm, the current status of NIST standardization, and practical deployment challenges. The work introduces an innovative incremental migration architecture that integrates mathematical foundations with security engineering, bridging the gap from foundational understanding to strategic migration decisions. Notably, it delivers the first authoritative technical guide tailored for the Portuguese-speaking academic community, thereby supporting the development of a secure and coherent PQC governance framework.

Floating-point semidefinite programming (SDP) solvers for quantum noncommutative optimization problems often yield unreliable upper bounds, undermining certifiability. This paper introduces the first rational post-processing framework grounded in floating-point SDP outputs to compute strictly verifiable upper bounds. Our method formally extracts exact rational bounds from numerical solutions, leveraging sparsity, symmetry reduction, and block-diagonalization to enhance computational efficiency and scalability. Applied to canonical problems—including entanglement detection and quantum steering—we obtain the first rigorously certified rational bounds with controllable error margins; these bounds exhibit significantly higher precision than conventional floating-point results. By bridging numerical computation with exact arithmetic verification, our approach establishes genuinely falsifiable theoretical guarantees for quantum optimization, thereby advancing the reliability and interpretability of quantum information processing.

This study investigates the computability and computational complexity of quantum channel capacities. By leveraging tools from quantum complexity theory and undecidability analysis, it rigorously establishes for the first time that computing the quantum capacity of a general quantum channel is QMA-hard. Furthermore, it demonstrates that the zero-error one-shot classical capacity assisted by maximal entanglement is uncomputable. These results resolve a fundamental open question in quantum information theory concerning the computability of channel capacities and provide crucial theoretical underpinnings for quantum communication.

This work addresses the limitation of conventional confusability graphs in fully characterizing the indistinguishability of input states in quantum channels. To overcome this, the authors propose a novel “quantum confusability multigraph” model that embeds channel output information into a graph structure, where multiple edges between vertices recover classical confusability relations. Building upon Weaver’s theory of quantum relations, they establish a rigorous mathematical framework for this model. The study establishes, for the first time, a one-to-one correspondence between quantum confusability multigraphs and quantum channels, and provides necessary and sufficient conditions for the existence of such multigraphs. This result fully characterizes the structural properties of confusability multigraphs realizable by quantum channels, thereby extending the applicability of quantum graph theory to problems of information distinguishability.

This work addresses the quantum constraint satisfaction problem (qCSP) induced by the EPR Hamiltonian, aiming to improve the approximation ratio of near-term quantum algorithms. We derive the first nonlinear monogamy-of-entanglement bound for star graphs, overcoming fundamental limitations of conventional linear monogamy constraints. Leveraging this bound, we redesign the parameterization structure of shallow variational quantum circuits—integrating entanglement analysis, nonlinear inequality derivation, and circuit architecture optimization. Experimentally, our approach achieves an approximation ratio of 0.8395 on the EPR qCSP, establishing the current state-of-the-art. Moreover, we rigorously prove that all existing frameworks relying on linear monogamy are provably capped at 0.8402, thereby exposing an inherent barrier and underscoring the necessity of nonlinear entanglement characterization and novel circuit design principles for further progress.