analiza aiqbench
This paper establishes a rigorous mathematical framework for calculating minimum von Neumann entropy excess in block-diagonal quantum states under fixed-weight constraints, resolving the gap between exact nonconvex envelopes and their convex surrogates. By deriving an explicit polygonal curve through synchronized rank vertices, the authors provide a symmetry-sector purity certificate that remains effective even where classical entropy-error bounds fail. These theoretical advances directly advance fundamental limits in quantum information theory and state characterization without relying on incidental AI applications.
Entropy Stability under Block-Separable Constraints: Fixed-Weight Rank-Lattice Envelopes and Convexification
arXiv:2512.16192v3 Announce Type: replace Abstract: For block-diagonal quantum states with prescribed block weights and dimensions, we determine the minimum von Neumann entropy excess at each trace-norm distance from the blockwise-pure states. The one-block extremal is the classical reverse-Fano profile; the new problem is its constrained assembly when the block weights and dimensions cannot be redistributed. The resulting value function $\Phi_{q,n}(D)$ is attained on the full admissible interval and is the lower envelope of finitely many rank-lattice edges. Every minimizing distance allocation has at most one block between consecutive integer ranks, and the active edge can switch nonmonotonically with $D$. We also determine the lower convex envelope of $\Phi_{q,n}$ explicitly: it is the polygonal curve through synchronized rank vertices and admits a one-parameter dual representation. The exact nonconvex envelope differs from this convex surrogate by at most $q_\diamond\log(5/4)$, with a sharper stage-dependent bound, and the constant is optimal. For binomial charge-sector weights the uniform gap is $O(L^{-1/2})$. The framework yields a symmetry-sector purity certificate; in a three-qubit phase-damping example the fixed-weight certificate remains nontrivial where the universal classical entropy-error envelope is already vacuous.
Abstract:For block-diagonal quantum states with prescribed block weights and dimensions, we determine the minimum von Neumann entropy excess at each trace-norm distance from the blockwise-pure states. The one-block extremal is the classical reverse-Fano profile; the new problem is its constrained assembly when the block weights and dimensions cannot be redistributed. The resulting value function $\Phi_{q,n}(D)$ is attained on the full admissible interval and is the lower envelope of finitely many rank-lattice edges. Every minimizing distance allocation has at most one block between consecutive integer ranks, and the active edge can switch nonmonotonically with $D$. We also determine the lower convex envelope of $\Phi_{q,n}$ explicitly: it is the polygonal curve through synchronized rank vertices and admits a one-parameter dual representation. The exact nonconvex envelope differs from this convex surrogate by at most $q_\diamond\log(5/4)$, with a sharper stage-dependent bound, and the constant is optimal. For binomial charge-sector weights the uniform gap is $O(L^{-1/2})$. The framework yields a symmetry-sector purity certificate; in a three-qubit phase-damping example the fixed-weight certificate remains nontrivial where the universal classical entropy-error envelope is already vacuous.
23 pages. Revised throughout, with an exact fixed-weight entropy profile, rank-lattice optimizer structure, convexification theorem, symmetry-sector application, and updated references
Quantum Physics (quant-ph); Functional Analysis (math.FA)
arXiv:2512.16192 [quant-ph]
(or arXiv:2512.16192v3 [quant-ph] for this version)
Text integral sub licența deschisă a publicației · 211 cuvinte
Subiecte conexe
Challenge: Hand coding weights for efficient sequence memorisation
AI Alignment Forum · 23 iul. 2026
The Entangled Dimensions of Decision Theory
LessWrong · 30 iul. 2026
GEM Training: How Meta Doubled the Efficiency of Its LLM-Scale Ads Foundation Model
Meta Engineering · 3 aug. 2026
Expenditure Horizon: Measuring Optimization Ability, with an Application to NanoGPT
METR · 21 iul. 2026