Speaker
Description
Our work provides a resource-theoretic framework for asking when quantum states are easy or hard to learn because of the spectral organization of their entanglement and magic resources. We introduce antiflatness as a characterization of the entanglement spectrum. While standard entanglement measures, such as the von Neumann entropy or the purity, only capture average properties of the spectrum, antiflatness quantifies its internal fluctuations and therefore provides complementary information about the organization of quantum correlations. The main contribution of the paper is the construction of a resource theory of antiflatness. We introduce a partial ordering of quantum states based on the spread of Rényi entropies and define the corresponding class of Flatness-Preserving Operations, obtaining necessary conditions for state convertibility that are not detected by standard majorization. We show that the Capacity of Entanglement can be interpreted as the Quantum Fisher Information associated with escort deformations of the entanglement spectrum, identifying it as a susceptibility that quantifies the response of the spectrum to infinitesimal deformations. We further investigate the typical behavior of antiflatness in Haar, Bures-Hall and doped-Clifford random-state ensembles, showing how spectral fluctuations evolve from stabilizer-like to generic quantum states.
The complexity of learning a quantum state may depend not only on its Hilbert-space dimension or entanglement entropy. We suggest that it may also depends on the internal organization of its entanglement spectrum and so on the resources associated to it. In this sense, antiflatness provides a possible resource-theoretic framework for investigating the role of spectral structure in quantum learning problems.
| Topic | Theoretical foundations of quantum learning |
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