Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products

2025/07/14 by Johnston, Nathaniel, Plosker, Sarah, Torrance, Charles +1
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2507.10327

Abstract

We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: ‖v²‖‖w²‖ - \langlev²,w²⟩ ≤ ‖v‖²‖w‖² - \langlev,w⟩² for all v,w ∈ ℝⁿ. We present three new proofs of this inequality that better illustrate "why" it is true and generalize it in several different ways: we generalize from vectors to matrices, we explore which exponents other than 2 result in the inequality holding, and we derive a version of the inequality involving three or more vectors.

Citations

• On the Generalizations of the Cauchy-Schwarz-Bunyakovsky Inequality with Applications to Elasticity
• Entanglement detection
• Quantum entanglement
• Classical deterministic complexity of Edmonds' problem and Quantum Entanglement
• A matrix realignment method for recognizing entanglement

Cited by

• Dynamic Linear Attention

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