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

Abstract

We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: $\Vert {\mathbf{v}}^2\Vert \Vert {\mathbf{w}}^2\Vert -⟨{\mathbf{v}}^2,{\mathbf{w}}^2⟩\le \Vert \mathbf{v}{\Vert }^2\Vert \mathbf{w}{\Vert }^2-⟨\mathbf{v},\mathbf{w}{⟩}^2$ for all $\mathbf{v},\mathbf{w}\in {\mathbb{R}}^n$. 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.

15 pages, 2 figures

References in corpus (5)

• Quantum entanglement
• Entanglement detection
• Some Properties of the Computable Cross Norm Criterion for Separability
• Diagonal unitary and orthogonal symmetries in quantum theory
• Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States