Uncertainty relation and the constrained quadratic programming

Abstract

The uncertainty relation is a fundamental concept in quantum theory, plays a pivotal role in various quantum information processing tasks. In this study, we explore the additive uncertainty relation pertaining to two or more observables, in terms of their variance,by utilizing the generalized Gell-Mann representation in qudit systems. We find that the tight state-independent lower bound of the variance sum can be characterized as a quadratic programming problem with nonlinear constraints in optimization theory. As illustrative examples, we derive analytical solutions for these quadratic programming problems in lower-dimensional systems, which align with the state-independent lower bounds. Additionally, we introduce a numerical algorithm tailored for solving these quadratic programming instances, highlighting its efficiency and accuracy. The advantage of our approach lies in its potential ability to simultaneously achieve the optimal value of the quadratic programming problem with nonlinear constraints but also precisely identify the extremal state where this optimal value is attained. This enables us to establish a tight state-independent lower bound for the sum of variances, and further identify the extremal state at which this lower bound is realized.

35 pages, LaTeX

References in corpus (32)

• Separability Criterion for Density Matrices
• Advances in Quantum Metrology
• Entropic Uncertainty Relations and their Applications
• The Uncertainty Relation for Smooth Entropies
• The Bloch Vector for N-Level Systems
• Violation of local uncertainty relations as a signature of entanglement
• EPR Steering Inequalities from Entropic Uncertainty Relations
• Conjectured Strong Complementary Information Tradeoff
• Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation
• Quantum Randomness Certified by the Uncertainty Principle
• Universal Uncertainty Relations
• Majorization entropic uncertainty relations
• Entropic uncertainty relation for mutually unbiased bases
• Source-independent quantum random number generation
• A generalized Pancharatnam geometric phase formula for three level systems
• Geometry of the generalized Bloch sphere for qutrits
• Entanglement Detection in Optical Lattices of Bosonic Atoms with Collective Measurements
• An equality between entanglement and uncertainty
• State-independent Uncertainty Relations and Entanglement Detection in Noisy Systems
• Three-dimensional visualization of a qutrit
• Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices
• The Conditional Uncertainty Principle
• Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach
• Tighter uncertainty relations based on Wigner-Yanase skew information for observables and channels
• Improving sum uncertainty relations with the quantum Fisher information
• Conjectured strong complementary-correlations tradeoff
• Entanglement detection by violations of noisy uncertainty relations: A proof of principle
• Incompatibility of Observables as State-Independent Bound of Uncertainty Relations
• Conditional entropic uncertainty relations for Tsallis entropies
• State-independent Uncertainty Relations and Entanglement Detection
• The Bloch vectors formalism for a finite-dimensional quantum system
• Uncertainty regions of observables and state-independent uncertainty relations