Deriving quantum constraints and tight uncertainty relations

TL;DR

This paper introduces a systematic method to derive all quantum constraints on expectation values of qudit operators, enabling the precise characterization of allowed quantum states and the derivation of tight uncertainty relations.

Contribution

It provides an analytical framework to determine the complete set of quantum constraints and tight uncertainty relations for various observables in qudits.

Findings

Derived all necessary and sufficient quantum constraints for expectation values.

Defined the allowed region as a convex set characterized by pure states.

Obtained explicit tight uncertainty relations for qubits and spin observables.

Abstract

We present a systematic procedure to obtain all necessary and sufficient (quantum) constraints on the expectation values for any set of qudit's operators. These constraints---arise form Hermiticity, normalization, and positivity of a statistical operator and through Born's rule---analytically define an allowed region. A point outside the admissible region does not correspond to any quantum state, whereas every point in it come from a quantum state. For a set of observables, the allowed region is a compact and convex set in a real space, and all its extreme points come from pure quantum states. By defining appropriate concave functions on the permitted region and then finding their absolute minimum at the extreme points, we obtain different tight uncertainty relations for qubit's and spin observables. In addition, quantum constraints are explicitly given for the Weyl operators and the spin observables.