A Universal Quantum Certainty Relation for Arbitrary Number of Observables

TL;DR

This paper establishes a universal, state-independent certainty relation for multiple quantum observables using lattice theory, revealing fundamental limits on measurement outcomes and linking quantum uncertainty with coherence.

Contribution

It introduces a universal majorization-based certainty relation for any number of observables, extending quantum uncertainty principles beyond two observables.

Findings

Universal bounds for three mutually complementary observables in dimension 2.

Incompatibility limits the spread of probability vectors for observables.

A trade-off relation between quantum coherence and measurement bases.

Abstract

We derive by lattice theory a universal quantum certainty relation for arbitrary $M$ observables in $N$-dimensional system, which provides a state-independent maximum lower bound on the direct-sum of the probability vectors in terms of majorization relation. While the utmost lower bound coincides with $(1/N,...,1/N)$ for any two observables with orthogonal bases, the majorization certainty relation for $M\geqslant3$ is shown to be nontrivial. The universal majorization bounds for three mutually complementary observables and a more general set of observables in dimension-2 are achieved. It is found that one cannot prepare a quantum state with probability vectors of incompatible observables spreading out arbitrarily. Moreover, we also explore the connections between quantum uncertainty and quantum coherence, and obtain a complementary relation for the quantum coherence as well, which characterizes a trade-off relation of quantum coherence with different bases and is illustrated by an explicit example.