Certainty relations, mutual entanglement and non-displacable manifolds

TL;DR

This paper establishes bounds for quantum measurement entropies, introduces novel uncertainty and certainty relations, and explores entanglement properties across multiple system splittings, highlighting the role of mutually unbiased bases and non-displacable manifolds.

Contribution

It derives explicit bounds for measurement entropies, formulates universal certainty relations, and analyzes entanglement structures using geometric and algebraic methods in quantum systems.

Findings

Maximal average entropy saturates at log N for two bases.

Certainty relations are nontrivial for three or more measurements.

Existence of states mutually separable or entangled with respect to different splittings.

Abstract

We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size $N$ in $L$ orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For $L=2$ the maximal average entropy saturates at $\log N$ as there exists a mutually coherent state, but certainty relations are shown to be nontrivial for $L \ge 3$ measurements. In the case of a prime power dimension, $N=p^k$, and the number of measurements $L=N+1$, the upper bound for the average entropy becomes minimal for a collection of mutually unbiased bases. Analogous approach is used to study entanglement with respect to $L$ different splittings of a composite system, linked by bi-partite quantum gates. We show that for any two-qubit unitary gate $U\in \mathcal{U}(4)$ there exist states being mutually separable or mutually entangled with respect to both splittings (related by $U$) of the composite system. The latter statement follows from the fact that the real projective space $\mathbb{R}P^{3}\subset\mathbb{C}P^{3}$ is non-displacable. For $L=3$ splittings the maximal sum of $L$ entanglement entropies is conjectured to achieve its minimum for a collection of three mutually entangled bases, formed by two mutually entangling gates.