Certainty relations, mutual entanglement, and nondisplaceable manifolds

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 $logN$ because there exists a mutually coherent state, but certainty relations are shown to be nontrivial for $L\ensuremath{\ge}3$ 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. An analogous approach is used to study entanglement with respect to $L$ different splittings of a composite system linked by bipartite quantum gates. We show that, for any two-qubit unitary gate $U\ensuremath{\in}\text{U}(4)$ 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 $\mathbb{R}{P}^{3}\ensuremath{\subset}\mathbb{C}{P}^{3}$ is nondisplaceable by a unitary transformation. 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.