Abstract
We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size in orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For the maximal average entropy saturates at because there exists a mutually coherent state, but certainty relations are shown to be nontrivial for measurements. In the case of a prime power dimension, , and the number of measurements , 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 different splittings of a composite system linked by bipartite quantum gates. We show that, for any two-qubit unitary gate there exist states being mutually separable or mutually entangled with respect to both splittings (related by ) of the composite system. The latter statement follows from the fact that the real projective space is nondisplaceable by a unitary transformation. For splittings the maximal sum of entanglement entropies is conjectured to achieve its minimum for a collection of three mutually entangled bases, formed by two mutually entangling gates.
3 More- Received 15 July 2015
DOI:https://doi.org/10.1103/PhysRevA.92.032109
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