Gaussian distribution of a quantum state with continuous spectrum is known to maximize the Shannon entropy at a fixed variance. Applying it to a pair of canonically conjugate quantum observables $hat x$ and $hat p$, quantum entropic uncertainty relat
ion can take a suggestive form, where the standard deviations $sigma_x$ and $sigma_p$ are featured explicitly. From the construction, it follows in a transparent manner that: (i) the entropic uncertainty relation implies the Kennard-Robertson uncertainty relation in a modifed form, $sigma_xsigma_pgeqhbar e^{cal N}/2$; (ii) the additional factor ${cal N}$ quantifies the quantum non-Gaussianity of the probability distributions of two observables; (iii) the lower bound of the entropic uncertainty relation for non-gaussian continuous variable (CV) mixed state becomes stronger with purity. Optimality of specific non-gaussian CV states to the refined uncertainty relation has been investigated and the existance of new class of CV quantum state is identified.
In the history of quantum mechanics, various types of uncertainty relationships have been introduced to accommodate different operational meanings of Heisenberg uncertainty principle. We derive an optimized entropic uncertainty relation (EUR) that qu
antifies an amount of quantum uncertainty in the scenario of successive measurements. The EUR characterizes the limitation in the measurability of two different quantities of a quantum state when they are measured through successive measurements. We find that the bound quantifies the information between the two measurements and imposes a condition that is consistent with the recently-derived error-disturbance relationship.
We study a Hamiltonian system describing a three spin-1/2 cluster-like interaction competing with an Ising-like exchange. We show that the ground state in the cluster phase possesses symmetry protected topological order. A continuous quantum phase tr
ansition occurs as result of the competition between the cluster and Ising terms. At the critical point the Hamiltonian is self-dual. The geometric entanglement is also studied. Our findings in one dimension corroborate the analysis of the two dimensional generalization of the system, indicating, at a mean field level, the presence of a direct transition between an antiferromagnetic and a valence bond solid ground state.
