We prove a few novel state-dependent uncertainty relations for product as well the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Roberson-Schrödinger uncertainty relation and other ones existing in the current literature. Also, we derive state dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only we cannot prepare quantum states for which two incompatible observables can have sharp values, but also we have both, lower and upper limits on the variances of quantum mechanical observables at a fundamental level.Introduction.-Quantum mechanics has many distinguishing features from classical mechanics in the microscopic world. One of these distinguished features is the existence of incompatible observables. As a result of this incompatibility, we have the uncertainty principle and the uncertainty relations. Owing to the seminal works by Heisenberg [1], Robertson [2] and Schrödinger [3], lower bounds were shown to exist for the product of variances of two non-commuting observables. Recently, Maccone and Pati have shown stronger uncertainty relations for all incompatible observables [4]. The stronger uncertainty relations have also been experimentally tested [5]. In addition, the entropic uncertainty and the reverse uncertainty relations also capture the essence of quantum uncertainty [6][7][8][9][10][11] and the incompatibility between two observables, but in a state-independently way.

For bipartite pure and mixed quantum states, in addition to the quantum mutual information, there is another measure of total correlation, namely, the entanglement of purification. We study the monogamy, polygamy, and additivity properties of the entanglement of purification for pure and mixed states. In this paper, we show that, in contrast to the quantum mutual information which is strictly monogamous for any tripartite pure states, the entanglement of purification is polygamous for the same. This shows that there can be genuinely two types of total correlation across any bipartite cross in a pure tripartite state. Furthermore, we find the lower bound and actual values of the entanglement of purification for different classes of tripartite and higher-dimensional bipartite mixed states. Thereafter, we show that if entanglement of purification is not additive on tensor product states, it is actually subadditive. Using these results, we identify some states which are additive on tensor products for entanglement of purification. The implications of these findings on the quantum advantage of dense coding are briefly discussed, whereby we show that for tripartite pure states, it is strictly monogamous and if it is nonadditive, then it is superadditive on tensor product states.Comment: 12 pages, 2 figures, Published versio

We derive several uncertainty relations for two arbitrary unitary operators acting on physical states of a Hilbert space. We show that our bounds are tighter in various cases than the ones existing in the current literature. Using the uncertainty relation for the unitary operators we obtain the tight state-independent lower bound for the uncertainty of two Pauli observables and anticommuting observables in higher dimensions. With regard to the minimum uncertainty states, we derive the minimum uncertainty state equation by analytic method, and relate this to the ground-state problem of the Harper Hamiltonian. Furthermore, the higher dimensional limit of the uncertainty relations and minimum uncertainty states are explored. From an operational point of view, we show that the uncertainty in the unitary operator is directly related to the visibility of quantum interference in an interferometer where one arm of the interferometer is affected by a unitary operator. This shows a principle of preparation uncertainty, i.e., for any quantum system, the amount of visibility for two general non-commuting unitary operators is non-trivially upper bounded.