Quantum Uncertainty Principle And Quantum Side Information Theory

In the late 1920 s,W.Heisenberg's Uncertainty Principle and matrix mechanics laid a solid foundation for modern quantum mechanics and revealed a fundamental difference between the classical and quantum mechanics.In classical mechanics,the measurement of position and momentum can be done at the same time.In quantum mechanics,it is impossible to simultaneously measure two complementary variables of a particle in precision,such as the position and momentum of a particle.The more precisely the position is determined,the less precisely the momentum is known in this instant,and vice versa.Because the position and momentum are connected with Fourier transform,which is now also called ”unbiased base”(mutually unbiased bases).Within 90 years after the first uncertainty relation,the formulations of uncertainty relations have been divided into variance-based uncertainty relations and entropic uncertainty relations.One of the important recent advances on uncertainty relations is to allow the measured quantum system to be correlated with its environment in a non-classical way,entropic uncertainty relations in the presence of quantum memory.Its division relies entirely on the presence of classical side information(expression of density matrices)or quantum side information(quantum correlations between observed system and quantum memory).The principle of uncertainty in the presence of quantum side information was proposed in 2010 by M.Berta,M.Christandl,R.Colbeck,J.M.Renes and R.Renner [53].This form of quantum side information is the so-called “quantum memory”.In [53],the authors point out that when the measured system is maximal entangled,then the uncertainty will disappear.The uncertainty principle in the presence of quantum memory shows a difference from the general entropic uncertainty relations,and its lower bound depends not only on the complementarity between the two sets of measurements,but also on the quantum correlation between the observed and quantum memory,which is measured by the conditional entropy H(A|B).Berta et al's uncertainty principle in the presence of quantum memory reveals uncertainties with quantum side information between the observables.In the recent important work of Coles and Piani,the entropic sum is controlled by the first and second maximum overlaps between the two projective measurements.We generalize the entropic uncertainty relation in the presence of quantum memory and find the exact dependence on all d largest overlaps between two measurements on any d-dimensional Hilbert space.Our bound is rigorously shown to be strictly tighter than previous entropic bounds in the presence of quantum memory.We propose stronger variance-based uncertainty relations than the existing ones for the product and sum of variances of two incompatible observables in a finite dimensional Hilbert space in the last chapter of our thesis.It is shown that the new uncertainty relations provide near-optimal state-dependent bounds,which can be useful for quantum metrology,entanglement detection etc.Next,we explain how to employ entropic uncertainty relations to derive lower bounds for the product of variances of incompatible observables for the first time.Finally,we give examples to show how the new bounds work compared with the recent bounds,which are some of the stronger ones for variance-based uncertainty relations.Moreover,we also formulate the concept of weighted uncertainty relations.The uncertainty principle is not only the watershed of quantum mechanics and classical mechanics,but also one of the specific expression for N.Bohr's complementarity principle.In the early development of quantum mechanics,N.Bohr put the complementarity principle as a foundation of Copenhagen interpretation.But when dealing with different problems,the complementary principle may have different formulation and meaning.However,the uncertainty principle always plays a central role.In modern times,the principle of uncertainty is more widely used in the determination of quantum entanglement,quantum key distribution and other related topics.Due to its wide applications in quantum theory and quantum information theory and its central role in the foundation of quantum theory,investigation of the various formulations of uncertainty principle,their corresponding lower bound and relations between quantum side information become the main topics of the thesis.