Uncertainty Principles In Quantum Information And Their Applications

In this dissertation,we discuss uncertainty principles in quantum information and their applications by using the knowledge of operator theory and operator algebra.The uncertainty principles with respect to the generalized metric adjusted skew information are established;The generalized Wigner-Yanase-Dyson(WYD)skew information,the generalized correlation measure and another quan-tities with respect to non-self adjoint operators are introduced,and several uncer-tainty relations are given;The uncertainty relations relating to the sum or product of variances with mixed state are proved,respectively;A new characterization of non-Markovian quantum evolution based on the covariance matrix from the uncertainty perspective is proposed;The monotonicity and convexity of the unified quantum(r,s)-entropy Ers(?)and the unified quantum(r,s)-mutual information Irs(?)with respect to parameters s and r are discussed,respectively.There are five chapters in this dissertation,in the following we shall show the contents.In the first chapter,we introduce some research background and status on our main contents,and list some notations,definitions and known results.In the second chapter,the measure F?,?(?ab)for correlations in terms of the WYD skew information is introduced and discussed.The following conclusions are obtained.For a classical-quantum state ?ab,F?,?(?ab)= 0 if and only if ?ab is a product state;F?,?(?ab)is locally unitary invariant and convex on the set of states with the fixed marginal ?a;F?,?(?ab)decreases under any local random unitary operation on B(Hb);For a quantum-classical state ?ab,F???(?ab)decreases under local operation on B(Hb);Lastly,F?,?(?ab)is computed for the pure states and the Bell-diagonal states,respectively.In the third chapter,several Schrodinger-type uncertainty relations for the gen-eralized metric adjusted skew information or the generalized metric adjusted corre-lation measure are established,respectively.Furthermore,we prove that some new uncertainty relations for skew information and WYD skew information hold by us-ing several canonical operator monotone functions.The concepts of the generalized WYD correlation measure,the generalized WYD skew information and the related quantities of non-self adjoint operators are introduced,and various properties of them are discussed.Moreover,we establish several generalizations of uncertainty relations expressed in terms of the generalized WYD skew information.Finally,we give two stronger uncertainty relations,relating to the sum of variances with re-spect to a mixed state,whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.Moreover,two stronger uncertainty relations in terms of the product of the variances of two ob-servables are established.Meanwhile,we also prove the more stringent uncertainty relations relating to the sum and product of the variances of three observables exist,respectively.In the fourth chapter,we propose a new characterization of non-Markovian quantum evolution based on covariance matrix.The fundamental properties of covariance matrices are elucidated.The measure captures quite directly the charac-teristics of non-Markovianity from the uncertainty perspective.We consider several typical examples and compare the covariance matrix characterization of quantum non-Markovianity with Fisher-information matrix,divisibility and the Breuer-Laine-Piilo(BLP)characterization of quantum non-Markoviantity.In the last chapter,monotonicity and convexity of the unified quantum(r,s)-entropy Ers(?)and the unified quantum(r,s)-mutual information Irs(?)are discussed.Some basic properties of them are explored and the following conclusions are estab-lished.(i)For any 0<r<1,Ers(?)is increasing with respect to s ?(-?,+?),and for any r>1,Ers(?)is decreasing with respect to s ?(-?,+?);(ii)For any s>0,Ers(?)is decreasing with respect to r ?(0,+?);(iii)For a product state pab,there are two real numbers a and b such that Irs(?ab)is increasing with respect to s ?[0,a]when r>1,and it is decreasing with respect to s ?[b,0]when 0<r<1;(iv)For a product state ?ab,Irs(?ab)is decreasing with respect to r ?(rs,+?)for each s>0,where rs=—max{as,bs},m>2 with m-21nm = 1 and tr?aas= tr?bbs = m-1/s(v)For any r>0,Ers(?)is convex with respect to s ?(-?,+?).