| The topic of quantum uncertainty relations and entanglement is one of the basic charac-teristics of quantum mechanics,they are the core problems of quantum information theory to generalize quantum uncertainty relations and build the entanglement c.riterion,and have been focused on by many scholars.In this paper,we devote to generalizing the existing uncertainty relations for pure states to mixed states,and constructing a class of entanglement criterion for multipartite Gaussian states based on uncertainty relations.The main results were as follows:1.To generalize the existing uncertainty relations for pure states to mixed states:(1)Characterization of the monotonicity of the standard deviation holds true for mixed states.Let A,B be observables and ρ is arbitrary quantum state,Then △ρ(A + B)<△ρ(A)+ △ρ(B).In particularly,noting that △ρ(tA)= t△ρ(A)for the nonnegative scalart,then we have △ρ(tA +(1-t)B)= t△ρ(A)+(1-t)△ρ(B)for anyt ∈[0,1].(2)Characterization the sum uncertainty relation for the mixed states.Let A,Bbe observables,△2(A)denotes the square of △(A)that is,the variance ofA,we have the uncer-tainty relations for mixed states:△ρ2(A)+△ρ2(B)≥±itr([A,B]ρ)-[tr[A(?)ib-<A>±i<B>)(?),and△ρ2(A)+ △ρ2+(B)≥1/2△ρ2(A+ B).(3)Characterization of the sum uncertainty relations induced by Monotonicity,and give the sum uncertainty relations for multi-observables.Let A,Bbe observables and ρ is arbitrary quantum states,we have the following inequality:△ρ2(A)+ △ρ2(B)≥△ρ2(A+ B)-2△ρ(A)△ρ(B).Let Ai be quantum observables,i=1,2,…,N,and p arbitrary a quantum state.we have △ρ2(∑i=1N Ai)≤∑i=1N△ρ2(Ai)+2∑i>j△ρ(Ai)△ρ(Aj).Noting that ∑i=1N△ρ2(Ai)≥2∑i>j△ρ(Ai)△ρ(Sj),so ∑i=1N△ρ2)Ai)≥1/2△ρ2(∑i=1N Ai).2.Characterization of entanglement eriterion independent on observables for multipartite Gaussian states based on uncertainty principle.Let ρ ∈ S(H1(?)H2…(?)Ha)with its covari-ance matrix Mρ=(mij)(2∑sj)×(2∑sj),ρ is fully separable,then for two set of arbitrary real numbers {αj(i)} and{βj(i)}(i=1,…,n and j=1,…si),Where(?)... |
