Bernoulli泛函上基于典则酉对合的量子熵.pdf

Pure Mathematics n,2023,13(8),2231-2239Published Online August 2023 in Hans.https:/www.hanspub.org/journal/pmhttps:/doi.org/10.12677/pm.2023.138229Bernoulliu;Kj?f?444)?“O?=vF2023c6?26FF2023c7?27FuF2023c8?3F f Bernoulli D(QBNs)uBernoulli m?O)fxv?;K?X(CAR)?O)f?fBernoulli m?gfBernoulli?;Kj?uBernoulli m?fm?;Kj?E?af?Tf?f?9f?eZ5cfBernoulli D(jf?Quantum Entropy Based onCanonical Unitary Involutionon Bernoulli FunctionalShengsheng LiuCollege of Mathematics and statistics,Northwest Normal University,Lanzhou GansuReceived:Jun.26th,2023;accepted:Jul.27th,2023;published:Aug.3rd,2023:4).Bernoulliu;Kj?f?J.n,2023,13(8):2231-2239.DOI:10.12677/pm.2023.1382294)AbstractQuantum Bernoulli noises(QBNs)are the family of annihilation and creation operatorsacting on the space of square integrable Bernoulli functional,which satisfy a canonicalanti-commutation relation(CAR)in equal time.The sum operator of annihilationand creation operator is a series of self-adjoint operator on Bernoulli functional space,which is called canonical unitary involution on Bernoulli functional.In this paper,based on the canonical unitary involution on the subspace of the Bernoulli functionalspace,we construct a class of density operators,and consider the quantum entropy ofthe density operator and some properties of the quantum entropy.KeywordsQuantum Bernoulli Noises,Unitary Involution,Quantum EntropyCopyright c?2023 by author(s)and Hans Publishers Inc.This work is licensed under the Creative Commons Attribution International License(CC BY 4.0).http:/creativecommons.org/licenses/by/4.0/1.f&Eny“n?:1,2,f?Kf&En?-,3nf?=k-?nd,?ykX2?Ac.VonNeumann?-?f?,5&EX?(5.f Bernoulli D(QBNs)uBernoulli m?O)fxk,kk0,v?;K?X(CAR),3mfXk-3.Cc5,QBNs?2 49.Pk=k+k(k 0),=Bernoulli?O)f?f,K?gf,Bernoulli?;Kj.z 10;Kj,y?fi?4n,z 11?;Kj?6,?da6fzf?fi?.?ulmfBernoulli D(,af?f?,LXe:?;Kjk,?E?Bernoulli mh?kfmhn?XeagDOI:10.12677/pm.2023.1382292232n4)fk+tI,0 k n,t 1,I hn?f,y?afkKA?,?hn?,af,dd?E?af,?af?f?9f?eZ5.?(?SXe:312,?ufBernoulli D(?,0?A-?n;13?,u;Kj?E?f?f?9eZ5.2.f Bernoulli D(3?!,/?f Bernoulli D(QBNs)?Vg,P9(.SNz 3.?N K?8,LN?k8,=|N#,#L8?.?=1,1NLkN?:N 7 1,1?8,(n)n0L3?;KKS?,zn 0,kn()=(n),.F=(n;n 0)dS?(n)n0)?;?(pn)n0?S?,0 pn 1,n 0.o3m(,F)3?VP,?P (n1,n2,nk)1(?1,?2,?k)=kYj=1p1+?j2njp1?j2nj,k 1,?j 1,1,nj N(1 j k)v:?i 6=j,ni6=nj.d?Vm(,F,P),Bernoulli m,Tm?EC Bernoulli.?Z=(Zn)n0CS?(n)n0)?Bernoulli,=Zn=n+qn pn2pnqn,n 0,qn=1 pn.w,Z=(Zn)n0Vm(,F,P)?C.?L2(,F,P)Z=(Zn)n0?Bernoulli?m,h L,=h=L2(,F,P).h,i k k Lh?S,?h,i u1C?5,u1?C5.dz 12,Z kbL5.KZ=Z|h?IO?(ONB),Z=1,DOI:10.12677/pm.2023.1382292233n4)Z=YjZj,6=.w,h?E Hilbert m.,uzn 0,Zn=Znh?;K ONB?.n 2.1 3?k 0,3h 3k.fk:h h,vk kk=1 kZ=1(k)Zk,kZ=1 1(k)Zk,kk?f,k=k,k=k,1(k)N?f8?5.fk?fkO3 Bernoulli m?fO)f.2.1 3?O)fxk,kk0f Bernoulli D(.enLf Bernoulli D(v?;K?X(CAR).n 2.2 3?k,l N,Ke?kl=lk,kl=lk,kl=lk,k 6=lkk=kk=0,kk+kk=I.I mh?f.PNn=0,1,2,n,nLNn?8,Khn=span Z|nh?fm.N?yhn?2n+1.n 2.3 3?k 0,Kk=k+kh?gjf,Bernoulli?;Kj.5?0 k n,hnkk?Cfm,l?khn?f.n 2.4 eZ|n hn?IO?,K12(Z+Zk)|n,k hn?IO?.y.,n,k T,w,12(Z+Zk)hn,?6=,kh12(Z+Zk),12(Z+Zk)i=0,?=,kDOI:10.12677/pm.2023.1382292234n4)h12(Z+Zk),12(Z+Zk)i=12?Z+Zk,Z+Zk?=12?hZ,Zi+?Zk,Zk?=1.L12(Z+Zk)|n,k hn?IO?.?n 2.5 n,k ,khn?fkA?1,1,A?A?OZ+ZkZ Zk.y.k ,dk?,kk?Z+Zk?=(k+k)Z+(k+k)Zk=Zk+Z,k?Z Zk?=(k+k)Z(k+k)Zk=Zk Z=(Z Zk).L1 1 fk?A?,?Z+ZkZ ZkA?A?.?n 2.6 0 k n,t R,k+tI hn?gfkA?t 1,t+1.y.dn2.5?.5 5?0 k n,t 1,k+tI?kA?K,(fk+tI?g5,k+tI hn?f,?,hn?,af.3.f?e?,af vTr()=1,K f.3f,f?.3!,Bernoulli mu;Kj?f?f?9eZ5,d,kf?.3.1 1 f?Von Neumann?S()=Tr(log),(3.1)?f?S(|0)=Tr(log)Tr(log0),6=0.(3.2)p?2.?.e0?-?n.n 3.1?n,k ,t 1,kTr(k+tI2n+1(1+t)=1.=k+tI2n+1(1+t)hn?f.DOI:10.12677/pm.2023.1382292235n4)y.L?O,kTr(k+tI)=Xnh12?Z+Zk?,(k+tI)12?Z+Zk)i=Xnh12(Z+Zk),(1+t)12?Z+Zk)i=(1+t)Xnh12(Z+Zk),12?Z+Zk)i=2n+1(1+t).XTr(k+tI2n+1(1+t)=1.?e,B,k=k+tI2n+1(1+t)d:Kj(?f,w,kgf,e?nd:Kj(?f?Von Neumann?.n 3.2 k Nn,t 1,d:Kj(?f?Von Neumann?S(k)=n+1 12n+1Tr?logk+tI1+t?,?f?S(k|j)=12n+1Tr(logk+tIj+tI),k 6=j.y.Von Neumann?9fk?g5,kS(k)=Tr?k+tI2n+1(1+t)logk+tI2n+1(1+t)?=n+1+12n+1S?k+tI1+t?,S?k+tI1+t?=Tr?k+tI1+tlogk+tI1+t?=Xn?12?Z+Zk?,k+tI1+tlogk+tI1+t12?Z+Zk?=Xn?k+tI1+t12?Z+Zk?,logk+tI1+t12?Z+Zk?=Xn?12(Z+Zk),logk+tI1+t12?Z+Zk?=Tr?logk+tI1+t?.l?S(k)=n+1 12n+1Tr?logk+tI1+t?.uf?5N?yTr(klogk)=12n+1Tr(logk),Tr(klogj)=12n+1Tr(logj).DOI:10.12677/pm.2023.1382292236n4)l?S(k|j)=Tr(klogk)Tr(klogj)=12n+1Tr(logk)12n+1Tr(logj)=12n+1Tr(logkj)=12n+1Tr(logk+tIj+tI).?e?d:Kj(?f?Von Neumann?eZ5.n 3.2 n,k,j ,k 6=j,kS(kj)=12n+1(S(k)+S(j).y.Von Neumann?9,$?5,kS(kj)=Tr(kjlogkj)=Tr(jklogk)Tr(kjlogj)=Xn?12?Z+Zj?,jklogk12?Z+Zj?Xn?12?Z+Zk?,kjlogj12?Z+Zk?=Xn?j12?Z+Zj),klogk12?Z+Zj?Xn?k12?Z+Zk?,jlogj12?Z+Zk)?=12n+1Xn?12?Z+Zj?,klogk12?Z+Zj?12n+1Xn?12?Z+Zk?,jlogj12?Z+Zk?=12n+1(S(k)+S(j).?3.3 n,k,j ,k 6=j,kS(kj)S(k)+S(j).n 3.4 n,k,j ,k 6=j,XJ p+q=1,oS(pk+qj)pS(k)+qS(j).y.?(x)=xlogx,v(px+qy)p(x)+q(y),ep+q=1,d,DOI:10.12677/pm.2023.1382292237n4)kS(pk+qj)=Tr(pk+qj)logp(k+qj)=Tr(pk+qj)Trp(k)+q(j)=pS(k)+qS(j).?z1 Nielsen,M.A.and Chuang,I.L.(2000)Quantum Computation and Quantum Information.Cambridge University Press,Cambridge.2 Watrous,J.(2018)Theory of