中国科学院：从量子信息观点看量子统计和热力学（理论物理研究所：孙昌璞）

从量子信息观点看量子统计和热力学 孙昌璞 中国科学院理论物理研究所 http:/power.itp.ac.cn/-suncp/quantum.htm ITP

从量子信息观点看量子统计和热力学 孙昌璞 中国科学院理论物理研究所 http://power.itp.ac.cn/~suncp/quantum.htm

1量子纠缠与量子统计力学的基础 Temperature and Quantum Phase transition emerge due to entanglement H. Dong, S. Yang, X F Liu,, CPS, 2007 H. Quan, Z Song, X Liu, P. Zanardi, CPs, Phys. Rev. Lett. 96, 140604 (2006) Y B. Gao and CPS, Phys. Rev. E75, 011105 (2007) 2信息擦除、麦克斯韦妖与量子热机 H. Quan, Y Wang, Y Liu, CPS, F Nori Phys. Rev. Lett. 97, 180402(2006) H. T Quan, P. Zhang, CPS, Phys. Rev. E 73, 036122(2006) 3.量子控制、纳米机械及其它 P. Zhang, Y Wang, CPS, Phys. Rev. Lett. 95, 097204(2005) F Xue, L. Zhong, Yong Li, and CPS, Phys. Rev. B75, 033407(2007) ITP

1.量子纠缠与量子统计力学的基础 2.信息擦除、麦克斯韦妖与量子热机 3. 量子控制、纳米机械及其它 Temperature and Quantum Phase transition emerge due to entanglement H. Dong, S. Yang, X.F. Liu, , CPS , 2007 H. Quan, Y. Wang, Y Liu, CPS, F. Nori Phys. Rev. Lett. 97, 180402(2006) H. Quan, Z. Song, X. Liu, P. Zanardi, CPS , Phys. Rev. Lett. 96, 140604 (2006) Y. B. Gao and CPS, Phys. Rev. E 75, 011105 (2007) H. T. Quan, P. Zhang, CPS, Phys. Rev. E 73, 036122 (2006) F. Xue, L. Zhong, Yong Li, and CPS, Phys. Rev. B 75, 033407 (2007) P. Zhang, Y. Wang, CPS, Phys. Rev. Lett . 95, 097204 (2005 )

Quantum Physics and Quantum Information Processing QPQIP in ITP CAS Previous members Dr. Y.X. Liu,(Riken) Dr. P. Zhang,(Georgia Tech) Dr. Yong Li,(Basel) Dr. Y D. Wang, (NTT) Dr F. Xue,(Riken) Dr Y.B. Gao (BPTU) Prof. Sixia Yu(UCST) Prof. xi ng Main Collaborators Present Members Dr Lan zhou Prof Zi Song(Nankai), Hai-Tao Quan Prof. Xue-Feng Liu(PKU) Liang He, Zi-Ri Gong Hui Dong, Tao Shi Prof. Franco Nori(UM, Riken) Shuo Yang, Jing-Ning Zhang Prof. Paolo Zanardi (Isi) Prof. su yi Prof. Chang-Pu Sun Prof J.Q. You(Fudan) ITP Prof. Li You(Georgia Tech)

Main Collaborators: Prof. Zi Song (Nankai), Prof. Xue-Feng Liu (PKU), Prof. Franco Nori (UM , Riken) Prof. Paolo Zanardi (ISI), Prof. J.Q. You (Fudan) Prof. Li You (Georgia Tech) Present Members Dr. Lan Zhou Hai-Tao Quan Liang He, Zi-Ri Gong Hui Dong, Tao Shi Shuo Yang, Jing-Ning Zhang Prof. Su YI Prof. Chang-Pu Sun QPQIP in ITP CAS Quantum Physics and Quantum Information Processing Previous Members: Dr. Y.X. Liu, (Riken) Dr. P. Zhang, (Georgia Tech) Dr. Yong Li , (Basel) Dr. Y. D. Wang, (NTT) Dr. F. Xue, (Riken) Dr. Y.B. Gao (BPTU) Prof. Sixia Yu (UCST) Prof. Xiaoguang Wang (ZJU )

等几率假设 统计力学 微正则、正则系综 等几率假设 ITP

等几率假设 等几率假设 统 计 力 学 微正则、正则系综 ？

统计物理基础 平衡态情况 等几率假说 对应于相同的宏观量(能量E)的Ω2(微观态中每 个态出现的几率相等 E P E 2(E,δ EsEn+∑s≤E+O Energy shell ITP

1 ( , ) P E  =  统计物理基础 平衡态情况 等几率假说 对应于相同的宏观量（能量 E）的 微观态中每 个态出现的几率相等 ( , ) E  E  E + 1 N n j j E E E   =   + +  Energy Shell

微正则系综密度矩阵 Sub Energy shell 热库的约束 E+6 E-E E-E E-E≤ e-e+ DOZ,恩 m 4tn,> N 4D0,.⑧ ITP

E +  E,   1 DN1E,   n,nj→E, |nn|  j1 N |njnj |  1 DN1E,   n |nn|  nj→En  j1 N |njnj | 1 N n j n j E E E E   = −   − +  Sub Energy Shell 微正则系综密度矩阵 热库的约束 E E n − + E E E − n

微正则系综刚系综 Ps=T(p)=∑Im|p(E,)m D入(E,δ) ∑|nxn1∑∑Is(m,n) im,AIn,IE -nod J=I D(E,8) ∑|nn|∑ Im, lE-nos D、(E-m0o)1m(n=∑P(o)1m(n Dv cE,8 统计熵 S(E): =In[D(E, S) ITP

{ } 1 1 { }[ ] 1 1 [ ] 1 ( ) | ( , ) | 1 | | ( , ) ( , ) 1 | | 1 ( , ) ( , ) | | ( ) | | ( , ) j j j E n j E n N S E j j m j N j j N n m n j N n m N n n n N Tr m E m n n m n D E n n D E D E n n n P n n D E              − − = + = + + =   =  =  − =    =         微正则系综 正则系综 ( ) : ln[ ( , )] N S E D E =  统计熵

热力学极限 D(E,S)DN+(E,8 nO<E DN(E-no,8) S(E-nO=S(E) IS(E) D+1(E,) no dE DME-no,8) S(E)-Bna D+1(E,) noo B dS(e dE P(o)≈e ITP

热力学极限 ( ) ( ) ( ) ( ) dS E S E n S E n dE S E n     − = − = − dS E( ) dE  = n E  1 ( , ) ( , ) D E D E N N    + 1 1 1 ( , ) ( ) ( , ) ( , ) ( , ) N n N N N n D E n P D E D E n D E e          + + + − − = − =  ( ) n P e n    − 

广义热化定理 Generalized thermalization Almost all the pure states of the"Universe can give the Canonic Equilibrium State by tracing over the Environment S Popescu et al, Nature Physics 2, 754(2006) S Goldstein et al phys. Rev. Lett. 96, 050403(2006) C(n, n) E>=∑ n)8∏ (E, =1 =m(甲5X5aD=∑ D、(E-m,6) InXi D31(E,6) ITP Based on The Law of Large Numbers(大数定律)

广义热化定理 Almost all the pure states of the “Universe ” can give the Canonic Equilibrium State by tracing over the Environment S. Popescu et al , Nature Physics 2, 754 (2006) S. Goldstein et al., Phys. Rev. Lett. 96, 050403 (2006) , , [ , ] 1 1 ( , ) | | | ( , ) j E N j E j n n j N C n n n n  D E  +  =   =      , , 1 ( , ) (| |) | | ( , ) N S B E E n N D E n Tr n n D E       + − =   =   Based on The Law of Large Numbers (大数定律) Generalized Thermalization

大数定律 Law of large numbers 当随机事件发生的次数很大时,偶然性会互相抵消。 使这些事件的结果的算术平均值在概率意义下十分接 近其数学期望或“真实值” 大数定律的不同表述: Cx2==∑X ITP

当随机事件发生的次数很大时，偶然性会互相抵消。 使这些事件的结果的算术平均值在概率意义下十分接 近其数学期望或 “真实值”. 大数定律的不同表述: 弱大数定律(1) 伯努利大数定律(2)、 辛钦-马尔可夫大数定律(3) 大数定律 Law of large numbers 大数定律的一个推论 1, 2, , :{ ...., } U X X X N , , , ( ) :{ ...., } i j k S U X X X  给定的一个大的足够随机的数集合 对于其有限子集 , 1 S U U j U j X X X X N   =    = 