Optimizing pointer states for dynamical evolution of quantum correlations under decoherence Bo You, Li-Xiang Cen Department of Physics, SiChuan University
Optimizing pointer states for dynamical evolution of quantum correlations under decoherence Bo You,Li-xiang Cen Department of Physics, SiChuan University
Outline 1. Dissipative channels and pointer bases 2. Two different optimizations in relation to quantum discord 2. 1. Bell-diagonal state 2.2. two-qubit state of two ranks 3. Optimizing pointer bases to achieve maximal condition entropy
Outline 1.Dissipative channels and pointer bases 2. Two different optimizations in relation to quantum discord 2.1.Bell-diagonal state 2.2.two-qubit state of two ranks 3. Optimizing pointer bases to achieve maximal condition entropy
Dissipative channels and pointer bases .o Dynamical evolution under dissipative channel E(p()=∑E(O)p(0)E2() In this case of dephasing channel E1=m+)(+|+ypn)(n,E,= Pn-)(n2-| e o Pointer bases: selected by the system-environment coupling COS +e sin n=sin e cos 202
Dissipative channels and pointer bases ❖ Dynamical evolution under dissipative channel ❖ In this case of dephasing channel ❖ Pointer bases: selected by the system-environment coupling 1 2 , 1 , 1 .t E n n p n n E p n n p e + + − − − − − = + = − = − ( ) † ( ) ( ) (0) ( ) i i i = t E t E t cos 0 sin 1 2 2 sin 0 cos 1 . 2 2 i i n e n e + − = + = −
≈ Quantum discord: OD (PAB)=S(P)+min p s(pBliA-S(PAB Where S(p)=-Tr(plog2 p o Geometry measure of Quantum Discord min AB AB ZAB O e Where 2o is the set of zero-discord Entanglement of formation E(p)=min、E(v) (Pi 4 s where E(AB))=S(pA) H Ollivier and W.H. Zurek, PhysRevLett 88.017901 B. Dakic, V Vedral, and C. Brukner, Phys. Rev. Lett. 105, 190502
❖ Quantum discord: ❖ Where ❖ Geometry measure of Quantum Discord: ❖ Where is the set of zero-discord. ❖ Entanglement of formation: ❖ where ( ) ( ) min ( ) ( ) A i A A AB A i B A AB i QD S p S i S = + − S Tr ( ) = − ( log2 ) ( ) 0 2 min AB DA AB AB AB = − 0 H. Ollivier and W. H. Zurek, PhysRevLett.88.017901 B.Dakic,V.Vedral, and C. Brukner,Phys. Rev. Lett. 105, 190502 ( ) ( ) , min i i AB i AB i AB p E p E = ( ) ( ) i i E S AB A =
Schematic illustration: dependence of decoherence dynamics on pointer bases pAB=plv)(v1+(1-p)v2)(v2|12)=2(00±1) a Discord A 20 0.006 0004 0.000 0.00.5 01.520 b Information C EoF 1.0 2.0 0.8 0.6 0010 0.4 0.005 000.5 0.000 P15 2.0 0.00.5 1.5 2.0
Schematic illustration: dependence of decoherence dynamics on pointer bases
Motivation and main concerns go We investigate the dynamics of decorrelation under different choices of the pointer states of system reservoir couplings. In detail, we consider a two qubit system, initially prepared in certain states with non-zero quantum correlations, subjected to local dissipative channels responsible for various pointer states. Dynamical evolution of entanglement, quantum discord, and the mutual information sharing between the two qubits, is depicted. We elucidate various optimizations of the pointer states e.g., minimization and maximization of the conditional entropy, as well as the geometric optimization via minimizing discord, and analyze the properties of the corresponding behavior of decorrelation
❖ We investigate the dynamics of decorrelation under different choices of the pointer states of systemreservoir couplings. In detail, we consider a twoqubit system,initially prepared in certain states with non-zero quantum correlations, subjected to local dissipative channels responsible for various pointer states. Dynamical evolution of entanglement, quantum discord,and the mutual information sharing between the two qubits, is depicted. We elucidate various optimizations of the pointer states, e.g., minimization and maximization of the conditional entropy, as well as the geometric optimization via minimizing discord,and analyze the properties of the corresponding behavior of decorrelation. Motivation and main concerns
Case 1: Two different optimizations in relation to quantum discord It is proved that quantum discord and its geometry measure can be rewritten as OD, (PAB)=S(PA)+minXp s(p8lia-S(PAB AB PAB DA(PAB)=min PAB -xaB AB AB Where IIi is the projector state achieving the minimize herefore we could choose the two projector states as our optimal bases, so that we could get some physics about dissipate process with different optimizations S.L. Luo andss Fu, PhysRevA. 82.034302
Case 1: Two different optimizations in relation to quantum discord It is proved that quantum discord and its geometry measure can be rewritten as: Where is the projector state achieving the minimize. S.L. Luo and S.S. Fu,PhysRevA.82.034302 A i ( ) ( ) ( ) ( ) ( ) ( ) min A i A A AB A i B A AB i A A i i AB AB QD S p S i S I I = + − = − Therefore we could choose the two projector states as our optimal bases,so that we could get some physics about dissipate process with different optimizations. ( ) 0 2 2 min AB A AB AB AB A A i i AB AB D = − = −
Bell-diagonal states g We can express the state as o For Bell-diagonal state, the measurements for minimizing the quantum discord and geometry measure of QD can be easily calculated, which are the same that is (±a)/2,|=max() I±a,)/2 maX lc (±a3)/2,!=max(c Thus. the dynamic with the two optimizations is the same for Bell-diagonal states S.L. Luo, Phys. Rev. A77, 042303(2008)
Bell-diagonal states ❖ We can express the state as: ❖ For Bell-diagonal state, the measurements for minimizing the quantum discord and geometry measure of QD can be easily calculated, which are the same, that is: S.L.Luo,Phys. Rev. A 77, 042303 (2008) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 3 3 / 2, max / 2, max / 2, max i A i i I c c I c c I c c = = = = Thus, the dynamic with the two optimizations is the same for Bell-diagonal states
Two qubits state of two ranks The state express as PAC=|1)(1|+|2)(q2 Where 1 1)=Ao100 2}=A1ep0o)+A2|01+Aa210)+A4|1 o The minimizing discord can be obtained by purifying the state to a three-qubit pure state ABC)=A0|000+A1e010+A211) +Aa2110+1|11 Go In the state, the minimal conditional entropy of Ac system is equal to the entanglement of BC-system E(PBc)=min S(pcl L.X. Cen, et al., Phys. Rev. A 83, 054101(2011
❖ The state express as ❖ Where ❖ The minimizing discord can be obtained by purifying the state to a three-qubit pure state: ❖ In the state, the minimal conditional entropy of ACsystem is equal to the entanglement of BC-system: ( ) min ( ) BC C A A E S i = L.X.Cen, et.al., Phys. Rev. A 83, 054101 (2011) Two qubits state of two ranks
whErefore, the minimizing conditional entropy of Ac system could be obtained by calculating the EoF of BC- system Thus through the method conducted by l= Wootters(phys. rev. let 80.2245), the decomposition achieving the EoF can be deduced oo So the measurement corresponding to minimizing conditional entropy can be determined by 平A)=)= 令 Thus the measurement is:∏4=1(±e:o) , where e is the normalized vector and 0 2入2(X2+ =12×2(1c089+2入4) e3
❖ Therefore, the minimizing conditional entropy of ACsystem could be obtained by calculating the EoF of BCsystem. ❖ Thus through the method conducted by Wootters(phys.rev.let.80.2245),the decomposition achieving the EoF can be deduced. ❖ So the measurement corresponding to minimizing conditional entropy can be determined by: ❖ Thus the measurement is: ❖ where is the normalized vector , and i BC z i ABC A BC i = i z ( ) 1 2 A = I e e
