Section 2.5 Renyi entropy and entanglement entropy

S_n(L,ϕ) and S(L,ϕ)

SnLϕ=((c (1+n) Log[ℓ])/(12 n))-((((c-24 hϕ) (1+n) (π^2)) (ℓ^2))/(72 ((L^2) n)))-((((1+n) ((c^2) (n^2)-24 c hϕ (1+(n^2))+48 ((hϕ)^2) (11+(n^2))) (π^4)) (ℓ^4))/(2160 (c (L^4) (n^3))))-(1/(34020 ((c^2) (L^6) (n^5))))((1+n) ((c^3) (n^4)+96 ((hϕ)^3) (-2+n) (2+n) (47+(n^2))-24 (c^2) hϕ (1+(n^2)+(n^4))+36 c ((hϕ)^2) (37+9 (n^2)+2 (n^4))) (π^6)) (ℓ^6)-(1/(453600 ((c^3) (22+5 c) (L^8) (n^7))))((1+n) (5 (c^5) (n^6)+5632 ((hϕ)^4) (-3+n) (-2+n) (2+n) (3+n) (119+(n^2))+64 (c^2) ((hϕ)^2) (1078-5863 hϕ+(268+1227 hϕ) (n^2)+(100+303 hϕ) (n^4)+(30+13 hϕ) (n^6))-2 (c^4) (-11 (n^6)+60 hϕ (1+(n^2)) (1+(n^4)))+48 (c^3) hϕ (-11 (1+(n^2)) (1+(n^4))+hϕ (251+71 (n^2)+29 (n^4)+9 (n^6)))-64 c ((hϕ)^3) (-44 (-2+n) (2+n) (11+(n^2)) (19+(n^2))+hϕ (-58213+33927 (n^2)-1647 (n^4)+13 (n^6)))) (π^8)) (ℓ^8)+((O[ℓ])^10);

SLϕ=(1/6) c Log[ℓ]-((((c-24 hϕ) (π^2)) (ℓ^2))/(36 (L^2)))-((((((c-24 hϕ))^2) (π^4)) (ℓ^4))/(1080 (c (L^4))))-((((((c-24 hϕ))^3) (π^6)) (ℓ^6))/(17010 ((c^2) (L^6))))-((((c^4) (22+5 c)-96 (c^3) (22+5 c) hϕ+1152 (c^2) (82+15 c) ((hϕ)^2)-92160 c (22+3 c) ((hϕ)^3)+184320 (88+9 c) ((hϕ)^4)) (π^8)) (ℓ^8))/(226800 ((c^3) (22+5 c) (L^8)))+((O[ℓ])^10);

S_n(L,q) and S(L,q)

SnLq=SnLLq+SnNLLq+SnNNLLq;

SnLLq=((c (1+n) Log[ℓ])/(12 n))-(((c (1+n) (π^2)) (ℓ^2))/(72 ((L^2) n)))+(-((c (1+n) (π^4))/(2160 ((L^4) n)))-(((c (-1+n) (((1+n))^2) (π^4)) (q^2))/(18 ((L^4) (n^3))))-((2 (c (-1+n) (((1+n))^2) (π^4)) (q^3))/(9 ((L^4) (n^3))))-((11 (c (-1+n) (((1+n))^2) (π^4)) (q^4))/(18 ((L^4) (n^3))))+((O[q])^5)) (ℓ^4)+(-((c (1+n) (π^6))/(34020 ((L^6) n)))+((c (-1+n) (((1+n))^2) (π^6) (q^2))/(27 (L^6) (n^3)))+((10 c (-1+n) (((1+n))^2) (π^6) (q^3))/(27 (L^6) (n^3)))+((c (1+n) (1-49 (n^2)+48 (n^4)) (π^6) (q^4))/(27 (L^6) (n^5)))+((O[q])^5)) (ℓ^6)+(-((c (1+n) (π^8))/(453600 ((L^8) n)))-(((c (-1+n) (((1+n))^2) (π^8)) (q^2))/(90 ((L^8) (n^3))))-((4 (c (-1+n) (((1+n))^2) (π^8)) (q^3))/(15 ((L^8) (n^3))))-(((c (-1+n) (((1+n))^2) (-11-62 (n^2)+3727 (n^4)) (π^8)) (q^4))/(1620 ((L^8) (n^7))))+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

SnNLLq=(((2 (1+n) (π^2) (q^2))/(3 (L^2) n))+(((1+n) (π^2) (q^3))/((L^2) n))+((2 (1+n) (π^2) (q^4))/((L^2) n))+((O[q])^5)) (ℓ^2)+(-((((1+n) (-11+9 (n^2)) (π^4)) (q^2))/(45 ((L^4) (n^3))))-((((1+n) (-44+41 (n^2)) (π^4)) (q^3))/(45 ((L^4) (n^3))))-((((1+n) (-51+49 (n^2)) (π^4)) (q^4))/(15 ((L^4) (n^3))))+((O[q])^5)) (ℓ^4)+(((2 (1+n) (31-46 (n^2)+17 (n^4)) (π^6) (q^2))/(945 (L^6) (n^5)))+(((1+n) (527-1013 (n^2)+492 (n^4)) (π^6) (q^3))/(945 (L^6) (n^5)))+((2 (1+n) (1255-2903 (n^2)+1654 (n^4)) (π^6) (q^4))/(945 (L^6) (n^5)))+((O[q])^5)) (ℓ^6)+(-((((1+n) (-205+415 (n^2)-278 (n^4)+62 (n^6)) (π^8)) (q^2))/(14175 ((L^8) (n^7))))-((((1+n) (-1025+2850 (n^2)-2694 (n^4)+866 (n^6)) (π^8)) (q^3))/(4725 ((L^8) (n^7))))-((((1+n) (-36257+139223 (n^2)-163681 (n^4)+66439 (n^6)) (π^8)) (q^4))/(28350 ((L^8) (n^7))))+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

SnNNLLq=(-((4 ((1+n) (11+(n^2)) (π^4)) (q^4))/(45 (c (L^4) (n^3))))+((O[q])^5)) (ℓ^4)+(((4 (1+n) (-345+271 (n^2)+26 (n^4)) (π^6) (q^4))/(945 c (L^6) (n^5)))+((O[q])^5)) (ℓ^6)+(-((8 ((1+n) (3398-3017 (n^2)+1141 (n^4)+116 (n^6)) (π^8)) (q^4))/(14175 (c (L^8) (n^7))))+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

SLq=(1/6) c Log[ℓ]+(-((c (π^2))/(36 (L^2)))+((4 (π^2) (q^2))/(3 (L^2)))+((2 (π^2) (q^3))/(L^2))+((4 (π^2) (q^4))/(L^2))+((O[q])^5)) (ℓ^2)+(-((c (π^4))/(1080 (L^4)))+((4 (π^4) (q^2))/(45 (L^4)))+((2 (π^4) (q^3))/(15 (L^4)))+((4 (-8+c) (π^4) (q^4))/(15 c (L^4)))+((O[q])^5)) (ℓ^4)+(-((c (π^6))/(17010 (L^6)))+((8 (π^6) (q^2))/(945 (L^6)))+((4 (π^6) (q^3))/(315 (L^6)))+((8 (-16+c) (π^6) (q^4))/(315 c (L^6)))+((O[q])^5)) (ℓ^6)+(-((c (π^8))/(226800 (L^8)))+((4 (π^8) (q^2))/(4725 (L^8)))+((2 (π^8) (q^3))/(1575 (L^8)))-((4 ((728+159 c) (π^8)) (q^4))/(1575 (c (L^8))))+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

Section 3.1 Relative entropy

S(ρ_A(L1,ϕ1)||ρ_A(L2,ϕ2))

SL1ϕ1L2ϕ2=((((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)))^2) (π^4) (ℓ^4))/(1080 c (L1^4) (L2^4)))+((((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)))^2) (2 (c-24 hϕ2) (L1^2)+(c-24 hϕ1) (L2^2)) (π^6) (ℓ^6))/(17010 (c^2) (L1^6) (L2^6)))+(1/(226800 (c^3) (22+5 c) (L1^8) (L2^8)))((3 (c^4) (22+5 c)-352 (c^3) (22+5 c) hϕ2+512 (c^2) (674+145 c) ((hϕ2)^2)-1351680 c (5+c) ((hϕ2)^3)+184320 (264+49 c) ((hϕ2)^4)) (L1^8)-4 (22+5 c) (c-24 hϕ1) (c-24 hϕ2) ((c^2)-80 c hϕ2+1280 ((hϕ2)^2)) (L1^6) (L2^2)-64 c (c (22+5 c)-240 (2+c) hϕ1+2880 ((hϕ1)^2)) (c-22 hϕ2) hϕ2 (L1^4) (L2^4)+((c^4) (22+5 c)-96 (c^3) (22+5 c) hϕ1+1152 (c^2) (82+15 c) ((hϕ1)^2)-92160 c (22+3 c) ((hϕ1)^3)+184320 (88+9 c) ((hϕ1)^4)) (L2^8)) (π^8) (ℓ^8)+((O[ℓ])^10);

S(ρ_A(L1,ϕ)||ρ_A(L2,q))

SL1ϕL2q=((((((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^4))/(1080 c (L1^4) (L2^4)))-((4 ((24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^4)) (q^2))/(45 (c (L1^2) (L2^4))))-((2 ((24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^4)) (q^3))/(15 (c (L1^2) (L2^4))))+((4 (-(-8+c) (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^4))/(15 c (L1^2) (L2^4)))+((O[q])^5)) (ℓ^4)+((((2 c (L1^2)+(c-24 hϕ) (L2^2)) (((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^6))/(17010 (c^2) (L1^6) (L2^6)))-((16 ((24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^6)) (q^2))/(945 (c (L1^2) (L2^6))))-((8 ((24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^6)) (q^3))/(315 (c (L1^2) (L2^6))))-((16 (((-16+c) c (L1^2)-(-8+c) (c-24 hϕ) (L2^2)) (π^6)) (q^4))/(315 ((c^2) (L1^2) (L2^6))))+((O[q])^5)) (ℓ^6)+((1/(226800 (c^3) (22+5 c) (L1^8) (L2^8)))(3 (c^4) (22+5 c) (L1^8)-4 (c^3) (22+5 c) (c-24 hϕ) (L1^6) (L2^2)+((c^4) (22+5 c)-96 (c^3) (22+5 c) hϕ+1152 (c^2) (82+15 c) ((hϕ)^2)-92160 c (22+3 c) ((hϕ)^3)+184320 (88+9 c) ((hϕ)^4)) (L2^8)) (π^8)-((4 ((c (31+5 c) (L1^4)-(53+10 c) (c-24 hϕ) (L1^2) (L2^2)+(c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L2^4)) (π^8)) (q^2))/(14175 (c (L1^4) (L2^8))))-((2 ((c (203+40 c) (L1^4)-(379+80 c) (c-24 hϕ) (L1^2) (L2^2)+8 (c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L2^4)) (π^8)) (q^3))/(14175 (c (L1^4) (L2^8))))-(1/(14175 ((c^2) (22+5 c) (L1^4) (L2^8))))4 ((c (22+5 c) (-9184+c (-1587+55 c)) (L1^4)-(22+5 c) (-2720+c (159+110 c)) (c-24 hϕ) (L1^2) (L2^2)+(-88+c (306+55 c)) (c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L2^4)) (π^8)) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

S(ρ_A(L1,q)||ρ_A(L2,ϕ))

SL1qL2ϕ=(((((((c-24 hϕ) (L1^2)-c (L2^2)))^2) (π^4))/(1080 c (L1^4) (L2^4)))+((4 ((c-24 hϕ) (L1^2)-c (L2^2)) (π^4) (q^2))/(45 c (L1^4) (L2^2)))+((2 ((c-24 hϕ) (L1^2)-c (L2^2)) (π^4) (q^3))/(15 c (L1^4) (L2^2)))+((4 ((c-24 hϕ) (L1^2)-(-8+c) (L2^2)) (π^4) (q^4))/(15 c (L1^4) (L2^2)))+((O[q])^5)) (ℓ^4)+(((((((c-24 hϕ) (L1^2)-c (L2^2)))^2) (2 (c-24 hϕ) (L1^2)+c (L2^2)) (π^6))/(17010 (c^2) (L1^6) (L2^6)))+((8 ((((c-24 hϕ))^2) (L1^4)-(c^2) (L2^4)) (π^6) (q^2))/(945 (c^2) (L1^6) (L2^4)))+((4 ((((c-24 hϕ))^2) (L1^4)-(c^2) (L2^4)) (π^6) (q^3))/(315 (c^2) (L1^6) (L2^4)))+((8 ((((c-24 hϕ))^2) (L1^4)-(-16+c) c (L2^4)) (π^6) (q^4))/(315 (c^2) (L1^6) (L2^4)))+((O[q])^5)) (ℓ^6)+((1/(226800 (c^3) (22+5 c) (L1^8) (L2^8)))((3 (c^4) (22+5 c)-352 (c^3) (22+5 c) hϕ+512 (c^2) (674+145 c) ((hϕ)^2)-1351680 c (5+c) ((hϕ)^3)+184320 (264+49 c) ((hϕ)^4)) (L1^8)-4 c (22+5 c) (c-24 hϕ) ((c^2)-80 c hϕ+1280 ((hϕ)^2)) (L1^6) (L2^2)-64 (c^2) (22+5 c) (c-22 hϕ) hϕ (L1^4) (L2^4)+(c^4) (22+5 c) (L2^8)) (π^8)+((4 ((c-24 hϕ) ((c^2)-80 c hϕ+1280 ((hϕ)^2)) (L1^6)-160 c (c-22 hϕ) hϕ (L1^4) (L2^2)-(c^3) (L2^6)) (π^8) (q^2))/(4725 (c^3) (L1^8) (L2^6)))+((2 ((c-24 hϕ) ((c^2)-80 c hϕ+1280 ((hϕ)^2)) (L1^6)-480 c (c-22 hϕ) hϕ (L1^4) (L2^2)-(c^3) (L2^6)) (π^8) (q^3))/(1575 (c^3) (L1^8) (L2^6)))+((4 ((22+5 c) (c-24 hϕ) ((c^2)-80 c hϕ+1280 ((hϕ)^2)) (L1^6)-3360 c (6+c) (c-22 hϕ) hϕ (L1^4) (L2^2)+(c^2) (22+5 c) (728+159 c) (L2^6)) (π^8) (q^4))/(1575 (c^3) (22+5 c) (L1^8) (L2^6)))+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

S(ρ_A(L1,q1)||ρ_A(L2,q2))

SL1q1L2q2=(((c ((((L1^2)-(L2^2)))^2) (π^4))/(1080 (L1^4) (L2^4)))-((4 ((L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^2)+(L1^2) (q2^2))) (q^2))/(45 ((L1^4) (L2^4))))-((2 ((L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^3)+(L1^2) (q2^3))) (q^3))/(15 ((L1^4) (L2^4))))+((4 (π^4) (8 ((((L2^2) (q1^2)-(L1^2) (q2^2)))^2)+c (L1-L2) (L1+L2) ((L2^2) (q1^4)-(L1^2) (q2^4))) (q^4))/(15 c (L1^4) (L2^4)))+((O[q])^5)) (ℓ^4)+(((c (2 (L1^6)-3 (L1^4) (L2^2)+(L2^6)) (π^6))/(17010 (L1^6) (L2^6)))-((8 ((π^6) ((L2^6) (q1^2)+2 (L1^6) (q2^2)-(L1^4) (L2^2) ((q1^2)+2 (q2^2)))) (q^2))/(945 ((L1^6) (L2^6))))-((4 ((π^6) ((L2^6) (q1^3)+2 (L1^6) (q2^3)-(L1^4) (L2^2) ((q1^3)+2 (q2^3)))) (q^3))/(315 ((L1^6) (L2^6))))+((8 (π^6) ((L2^2) (c (L1^4)-(-16+c) (L2^4)) (q1^4)-32 (L1^4) (L2^2) (q1^2) (q2^2)+2 (L1^4) (-(-16+c) (L1^2)+(-8+c) (L2^2)) (q2^4)) (q^4))/(315 c (L1^6) (L2^6)))+((O[q])^5)) (ℓ^6)+(((c (3 (L1^8)-4 (L1^6) (L2^2)+(L2^8)) (π^8))/(226800 (L1^8) (L2^8)))+((4 (π^8) (3 (L2^2) ((L1^6)-(L2^6)) (q1^2)-(L1^4) (L1-L2) (L1+L2) ((31+5 c) (L1^2)-(22+5 c) (L2^2)) (q2^2)) (q^2))/(14175 (L1^8) (L2^8)))+((2 (π^8) (9 (L2^2) ((L1^6)-(L2^6)) (q1^3)-(L1^4) (L1-L2) (L1+L2) ((203+40 c) (L1^2)-8 (22+5 c) (L2^2)) (q2^3)) (q^3))/(14175 (L1^8) (L2^8)))+(1/(14175 c (L1^8) (L2^8)))4 (π^8) (9 (L2^2) (c (L1^6)+(728+159 c) (L2^6)) (q1^4)-48 (L1^4) (L2^2) ((53+10 c) (L1^2)+10 (22+5 c) (L2^2)) (q1^2) (q2^2)+(L1^4) ((9184+(1587-55 c) c) (L1^4)+(-2720+c (159+110 c)) (L1^2) (L2^2)+(88-c (306+55 c)) (L2^4)) (q2^4)) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

Section 3.1 Symmetrized relative entropy

S(ρ_A(L1,ϕ1),ρ_A(L2,ϕ2))

SL1ϕ1L2ϕ2=((((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)))^2) (π^4) (ℓ^4))/(540 c (L1^4) (L2^4)))+((((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)))^2) ((c-24 hϕ2) (L1^2)+(c-24 hϕ1) (L2^2)) (π^6) (ℓ^6))/(5670 (c^2) (L1^6) (L2^6)))+(1/(56700 (c^3) (22+5 c) (L1^8) (L2^8)))(22 (c^4) (L1^8)+5 (c^5) (L1^8)-2464 (c^3) hϕ2 (L1^8)-560 (c^4) hϕ2 (L1^8)+109888 (c^2) ((hϕ2)^2) (L1^8)+22880 (c^3) ((hϕ2)^2) (L1^8)-2196480 c ((hϕ2)^3) (L1^8)-407040 (c^2) ((hϕ2)^3) (L1^8)+16220160 ((hϕ2)^4) (L1^8)+2672640 c ((hϕ2)^4) (L1^8)-(22+5 c) (c-24 hϕ1) (c-24 hϕ2) ((c^2)-80 c hϕ2+1280 ((hϕ2)^2)) (L1^6) (L2^2)-16 c (c (22+5 c)-240 (2+c) hϕ1+2880 ((hϕ1)^2)) (c-22 hϕ2) hϕ2 (L1^4) (L2^4)-16 c (c-22 hϕ1) hϕ1 (c (22+5 c)-240 (2+c) hϕ2+2880 ((hϕ2)^2)) (L1^4) (L2^4)-(22+5 c) (c-24 hϕ1) ((c^2)-80 c hϕ1+1280 ((hϕ1)^2)) (c-24 hϕ2) (L1^2) (L2^6)+22 (c^4) (L2^8)+5 (c^5) (L2^8)-2464 (c^3) hϕ1 (L2^8)-560 (c^4) hϕ1 (L2^8)+109888 (c^2) ((hϕ1)^2) (L2^8)+22880 (c^3) ((hϕ1)^2) (L2^8)-2196480 c ((hϕ1)^3) (L2^8)-407040 (c^2) ((hϕ1)^3) (L2^8)+16220160 ((hϕ1)^4) (L2^8)+2672640 c ((hϕ1)^4) (L2^8)) (π^8) (ℓ^8)+((O[ℓ])^10);

S(ρ_A(L1,ϕ),ρ_A(L2,q))

SL1ϕL2q=((((((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^4))/(540 c (L1^4) (L2^4)))-((8 ((24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^4)) (q^2))/(45 (c (L1^2) (L2^4))))-((4 ((24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^4)) (q^3))/(15 (c (L1^2) (L2^4))))+((8 (-(-8+c) (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^4))/(15 c (L1^2) (L2^4)))+((O[q])^5)) (ℓ^4)+((((c (L1^2)+(c-24 hϕ) (L2^2)) (((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^6))/(5670 (c^2) (L1^6) (L2^6)))-((8 ((3 c (L1^2)+(c-24 hϕ) (L2^2)) (24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^6)) (q^2))/(945 ((c^2) (L1^4) (L2^6))))-((4 ((3 c (L1^2)+(c-24 hϕ) (L2^2)) (24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^6)) (q^3))/(315 ((c^2) (L1^4) (L2^6))))-((8 ((3 (-16+c) c (L1^4)-2 (-8+c) (c-24 hϕ) (L1^2) (L2^2)-(((c-24 hϕ))^2) (L2^4)) (π^6)) (q^4))/(315 ((c^2) (L1^4) (L2^6))))+((O[q])^5)) (ℓ^6)+((1/(56700 (c^3) (22+5 c) (L1^8) (L2^8)))((c^4) (22+5 c) (L1^8)-(c^3) (22+5 c) (c-24 hϕ) (L1^6) (L2^2)-16 (c^2) (22+5 c) (c-22 hϕ) hϕ (L1^4) (L2^4)-c (22+5 c) (c-24 hϕ) ((c^2)-80 c hϕ+1280 ((hϕ)^2)) (L1^2) (L2^6)+((c^4) (22+5 c)-112 (c^3) (22+5 c) hϕ+32 (c^2) (3434+715 c) ((hϕ)^2)-7680 c (286+53 c) ((hϕ)^3)+92160 (176+29 c) ((hϕ)^4)) (L2^8)) (π^8)-(1/(14175 ((c^3) (L1^6) (L2^8))))4 (((c^3) (34+5 c) (L1^6)-(c^2) (53+10 c) (c-24 hϕ) (L1^4) (L2^2)+c ((c^2) (22+5 c)-240 (c^2) hϕ+960 (-11+3 c) ((hϕ)^2)) (L1^2) (L2^4)-3 (c-24 hϕ) ((c^2)-80 c hϕ+1280 ((hϕ)^2)) (L2^6)) (π^8)) (q^2)-(1/(14175 ((c^3) (L1^6) (L2^8))))2 ((4 (c^3) (53+10 c) (L1^6)-(c^2) (379+80 c) (c-24 hϕ) (L1^4) (L2^2)+8 c ((c^2) (22+5 c)+60 (1-4 c) c hϕ+360 (-33+8 c) ((hϕ)^2)) (L1^2) (L2^4)-9 (c-24 hϕ) ((c^2)-80 c hϕ+1280 ((hϕ)^2)) (L2^6)) (π^8)) (q^3)+(1/(14175 (c^3) (22+5 c) (L1^6) (L2^8)))4 (-c (L1^2) (c (22+5 c) (-9184+c (-1587+55 c)) (L1^4)-(22+5 c) (-2720+c (159+110 c)) (c-24 hϕ) (L1^2) (L2^2)+(-88+c (306+55 c)) (c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L2^4))+9 ((c^2) (22+5 c) (728+159 c) (L1^6)-3360 c (6+c) (c-22 hϕ) hϕ (L1^2) (L2^4)+(22+5 c) (c-24 hϕ) ((c^2)-80 c hϕ+1280 ((hϕ)^2)) (L2^6))) (π^8) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

S(ρ_A(L1,q1),ρ_A(L2,q2))

SL1q1L2q2=(((c ((((L1^2)-(L2^2)))^2) (π^4))/(540 (L1^4) (L2^4)))-((8 ((L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^2)+(L1^2) (q2^2))) (q^2))/(45 ((L1^4) (L2^4))))-((4 ((L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^3)+(L1^2) (q2^3))) (q^3))/(15 ((L1^4) (L2^4))))+((8 (π^4) (8 ((((L2^2) (q1^2)-(L1^2) (q2^2)))^2)+c (L1-L2) (L1+L2) ((L2^2) (q1^4)-(L1^2) (q2^4))) (q^4))/(15 c (L1^4) (L2^4)))+((O[q])^5)) (ℓ^4)+(((c ((((L1^2)-(L2^2)))^2) ((L1^2)+(L2^2)) (π^6))/(5670 (L1^6) (L2^6)))-((8 ((L1-L2) (L1+L2) (π^6) (-3 (L2^4) (q1^2)+3 (L1^4) (q2^2)+(L1^2) (L2^2) (-(q1^2)+(q2^2)))) (q^2))/(945 ((L1^6) (L2^6))))-((4 ((L1-L2) (L1+L2) (π^6) (-3 (L2^4) (q1^3)+3 (L1^4) (q2^3)+(L1^2) (L2^2) (-(q1^3)+(q2^3)))) (q^3))/(315 ((L1^6) (L2^6))))+((8 (π^6) ((L2^2) (c (L1^4)+2 (-8+c) (L1^2) (L2^2)-3 (-16+c) (L2^4)) (q1^4)-32 (L1^2) (L2^2) ((L1^2)+(L2^2)) (q1^2) (q2^2)+(L1^2) (-3 (-16+c) (L1^4)+2 (-8+c) (L1^2) (L2^2)+c (L2^4)) (q2^4)) (q^4))/(315 c (L1^6) (L2^6)))+((O[q])^5)) (ℓ^6)+(((c ((L1^8)-(L1^6) (L2^2)-(L1^2) (L2^6)+(L2^8)) (π^8))/(56700 (L1^8) (L2^8)))+((4 (L1-L2) (L1+L2) (π^8) ((34+5 c) (L2^6) (q1^2)-(34+5 c) (L1^6) (q2^2)-(L1^2) (L2^4) ((19+5 c) (q1^2)+3 (q2^2))+(L1^4) (L2^2) (3 (q1^2)+(19+5 c) (q2^2))) (q^2))/(14175 (L1^8) (L2^8)))+((2 (L1-L2) (L1+L2) (π^8) (4 (53+10 c) (L2^6) (q1^3)-4 (53+10 c) (L1^6) (q2^3)-(L1^2) (L2^4) ((167+40 c) (q1^3)+9 (q2^3))+(L1^4) (L2^2) (9 (q1^3)+(167+40 c) (q2^3))) (q^3))/(14175 (L1^8) (L2^8)))+(1/(14175 c (L1^8) (L2^8)))4 (π^8) ((L2^2) (9 c (L1^6)+(88-c (306+55 c)) (L1^4) (L2^2)+(-2720+c (159+110 c)) (L1^2) (L2^4)+(15736+(3018-55 c) c) (L2^6)) (q1^4)-48 (L1^2) (L2^2) ((53+10 c) (L1^4)+20 (22+5 c) (L1^2) (L2^2)+(53+10 c) (L2^4)) (q1^2) (q2^2)+(L1^2) ((15736+(3018-55 c) c) (L1^6)+(-2720+c (159+110 c)) (L1^4) (L2^2)+(88-c (306+55 c)) (L1^2) (L2^4)+9 c (L2^6)) (q2^4)) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

Section 3.1 2nd symmetrized relative entropy

S_2(ρ_A(L1,ϕ1),ρ_A(L2,ϕ2))

S2L1ϕ1L2ϕ2=((((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)))^2) (π^4) (ℓ^4))/(4608 c (L1^4) (L2^4)))+((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)) (11 c ((L1^4)-(L2^4))+240 (-hϕ2 (L1^4)+hϕ1 (L2^4))) (π^6) (ℓ^6))/(552960 c (L1^6) (L2^6)))+(1/(1238630400 (c^2) (22+5 c) (L1^8) (L2^8)))((2153 (c^3) (22+5 c)-95520 (c^2) (22+5 c) hϕ2+967680 c (24+5 c) ((hϕ2)^2)+19353600 c ((hϕ2)^3)-212889600 ((hϕ2)^4)) (L1^8)-20 c (22+5 c) (c-24 hϕ1) (73 c-1512 hϕ2) (L1^6) (L2^2)-126 ((c^2) (22+5 c) (11 c-240 hϕ1)-240 c (c (22+5 c)-80 (6+c) hϕ1-640 ((hϕ1)^2)) hϕ2+153600 (c-22 hϕ1) hϕ1 ((hϕ2)^2)) (L1^4) (L2^4)-20 c (22+5 c) (73 c-1512 hϕ1) (c-24 hϕ2) (L1^2) (L2^6)+(2153 (c^3) (22+5 c)-95520 (c^2) (22+5 c) hϕ1+967680 c (24+5 c) ((hϕ1)^2)+19353600 c ((hϕ1)^3)-212889600 ((hϕ1)^4)) (L2^8)) (π^8) (ℓ^8)+((O[ℓ])^10);

S_2(ρ_A(L1,ϕ),ρ_A(L2,q))

S2L1ϕL2q=((((((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^4))/(4608 c (L1^4) (L2^4)))+(((-c (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^2))/(48 c (L1^2) (L2^4)))+(((-c (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^3))/(32 c (L1^2) (L2^4)))+(((-(-8+c) (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^4))/(16 c (L1^2) (L2^4)))+((O[q])^5)) (ℓ^4)+((((24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (240 hϕ (L2^4)+11 c ((L1^4)-(L2^4))) (π^6))/(552960 c (L1^6) (L2^6)))+(((3 c (33+10 c) (L1^4)-10 (11+3 c) (c-24 hϕ) (L1^2) (L2^2)+(11 c-240 hϕ) (L2^4)) (π^6) (q^2))/(11520 c (L1^4) (L2^6)))+(((c (319+80 c) (L1^4)-10 (33+8 c) (c-24 hϕ) (L1^2) (L2^2)+(11 c-240 hϕ) (L2^4)) (π^6) (q^3))/(7680 c (L1^4) (L2^6)))+(((11 (-160+c (9+10 c)) (L1^4)-10 (59+11 c) (c-24 hϕ) (L1^2) (L2^2)+(11 c-240 hϕ) (L2^4)) (π^6) (q^4))/(3840 c (L1^4) (L2^6)))+((O[q])^5)) (ℓ^6)+((1/(1238630400 (c^2) (22+5 c) (L1^8) (L2^8)))(2153 (c^3) (22+5 c) (L1^8)-1460 (c^2) (22+5 c) (c-24 hϕ) (L1^6) (L2^2)-126 (c^2) (22+5 c) (11 c-240 hϕ) (L1^4) (L2^4)-20 (c^2) (22+5 c) (73 c-1512 hϕ) (L1^2) (L2^6)+(2153 (c^3) (22+5 c)-95520 (c^2) (22+5 c) hϕ+967680 c (24+5 c) ((hϕ)^2)+19353600 c ((hϕ)^3)-212889600 ((hϕ)^4)) (L2^8)) (π^8)-((((c (1963+1701 c) (L1^6)-63 (41+30 c) (c-24 hϕ) (L1^4) (L2^2)+63 (c (11+3 c)-80 (3+c) hϕ+320 ((hϕ)^2)) (L1^2) (L2^4)+(-73 c+1512 hϕ) (L2^6)) (π^8)) (q^2))/(1290240 (c (L1^6) (L2^8))))-((((c (21157+12376 c) (L1^6)-7 (3309+1840 c) (c-24 hϕ) (L1^4) (L2^2)+21 (3 c (33+8 c)-80 (27+8 c) hϕ+2560 ((hϕ)^2)) (L1^2) (L2^4)+(-73 c+1512 hϕ) (L2^6)) (π^8)) (q^3))/(860160 (c (L1^6) (L2^8))))+(1/(430080 (c^2) (22+5 c) (L1^6) (L2^8)))(-c (22+5 c) (-96768+c (14059+27237 c)) (L1^6)+21 c (22+5 c) (3563+1490 c) (c-24 hϕ) (L1^4) (L2^2)-21 (3 (c^2) (22+5 c) (59+11 c)-80 c (1046+55 c (9+c)) hϕ+320 (-88+c (306+55 c)) ((hϕ)^2)) (L1^2) (L2^4)+c (22+5 c) (73 c-1512 hϕ) (L2^6)) (π^8) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

S_2(ρ_A(L1,q1),ρ_A(L2,q2))

S2L1q1L2q2=(((c ((((L1^2)-(L2^2)))^2) (π^4))/(4608 (L1^4) (L2^4)))-((((L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^2)+(L1^2) (q2^2))) (q^2))/(48 ((L1^4) (L2^4))))-((((L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^3)+(L1^2) (q2^3))) (q^3))/(32 ((L1^4) (L2^4))))+(((π^4) (8 ((((L2^2) (q1^2)-(L1^2) (q2^2)))^2)+c (L1-L2) (L1+L2) ((L2^2) (q1^4)-(L1^2) (q2^4))) (q^4))/(16 c (L1^4) (L2^4)))+((O[q])^5)) (ℓ^4)+(((11 c ((((L1^2)-(L2^2)))^2) ((L1^2)+(L2^2)) (π^6))/(552960 (L1^6) (L2^6)))+(((L1-L2) (L1+L2) (π^6) (-3 (33+10 c) (L2^4) (q1^2)+3 (33+10 c) (L1^4) (q2^2)+11 (L1^2) (L2^2) (q1-q2) (q1+q2)) (q^2))/(11520 (L1^6) (L2^6)))+(((L1-L2) (L1+L2) (π^6) (-(319+80 c) (L2^4) (q1^3)+(319+80 c) (L1^4) (q2^3)+11 (L1^2) (L2^2) ((q1^3)-(q2^3))) (q^3))/(7680 (L1^6) (L2^6)))+(1/(3840 c (L1^6) (L2^6)))(π^6) ((L2^2) (11 c (L1^4)-10 c (59+11 c) (L1^2) (L2^2)+11 (-160+c (9+10 c)) (L2^4)) (q1^4)+160 (11+3 c) (L1^2) (L2^2) ((L1^2)+(L2^2)) (q1^2) (q2^2)+(L1^2) (11 (-160+c (9+10 c)) (L1^4)-10 c (59+11 c) (L1^2) (L2^2)+11 c (L2^4)) (q2^4)) (q^4)+((O[q])^5)) (ℓ^6)+(((c ((((L1^2)-(L2^2)))^2) (2153 (L1^4)+2846 (L1^2) (L2^2)+2153 (L2^4)) (π^8))/(1238630400 (L1^8) (L2^8)))+(((L1-L2) (L1+L2) (π^8) ((1963+1701 c) (L2^6) (q1^2)-(1963+1701 c) (L1^6) (q2^2)-(L1^2) (L2^4) ((620+189 c) (q1^2)+73 (q2^2))+(L1^4) (L2^2) (73 (q1^2)+(620+189 c) (q2^2))) (q^2))/(1290240 (L1^8) (L2^8)))+(1/(860160 (L1^8) (L2^8)))(L1-L2) (L1+L2) (π^8) ((21157+12376 c) (L2^6) (q1^3)-(21157+12376 c) (L1^6) (q2^3)-(L1^2) (L2^4) ((2006+504 c) (q1^3)+73 (q2^3))+(L1^4) (L2^2) (73 (q1^3)+2 (1003+252 c) (q2^3))) (q^3)+(1/(430080 c (L1^8) (L2^8)))(π^8) ((L2^2) (73 c (L1^6)-63 c (59+11 c) (L1^4) (L2^2)+21 c (3563+1490 c) (L1^2) (L2^4)+(96768-c (14059+27237 c)) (L2^6)) (q1^4)-336 (L1^2) (L2^2) (3 (41+30 c) (L1^4)+10 (33+2 c (8+c)) (L1^2) (L2^2)+3 (41+30 c) (L2^4)) (q1^2) (q2^2)+(L1^2) ((96768-c (14059+27237 c)) (L1^6)+21 c (3563+1490 c) (L1^4) (L2^2)-63 c (59+11 c) (L1^2) (L2^4)+73 c (L2^6)) (q2^4)) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

Section 3.2 Jensen-Shannon divergence

JS(ρ_A(L1,ϕ1),ρ_A(L2,ϕ2))

JSL1ϕ1L2ϕ2=((((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)))^2) (π^4) (ℓ^4))/(4320 c (L1^4) (L2^4)))+((((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)))^2) ((c-24 hϕ2) (L1^2)+(c-24 hϕ1) (L2^2)) (π^6) (ℓ^6))/(45360 (c^2) (L1^6) (L2^6)))+(1/(65318400 (c^3) (22+5 c) (L1^8) (L2^8)))(-25 (c^6) ((((L1^2)-(L2^2)))^4)+291962880 (((hϕ2 (L1^2)-hϕ1 (L2^2)))^2) (7 ((hϕ2)^2) (L1^4)+10 hϕ1 hϕ2 (L1^2) (L2^2)+7 ((hϕ1)^2) (L2^4))+10 (c^5) ((((L1^2)-(L2^2)))^2) ((41+240 hϕ2) (L1^4)-2 (-67+120 hϕ1+120 hϕ2) (L1^2) (L2^2)+(41+240 hϕ1) (L2^4))-16 (c^4) (L1-L2) (L1+L2) ((-143+60 hϕ2 (53+90 hϕ2)) (L1^6)-15 (11+8 hϕ2 (-34+45 hϕ2)+hϕ1 (-52+720 hϕ2)) (L1^4) (L2^2)+15 (11-52 hϕ2+8 hϕ1 (-34+45 hϕ1+90 hϕ2)) (L1^2) (L2^4)+(143-60 hϕ1 (53+90 hϕ1)) (L2^6))+1105920 c (((hϕ2)^3) (-242+255 hϕ2) (L1^8)+4 ((hϕ2)^2) (hϕ1 (22-27 hϕ2)+11 hϕ2) (L1^6) (L2^2)+2 hϕ1 hϕ2 (hϕ1 (55-147 hϕ2)+55 hϕ2) (L1^4) (L2^4)+4 ((hϕ1)^2) (hϕ1 (11-27 hϕ2)+22 hϕ2) (L1^2) (L2^6)+((hϕ1)^3) (-242+255 hϕ1) (L2^8))-9216 (c^2) (((hϕ2)^2) (-1387+900 hϕ2 (5+hϕ2)) (L1^8)-2 hϕ2 (10 hϕ2 (-22+27 hϕ2)+3 hϕ1 (3+10 hϕ2) (-11+60 hϕ2)) (L1^6) (L2^2)+(60 hϕ1 (18-59 hϕ2) hϕ2+209 ((hϕ2)^2)+((hϕ1)^2) (209+60 hϕ2 (-59+90 hϕ2))) (L1^4) (L2^4)-2 hϕ1 (-99 hϕ2+10 hϕ1 (-22+21 hϕ2+9 hϕ1 (3+20 hϕ2))) (L1^2) (L2^6)+((hϕ1)^2) (-1387+900 hϕ1 (5+hϕ1)) (L2^8))+768 (c^3) (hϕ2 (-352+15 hϕ2 (193+120 hϕ2)) (L1^8)-(3 hϕ2 (3+10 hϕ2) (-11+60 hϕ2)+hϕ1 (11+450 hϕ2 (-1+12 hϕ2))) (L1^6) (L2^2)+3 ((88-355 hϕ1) hϕ1+88 hϕ2+60 hϕ1 (-23+30 hϕ1) hϕ2+5 (-71+360 hϕ1) ((hϕ2)^2)) (L1^4) (L2^4)-(11 hϕ2+3 hϕ1 (-3 (11+50 hϕ2)+10 hϕ1 (7+60 hϕ1+180 hϕ2))) (L1^2) (L2^6)+hϕ1 (-352+15 hϕ1 (193+120 hϕ1)) (L2^8))) (π^8) (ℓ^8)+((O[ℓ])^10);

JS(ρ_A(L1,ϕ),ρ_A(L2,q))

JSL1ϕL2q=((((((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^4))/(4320 c (L1^4) (L2^4)))+(((-c (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^2))/(45 c (L1^2) (L2^4)))+(((-c (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^3))/(30 c (L1^2) (L2^4)))+(((-(-8+c) (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^4))/(15 c (L1^2) (L2^4)))+((O[q])^5)) (ℓ^4)+((((c (L1^2)+(c-24 hϕ) (L2^2)) (((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^6))/(45360 (c^2) (L1^6) (L2^6)))-((((3 c (L1^2)+(c-24 hϕ) (L2^2)) (24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^6)) (q^2))/(945 ((c^2) (L1^4) (L2^6))))-((((3 c (L1^2)+(c-24 hϕ) (L2^2)) (24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^6)) (q^3))/(630 ((c^2) (L1^4) (L2^6))))+(((-3 (-16+c) c (L1^4)+2 (-8+c) (c-24 hϕ) (L1^2) (L2^2)+(((c-24 hϕ))^2) (L2^4)) (π^6) (q^4))/(315 (c^2) (L1^4) (L2^6)))+((O[q])^5)) (ℓ^6)+((1/(65318400 (c^3) (22+5 c) (L1^8) (L2^8)))(-(c^4) (-104+5 c) (22+5 c) (L1^8)+4 (c^3) (4+5 c) (22+5 c) (c-24 hϕ) (L1^6) (L2^2)-6 (c^2) (22+5 c) (5 (c^2) (8+c)-48 c (32+5 c) hϕ+192 (76+15 c) ((hϕ)^2)) (L1^4) (L2^4)+4 c (22+5 c) (c-24 hϕ) ((c^2) (4+5 c)-240 (-4+c) c hϕ+2880 (-8+c) ((hϕ)^2)) (L1^2) (L2^6)+(-(c^4) (-104+5 c) (22+5 c)+96 (c^3) (-128+5 c) (22+5 c) hϕ-1152 (c^2) (-11096+5 c (-386+15 c)) ((hϕ)^2)+276480 c (-968+5 (-30+c) c) ((hϕ)^3)-1658880 (-1232+5 (-34+c) c) ((hϕ)^4)) (L2^8)) (π^8)-(1/(340200 ((c^3) (L1^6) (L2^8))))(((c^3) (368+55 c) (L1^6)-3 (c^2) (172+35 c) (c-24 hϕ) (L1^4) (L2^2)+9 c ((c^2) (16+5 c)+80 (8-3 c) c hϕ+320 (-68+9 c) ((hϕ)^2)) (L1^2) (L2^4)+(c-24 hϕ) ((c^2) (4+5 c)-240 (-4+c) c hϕ+2880 (-8+c) ((hϕ)^2)) (L2^6)) (π^8)) (q^2)-(1/(226800 ((c^3) (L1^6) (L2^8))))(((c^3) (808+155 c) (L1^6)-(c^2) (1396+305 c) (c-24 hϕ) (L1^4) (L2^2)+c (145 (c^3)+8 (c^2) (73-870 hϕ)-449280 ((hϕ)^2)+960 c hϕ (8+87 hϕ)) (L1^2) (L2^4)+(c-24 hϕ) ((c^2) (4+5 c)-240 (-4+c) c hϕ+2880 (-8+c) ((hϕ)^2)) (L2^6)) (π^8)) (q^3)-(1/(113400 ((c^3) (22+5 c) (L1^6) (L2^8))))(((c^2) (22+5 c) (-61984+c (-11992+215 c)) (L1^6)-c (22+5 c) (-8960+c (756+425 c)) (c-24 hϕ) (L1^4) (L2^2)+((c^2) (22+5 c) (608+c (1224+205 c))-240 c (3520+c (32+c (1094+205 c))) hϕ+2880 (4224+c (-4936+c (234+205 c))) ((hϕ)^2)) (L1^2) (L2^4)+(22+5 c) (c-24 hϕ) ((c^2) (4+5 c)-240 (-4+c) c hϕ+2880 (-8+c) ((hϕ)^2)) (L2^6)) (π^8)) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

JS(ρ_A(L1,q1),ρ_A(L2,q2))

JSL1q1L2q2=(((c ((((L1^2)-(L2^2)))^2) (π^4))/(4320 (L1^4) (L2^4)))-((((L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^2)+(L1^2) (q2^2))) (q^2))/(45 ((L1^4) (L2^4))))-((((L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^3)+(L1^2) (q2^3))) (q^3))/(30 ((L1^4) (L2^4))))+(((π^4) (8 ((((L2^2) (q1^2)-(L1^2) (q2^2)))^2)+c (L1-L2) (L1+L2) ((L2^2) (q1^4)-(L1^2) (q2^4))) (q^4))/(15 c (L1^4) (L2^4)))+((O[q])^5)) (ℓ^4)+(((c ((((L1^2)-(L2^2)))^2) ((L1^2)+(L2^2)) (π^6))/(45360 (L1^6) (L2^6)))-((((L1-L2) (L1+L2) (π^6) (-3 (L2^4) (q1^2)+3 (L1^4) (q2^2)+(L1^2) (L2^2) (-(q1^2)+(q2^2)))) (q^2))/(945 ((L1^6) (L2^6))))-((((L1-L2) (L1+L2) (π^6) (-3 (L2^4) (q1^3)+3 (L1^4) (q2^3)+(L1^2) (L2^2) (-(q1^3)+(q2^3)))) (q^3))/(630 ((L1^6) (L2^6))))+(((π^6) ((L2^2) (c (L1^4)+2 (-8+c) (L1^2) (L2^2)-3 (-16+c) (L2^4)) (q1^4)-32 (L1^2) (L2^2) ((L1^2)+(L2^2)) (q1^2) (q2^2)+(L1^2) (-3 (-16+c) (L1^4)+2 (-8+c) (L1^2) (L2^2)+c (L2^4)) (q2^4)) (q^4))/(315 c (L1^6) (L2^6)))+((O[q])^5)) (ℓ^6)+(-((c ((((L1^2)-(L2^2)))^2) ((-104+5 c) (L1^4)-2 (112+5 c) (L1^2) (L2^2)+(-104+5 c) (L2^4)) (π^8))/(65318400 ((L1^8) (L2^8))))+(((L1-L2) (L1+L2) (π^8) ((368+55 c) (L2^6) (q1^2)-(368+55 c) (L1^6) (q2^2)+(L1^2) (L2^4) (-2 (74+25 c) (q1^2)+(4+5 c) (q2^2))+(L1^4) (L2^2) (-(4+5 c) (q1^2)+2 (74+25 c) (q2^2))) (q^2))/(340200 (L1^8) (L2^8)))+(((L1-L2) (L1+L2) (π^8) ((808+155 c) (L2^6) (q1^3)-(808+155 c) (L1^6) (q2^3)+(L1^2) (L2^4) (-6 (98+25 c) (q1^3)+(4+5 c) (q2^3))+(L1^4) (L2^2) (-(4+5 c) (q1^3)+6 (98+25 c) (q2^3))) (q^3))/(226800 (L1^8) (L2^8)))+(1/(113400 c (L1^8) (L2^8)))(π^8) (-(L2^2) (c (4+5 c) (L1^6)+(608+c (1224+205 c)) (L1^4) (L2^2)+(8960-c (756+425 c)) (L1^2) (L2^4)+(-61984+c (-11992+215 c)) (L2^6)) (q1^4)-48 (L1^2) (L2^2) ((172+35 c) (L1^4)+10 (184+41 c) (L1^2) (L2^2)+(172+35 c) (L2^4)) (q1^2) (q2^2)-(L1^2) ((-61984+c (-11992+215 c)) (L1^6)+(8960-c (756+425 c)) (L1^4) (L2^2)+(608+c (1224+205 c)) (L1^2) (L2^4)+c (4+5 c) (L2^6)) (q2^4)) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);

Section 3.3 Schatten 2-norm

||ρ_A(L1,ϕ1)-ρ_A(L2,ϕ2)||_2^2

L1ϕ1L2ϕ222=(ℓ^(-c/8))((((2+c) ((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)))^2) (π^4) (ℓ^4))/(9216 c (L1^4) (L2^4)))+(((4+c) ((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)) ((c (22+5 c)-240 (2+c) hϕ2+2880 ((hϕ2)^2)) (L1^4)+(-c (22+5 c)+240 (2+c) hϕ1-2880 ((hϕ1)^2)) (L2^4)) (π^6) (ℓ^6))/(4423680 c (L1^6) (L2^6)))+(1/(178362777600 c))((1/(29+70 c))700 (6+c) (((c (-1+2 c) (22+5 c) (68+7 c)-72 (-248+c (938+c (617+70 c))) hϕ1+1728 (4+c) (29+70 c) ((hϕ1)^2)-13824 (29+70 c) ((hϕ1)^3))/(L1^6))+((-c (-1+2 c) (22+5 c) (68+7 c)+72 (-248+c (938+c (617+70 c))) hϕ2-1728 (4+c) (29+70 c) ((hϕ2)^2)+13824 (29+70 c) ((hϕ2)^3))/(L2^6))) (((c-24 hϕ1)/(L1^2))-((c-24 hϕ2)/(L2^2)))+((21 (216+c (62+5 c)) ((((c (22+5 c)-240 (2+c) hϕ2+2880 ((hϕ2)^2)) (L1^4)+(-c (22+5 c)+240 (2+c) hϕ1-2880 ((hϕ1)^2)) (L2^4)))^2))/((22+5 c) (L1^8) (L2^8)))+((960 (416+c (275+56 c)) ((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)) (31 c ((L1^6)-(L2^6))+504 (-hϕ2 (L1^6)+hϕ1 (L2^6))))/((29+70 c) (L1^8) (L2^8)))) (π^8) (ℓ^8)+((O[ℓ])^10));

||ρ_A(L1,ϕ),ρ_A(L2,q)||_2^2

L1ϕL2q22=(ℓ^(-c/8))(((((2+c) (((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^4))/(9216 c (L1^4) (L2^4)))+(((2+c) (-c (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^2))/(96 c (L1^2) (L2^4)))+(((2+c) (-c (L1^2)+(c-24 hϕ) (L2^2)) (π^4) (q^3))/(64 c (L1^2) (L2^4)))-((((2+c) ((-8+c) (L1^2)-(c-24 hϕ) (L2^2)) (π^4)) (q^4))/(32 (c (L1^2) (L2^4))))+((O[q])^5)) (ℓ^4)+((((4+c) (c (22+5 c) (L1^4)+(-c (22+5 c)+240 (2+c) hϕ-2880 ((hϕ)^2)) (L2^4)) (24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (π^6))/(4423680 c (L1^6) (L2^6)))+(((4+c) (9 c (22+5 c) (L1^4)-10 (22+5 c) (c-24 hϕ) (L1^2) (L2^2)+(c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L2^4)) (π^6) (q^2))/(92160 c (L1^4) (L2^6)))+(((4+c) (29 c (22+5 c) (L1^4)-30 (22+5 c) (c-24 hϕ) (L1^2) (L2^2)+(c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L2^4)) (π^6) (q^3))/(61440 c (L1^4) (L2^6)))+(((4+c) ((-3520+c (438+205 c)) (L1^4)-210 (6+c) (c-24 hϕ) (L1^2) (L2^2)+(c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L2^4)) (π^6) (q^4))/(30720 c (L1^4) (L2^6)))+((O[q])^5)) (ℓ^6)+((1/(178362777600 c))(((700 (6+c) (((c (-1+2 c) (22+5 c) (68+7 c)-72 (-248+c (938+c (617+70 c))) hϕ+1728 (4+c) (29+70 c) ((hϕ)^2)-13824 (29+70 c) ((hϕ)^3))/(L1^6))-((c (-1+2 c) (22+5 c) (68+7 c))/(L2^6))) (((c-24 hϕ)/(L1^2))-(c/(L2^2))))/(29+70 c))+((21 (216+c (62+5 c)) (((c (22+5 c) (L1^4)+(-c (22+5 c)+240 (2+c) hϕ-2880 ((hϕ)^2)) (L2^4)))^2))/((22+5 c) (L1^8) (L2^8)))+((960 (416+c (275+56 c)) (24 hϕ (L2^2)+c (L1-L2) (L1+L2)) (504 hϕ (L2^6)+31 c ((L1^6)-(L2^6))))/((29+70 c) (L1^8) (L2^8)))) (π^8)+(1/(185794560 c (L1^6) (L2^8)))(c (-282672+c (-263236+7 c (-9492+235 c))) (L1^6)-21 (-17712+c (-14740+c (-3552+55 c))) (c-24 hϕ) (L1^4) (L2^2)-21 (216+c (62+5 c)) (c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L1^2) (L2^4)+(c (10512+c (5020+7 c (96+5 c)))-504 (2+c) (216+c (62+5 c)) hϕ+60480 (4+c) (6+c) ((hϕ)^2)-483840 (6+c) ((hϕ)^3)) (L2^6)) (π^8) (q^2)+(1/(123863040 c (L1^6) (L2^8)))(c (-3046608+c (-2235388+7 c (-70332+685 c))) (L1^6)-21 (-158832+c (-113540+c (-24672+155 c))) (c-24 hϕ) (L1^4) (L2^2)-63 (216+c (62+5 c)) (c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L1^2) (L2^4)+(c (10512+c (5020+7 c (96+5 c)))-504 (2+c) (216+c (62+5 c)) hϕ+60480 (4+c) (6+c) ((hϕ)^2)-483840 (6+c) ((hϕ)^3)) (L2^6)) (π^8) (q^3)+(1/(61931520 c (22+5 c) (L1^6) (L2^8)))((22+5 c) (13934592+c (-1598448+c (-4469620+7 c (-172236+955 c)))) (L1^6)-21 (22+5 c) (-534192+c (-315380+c (-60432+215 c))) (c-24 hϕ) (L1^4) (L2^2)-441 (6+c) (216+c (62+5 c)) (c (22+5 c)-240 (2+c) hϕ+2880 ((hϕ)^2)) (L1^2) (L2^4)+(22+5 c) (c (10512+c (5020+7 c (96+5 c)))-504 (2+c) (216+c (62+5 c)) hϕ+60480 (4+c) (6+c) ((hϕ)^2)-483840 (6+c) ((hϕ)^3)) (L2^6)) (π^8) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10));

||ρ_A(L1,q1)-ρ_A(L2,q2)||_2^2

L1q1L2q222=(ℓ^(-c/8))((((c (2+c) ((((L1^2)-(L2^2)))^2) (π^4))/(9216 (L1^4) (L2^4)))-((((2+c) (L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^2)+(L1^2) (q2^2))) (q^2))/(96 ((L1^4) (L2^4))))-((((2+c) (L1-L2) (L1+L2) (π^4) (-(L2^2) (q1^3)+(L1^2) (q2^3))) (q^3))/(64 ((L1^4) (L2^4))))-((((2+c) (π^4) (-8 ((((L2^2) (q1^2)-(L1^2) (q2^2)))^2)+c (L1-L2) (L1+L2) (-(L2^2) (q1^4)+(L1^2) (q2^4)))) (q^4))/(32 (c (L1^4) (L2^4))))+((O[q])^5)) (ℓ^4)+(((c (4+c) (22+5 c) ((((L1^2)-(L2^2)))^2) ((L1^2)+(L2^2)) (π^6))/(4423680 (L1^6) (L2^6)))+(((4+c) (22+5 c) (L1-L2) (L1+L2) (π^6) (-9 (L2^4) (q1^2)+9 (L1^4) (q2^2)+(L1^2) (L2^2) (q1-q2) (q1+q2)) (q^2))/(92160 (L1^6) (L2^6)))+(((4+c) (22+5 c) (L1-L2) (L1+L2) (π^6) (-29 (L2^4) (q1^3)+29 (L1^4) (q2^3)+(L1^2) (L2^2) ((q1^3)-(q2^3))) (q^3))/(61440 (L1^6) (L2^6)))+(1/(30720 c (L1^6) (L2^6)))(4+c) (π^6) ((L2^2) (c (22+5 c) (L1^4)-210 c (6+c) (L1^2) (L2^2)+(-3520+c (438+205 c)) (L2^4)) (q1^4)+160 (22+5 c) (L1^2) (L2^2) ((L1^2)+(L2^2)) (q1^2) (q2^2)+(L1^2) ((-3520+c (438+205 c)) (L1^4)-210 c (6+c) (L1^2) (L2^2)+c (22+5 c) (L2^4)) (q2^4)) (q^4)+((O[q])^5)) (ℓ^6)+((1/(178362777600 (L1^8) (L2^8)))c (((L1-L2))^2) (((L1+L2))^2) ((310032+c (151724+35 c (636+35 c))) (L1^4)+2 (18+5 c) (11384+7 c (354+25 c)) (L1^2) (L2^2)+(310032+c (151724+35 c (636+35 c))) (L2^4)) (π^8)+(1/(185794560 (L1^8) (L2^8)))(L1-L2) (L1+L2) (π^8) ((282672+c (263236+7 (9492-235 c) c)) (L2^6) (q1^2)+(-282672+c (-263236+7 c (-9492+235 c))) (L1^6) (q2^2)+(L1^4) (L2^2) ((10512+c (5020+7 c (96+5 c))) (q1^2)+2 (36+7 c) (1240+c (402+35 c)) (q2^2))+(L1^2) (L2^4) (-2 (36+7 c) (1240+c (402+35 c)) (q1^2)-(10512+c (5020+7 c (96+5 c))) (q2^2))) (q^2)+(1/(123863040 (L1^8) (L2^8)))(L1-L2) (L1+L2) (π^8) ((3046608+c (2235388+7 (70332-685 c) c)) (L2^6) (q1^3)+(-3046608+c (-2235388+7 c (-70332+685 c))) (L1^6) (q2^3)+(L1^2) (L2^4) (-4 (72216+c (37238+7 c (921+55 c))) (q1^3)-(10512+c (5020+7 c (96+5 c))) (q2^3))+(L1^4) (L2^2) ((10512+c (5020+7 c (96+5 c))) (q1^3)+4 (72216+c (37238+7 c (921+55 c))) (q2^3))) (q^3)+(1/(61931520 c (L1^8) (L2^8)))(π^8) (21 (216+c (62+5 c)) ((L2^4) (-21 c (6+c) (L1^4)+(1760+c (526+21 c)) (L2^4)) (q1^4)-160 (22+5 c) (L1^4) (L2^4) (q1^2) (q2^2)+(L1^4) ((1760+c (526+21 c)) (L1^4)-21 c (6+c) (L2^4)) (q2^4))-(1/(29+70 c))48 (416+c (275+56 c)) (-(L2^2) (31 c (L1^6)+21 c (841+1640 c) (L1^2) (L2^4)+4 (84+(5657-8610 c) c) (L2^6)) (q1^4)+336 (1+120 c) (L1^2) (L2^2) ((L1^4)+(L2^4)) (q1^2) (q2^2)+(L1^2) (4 (-84+c (-5657+8610 c)) (L1^6)-21 c (841+1640 c) (L1^4) (L2^2)-31 c (L2^6)) (q2^4))+(1/(29+70 c))35 (6+c) (-1+2 c) (68+7 c) ((L2^2) (c (22+5 c) (L1^6)+3 (638-215 c) c (L1^2) (L2^4)+32 (-363+c (-143+20 c)) (L2^6)) (q1^4)+528 (22+5 c) (L1^2) (L2^2) ((L1^4)+(L2^4)) (q1^2) (q2^2)+(L1^2) (32 (-363+c (-143+20 c)) (L1^6)+3 (638-215 c) c (L1^4) (L2^2)+c (22+5 c) (L2^6)) (q2^4))) (q^4)+((O[q])^5)) (ℓ^8)+((O[ℓ])^10));

Section 3.3 Schatten 4-norm

||ρ_A(L1,ϕ1)-ρ_A(L2,ϕ2)||_4^4

L1ϕ1L2ϕ244=(ℓ^(-5c/16))((((2+c) (192+25 c (-14+25 c)) ((((c-24 hϕ2) (L1^2)-(c-24 hϕ1) (L2^2)))^4) (π^8))/(21743271936 (c^3) (L1^8) (L2^8)))(ℓ^8)+((O[ℓ])^10));

||ρ_A(L1,ϕ),ρ_A(L2,q)||_4^4

L1ϕL2q44=(ℓ^(-5c/16))(((((2+c) (192+25 c (-14+25 c)) (((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^4) (π^8))/(21743271936 (c^3) (L1^8) (L2^8)))-((((2+c) (192+25 c (-14+25 c)) (((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^3) (π^8)) (q^2))/(113246208 ((c^3) (L1^6) (L2^8))))-((((2+c) (192+25 c (-14+25 c)) (((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^3) (π^8)) (q^3))/(75497472 ((c^3) (L1^6) (L2^8))))-((((2+c) (192+25 c (-14+25 c)) ((-24+c) (L1^2)-(c-24 hϕ) (L2^2)) (((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^8)) (q^4))/(37748736 ((c^3) (L1^6) (L2^8))))+((O[q])^5))(ℓ^8)+((O[ℓ])^10));

||ρ_A(L1,q1)-ρ_A(L2,q2)||_4^4

L1q1L2q244=(ℓ^(-5c/16))((((c (2+c) (192+25 c (-14+25 c)) (((L1-L2))^4) (((L1+L2))^4) (π^8))/(21743271936 (L1^8) (L2^8)))-((((2+c) (192+25 c (-14+25 c)) (((L1-L2))^3) (((L1+L2))^3) (π^8) (-(L2^2) (q1^2)+(L1^2) (q2^2))) (q^2))/(113246208 ((L1^8) (L2^8))))-((((2+c) (192+25 c (-14+25 c)) (((L1-L2))^3) (((L1+L2))^3) (π^8) (-(L2^2) (q1^3)+(L1^2) (q2^3))) (q^3))/(75497472 ((L1^8) (L2^8))))-((((2+c) (192+25 c (-14+25 c)) (((L1-L2))^2) (((L1+L2))^2) (π^8) (-24 ((((L2^2) (q1^2)-(L1^2) (q2^2)))^2)+c (L1-L2) (L1+L2) (-(L2^2) (q1^4)+(L1^2) (q2^4)))) (q^4))/(37748736 (c (L1^8) (L2^8))))+((O[q])^5))(ℓ^8)+((O[ℓ])^10));

Section 4 Check of ETH

difference of Renyi entropy

((c (-1+n) (((1+n))^2) (π^4) (Subscript[ϵ, ϕ]) (-1+22 (Subscript[ϵ, ϕ])) (ℓ^4))/(90 (L^4) (n^3)))-(((c (-1+n) (((1+n))^2) (π^6) (Subscript[ϵ, ϕ]) (2+4 (n^2)-3 (37+46 (n^2)) (Subscript[ϵ, ϕ])+8 (188+145 (n^2)) (Subscript[ϵ, ϕ]^2))) (ℓ^6))/(2835 ((L^6) (n^5))))+(1/(56700 (22+5 c) (L^8) (n^7)))c (1+n) (π^8) (Subscript[ϵ, ϕ]) (3 (22+5 c) (1+(n^2)+(n^4)-3 (n^6))+2 (Subscript[ϵ, ϕ]) (-4312-753 c-(1072+213 c) (n^2)-(400+87 c) (n^4)+3 (1544+351 c) (n^6)+4 (Subscript[ϵ, ϕ]) (-c (-1+n) (1+n) (5863+4636 (n^2)+4333 (n^4))-44 (-836+89 (n^2)+26 (n^4)+433 (n^6))+(c (-1+n) (1+n) (58213+24286 (n^2)+25933 (n^4))+88 (-4284+1511 (n^2)-106 (n^4)+1295 (n^6))) (Subscript[ϵ, ϕ])))) (ℓ^8)+((O[ℓ])^10);

difference of entanglement entropy

-((128 (c (π^8) (((1-22 (Subscript[ϵ, ϕ])))^2) (Subscript[ϵ, ϕ]^2)) (ℓ^8))/(1575 ((22+5 c) (L^8))))+((O[ℓ])^10);

Relative entropy S(ρ_A(L,ϕ)||ρ_A(β))

((128 c (π^8) (((1-22 (Subscript[ϵ, ϕ])))^2) (Subscript[ϵ, ϕ]^2) (ℓ^8))/(1575 (22+5 c) (L^8)))+((O[ℓ])^10);

Relative entropy S(ρ_A(β)||ρ_A(L,ϕ))

((128 c (π^8) (((1-22 (Subscript[ϵ, ϕ])))^2) (Subscript[ϵ, ϕ]^2) (ℓ^8))/(1575 (22+5 c) (L^8)))+((O[ℓ])^10);

Symmetrized relative entropy S(ρ_A(L,ϕ),ρ_A(β))

((256 c (π^8) (((1-22 (Subscript[ϵ, ϕ])))^2) (Subscript[ϵ, ϕ]^2) (ℓ^8))/(1575 (22+5 c) (L^8)))+((O[ℓ])^10);

2nd symmetrized relative entropy S_2(ρ_A(L,ϕ),ρ_A(β))

((c (27+5 c) (π^8) (((1-22 (Subscript[ϵ, ϕ])))^2) (Subscript[ϵ, ϕ]^2) (ℓ^8))/(3200 (22+5 c) (L^8)))+((O[ℓ])^10);

Jensen-Renyi divergence JR_n(ρ_A(L,ϕ),ρ_A(β))

-(((c (1+n) (175 (c^2) (((-1+(n^2)))^3)+70 c (((-1+(n^2)))^2) (-55+7 (n^2))-8 (11+(n^2)) (381-298 (n^2)+157 (n^4))) (π^8) (((1-22 (Subscript[ϵ, ϕ])))^2) (Subscript[ϵ, ϕ]^2)) (ℓ^8))/(2268000 ((22+5 c) (L^8) (n^7))))+((O[ℓ])^10);

Jensen-Shannon divergence JS(ρ_A(L,ϕ),ρ_A(β))

((32 c (π^8) (((1-22 (Subscript[ϵ, ϕ])))^2) (Subscript[ϵ, ϕ]^2) (ℓ^8))/(1575 (22+5 c) (L^8)))+((O[ℓ])^10);

Schatten 2-norm ||ρ_A(L,ϕ)-ρ_A(β)||_2^2

(ℓ^(-c/8))(((c (216+c (62+5 c)) (π^8) (((1-22 (Subscript[ϵ, ϕ])))^2) (Subscript[ϵ, ϕ]^2) (ℓ^8))/(25600 (22+5 c) (L^8)))+((O[ℓ])^10));

Appendix A Relative entropy from modular Hamiltonian

S(ρ_A(L1,ϕ)||ρ_A(L2))

SL1ϕL2ϕ=(((((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^4) (ℓ^4))/(1080 c (L1^4) (L2^4)))+(((2 c (L1^2)+(c-24 hϕ) (L2^2)) (((24 hϕ (L2^2)+c (L1-L2) (L1+L2)))^2) (π^6) (ℓ^6))/(17010 (c^2) (L1^6) (L2^6)))+((((((c^4) (22+5 c)-96 (c^3) (22+5 c) hϕ+1152 (c^2) (82+15 c) ((hϕ)^2)-92160 c (22+3 c) ((hϕ)^3)+184320 (88+9 c) ((hϕ)^4))/((c^3) (22+5 c) (L1^8)))+((3 c)/(L2^8))-((4 c)/((L1^2) (L2^6)))+((96 hϕ)/((L1^2) (L2^6)))) (π^8) (ℓ^8))/226800)+((O[ℓ])^10);

S(ρ_A(L1,q)||ρ_A(L2))

SL1qL2=(((c ((((L1^2)-(L2^2)))^2) (π^4))/(1080 (L1^4) (L2^4)))+((4 (L1-L2) (L1+L2) (π^4) (q^2))/(45 (L1^4) (L2^2)))+((2 (L1-L2) (L1+L2) (π^4) (q^3))/(15 (L1^4) (L2^2)))+((4 (c (L1^2)-(-8+c) (L2^2)) (π^4) (q^4))/(15 c (L1^4) (L2^2)))+((O[q])^5)) (ℓ^4)+(((c (2 (L1^6)-3 (L1^4) (L2^2)+(L2^6)) (π^6))/(17010 (L1^6) (L2^6)))+((8 ((L1^4)-(L2^4)) (π^6) (q^2))/(945 (L1^6) (L2^4)))+((4 ((L1^4)-(L2^4)) (π^6) (q^3))/(315 (L1^6) (L2^4)))+((8 (c (L1^4)-(-16+c) (L2^4)) (π^6) (q^4))/(315 c (L1^6) (L2^4)))+((O[q])^5)) (ℓ^6)+(((c (3 (L1^8)-4 (L1^6) (L2^2)+(L2^8)) (π^8))/(226800 (L1^8) (L2^8)))+((4 ((L1^6)-(L2^6)) (π^8) (q^2))/(4725 (L1^8) (L2^6)))+((2 ((L1^6)-(L2^6)) (π^8) (q^3))/(1575 (L1^8) (L2^6)))+((4 (c (L1^6)+(728+159 c) (L2^6)) (π^8) (q^4))/(1575 c (L1^8) (L2^6)))+((O[q])^5)) (ℓ^8)+((O[ℓ])^10);