text: The exact dCS metric for a spinning BH possesses a fourth constant of motion: ...20 destroyed. This allowed us to conjecture that geodesics of an exact dCS BH metric would not be chaotic. Our numerical ﬁndings also suggest that geodesics in an exact dCS background may have a fourth constant of the motion. At second order in rotation, just a constant of the motion could not be found in [24], who suggested that an exact dCS background may be of Petrov type I. Our results, however, suggest that as terms of higher and higher order in the slow-rotation expansion are included, the symmetry that gives rise to a fourth constant of the motion seems to become restored, and the exact dCS metric for a BH may be of Petrov type D after all. These ﬁndings have implications on ongoing eﬀorts to ﬁnd an exact solution in dCS gravity for spinning BHs. The analysis we have carried out, however, is only an approximation to the motion of extreme mass-ratio inspirals in dCS gravity. Needless to say, we have not included dissipative eﬀects in the orbits, which is why we were able to study bounded motion. The inclusion of such dissipative eﬀects with the resummed dCS metric could be the topic of future studies. Furthermore, we have here neglected non-geodesic forces induced by the scalar (magnetic dipole) charge of the small object. Such a force may lead to interesting corrections on the motion found here that could also be studied further. Ultimately, once such studies have been carried out, one can investigate the signatures of dCS gravity on extreme mass-ratio inspirals. The only studies on this topic are those of [36, 48], which did not considered a resummed metric and did not include the magnetic-dipole force due to the small object. Redoing such an analysis would allow us to estimate the accuracy to which dCS gravity could be constrained by future gravitational wave observations with LISA. Acknowledgements We thank Neil Cornish, Dimitry Ayzenberg, Kent Yagi, Paolo Pani, Andrea Masselli, Charles Kankelborg, Thomas Sotiriou, Olivier Sarbach and Houri Tsuyoshi for very useful suggestions and comments. A.C.-A. acknowledges funding from the Fundaci´on Universitaria Konrad Lorenz (Project 5INV1). N.Y. and A.C.- A. acknowledge ﬁnancial support through NSF CAREER grant PHY-1250636 and NASA grants NNX16AB98G and 80NSSC17M0041. L.A.P. acknowledges funding from the Comit´e para el Desarrollo de la Investigaci´on -CODI- of Universidad de Antioquia, Colombia under the Estrateg´ıa de Sostenibilidad. Computational eﬀorts were performed on the Hyalite High-Performance Computing System, operated and supported by Montana State University’s Information Technology Center. Appendix A. BH Solutions in dCS gravity to ﬁfth order in Boyer Lindquist Coordinates The dCS metric in Boyer-Lindquist coordinates (t, r, θ, φ) is obtained from the solution given in Ref. [25] in Hartle-Thorne coordinates (t, r , θ , φ), after making HT HT a coordinate transformation im plicitly given by,  M4 − − − r = χ4 8M4 + 72M(cid:0) 3r + 36M2r2 15M(cid:1)r3 5r4 HT 40r7 − + 10 cos2(θ)(3M + r) 12M3 18M2r + r3 ##LTLine##

text: The

text: exact

text: dCS

text: metric

text: for

text: a

text: spinning

text: BH

text: possesses

text: a

text: fourth

text: constant

text: of

text: motion:

text: destroyed.

text: This

text: allowed

text: us

text: to

text: conjecture

text: that

text: geodesics

text: of

text: an

text: exact

text: dCS

text: BH

text: metric

text: would

text: not

text: be

text: chaotic.

text: Our

text: numerical

text: ﬁndings

text: also

text: suggest

text: that

text: geodesics

text: in

text: an

text: exact

text: dCS

text: background

text: may

text: have

text: a

text: fourth

text: constant

text: of

text: the

text: motion.

text: At

text: second

text: order

text: in

text: rotation,

text: just

text: a

text: constant

text: of

text: the

text: motion

text: could

text: not

text: be

text: found

text: in

text: who

text: suggested

text: that

text: an

text: exact

text: dCS

text: background

text: may

text: be

text: of

text: Petrov

text: type

text: I.

text: Our

text: results,

text: however,

text: suggest

text: that

text: as

text: terms

text: of

text: higher

text: and

text: higher

text: order

text: in

text: the

text: slow-rotation

text: expansion

text: are

text: included,

text: the

text: symmetry

text: that

text: gives

text: rise

text: to

text: a

text: fourth

text: constant

text: of

text: the

text: motion

text: seems

text: to

text: become

text: restored,

text: and

text: the

text: exact

text: dCS

text: metric

text: for

text: a

text: BH

text: may

text: be

text: of

text: Petrov

text: type

text: D

text: after

text: all.

text: These

text: ﬁndings

text: have

text: implications

text: on

text: ongoing

text: eﬀorts

text: to

text: ﬁnd

text: an

text: exact

text: solution

text: in

text: dCS

text: gravity

text: for

text: spinning

text: BHs.

text: The

text: analysis

text: we

text: have

text: carried

text: out,

text: however,

text: is

text: only

text: an

text: approximation

text: to

text: the

text: motion

text: of

text: extreme

text: mass-ratio

text: inspirals

text: in

text: dCS

text: gravity.

text: Needless

text: to

text: say,

text: we

text: have

text: not

text: included

text: dissipative

text: eﬀects

text: in

text: the

text: orbits,

text: which

text: is

text: why

text: we

text: were

text: able

text: to

text: study

text: bounded

text: motion.

text: The

text: inclusion

text: of

text: such

text: dissipative

text: eﬀects

text: with

text: the

text: resummed

text: dCS

text: metric

text: could

text: be

text: the

text: topic

text: of

text: future

text: studies.

text: Furthermore,

text: we

text: have

text: here

text: neglected

text: non-geodesic

text: forces

text: induced

text: by

text: the

text: scalar

text: (magnetic

text: dipole)

text: charge

text: of

text: the

text: small

text: object.

text: Such

text: a

text: force

text: may

text: lead

text: to

text: interesting

text: corrections

text: on

text: the

text: motion

text: found

text: here

text: that

text: could

text: also

text: be

text: studied

text: further.

text: Ultimately,

text: once

text: such

text: studies

text: have

text: been

text: carried

text: out,

text: one

text: can

text: investigate

text: the

text: signatures

text: of

text: dCS

text: gravity

text: on

text: extreme

text: mass-ratio

text: inspirals.

text: The

text: only

text: studies

text: on

text: this

text: topic

text: are

text: those

text: of

text: which

text: did

text: not

text: considered

text: a

text: resummed

text: metric

text: and

text: did

text: not

text: include

text: the

text: magnetic-dipole

text: force

text: due

text: to

text: the

text: small

text: object.

text: Redoing

text: such

text: an

text: analysis

text: would

text: allow

text: us

text: to

text: estimate

text: the

text: accuracy

text: to

text: which

text: dCS

text: gravity

text: could

text: be

text: constrained

text: by

text: future

text: gravitational

text: wave

text: observations

text: with

text: LISA.

text: Acknowledgements

text: We

text: thank

text: Neil

text: Cornish,

text: Dimitry

text: Ayzenberg,

text: Kent

text: Yagi,

text: Paolo

text: Pani,

text: Andrea

text: Masselli,

text: Charles

text: Kankelborg,

text: Thomas

text: Sotiriou,

text: Olivier

text: Sarbach

text: and

text: Houri

text: Tsuyoshi

text: for

text: very

text: useful

text: suggestions

text: and

text: comments.

text: A.C.-A.

text: acknowledges

text: funding

text: from

text: the

text: Fundaci´on

text: Universitaria

text: Konrad

text: Lorenz

text: (Project

text: N.Y.

text: and

text: A.C.-

text: A.

text: acknowledge

text: ﬁnancial

text: support

text: through

text: NSF

text: CAREER

text: grant

text: and

text: NASA

text: grants

text: and

text: L.A.P.

text: acknowledges

text: funding

text: from

text: the

text: Comit´e

text: para

text: el

text: Desarrollo

text: de

text: la

text: Investigaci´on

text: -CODI-

text: of

text: Universidad

text: de

text: Antioquia,

text: Colombia

text: under

text: the

text: Estrateg´ıa

text: de

text: Sostenibilidad.

text: Computational

text: eﬀorts

text: were

text: performed

text: on

text: the

text: Hyalite

text: High-Performance

text: Computing

text: System,

text: operated

text: and

text: supported

text: by

text: Montana

text: State

text: University’s

text: Information

text: Technology

text: Center.

text: Appendix

text: A.

text: BH

text: Solutions

text: in

text: dCS

text: gravity

text: to

text: ﬁfth

text: order

text: in

text: Boyer

text: Lindquist

text: Coordinates

text: The

text: dCS

text: metric

text: in

text: Boyer-Lindquist

text: coordinates

text: (t,

text: r,

text: θ,

text: φ)

text: is

text: obtained

text: from

text: the

text: solution

text: given

text: in

text: Ref.

text: in

text: Hartle-Thorne

text: coordinates

text: (t,

text: r

text: θ

text: φ),

text: after

text: making

text: HT

text: HT

text: a

text: coordinate

text: transformation

text: im

text: plicitly

text: given

text: by,

text: r

text: χ4

text: HT

text: r)

text: ##LTLine##


text:  The exact dCS metric for a spinning BH possesses a fourth constant of motion: ...21 (cid:0) (cid:1)  − − − 5 cos4(θ) 192M4 60M3r 28M2r2 + 3Mr3 + r4 (cid:0)  M6 cos(2θ)(3M + r) + ζχ4 2 5757696M6 + 1656648M5r 602112r12 (cid:1) (cid:0) − − − 275748M4r2 709360M3r3 173364M2r4 − − − 14154Mr5 4627r6 + 3(2M r) cos(2θ) 2878848M5 +553140M4r + 226(cid:1)560M3r2 − 144380M2r3 − − 1092Mr4 273r5 (A.1)  M4 sin(θ) cos(θ) − θ = χ4 3r(2M + r)2  HT 8r7 (cid:0) (cid:1)  − + 2 cos2(θ) 12M3 + 2M2r + 4Mr2 + r3 . (A.2) O The extra terms of the metric (4) are, to (ζ, χ5), (cid:18)  1 M5 2624 M 492831 M2 1771487 M3 gdCS = ζχ4 1 + + + tt 384 r5 35 r 3920 r2 1680 r3 330775 M4 4430511 M(cid:19)5 − 6957813 M(cid:0)6 6488861(cid:1)M(cid:18)7 + + + 168 r4 980 r5 980 r6 980 r7 667071 M8 15984 M9 − 1819 M3 − M + + 3 cos2(θ) 1 1 + 70 r8 5 r9 56448 r3 r 51806 M2 135383 M3 309664 M4 − 36264049 M5 + + + 12733 r2 63665 r3 38199 r4 381990 r5 −7873793 M6 − 32533551 M7(cid:19) 73025558 M8 −(cid:18)8708988 M9 + 38199 r6 63665 r7 63665 r8 12733 r9 −433800 M10 − 3483648 M11 − 701429 M5 1013451 M 1 + 1819 r10 1819 r11 23708160 r5 701429 r 1154835 M2 − 3346744 M3 3992148 M4 − 9591516 M5 + + 701429 r2 701429 r3 701429 r4 701429 r5  94091244 M6 −(cid:19)103967604 M7 − 41345640 M8 + (cid:0) (cid:1) 701429 r6 701429 r7 701429 r8 109734912 M9 − +  (cid:18) 35 cos4(θ) 30 cos2(θ) + 3 , (A.3) 701429 r9  5 M4 577 M 8113 M2 − 109309 M3 gdCS = ζχ4 1 + + rr 384 r4f (r)3 175 r 1200 r2 11760 r3 2125311 M4 267403 M5 2001821 M6 − 19927289 M7 (cid:19) + + + 3920 r4 1470 r5 420 r6 980 r7 22161021 M8 − 12553726 M9 249993 M10 735264 M11 + + + 980 r8 245 r9 5 r10 25 r11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: The

label: paragraph

text: exact

text: dCS

text: metric

text: for

text: a

text: spinning

text: BH

text: possesses

text: a

text: fourth

text: constant

text: of

text: motion:

text: r)

text: ζχ4

text: r)

text: sin(θ)

text: cos(θ)

text: θ

text: χ4

text: HT

text: O

text: The

text: extra

text: terms

text: of

text: the

text: metric

text: are,

text: to

text: (ζ,

text: M

text: gdCS

text: ζχ4

text: tt

text: r

text: M

text: r

text: M

text: r

text: M

text: gdCS

text: ζχ4

text: rr

text: r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The exact dCS metric for a spinning BH possesses a fourth constant of motion: ...25 [44] Lukes-Gerakopoulos G 2012 Phys. Rev. D86 044013 (Preprint 1206.0660) [45] Zelenka O and Loukes-Gerakopoulos G 2017 Chaotic motion in the Johannsen-Psaltis spacetime (Preprint 1711.02442) URL https://inspirehep.net/record/1634827/files/arXiv:1711. 02442.pdf [46] Lukes-Gerakopoulos G and Kopa´ˇcek O 2017 Int. J. Mod. Phys. D27 1850010 (Preprint 1709.08446) [47] Lukes-Gerakopoulos G and Contopoulos G 2013 Mind the resonances: Final stages of accretion into bumpy black holes Journal of Physics: Conference Series vol 453 (IOP Publishing) p 012005 [48] Canizares P, Gair J R and Sopuerta C F 2012 Phys. Rev. D86 044010 (Preprint 1205.1253) [49] Stein L C and Yagi K 2014 Physical Review D 89 044026 [50] Vincent F 2013 Classical and Quantum Gravity 31 025010 [51] Ayzenberg D and Yunes N 2017 Classical and Quantum Gravity 34 115003 [52] Dormand J R and Prince P J 1980 Journal of computational and applied mathematics 6 19–26 [53] Bodenheimer P, Laughlin G P, Rozyczka M, Plewa T, Yorke H W and Yorke H W 2006 Numerical methods in astrophysics: an introduction (Taylor & Francis) [54] Apostolatos T A, Lukes-Gerakopoulos G and Contopoulos G 2009 Phys. Rev. Lett. 103 111101 (Preprint 0906.0093) [55] Brink J, Geyer M and Hinderer T 2015 Phys. Rev. D91 083001 (Preprint 1501.07728)

text: The

text: exact

text: dCS

text: metric

text: for

text: a

text: spinning

text: BH

text: possesses

text: a

text: fourth

text: constant

text: of

text: motion:

label: reference

text: Lukes-Gerakopoulos

text: G

text: Phys.

text: Rev.

text: (Preprint

text: Zelenka

text: O

text: and

text: Loukes-Gerakopoulos

text: G

text: Chaotic

text: motion

text: in

text: the

text: Johannsen-Psaltis

text: spacetime

text: (Preprint

text: URL

text: Lukes-Gerakopoulos

text: G

text: and

text: Kopa´ˇcek

text: O

text: Int.

text: J.

text: Mod.

text: Phys.

text: (Preprint

text: Lukes-Gerakopoulos

text: G

text: and

text: Contopoulos

text: G

text: Mind

text: the

text: resonances:

text: Final

text: stages

text: of

text: accretion

text: into

text: bumpy

text: black

text: holes

text: Journal

text: of

text: Physics:

text: Conference

text: Series

text: vol

text: (IOP

text: Publishing)

text: p

text: Canizares

text: P,

text: Gair

text: J

text: R

text: and

text: Sopuerta

text: C

text: F

text: Phys.

text: Rev.

text: (Preprint

text: Stein

text: L

text: C

text: and

text: Yagi

text: K

text: Physical

text: Review

text: D

text: Vincent

text: F

text: Classical

text: and

text: Quantum

text: Gravity

text: Ayzenberg

text: D

text: and

text: Yunes

text: N

text: Classical

text: and

text: Quantum

text: Gravity

text: Dormand

text: J

text: R

text: and

text: Prince

text: P

text: J

text: Journal

text: of

text: computational

text: and

text: applied

text: mathematics

text: Bodenheimer

text: P,

text: Laughlin

text: G

text: P,

text: Rozyczka

text: M,

text: Plewa

text: T,

text: Yorke

text: H

text: W

text: and

text: Yorke

text: H

text: W

text: Numerical

text: methods

text: in

text: astrophysics:

text: an

text: introduction

text: (Taylor

text: Francis)

text: Apostolatos

text: T

text: A,

text: Lukes-Gerakopoulos

text: G

text: and

text: Contopoulos

text: G

text: Phys.

text: Rev.

text: Lett.

text: (Preprint

text: Brink

text: J,

text: Geyer

text: M

text: and

text: Hinderer

text: T

text: Phys.

text: Rev.

text: (Preprint


text: 4 HAIAN HE ΑΑ22 ΑΑ77 ΑΑ66 ΑΑ55 ΑΑ44 ΑΑ33 ΑΑ11 Figure 2. Dynkin diagram of E . 7 ω(H ) = H , ω(X± ) = X± , α1 α6 α1 α6 ω(H ) = H , ω(X± ) = X± , α2 α2 α2 α2 ω(H ) = H , ω(X± ) = X± , α3 α5 α3 α5 ω(H ) = H , ω(X± ) = X± . √ α4 α4√ α4 α4 √ − − − Let τ = exp( 1πH ), τ = exp( 1π(H + H )), τ = ω, τ = ωexp( 1πH ), 1 α2 2 α1 α6 3 4 α2 where exp represents the exponential map from e − to Aute − . Then τ , τ , τ , and 6( 78) 6( 78) 1 2 3 τ represent all conjugacy classes of involutions in Aute − , which correspond to real forms 4 6( 78) e6(2), e6(−14), e6(−26), and e6(6). ∼ From [5], it is known that (Inte6(−78))τ3 = F4(−52), the compact Lie gr∼oup of t⊕ype F4, and there exist an involutive automorphism η of F4(−52) such that fη4(−52) = sp(3) sp(1∼), where×f4(−52) d(cid:104)e−notes−th(cid:105)e compact Lie algebra of type F4. Moreover, ((Inte6(−78))τ3)η = Sp(3) Sp(1)/ ( I , 1) . Let i, j, and k denote the fundamental quaternion units, and then 3 − 1 0 0   − − set x = τ , x = η = (I , 1), x = (iI , i), x = (jI , j), x = ( 0 1 0 , 1), and 0 3 1 3 2 3 3 3 4   0 0 1 − 1 0 0 x = ( 0 1 0 , 1). 5 − 0 0 1 ≤ ≤ For a pair (r, s) of integers with r 2 and s 3, deﬁne (cid:104) · · · · · · (cid:105) F := x , x , , x , x , x , , x r,s 0 1 s 4 5 r+3 and (cid:48) (cid:104)  · · · · ·· (cid:105)   F := x , x , , x , x , x , , x . (cid:18) (cid:19) (cid:18) r,s(cid:19) 1∼ 2 (cid:104)s− 4(cid:105) 5 r+3 On the other hand, (Inte6(−78))τ4 =Sp(4)/ I4 . Set y0 = σ4, y1 = iI4, y2 = jI4, y3 =  1 0 0 0   0 1 0 0  − − I 0 0 I 0 1 0 0 1 0 0 0 2 , y = 2 , y = , and y = . 0 I 4 I 0 5 0 0 1 0 6 0 0 0 1 2 2 − 0 0 0 1 0 0 1 0 ≤ ≤ For a quadruplet (u, v, r, s) of integers with u + v 1 and r + s 2, deﬁne (cid:104) · · · · · · · · · (cid:105) F := y , y , , y , y , y , , y , y , y , , y u,v,r,s 0 1 u+2v 3 4 2+2s 3+2s 5+2s 1+2r+2s and (cid:48) (cid:104) · · · · · · · · · (cid:105) F := y , y , , y , y , y , , y , y , y , , y . u,v,r,s 1 2 u+2v 3 4 2+2s 3+2s 5+2s 1+2r+2s According to [12, Proposition 6.3 & Proposition 6.5], each elementary abelian 2-subgroup in (cid:48) (cid:48) Autu is conjugate to one of the groups in the one family of F , F , F , and F , 0 (cid:48) (cid:48) r,s r,s u,v,r,s u,v,r,s and all groups in the families of F , F , F , and F are pairwisely non-conjugate. r,s r,s u,v,r,s u,v,r,s Let g = e , the complex simple Lie algebra of type E . Fix a Cartan subalgebra of e and 7 { | ≤ ≤ } 7 7 a simple root system α 1 i 7 , the Dynkin diagram of which is given in Figure 2. Let i u = e − , the compact simple Lie algebra of type E , and G = Autu . By [12], there are 0 7( 133) 7 0 exactly three conjugacy classes of involutive automorphisms in G with representatives σ , σ , 1 2 and σ3 such that ∼ × (cid:104) − − (cid:105) Gσ1 = (Spin(12) Sp(1))/ (c, 1), ( c, 1) , Gσ2 ∼= ((E6(−78) × U(1))/(cid:104)(c(cid:48), e2π√3−1 )(cid:105)) (cid:111) (cid:104)ω(cid:105), ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: HAIAN

text: HE

text: ΑΑ22

font: default

text: ΑΑ77

text: ΑΑ66

text: ΑΑ55

text: ΑΑ44

text: ΑΑ33

text: ΑΑ11

text: Figure

text: Dynkin

label: caption

text: diagram

text: of

text: E

text: ω(H

text: H

text: ω(X±

text: X±

text: α1

text: α6

text: α1

text: α6

text: ω(H

text: H

text: ω(X±

text: X±

text: α2

text: α2

text: α2

text: α2

text: ω(H

text: H

text: ω(X±

text: X±

text: α3

text: α5

text: α3

text: α5

text: ω(H

text: H

text: ω(X±

text: X±

text: α4

text: α4√

text: α4

text: α4

text: Let

text: τ

text: exp(

text: τ

text: exp(

text: H

text: τ

text: ω,

text: τ

text: ωexp(

text: α2

text: α1

text: α6

text: α2

text: where

text: exp

text: represents

text: the

text: exponential

text: map

text: from

text: e

text: to

text: Aute

text: Then

text: τ

text: τ

text: τ

text: and

text: τ

text: represent

text: all

text: conjugacy

text: classes

text: of

text: involutions

text: in

text: Aute

text: which

text: correspond

text: to

text: real

text: forms

text: and

text: From

text: it

text: is

text: known

text: that

text: the

text: compact

text: Lie

text: gr∼oup

text: of

text: t⊕ype

text: and

text: there

text: exist

text: an

text: involutive

text: automorphism

text: η

text: of

text: such

text: that

text: compact

text: Lie

text: algebra

text: of

text: type

text: Moreover,

text: I

text: Let

text: i,

text: j,

text: and

text: k

text: denote

text: the

text: fundamental

text: quaternion

text: units,

text: and

text: then

text: set

text: x

text: τ

text: x

text: η

text: (I

text: x

text: (iI

text: i),

text: x

text: (jI

text: j),

text: x

text: and

text: x

text: For

text: a

text: pair

text: (r,

text: s)

text: of

text: integers

text: with

text: r

text: and

text: s

text: deﬁne

text: F

text: x

text: x

text: x

text: x

text: x

text: x

text: r,s

text: s

text: and

text: F

text: x

text: x

text: x

text: x

text: x

text: x

text: 1∼

text: On

text: the

text: other

text: hand,

text: Set

text: y0

text: I

text: I

text: y

text: y

text: and

text: y

text: I

text: I

text: For

text: a

text: quadruplet

text: (u,

text: v,

text: r,

text: s)

text: of

text: integers

text: with

text: u

text: v

text: and

text: r

text: s

text: deﬁne

text: F

text: y

text: y

text: y

text: y

text: y

text: y

text: y

text: y

text: y

text: u,v,r,s

text: and

text: F

text: y

text: y

text: y

text: y

text: y

text: y

text: y

text: y

text: y

text: u,v,r,s

text: According

text: to

text: Proposition

text: Proposition

text: each

text: elementary

text: abelian

text: in

text: Autu

text: is

text: conjugate

text: to

text: one

text: of

text: the

text: groups

text: in

text: the

text: one

text: family

text: of

text: F

text: F

text: F

text: and

text: F

text: r,s

text: r,s

text: u,v,r,s

text: u,v,r,s

text: and

text: all

text: groups

text: in

text: the

text: families

text: of

text: F

text: F

text: F

text: and

text: F

text: are

text: pairwisely

text: non-conjugate.

text: r,s

text: r,s

text: u,v,r,s

text: u,v,r,s

text: Let

text: g

text: e

text: the

text: complex

text: simple

text: Lie

text: algebra

text: of

text: type

text: E

text: Fix

text: a

text: Cartan

text: subalgebra

text: of

text: e

text: and

text: a

text: simple

text: root

text: system

text: α

text: i

text: the

text: Dynkin

text: diagram

text: of

text: which

text: is

text: given

text: in

text: Figure

text: Let

text: i

text: u

text: e

text: the

text: compact

text: simple

text: Lie

text: algebra

text: of

text: type

text: E

text: and

text: G

text: Autu

text: By

text: there

text: are

text: exactly

text: three

text: conjugacy

text: classes

text: of

text: involutive

text: automorphisms

text: in

text: G

text: with

text: representatives

text: σ

text: σ

text: and

text: σ3

text: such

text: that

text: (c,

text: c,

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: CLASSIFICATION OF KLEIN FOUR SYMMETRIC√PAIRS OF HOLOMORPHIC TYPE FOR E7(−25) 5 ∼ (cid:104) − (cid:105) (cid:111) (cid:104) (cid:105) Gσ3 = (SU(8)/ 1I ) ω , 8 (cid:48) where c is a certain element in the center of Spin(12), c is a nontrivial element in the center ⊕ o√f the compact Lie group E − , and ω is an involutive automorphism such that (e − − R ∼ ⊕ 6( 78) 6( 78) 1 )ω = f 0, su(8)ω = sp(4). 4 The author needs to ﬁnd all the elementary abelian 2-subgroups in Autu containing σ 0 2 because the noncompact dual of u corresponding to σ is just e − . By [12], the elementary 0 2 7( 25) abelian 2-subgroups in Autu containing σ are all induced from Aute − as follows. 0 2 6( 78) → Denote by Gσ2 the subgroup of Gσ2 genera→ted by E6(−78) and ω. Let p : Gσ2 Aute6(−78) be the adjoint homomorphism, and let i : Gσ2 Gσ2 be the inclusion map−. For ×an(cid:104)y el(cid:105)ementary abelian 2-subgroup F˜ of Aute − , let F be the sylow 2-subgroup of i(p 1F˜) σ . By [12, 6( 78) (cid:55)→ 2 Proposition 7.7&Proposition 7.10], F˜ F gives one to one correspondence from the conjugacy classes of the elementary abelian 2-subgroups of Aute − onto the conjugacy classes of the 6( 78) elementary abelian 2-subgroups of Autu containing σ . 0 ≤ ≤ 2 Retain the previous notations, and τi for 1 i 4 lie in Gσ2. By [12], one has ∼ ∼ τ τ σ , 1 G 2 G 1 ∼ ∼ ∼ τ σ τ τ σ σ , 2 2 G 3 G 3 2 G 2 ∼ ∼ ∼ τ σ τ τ σ σ . 1 2 G 4 G 4 2 G 3 Lemma 5. There are exactly 8 conjugacy classes of the elementary abelian 2-subgroups of (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) rank 3 of Autu containing σ : x , x , σ , x , x , σ , x , x , σ , x , x , σ , x , x , σ , (cid:104) (cid:105) (cid:104) 0 (cid:105) (cid:104)2 0 (cid:105)1 2 0 4 2 1 2 2 1 4 2 4 5 2 y , y , σ , y , y , σ , and y , y , σ . 0 1 2 0 3 2 3 4 2 Proof. The conjugacy classes of the Klein four subgroups of Aute − are given by [12, Propo- 6( 78) sition 6.3&Proposition 6.5], and then the conclusion follows from [12, Proposition 7.7&Propo- (cid:3) sition 7.10]. √ Let g be a non-compact simple Lie algebra. For each involutive automorphism σ of g , 0 − 0 √ there exists a Cartan involution θ of g , which commutes with σ. Denote by g := g + 1g 0 − − 0 0 the complexiﬁcation of g , and write u := gθ + 1g θ for the compact dual of g . Extend 0 0 0 0 0 σ to the unique holomorphic involutive automorphism of g and restrict it to u , and then 0 σ becomes an involutive automorphism of u . Thus, any involutive automorphism σ of g 0 0 gives a pair (u , σ), where u is a compact real form of g and σ is regarded as an involutive 0 (cid:48) 0 (cid:48) (cid:48) automorphism of u . If σ is conjugate to σ in Autg , which gives another pair (u , σ ), then 0 ∈ · 0(cid:48) − (cid:48) 0 there exists an element g Autg such that g u = u and g 1σg = σ ; namely, the two pairs (cid:48) (cid:48) 0 0 (u , σ) and (u , σ ) are conjugate by elements in Autg. Thus, up to conjugations of compact 0 0 real forms of g, there exists a map { } −→ { } π : Conjugacy classes of Autg Conjugacy classes of Autu 0 0 which is surjective but not necessarily injective. ∈ If Γ is a ﬁnite abelian subgroup of Autg , there exists a Cartan involution θ Autg 0 0 centralizing Γ by [4, Proposition 2]. In the same way, Γ becomes a ﬁnite abelian subgroup of Autu . Thus, any ﬁnite abelian subgroup of Autg is obtained from a ﬁnite abelian subgroup 0 0 of Autu . 0 On the other hand, ﬁx an involutive automorphism θ of u , and by holomorphic extension 0 → and restriction, θ is a Cartan involution of a noncompact dual g of u . Let Θ : Autu Autu (cid:55)→ − 0 0 0 0 be given by f θ 1fθ, whose diﬀerential is θ on u . Suppose that Γ and Γ are two ﬁnite 0 − 1 2 ∈ abelian subgroups of Autu , which are centralized by θ. If Γ = g 1Γ g for some g (Autu )Θ, 0 2 1 0 then Γ and Γ are conjugate in Autg because (Autu )Θ is contained in Autg . 1 2 0 0 0 Thus, for a compact Lie algebra u , a pair (θ, Γ) with θ an involutive automorphism of u 0 ∈ 0 and Γ a Klein four subgroup of Autu such that θ / Γ and θ centralizes Γ, gives a Klein four 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CLASSIFICATION

text: OF

text: KLEIN

text: FOUR

text: SYMMETRIC√PAIRS

text: OF

text: HOLOMORPHIC

text: TYPE

text: FOR

text: ω

text: where

text: c

text: is

text: a

text: certain

text: element

text: in

text: the

text: center

text: of

text: c

text: is

text: a

text: nontrivial

text: element

text: in

text: the

text: center

text: o√f

text: the

text: compact

text: Lie

text: group

text: E

text: and

text: ω

text: is

text: an

text: involutive

text: automorphism

text: such

text: that

text: (e

text: R

text: )ω

text: f

text: The

text: author

text: needs

text: to

text: ﬁnd

text: all

text: the

text: elementary

text: abelian

text: in

text: Autu

text: containing

text: σ

text: because

text: the

text: noncompact

text: dual

text: of

text: u

text: corresponding

text: to

text: σ

text: is

text: just

text: e

text: By

text: the

text: elementary

text: abelian

text: in

text: Autu

text: containing

text: σ

text: are

text: all

text: induced

text: from

text: Aute

text: as

text: follows.

text: Denote

text: by

text: the

text: subgroup

text: of

text: genera→ted

text: by

text: and

text: ω.

text: Let

text: p

text: be

text: the

text: adjoint

text: homomorphism,

text: and

text: let

text: i

text: be

text: the

text: inclusion

text: map−.

text: For

text: abelian

text: F˜

text: of

text: Aute

text: let

text: F

text: be

text: the

text: sylow

text: of

text: i(p

text: σ

text: By

text: Proposition

text: F˜

text: F

text: gives

text: one

text: to

text: one

text: correspondence

text: from

text: the

text: conjugacy

text: classes

text: of

text: the

text: elementary

text: abelian

text: of

text: Aute

text: onto

text: the

text: conjugacy

text: classes

text: of

text: the

text: elementary

text: abelian

text: of

text: Autu

text: containing

text: σ

text: Retain

text: the

text: previous

text: notations,

text: and

text: τi

text: for

text: i

text: lie

text: in

text: By

text: one

text: has

text: τ

text: τ

text: σ

text: G

text: G

text: τ

text: σ

text: τ

text: τ

text: σ

text: σ

text: G

text: G

text: G

text: τ

text: σ

text: τ

text: τ

text: σ

text: σ

text: G

text: G

text: G

text: Lemma

text: There

text: are

text: exactly

text: conjugacy

text: classes

text: of

text: the

text: elementary

text: abelian

text: of

text: rank

text: of

text: Autu

text: containing

text: σ

text: x

text: x

text: σ

text: x

text: x

text: σ

text: x

text: x

text: σ

text: x

text: x

text: σ

text: x

text: x

text: σ

text: y

text: y

text: σ

text: y

text: y

text: σ

text: and

text: y

text: y

text: σ

text: Proof.

text: The

text: conjugacy

text: classes

text: of

text: the

text: Klein

text: four

text: subgroups

text: of

text: Aute

text: are

text: given

text: by

text: Propo-

text: sition

text: and

text: then

text: the

text: conclusion

text: follows

text: from

text: Proposition

text: sition

text: Let

text: g

text: be

text: a

text: non-compact

text: simple

text: Lie

text: algebra.

text: For

text: each

text: involutive

text: automorphism

text: σ

text: of

text: g

text: there

text: exists

text: a

text: Cartan

text: involution

text: θ

text: of

text: g

text: which

text: commutes

text: with

text: σ.

text: Denote

text: by

text: g

text: g

text: the

text: complexiﬁcation

text: of

text: g

text: and

text: write

text: u

text: gθ

text: θ

text: for

text: the

text: compact

text: dual

text: of

text: g

text: Extend

text: σ

text: to

text: the

text: unique

text: holomorphic

text: involutive

text: automorphism

text: of

text: g

text: and

text: restrict

text: it

text: to

text: u

text: and

text: then

text: σ

text: becomes

text: an

text: involutive

text: automorphism

text: of

text: u

text: Thus,

text: any

text: involutive

text: automorphism

text: σ

text: of

text: g

text: gives

text: a

text: pair

text: (u

text: σ),

text: where

text: u

text: is

text: a

text: compact

text: real

text: form

text: of

text: g

text: and

text: σ

text: is

text: regarded

text: as

text: an

text: involutive

text: automorphism

text: of

text: u

text: If

text: σ

text: is

text: conjugate

text: to

text: σ

text: in

text: Autg

text: which

text: gives

text: another

text: pair

text: (u

text: σ

text: then

text: there

text: exists

text: an

text: element

text: g

text: Autg

text: such

text: that

text: g

text: u

text: u

text: and

text: g

text: σ

text: namely,

text: the

text: two

text: pairs

text: (u

text: σ)

text: and

text: (u

text: σ

text: are

text: conjugate

text: by

text: elements

text: in

text: Autg.

text: Thus,

text: up

text: to

text: conjugations

text: of

text: compact

text: real

text: forms

text: of

text: g,

text: there

text: exists

text: a

text: map

text: π

text: Conjugacy

text: classes

text: of

text: Autg

text: Conjugacy

text: classes

text: of

text: Autu

text: which

text: is

text: surjective

text: but

text: not

text: necessarily

text: injective.

text: If

text: Γ

text: is

text: a

text: ﬁnite

text: abelian

text: subgroup

text: of

text: Autg

text: there

text: exists

text: a

text: Cartan

text: involution

text: θ

text: Autg

text: centralizing

text: Γ

text: by

text: Proposition

text: In

text: the

text: same

text: way,

text: Γ

text: becomes

text: a

text: ﬁnite

text: abelian

text: subgroup

text: of

text: Autu

text: Thus,

text: any

text: ﬁnite

text: abelian

text: subgroup

text: of

text: Autg

text: is

text: obtained

text: from

text: a

text: ﬁnite

text: abelian

text: subgroup

text: of

text: Autu

text: On

text: the

text: other

text: hand,

text: ﬁx

text: an

text: involutive

text: automorphism

text: θ

text: of

text: u

text: and

text: by

text: holomorphic

text: extension

text: and

text: restriction,

text: θ

text: is

text: a

text: Cartan

text: involution

text: of

text: a

text: noncompact

text: dual

text: g

text: of

text: u

text: Let

text: Θ

text: Autu

text: Autu

text: be

text: given

text: by

text: f

text: θ

text: whose

text: diﬀerential

text: is

text: θ

text: on

text: u

text: Suppose

text: that

text: Γ

text: and

text: Γ

text: are

text: two

text: ﬁnite

text: abelian

text: subgroups

text: of

text: Autu

text: which

text: are

text: centralized

text: by

text: θ.

text: If

text: Γ

text: g

text: 1Γ

text: g

text: for

text: some

text: g

text: (Autu

text: )Θ,

text: then

text: Γ

text: and

text: Γ

text: are

text: conjugate

text: in

text: Autg

text: because

text: (Autu

text: )Θ

text: is

text: contained

text: in

text: Autg

text: Thus,

text: for

text: a

text: compact

text: Lie

text: algebra

text: u

text: a

text: pair

text: (θ,

text: Γ)

text: with

text: θ

text: an

text: involutive

text: automorphism

text: of

text: u

text: and

text: Γ

text: a

text: Klein

text: four

text: subgroup

text: of

text: Autu

text: such

text: that

text: θ

text: Γ

text: and

text: θ

text: centralizes

text: Γ,

text: gives

text: a

text: Klein

text: four

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 HAIAN HE ∈ Finally, any element in those Γ is either in E − or of the form xσ with x E − , and 6( 78) 2 6( 78) (cid:3) it is obvious that it satisﬁes the requirement (iv) in either case. (cid:104) (cid:105) Lemma 11. Suppose that the subgroup generated by θ and Γ is conjugate to x , x , σ in G. 1 4 2 Then the pair (θ, Γ) satisﬁes all of the four requirements if and only if it is conjugate to one (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) of (σ , x , x ), (σ , x , x σ ), and (σ , x σ , x ). 2 1 4 2 1 4 2 2 1 2 4 Proof. Without loss of generality, assumes that the group generated by θ and Γ is equal (cid:104) (cid:105) ∼ ∼ ∼ ∼ to x , x , σ . It is known by [12] that x x x x σ , x σ σ , and 1∼ 4 2 ∼ 1 G 4 G 1 4 G 1 4 2 G 2 x σ x x σ σ . Hence, the elements conjugate to σ are exactly σ and x σ 1 (cid:104)2 G 1(cid:105) 4 2 G 3 2 (cid:104) (cid:105) 2(cid:104) 4(cid:105) 2 in x , x , σ . Thus, there are totally eight possible pairs: (σ , x , x ), (σ , x σ , x ), (cid:104)1 4 2(cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) 2 (cid:105) 1 4 (cid:104) 2 1 (cid:105)2 4 (σ , x , x σ ), (σ , x σ , x σ ), (x σ , x , x ), (x σ , x σ , x ), (x σ , x , x σ ), and 2 1(cid:104) 4 2 (cid:105)2 1 2 4 2 4 2 1 4 4 2 1 2 4 4 2 1 4 2 (x σ , x σ , x σ ). 4 2 1 2 4 2 (cid:104) (cid:105) Let f be a group automorphism from x , x , σ to itself given by f(σ ) = x σ , f(x ) = x , ∼1 2 2 ∈ (cid:104) (cid:105) 2 4 2 1 1 and f(x ) = x . It is obvious that f(g) g for all g x , x , σ . Moreover, it is obvious 4 4 G 1 4 2 that m(x , x ) = m(x , x x ) = m(x x , x ). Hence by [12, Proposition 7.25], f gives a 1 4 1 (cid:104)1 4 (cid:105) 1 4 (cid:104) 4 (cid:105) (cid:104) (cid:105) conjugation between (σ , x , x ) and (σ , x , x ). Similary, one shows that (σ , x σ , x ), (cid:104) (cid:105) 2 (cid:104) 1 4 (cid:105) 2 1 4 (cid:104) (cid:105) 2 (cid:104) 1 2 4(cid:105) (σ , x , x σ ), and (σ , x σ , x σ ) are conjugate to (x σ , x σ , x ), (x σ , x , x σ ), 2 1 4 (cid:104)2 (cid:105)2 1 2 4 2 (cid:104) 4 2 (cid:105)1 2 4 (cid:104)4 2 1 4(cid:105) 2 and (x σ , x σ , x σ ) respectively. Furthermore, (σ , x , x σ ) and (σ , x σ , x σ ) are 4 2 1 2 4 2 2 1 4 2 2 1 2 4 2 shown to be conjugate in the similar way. (cid:104) (cid:105) (cid:104) (cid:105) On the other hand, x , x contains exactly three elements conjugate to σ , x σ , x 1 4 1 (cid:104) 1 2 4(cid:105) contains exactly one element conjugate to σ and two elements conjugate to σ , and x , x σ 1 3 1 4 2 contains exactly one element conjugate to σ , one element conjugate to σ , and one element 1 2 (cid:104) (cid:105) conjugate to σ . Therefore, these Γ are pairwisely non-conjugate. It follows that (σ , x , x ), (cid:104) (cid:105)3 (cid:104) (cid:105) 2 1 4 (σ , x , x σ ), and (σ , x σ , x ) are pairwisely non-conjugate. 2 1 4 2 2 1 2 4 Finally, for the same reason as the proof for Lemma 10, all the pairs satisfy the requirement (cid:3) (iv). (cid:104) (cid:105) Lemma 12. Suppose that the subgroup generated by θ and Γ is conjugate to x , x , σ in G. 4 5 2 Then the pair (θ, Γ) satisﬁes all of the four requirements if and only if it is conjugate to either (cid:104) (cid:105) (cid:104) (cid:105) (σ , x , x ) or (σ , x , x σ ). 2 4 5 2 4 5 2 Proof. Without loss of generality, assumes that the group generated by θ and Γ is equal to (cid:104) (cid:105) x , x , σ . The proof is parallel to that for Lemma 11, so the author omits the detailed 4 5 2 ∼ ∼ ∼ ∼ calculation and just states the sketch. Firstly, x x x x σ and x σ ∼ ∼ 4 G 5 G 4 5 G 1 (cid:104) 4 2(cid:105) G x σ x x σ σ . Secondly, any pair (θ, Γ) is conjugate to either (σ , x , x ) or 5 2(cid:104) G 4 (cid:105)5 2 G 2 (cid:104) (cid:105) (cid:104) (cid:105) 2 4 5 (σ , x , x σ ). Thirdly, (σ , x , x ) and (σ , x , x σ ) are not conjugate. Finally, they 2 4 5 2 2 4 5 2 4 5 2 (cid:3) satisfy the four requirements. (cid:104) (cid:105) Lemma 13. Suppose that the subgroup generated by θ and Γ is conjugate to y , y , σ in G. 3 4 2 Then the pair (θ, Γ) satisﬁes all of the four requirements if and only if it is conjugate to one (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) of (σ , y , y ), (σ , y , y σ ), and (σ , y σ , y σ ). 2 3 4 2 3 4 2 2 3 2 4 2 Proof. Without loss of generality, assumes that the group generated by θ and Γ is equal to (cid:104) (cid:105) y , y , σ . The proof is parallel to that for Lemma 11, so the author omits the detailed 3 4 2 ∼ ∼ ∼ ∼ ∼ calculation and just states the sketch. Firstly, y y y y σ , y σ y σ σ , ∼ 3 G 4 G 3 4 G (cid:104)1 3 (cid:105)2 G (cid:104)4 2 G (cid:105)2 and y y σ σ . Secondly, any pair (θ, Γ) is conjugate to one of (σ , y , y ), (σ , y , y σ ), 3 4 (cid:104)2 G 3 (cid:105) 2 3 4 2 3 4 2 and (σ , y σ , y σ ). Thirdly, the three pairs are pairwisely non-conjugate. Finally, they 2 3 2 4 2 (cid:3) satisfy the four requirements.

text: HAIAN

text: HE

text: Finally,

text: any

text: element

text: in

text: those

text: Γ

text: is

text: either

text: in

text: E

text: or

text: of

text: the

text: form

text: xσ

text: with

text: x

text: E

text: and

text: it

text: is

text: obvious

text: that

text: it

text: satisﬁes

text: the

text: requirement

text: (iv)

text: in

text: either

text: case.

text: Lemma

text: Suppose

text: that

text: the

text: subgroup

text: generated

text: by

text: θ

text: and

text: Γ

text: is

text: conjugate

text: to

text: x

text: x

text: σ

text: in

text: G.

text: Then

text: the

text: pair

text: (θ,

text: Γ)

text: satisﬁes

text: all

text: of

text: the

text: four

text: requirements

text: if

text: and

text: only

text: if

text: it

text: is

text: conjugate

text: to

text: one

text: of

text: (σ

text: x

text: x

text: (σ

text: x

text: x

text: σ

text: and

text: (σ

text: x

text: σ

text: x

text: Proof.

text: Without

text: loss

text: of

text: generality,

text: assumes

text: that

text: the

text: group

text: generated

text: by

text: θ

text: and

text: Γ

text: is

text: equal

text: to

text: x

text: x

text: σ

text: It

text: is

text: known

text: by

text: that

text: x

text: x

text: x

text: x

text: σ

text: x

text: σ

text: σ

text: and

text: 1∼

text: G

text: G

text: G

text: G

text: x

text: σ

text: x

text: x

text: σ

text: σ

text: Hence,

text: the

text: elements

text: conjugate

text: to

text: σ

text: are

text: exactly

text: σ

text: and

text: x

text: σ

text: G

text: G

text: in

text: x

text: x

text: σ

text: Thus,

text: there

text: are

text: totally

text: eight

text: possible

text: pairs:

text: (σ

text: x

text: x

text: (σ

text: x

text: σ

text: x

text: (σ

text: x

text: x

text: σ

text: (σ

text: x

text: σ

text: x

text: σ

text: (x

text: σ

text: x

text: x

text: (x

text: σ

text: x

text: σ

text: x

text: (x

text: σ

text: x

text: x

text: σ

text: and

text: (x

text: σ

text: x

text: σ

text: x

text: σ

text: Let

text: f

text: be

text: a

text: group

text: automorphism

text: from

text: x

text: x

text: σ

text: to

text: itself

text: given

text: by

text: f(σ

text: x

text: σ

text: f(x

text: x

text: ∼1

text: and

text: f(x

text: x

text: It

text: is

text: obvious

text: that

text: f(g)

text: g

text: for

text: all

text: g

text: x

text: x

text: σ

text: Moreover,

text: it

text: is

text: obvious

text: G

text: that

text: m(x

text: x

text: m(x

text: x

text: x

text: m(x

text: x

text: x

text: Hence

text: by

text: Proposition

text: f

text: gives

text: a

text: conjugation

text: between

text: (σ

text: x

text: x

text: and

text: (σ

text: x

text: x

text: Similary,

text: one

text: shows

text: that

text: (σ

text: x

text: σ

text: x

text: (σ

text: x

text: x

text: σ

text: and

text: (σ

text: x

text: σ

text: x

text: σ

text: are

text: conjugate

text: to

text: (x

text: σ

text: x

text: σ

text: x

text: (x

text: σ

text: x

text: x

text: σ

text: and

text: (x

text: σ

text: x

text: σ

text: x

text: σ

text: respectively.

text: Furthermore,

text: (σ

text: x

text: x

text: σ

text: and

text: (σ

text: x

text: σ

text: x

text: σ

text: are

text: shown

text: to

text: be

text: conjugate

text: in

text: the

text: similar

text: way.

text: On

text: the

text: other

text: hand,

text: x

text: x

text: contains

text: exactly

text: three

text: elements

text: conjugate

text: to

text: σ

text: x

text: σ

text: x

text: contains

text: exactly

text: one

text: element

text: conjugate

text: to

text: σ

text: and

text: two

text: elements

text: conjugate

text: to

text: σ

text: and

text: x

text: x

text: σ

text: contains

text: exactly

text: one

text: element

text: conjugate

text: to

text: σ

text: one

text: element

text: conjugate

text: to

text: σ

text: and

text: one

text: element

text: conjugate

text: to

text: σ

text: Therefore,

text: these

text: Γ

text: are

text: pairwisely

text: non-conjugate.

text: It

text: follows

text: that

text: (σ

text: x

text: x

text: (σ

text: x

text: x

text: σ

text: and

text: (σ

text: x

text: σ

text: x

text: are

text: pairwisely

text: non-conjugate.

text: Finally,

text: for

text: the

text: same

text: reason

text: as

text: the

text: proof

text: for

text: Lemma

text: all

text: the

text: pairs

text: satisfy

text: the

text: requirement

text: (iv).

text: Lemma

text: Suppose

text: that

text: the

text: subgroup

text: generated

text: by

text: θ

text: and

text: Γ

text: is

text: conjugate

text: to

text: x

text: x

text: σ

text: in

text: G.

text: Then

text: the

text: pair

text: (θ,

text: Γ)

text: satisﬁes

text: all

text: of

text: the

text: four

text: requirements

text: if

text: and

text: only

text: if

text: it

text: is

text: conjugate

text: to

text: either

text: (σ

text: x

text: x

text: or

text: (σ

text: x

text: x

text: σ

text: Proof.

text: Without

text: loss

text: of

text: generality,

text: assumes

text: that

text: the

text: group

text: generated

text: by

text: θ

text: and

text: Γ

text: is

text: equal

text: to

text: x

text: x

text: σ

text: The

text: proof

text: is

text: parallel

text: to

text: that

text: for

text: Lemma

text: so

text: the

text: author

text: omits

text: the

text: detailed

text: calculation

text: and

text: just

text: states

text: the

text: sketch.

text: Firstly,

text: x

text: x

text: x

text: x

text: σ

text: and

text: x

text: σ

text: G

text: G

text: G

text: G

text: x

text: σ

text: x

text: x

text: σ

text: σ

text: Secondly,

text: any

text: pair

text: (θ,

text: Γ)

text: is

text: conjugate

text: to

text: either

text: (σ

text: x

text: x

text: or

text: G

text: G

text: (σ

text: x

text: x

text: σ

text: Thirdly,

text: (σ

text: x

text: x

text: and

text: (σ

text: x

text: x

text: σ

text: are

text: not

text: conjugate.

text: Finally,

text: they

text: satisfy

text: the

text: four

text: requirements.

text: Lemma

text: Suppose

text: that

text: the

text: subgroup

text: generated

text: by

text: θ

text: and

text: Γ

text: is

text: conjugate

text: to

text: y

text: y

text: σ

text: in

text: G.

text: Then

text: the

text: pair

text: (θ,

text: Γ)

text: satisﬁes

text: all

text: of

text: the

text: four

text: requirements

text: if

text: and

text: only

text: if

text: it

text: is

text: conjugate

text: to

text: one

text: of

text: (σ

text: y

text: y

text: (σ

text: y

text: y

text: σ

text: and

text: (σ

text: y

text: σ

text: y

text: σ

text: Proof.

text: Without

text: loss

text: of

text: generality,

text: assumes

text: that

text: the

text: group

text: generated

text: by

text: θ

text: and

text: Γ

text: is

text: equal

text: to

text: y

text: y

text: σ

text: The

text: proof

text: is

text: parallel

text: to

text: that

text: for

text: Lemma

text: so

text: the

text: author

text: omits

text: the

text: detailed

text: calculation

text: and

text: just

text: states

text: the

text: sketch.

text: Firstly,

text: y

text: y

text: y

text: y

text: σ

text: y

text: σ

text: y

text: σ

text: σ

text: G

text: G

text: G

text: G

text: G

text: and

text: y

text: y

text: σ

text: σ

text: Secondly,

text: any

text: pair

text: (θ,

text: Γ)

text: is

text: conjugate

text: to

text: one

text: of

text: (σ

text: y

text: y

text: (σ

text: y

text: y

text: σ

text: G

text: and

text: (σ

text: y

text: σ

text: y

text: σ

text: Thirdly,

text: the

text: three

text: pairs

text: are

text: pairwisely

text: non-conjugate.

text: Finally,

text: they

text: satisfy

text: the

text: four

text: requirements.


text: 12 where g(FS) (θ) denotes the Fubini-Study metric. The inﬁnitesimal line element ds2 corresponding to the Fubini- ij FS Study metric tensor g(FS) (θ) is given by, ij ds2 = g(FS) (θ) dθidθj. (35) FS ij The metric tensor components g(FS) (θ) must be such that [49]: (1) they transform properly under a change of the ij → (cid:48) (cid:48) → (cid:48) coordinates θ θ = θ (θ), (2) they are invariant under gauge transformations, ψ (θ) ψ (θ) = eiα(θ)ψ (θ), and (3) they deﬁne a positive deﬁnite metric tensor. Imposing these conditions, it can be shown that ds2 can be deﬁned as, FS (cid:107) (cid:107) − |(cid:104) | (cid:105)| (cid:104) | (cid:105) − (cid:104) | (cid:105) (cid:104) | (cid:105) (cid:104) | (cid:105) − | (cid:104) (cid:48)| (cid:105) | ds2 d=ef dψ 2 ψ dψ 2 = dψ dψ dψ ψ ψ dψ = dψ⊥ dψ⊥ = 1 ψ ψ 2, (36) FS | (cid:105) | (cid:105) where dψ and dψ⊥ are given by, | (cid:105) | (cid:48)(cid:105) − | (cid:105) | (cid:105) | (cid:105) − | (cid:105) (cid:104) | (cid:105) dψ d=ef ψ ψ , and dψ⊥ d=ef dψ ψ ψ dψ , (37) | (cid:105) | (cid:48)(cid:105) | (cid:105) respectively. For the sake of clarity, we remark that ψ and ψ are two neighboring normalized pure states, dψ | (cid:105) | (cid:105) | (cid:105) | (cid:105) | (cid:48)(cid:105) is the diﬀerence between them, and dψ⊥ is the projection of dψ orthogonal to ψ . Expanding ψ and ψ with {| (cid:105)} ∈ { } respect to an orthonormal basis m with m 1,..., N , we obtain (cid:88) (cid:112) (cid:88) (cid:112) | (cid:105) N | (cid:105) | (cid:48)(cid:105) N | (cid:105) ψ d=ef p (θ)eiφm(θ) m and ψ d=ef p + dp ei(φm+dφm) m , (38) m m m m=1 m=1 respectively. Substituting Eq. (38) into Eq. (36) and recalling Eq. (34), after some tedious but straightforward algebra [35], the inﬁnitesimal Wigner-Yanase line element ds2 = 4ds2 becomes, WY FS   (cid:32) (cid:33)  (cid:88) (cid:88) (cid:88)      2 N p˙2 N − N ds2 = m + 4 p φ˙2 p φ˙ dθ2, (39) WY p m m m m m m=1 m=1 m=1 ##LTLine## ##LTLine## ##LTLine##

text: where

text: g(FS)

text: (θ)

text: denotes

text: the

text: Fubini-Study

text: metric.

text: The

text: inﬁnitesimal

text: line

text: element

text: corresponding

text: to

text: the

text: Fubini-

text: ij

text: FS

text: Study

text: metric

text: tensor

text: g(FS)

text: (θ)

text: is

text: given

text: by,

text: ij

text: g(FS)

text: (θ)

text: dθidθj.

text: FS

text: ij

text: The

text: metric

text: tensor

text: components

text: g(FS)

text: (θ)

text: must

text: be

text: such

text: that

text: they

text: transform

text: properly

text: under

text: a

text: change

text: of

text: the

text: ij

text: coordinates

text: θ

text: θ

text: θ

text: (θ),

text: they

text: are

text: invariant

text: under

text: gauge

text: transformations,

text: ψ

text: (θ)

text: ψ

text: (θ)

text: eiα(θ)ψ

text: (θ),

text: and

text: they

text: deﬁne

text: a

text: positive

text: deﬁnite

text: metric

text: tensor.

text: Imposing

text: these

text: conditions,

text: it

text: can

text: be

text: shown

text: that

text: can

text: be

text: deﬁned

text: as,

text: FS

text: d=ef

text: dψ

text: ψ

text: dψ

text: dψ

text: dψ

text: dψ

text: ψ

text: ψ

text: dψ

text: dψ⊥

text: dψ⊥

text: ψ

text: ψ

text: FS

text: where

text: dψ

text: and

text: dψ⊥

text: are

text: given

text: by,

text: dψ

text: d=ef

text: ψ

text: ψ

text: and

text: dψ⊥

text: d=ef

text: dψ

text: ψ

text: ψ

text: dψ

text: respectively.

text: For

text: the

text: sake

text: of

text: clarity,

text: we

text: remark

text: that

text: ψ

text: and

text: ψ

text: are

text: two

text: neighboring

text: normalized

text: pure

text: states,

text: dψ

text: is

text: the

text: diﬀerence

text: between

text: them,

text: and

text: dψ⊥

text: is

text: the

text: projection

text: of

text: dψ

text: orthogonal

text: to

text: ψ

text: Expanding

text: ψ

text: and

text: ψ

text: with

text: respect

text: to

text: an

text: orthonormal

text: basis

text: m

text: with

text: m

text: N

text: we

text: obtain

text: N

text: N

text: ψ

text: d=ef

text: p

text: (θ)eiφm(θ)

text: m

text: and

text: ψ

text: d=ef

text: p

text: dp

text: ei(φm+dφm)

text: m

text: m

text: m

text: m

text: respectively.

text: Substituting

text: Eq.

text: into

text: Eq.

text: and

text: recalling

text: Eq.

text: after

text: some

text: tedious

text: but

text: straightforward

text: algebra

text: the

text: inﬁnitesimal

text: Wigner-Yanase

text: line

text: element

text: becomes,

text: WY

text: FS

text: N

text: N

text: N

text: m

text: p

text: φ˙2

text: p

text: φ˙

text: WY

text: p

text: m

text: m

text: m

text: m

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 23 leads to the following Euler-Lagrange diﬀerential equations, (cid:18) (cid:19) L L d ∂ − ∂ new new = 0. (90) dθ ∂q˙k¯ ∂qk¯ A straightforward computation yields the following three equalities, (cid:16) (cid:17) L (cid:80) ∂ new = q˙k¯ , ∂q˙k¯ N q˙2 12 k=1 k (cid:18) (cid:19) (cid:16) (cid:17) (cid:16) (cid:17) L (cid:80) (cid:80) d ∂ new = q¨k¯ − q˙k¯2q¨k¯ , dθ ∂q˙k¯ N q˙2 12 N q˙2 32 k=1 k k=1 k L ∂ − ∂qnk¯ew = 2λqk¯. (91) Employing the three relations in Eq. (91), the Euler-Lagrange equations in Eq. (90) become (cid:16) (cid:17) (cid:16) (cid:17) (cid:80) (cid:80) q¨k¯ 1 − q˙k¯2q¨k¯ 3 + 2λqk¯ = 0, (92) N q˙2 2 N q˙2 2 k=1 k k=1 k that is, (cid:32) (cid:33) (cid:88) (cid:80) 1 q¨k¯ − qN˙k¯2q¨k¯q˙2 + 2λ N q˙k2 2 qk¯ = 0. (93) k=1 k k=1 L Observe that in terms of the Lagrangian-like function deﬁned in Eq. (77) expressed in terms of the probability L L L amplitudes q , we ﬁnd that (θ) and ˙ (θ) / (θ) are given by, k (cid:32) (cid:33) (cid:88) (cid:80) 1 L L (θ) = 12F 12 (θ) = N q˙k2 2 , and L˙ ((θθ)) = qN˙k¯q¨k¯q˙2 , (94) k=1 k=1 k ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: leads

text: to

text: the

text: following

text: Euler-Lagrange

text: diﬀerential

text: equations,

text: L

text: L

text: d

text: new

text: new

text: dθ

text: ∂q˙k¯

text: ∂qk¯

text: A

text: straightforward

text: computation

text: yields

text: the

text: following

text: three

text: equalities,

text: L

text: new

text: q˙k¯

text: ∂q˙k¯

text: N

text: k

text: L

text: d

text: new

text: q¨k¯

text: dθ

text: ∂q˙k¯

text: N

text: N

text: k

text: k

text: L

text: ∂qnk¯ew

text: Employing

text: the

text: three

text: relations

text: in

text: Eq.

text: the

text: Euler-Lagrange

text: equations

text: in

text: Eq.

text: become

text: q¨k¯

text: N

text: N

text: k

text: k

text: that

text: is,

text: q¨k¯

text: 2λ

text: N

text: qk¯

text: k

text: L

text: Observe

text: that

text: in

text: terms

text: of

text: the

text: Lagrangian-like

text: function

text: deﬁned

text: in

text: Eq.

text: expressed

text: in

text: terms

text: of

text: the

text: probability

text: L

text: L

text: L

text: amplitudes

text: q

text: we

text: ﬁnd

text: that

text: (θ)

text: and

text: (θ)

text: (θ)

text: are

text: given

text: by,

text: k

text: L

text: L

text: (θ)

text: (θ)

text: N

text: and

text: L˙

text: ((θθ))

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 46 In analogy to the ﬁrst example, the quantum process proceeds at constant speed and, thus, dissipates minimum availability. Moreover, the dissipated availability grows linearly with τ. However, comparing Eqs. (169) and (170) with Eqs. (173) and (174), we observe that while the computational speed of the process is smaller in this second case, the availability loss is also smaller. 3. Example three: Power-law decay F F def In this case, since (θ) = 0 , Eq. (165) becomes (1+Ωθ)4 (cid:18) (cid:19) d2θ − 2Ω dθ 2 = 0. (175) dt2 1 + Ωθ dt Assuming initial conditions θ (t ) = θ and θ˙ (t ) = θ˙ , integrating Eq. (175), we obtain 0 0 0 0 (cid:104) (cid:105) (cid:104) (cid:105) − − θ (t) = (1 + Ωθ0)2 + Ωθ˙0 (t t0) 1+ΩΩθ˙0θ0 . (176) − − Ω2θ˙0 1+ΩΩθ˙0θ0 (t t0) Furthermore, the availability loss Λ in Eq. (158) becomes dissipated F θ˙2 Λ (τ) = 0 0 τ. (177) dissipated 4 (1 + Ωθ )2 0 Finally, the computational speed v in Eq. (166) is given by (cid:112) 1 F 1 v = θ˙ . (178) 2 0 1 + Ωθ 0 0 In analogy to the ﬁrst and second examples, the quantum process proceeds at constant speed and, thus, dissipates minimum availability. In addition, the dissipated availability grows linearly with τ. However, comparing Eqs. (169) and (170) with Eqs. (177) and (178), we observe that while the computational speed of the process is smaller in this third case, the availability loss is also smaller. In Table I, we report the observed behavior of availability losses, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: analogy

text: to

text: the

text: ﬁrst

text: example,

text: the

text: quantum

text: process

text: proceeds

text: at

text: constant

text: speed

text: and,

text: thus,

text: dissipates

text: minimum

text: availability.

text: Moreover,

text: the

text: dissipated

text: availability

text: grows

text: linearly

text: with

text: τ.

text: However,

text: comparing

text: Eqs.

text: and

text: with

text: Eqs.

text: and

text: we

text: observe

text: that

text: while

text: the

text: computational

text: speed

text: of

text: the

text: process

text: is

text: smaller

text: in

text: this

text: second

text: case,

text: the

text: availability

text: loss

text: is

text: also

text: smaller.

label: section

text: Example

text: three:

text: Power-law

text: decay

text: F

text: F

text: def

text: In

text: this

text: case,

text: since

text: (θ)

text: Eq.

text: becomes

text: 2Ω

text: dθ

text: Ωθ

text: dt

text: Assuming

text: initial

text: conditions

text: θ

text: (t

text: θ

text: and

text: θ˙

text: (t

text: θ˙

text: integrating

text: Eq.

text: we

text: obtain

text: θ

text: (t)

text: Ωθ˙0

text: (t

text: 1+ΩΩθ˙0θ0

text: Ω2θ˙0

text: 1+ΩΩθ˙0θ0

text: (t

text: Furthermore,

text: the

text: availability

text: loss

text: Λ

text: in

text: Eq.

text: becomes

text: dissipated

text: F

text: θ˙2

text: Λ

text: (τ)

text: τ.

text: dissipated

text: Ωθ

text: Finally,

text: the

text: computational

text: speed

text: v

text: in

text: Eq.

text: is

text: given

text: by

text: F

text: v

text: θ˙

text: Ωθ

text: In

text: analogy

text: to

text: the

text: ﬁrst

text: and

text: second

text: examples,

text: the

text: quantum

text: process

text: proceeds

text: at

text: constant

text: speed

text: and,

text: thus,

text: dissipates

text: minimum

text: availability.

text: In

text: addition,

text: the

text: dissipated

text: availability

text: grows

text: linearly

text: with

text: τ.

text: However,

text: comparing

text: Eqs.

text: and

text: with

text: Eqs.

text: and

text: we

text: observe

text: that

text: while

text: the

text: computational

text: speed

text: of

text: the

text: process

text: is

text: smaller

text: in

text: this

text: third

text: case,

text: the

text: availability

text: loss

text: is

text: also

text: smaller.

text: In

text: Table

text: I,

text: we

text: report

text: the

text: observed

text: behavior

text: of

text: availability

text: losses,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 48 a diﬀerential equation describing a critically damped harmonic oscillator [see Eqs. (126), (135), and (137)]. The overall picture emerging from the analysis of these three cases inspired us to further investigate the connection between the Fisher information and dissipative eﬀects. 3. We used the Riemannian geometrization of thermodynamical concepts, including thermodynamic speed and dissipated availability, to study the behavior of both the availability loss [see Eq. (158)] and the computational speed [see Eq. (166)] of the quantum processes speciﬁed in terms of the previously mentioned output probability paths. Speciﬁcally, after ﬁnding the optimal parametrization of the statistical variable θ that speciﬁes our output probabilities pk¯ (θ), we evaluated both the availability loss and the computational speed along the optimal geodesic paths corresponding to the above mentioned three scenarios [see Table I together with Eqs. (169), (170), (173), (174), (177), and (178)]. Our main ﬁnding here is that a greater computational speed comes necessarily at the expense of a greater availability loss. As a ﬁnal remark, we recall that from a quantum mechanical standpoint, the output state in Grover’s quantum search algorithm follows a geodesic path obtained from the Fubini-Study metric on the manifold of Hilbert-space rays. In addition, Grover’s algorithm is speciﬁed by a constant Fisher information. A topic of great interest in quantum computing is the investigation of constructive uses of dissipation. For instance, in Ref. [31] it was shown that it is (cid:16)(cid:112) (cid:17) possible to modify Grover’s algorithm by introducing a suitable amount of dissipation in such a manner that the newly obtained algorithm, while preserving the typical number of queries O N/M (where N is the number of items and M is the number of target items), gains robustness by damping out the oscillations between the target and nontarget states. Furthermore, the problem of designing quantum algorithms that are both fast and thermodynamically eﬃcient is a very challenging and relevant problem [88]. To the best of our knowledge, there does not exist any conclusive investigation that concerns this type of issue. In Ref. [89], however, it was shown there that the faster one seeks to implement a shortcut, the higher is the thermodynamic cost of realizing the associated quantum process. Despite the limits of our investigation, we are conﬁdent that our information geometric analysis of the evolution of quantum systems combined with thermodynamical considerations can be especially relevant to information physicists and, more speciﬁcally, quantum information theorists with particular interest in thermodynamical aspects of quantum information. We also strongly believe that the signiﬁcance of our work runs far deeper than what is presently understood. However, signiﬁcant further exploration is needed to make a precise formal connection among parameter- dependent probe Hamiltonians, Fisher information, and optimal cooling paths on the underlying parameter manifold. In conclusion, based also on our recent ﬁndings in quantum computing [36], statistical mechanics [90, 91], and ##LTLine##

text: a

label: list

text: diﬀerential

text: equation

text: describing

text: a

text: critically

text: damped

text: harmonic

text: oscillator

text: [see

text: Eqs.

text: and

text: The

text: overall

text: picture

text: emerging

text: from

text: the

text: analysis

text: of

text: these

text: three

text: cases

text: inspired

text: us

text: to

text: further

text: investigate

text: the

text: connection

text: between

text: the

text: Fisher

text: information

text: and

text: dissipative

text: eﬀects.

text: We

text: used

text: the

text: Riemannian

text: geometrization

text: of

text: thermodynamical

text: concepts,

text: including

text: thermodynamic

text: speed

text: and

text: dissipated

text: availability,

text: to

text: study

text: the

text: behavior

text: of

text: both

text: the

text: availability

text: loss

text: [see

text: Eq.

text: and

text: the

text: computational

text: speed

text: [see

text: Eq.

text: of

text: the

text: quantum

text: processes

text: speciﬁed

text: in

text: terms

text: of

text: the

text: previously

text: mentioned

text: output

text: probability

text: paths.

text: Speciﬁcally,

text: after

text: ﬁnding

text: the

text: optimal

text: parametrization

text: of

text: the

text: statistical

text: variable

text: θ

text: that

text: speciﬁes

text: our

text: output

text: probabilities

text: pk¯

text: (θ),

text: we

text: evaluated

text: both

text: the

text: availability

text: loss

text: and

text: the

text: computational

text: speed

text: along

text: the

text: optimal

text: geodesic

text: paths

text: corresponding

text: to

text: the

text: above

text: mentioned

text: three

text: scenarios

text: [see

text: Table

text: I

text: together

text: with

text: Eqs.

text: and

text: Our

text: main

text: ﬁnding

text: here

text: is

text: that

text: a

text: greater

text: computational

text: speed

text: comes

text: necessarily

text: at

text: the

text: expense

text: of

text: a

text: greater

text: availability

text: loss.

text: As

text: a

text: ﬁnal

text: remark,

text: we

text: recall

text: that

text: from

text: a

text: quantum

text: mechanical

text: standpoint,

text: the

text: output

text: state

text: in

text: Grover’s

text: quantum

text: search

text: algorithm

text: follows

text: a

text: geodesic

text: path

text: obtained

text: from

text: the

text: Fubini-Study

text: metric

text: on

text: the

text: manifold

text: of

text: Hilbert-space

text: rays.

text: In

text: addition,

text: Grover’s

text: algorithm

text: is

text: speciﬁed

text: by

text: a

text: constant

text: Fisher

text: information.

text: A

text: topic

text: of

text: great

text: interest

text: in

text: quantum

text: computing

text: is

text: the

text: investigation

text: of

text: constructive

text: uses

text: of

text: dissipation.

text: For

text: instance,

text: in

text: Ref.

text: it

text: was

text: shown

text: that

text: it

text: is

text: possible

text: to

text: modify

text: Grover’s

text: algorithm

text: by

text: introducing

text: a

text: suitable

text: amount

text: of

text: dissipation

text: in

text: such

text: a

text: manner

text: that

text: the

text: newly

text: obtained

text: algorithm,

text: while

text: preserving

text: the

text: typical

text: number

text: of

text: queries

text: O

text: N/M

text: (where

text: N

text: is

text: the

text: number

text: of

text: items

text: and

text: M

text: is

text: the

text: number

text: of

text: target

text: items),

text: gains

text: robustness

text: by

text: damping

text: out

text: the

text: oscillations

text: between

text: the

text: target

text: and

text: nontarget

text: states.

text: Furthermore,

text: the

text: problem

text: of

text: designing

text: quantum

text: algorithms

text: that

text: are

text: both

text: fast

text: and

text: thermodynamically

text: eﬃcient

text: is

text: a

text: very

text: challenging

text: and

text: relevant

text: problem

text: To

text: the

text: best

text: of

text: our

text: knowledge,

text: there

text: does

text: not

text: exist

text: any

text: conclusive

text: investigation

text: that

text: concerns

text: this

text: type

text: of

text: issue.

text: In

text: Ref.

text: however,

text: it

text: was

text: shown

text: there

text: that

text: the

text: faster

text: one

text: seeks

text: to

text: implement

text: a

text: shortcut,

text: the

text: higher

text: is

text: the

text: thermodynamic

text: cost

text: of

text: realizing

text: the

text: associated

text: quantum

text: process.

text: Despite

text: the

text: limits

text: of

text: our

text: investigation,

text: we

text: are

text: conﬁdent

text: that

text: our

text: information

text: geometric

text: analysis

text: of

text: the

text: evolution

text: of

text: quantum

text: systems

text: combined

text: with

text: thermodynamical

text: considerations

text: can

text: be

text: especially

text: relevant

text: to

text: information

text: physicists

text: and,

text: more

text: speciﬁcally,

text: quantum

text: information

text: theorists

text: with

text: particular

text: interest

text: in

text: thermodynamical

text: aspects

text: of

text: quantum

text: information.

text: We

text: also

text: strongly

text: believe

text: that

text: the

text: signiﬁcance

text: of

text: our

text: work

text: runs

text: far

text: deeper

text: than

text: what

text: is

text: presently

text: understood.

text: However,

text: signiﬁcant

text: further

text: exploration

text: is

text: needed

text: to

text: make

text: a

text: precise

text: formal

text: connection

text: among

text: parameter-

text: dependent

text: probe

text: Hamiltonians,

text: Fisher

text: information,

text: and

text: optimal

text: cooling

text: paths

text: on

text: the

text: underlying

text: parameter

text: manifold.

text: In

text: conclusion,

text: based

text: also

text: on

text: our

text: recent

text: ﬁndings

text: in

text: quantum

text: computing

text: statistical

text: mechanics

text: and

text: ##LTLine##


text: 51 [19] M. Hubner, Explicit computation of the Bures distance for density matrices, Phys. Lett. A163, 239 (1992). [20] N. Margolus and L. B. Levitin, The maximum speed of quantum evolution, Physica D120, 188 (1998). [21] M. Andrecut and M. K. Ali, The adiabatic analog of the Margolus-Levitin theorem, J. Phys. A37, L157 (2004). [22] M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Quantum speed limits for physical processes, Phys. Rev. Lett. 110, 050402 (2013). [23] A. del Campo, I. L. Egusquiza, M. L. Plenio, and S. F. Huelga, Quantum speed limits in open quantum systems, Phys. Rev. Lett. 110, 050403 (2013). [24] S. Deﬀner and E. Lutz, Quantum speed limit for non-Markovian dynamics, Phys. Rev. Lett. 111, 010402 (2013). [25] P. J. Jones and P. Kok, Geometric derivation of the quantum speed limit, Phys. Rev. A82, 022107 (2010). [26] M. Zwierz, Comment on Geometric derivation of the quantum speed limit, Phys. Rev. A86, 016101 (2012). [27] F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum computation and quantum-state engineering driven by dissipation, Nature Phys. 5, 633 (2009). [28] R. J. C. Spreeuw and T. W. Hijmans, Robust quantum searching with spontaneously decaying qubits, Phys. Rev. A76, 022306 (2007). [29] M. H. S. Amin, P. J. Love, and C. J. S. Truncik, Thermally assisted adiabatic quantum computation, Phys. Rev. Lett. 100, 060503 (2008). [30] I. de Vega, M. C. Banuls, and A. Perez, Eﬀects of dissipation on an adiabatic quantum search algorithm, New. J. Phys. 12, 123010 (2010). [31] A. Mizel, Critically damped quantum search, Phys. Rev. Lett. 102, 150501 (2009). [32] J. J. Alvarez and C. Gomez, A comment on Fisher information and quantum algorithms, arXiv:quant-ph/9910115 (2000). [33] A. Miyake and M. Wadati, Geometric strategy for the optimal quantum search, Phys. Rev. A64, 042317 (2001). [34] C. Cafaro and S. Mancini, An information geometric viewpoint of algorithms in quantum computing, in Bayesian Inference and Maximum Entropy Methods in Science and Engineering, AIP Conf. Proc. 1443, 374 (2012). [35] C. Cafaro and S. Mancini, On Grover’s search algorithm from a quantum information geometry viewpoint, Physica A391, 1610 (2012). [36] C. Cafaro, Geometric algebra and information geometry for quantum computational software, Physica A470, 154 (2017). [37] P. Salamon and R. S. Berry, Thermodynamic length and dissipated availability, Phys. Rev. Lett. 51, 1127 (1983). [38] T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & Sons, New York, 2006). [39] G. E. Crooks, Fisher information and statistical mechanics, Technical note 008v4, http://threeplusone.com/sher (2012). [40] S. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994). [41] S. Boixo, S. T. Flammia, C. M. Caves, and JM Geremia, Generalized limits for single-parameter quantum estimation, Phys. Rev. Lett. 98, 090401 (2007).

text: M.

text: Hubner,

text: Explicit

text: computation

text: of

text: the

text: Bures

text: distance

text: for

text: density

text: matrices,

text: Phys.

text: Lett.

text: N.

text: Margolus

text: and

text: L.

text: B.

text: Levitin,

text: The

text: maximum

text: speed

text: of

text: quantum

text: evolution,

text: Physica

text: M.

text: Andrecut

text: and

text: M.

text: K.

text: Ali,

text: The

text: adiabatic

text: analog

text: of

text: the

text: Margolus-Levitin

text: theorem,

text: J.

text: Phys.

text: M.

text: M.

text: Taddei,

text: B.

text: M.

text: Escher,

text: L.

text: Davidovich,

text: and

text: R.

text: L.

text: de

text: Matos

text: Filho,

text: Quantum

text: speed

text: limits

text: for

text: physical

text: processes,

text: Phys.

text: Rev.

text: Lett.

text: A.

text: del

text: Campo,

text: I.

text: L.

text: Egusquiza,

text: M.

text: L.

text: Plenio,

text: and

text: S.

text: F.

text: Huelga,

text: Quantum

text: speed

text: limits

text: in

text: open

text: quantum

text: systems,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Deﬀner

text: and

text: E.

text: Lutz,

text: Quantum

text: speed

text: limit

text: for

text: non-Markovian

text: dynamics,

text: Phys.

text: Rev.

text: Lett.

text: P.

text: J.

text: Jones

text: and

text: P.

text: Kok,

text: Geometric

text: derivation

text: of

text: the

text: quantum

text: speed

text: limit,

text: Phys.

text: Rev.

text: M.

text: Zwierz,

text: Comment

text: on

text: Geometric

text: derivation

text: of

text: the

text: quantum

text: speed

text: limit,

text: Phys.

text: Rev.

text: F.

text: Verstraete,

text: M.

text: M.

text: Wolf,

text: and

text: J.

text: I.

text: Cirac,

text: Quantum

text: computation

text: and

text: quantum-state

text: engineering

text: driven

text: by

text: dissipation,

text: Nature

text: Phys.

text: R.

text: J.

text: C.

text: Spreeuw

text: and

text: T.

text: W.

text: Hijmans,

text: Robust

text: quantum

text: searching

text: with

text: spontaneously

text: decaying

text: qubits,

text: Phys.

text: Rev.

text: M.

text: H.

text: S.

text: Amin,

text: P.

text: J.

text: Love,

text: and

text: C.

text: J.

text: S.

text: Truncik,

text: Thermally

text: assisted

text: adiabatic

text: quantum

text: computation,

text: Phys.

text: Rev.

text: Lett.

text: I.

text: de

text: Vega,

text: M.

text: C.

text: Banuls,

text: and

text: A.

text: Perez,

text: Eﬀects

text: of

text: dissipation

text: on

text: an

text: adiabatic

text: quantum

text: search

text: algorithm,

text: New.

text: J.

text: Phys.

text: A.

text: Mizel,

text: Critically

text: damped

text: quantum

text: search,

text: Phys.

text: Rev.

text: Lett.

text: J.

text: J.

text: Alvarez

text: and

text: C.

text: Gomez,

text: A

text: comment

text: on

text: Fisher

text: information

text: and

text: quantum

text: algorithms,

text: A.

text: Miyake

text: and

text: M.

text: Wadati,

text: Geometric

text: strategy

text: for

text: the

text: optimal

text: quantum

text: search,

text: Phys.

text: Rev.

text: C.

text: Cafaro

text: and

text: S.

text: Mancini,

text: An

text: information

text: geometric

text: viewpoint

text: of

text: algorithms

text: in

text: quantum

text: computing,

text: in

text: Bayesian

text: Inference

text: and

text: Maximum

text: Entropy

text: Methods

text: in

text: Science

text: and

text: Engineering,

text: AIP

text: Conf.

text: Proc.

text: C.

text: Cafaro

text: and

text: S.

text: Mancini,

text: On

text: Grover’s

text: search

text: algorithm

text: from

text: a

text: quantum

text: information

text: geometry

text: viewpoint,

text: Physica

text: C.

text: Cafaro,

text: Geometric

text: algebra

text: and

text: information

text: geometry

text: for

text: quantum

text: computational

text: software,

text: Physica

text: P.

text: Salamon

text: and

text: R.

text: S.

text: Berry,

text: Thermodynamic

text: length

text: and

text: dissipated

text: availability,

text: Phys.

text: Rev.

text: Lett.

text: T.

text: M.

text: Cover

text: and

text: J.

text: A.

text: Thomas,

text: Elements

text: of

text: Information

text: Theory

text: (John

text: Wiley

text: Sons,

text: New

text: York,

text: G.

text: E.

text: Crooks,

text: Fisher

text: information

text: and

text: statistical

text: mechanics,

text: Technical

text: note

text: S.

text: Braunstein

text: and

text: C.

text: M.

text: Caves,

text: Statistical

text: distance

text: and

text: the

text: geometry

text: of

text: quantum

text: states,

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Boixo,

text: S.

text: T.

text: Flammia,

text: C.

text: M.

text: Caves,

text: and

text: JM

text: Geremia,

text: Generalized

text: limits

text: for

text: single-parameter

text: quantum

text: estimation,

text: Phys.

text: Rev.

text: Lett.


text: Published as a conference paper at ICLR 2018 A PROOF OF THE DETERMINIZATION THEOREM The proof for the determinization mentioned in the paper is achieved based on the following lem- mata. Lemma A.1. The absolu(cid:88)te sum of positive and negative d(cid:88)ifferences between the empiric distribution (deterministic frequency) and goal distribution (non-deterministic frequency) is equal. C (a) C (a) f (a) − = − f (a) − C C a:f(a)≥C(a) a:f(a)<C(a) C C (cid:88) (cid:88) Proof. Straightforward from the observation that C (a) f (a) = = 1 C a a (cid:26) (cid:27) Lemma A.2. For any t C (a) − ≤ 1 + b (t) max t f (a) a t t (cid:26) (cid:27) Proof. The proof of A.2 is done by induction. For t = 1 ∀ ∈ C (a) − C (a) − a A : t f (a) = max t f (a) t a t Hence we look at a ∈ A. C (a) − ≤ C (a) t f (a) t t t ≤ 1 + b (1) 1 assume the claim is true for t = T then for t = T + 1 There exists a such that C (a) /T − f (a) ≤ T b (t) which the algorithm chooses for this a. For it C (a) − C (a) + 1 − T+1 f (a) = T f (a) T + 1 T + 1 C (a) − 1 = T f (a) + T + 1 T + 1 − C (a) (T + 1) f (a) 1 = T + T + 1 T + 1 ≤ 1 + b (t) T + 1 It also holds that ∀a(cid:48) ∈ A s.t. a(cid:48) (cid:54)= a C (a) − C (a) − T+1 f (a) = T f (a) T + 1 T + 1 − C (a) (T + 1) f (a) = T T + 1 − C (a) Tf (a) < T T + 1 ≤ 1 + b (t) T + 1 Proof of 3.1. It holds from A.2 together with A.1 that in the step t in the worst case all but one of the actions have Ct(a) − f(a) = 1 and the last action has f(a) − Ct(a) = −|A|−1. So by the bound on sum of positivestand negativestwe get: t t C (a) T lim = f(a) T→∞ T 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Published

text: as

text: a

text: conference

text: paper

text: at

text: ICLR

text: A

text: PROOF

text: OF

text: THE

text: DETERMINIZATION

text: THEOREM

text: The

text: proof

text: for

text: the

text: determinization

text: mentioned

text: in

text: the

text: paper

text: is

text: achieved

text: based

text: on

text: the

text: following

text: lem-

text: mata.

text: Lemma

text: The

text: sum

text: of

text: positive

text: and

text: negative

text: between

text: the

text: empiric

text: distribution

text: (deterministic

text: frequency)

text: and

text: goal

text: distribution

text: (non-deterministic

text: frequency)

text: is

text: equal.

text: C

text: (a)

text: C

text: (a)

text: f

text: (a)

text: f

text: (a)

text: C

text: C

text: a:f(a)≥C(a)

text: a:f(a)<C(a)

text: C

text: C

text: Proof.

text: Straightforward

text: from

text: the

text: observation

text: that

text: C

text: (a)

text: f

text: (a)

text: C

text: a

text: a

text: Lemma

text: For

text: any

text: t

text: C

text: (a)

text: b

text: (t)

text: max

text: t

text: f

text: (a)

text: a

text: t

text: t

text: Proof.

text: The

text: proof

text: of

text: is

text: done

text: by

text: induction.

text: For

text: t

text: C

text: (a)

text: C

text: (a)

text: a

text: A

text: t

text: f

text: (a)

text: max

text: t

text: f

text: (a)

text: t

text: a

text: t

text: Hence

text: we

text: look

text: at

text: a

text: A.

text: C

text: (a)

text: C

text: (a)

text: t

text: f

text: (a)

text: t

text: t

text: t

text: b

text: assume

text: the

text: claim

text: is

text: true

text: for

text: t

text: T

text: then

text: for

text: t

text: T

text: There

text: exists

text: a

text: such

text: that

text: C

text: (a)

text: /T

text: f

text: (a)

text: T

text: b

text: (t)

text: which

text: the

text: algorithm

text: chooses

text: for

text: this

text: a.

text: For

text: it

text: C

text: (a)

text: C

text: (a)

text: f

text: (a)

text: T

text: f

text: (a)

text: T

text: T

text: C

text: (a)

text: T

text: f

text: (a)

text: T

text: T

text: C

text: (a)

text: (T

text: f

text: (a)

text: T

text: T

text: T

text: b

text: (t)

text: T

text: It

text: also

text: holds

text: that

text: A

text: s.t.

text: a

text: C

text: (a)

text: C

text: (a)

text: f

text: (a)

text: T

text: f

text: (a)

text: T

text: T

text: C

text: (a)

text: (T

text: f

text: (a)

text: T

text: T

text: C

text: (a)

text: Tf

text: (a)

text: T

text: T

text: b

text: (t)

text: T

text: Proof

text: of

text: It

text: holds

text: from

text: together

text: with

text: that

text: in

text: the

text: step

text: t

text: in

text: the

text: worst

text: case

text: all

text: but

text: one

text: of

text: the

text: actions

text: have

text: Ct(a)

text: f(a)

text: and

text: the

text: last

text: action

text: has

text: f(a)

text: Ct(a)

text: So

text: by

text: the

text: bound

text: on

text: sum

text: of

text: positivestand

text: negativestwe

text: get:

text: t

text: t

text: C

text: (a)

text: T

text: lim

text: f(a)

text: T→∞

text: T

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Published as a conference paper at ICLR 2018 We prove that given a stochastic action-selection rule f (a|s), every deterministic policy that does not choose an action that was visited too many times until now (with respect to the expected number according to the probability distribution) is a determinization of f. Formally, lets assume that given a certain Q function and state s we wish a certain ratio between(cid:80)different choices of actions a ∈ A to hold. We denote the frequency of this ratio f (a|s). For brevity we assume s and Q are constants Q and denote f (a|s) = f (a). We also assume a counter C (s, a) is kept denoting the number of Q choices of a in s. For brevity we denote C (s, a) = C (a) and C (s, a) = C. When we look at a the counters after T steps we use subscript C (a). Following this notation, note that C = T. T T Theorem 3.1. For any sub-linear function b (t) and for any deterministic policy which chooses at step T an action a such that CT(a) − f (a) ≤ b (t) it holds that ∀a ∈ A T C (a) T lim = f (a) T→∞ T Proof. For a full proof of the theorem see Appendix A in the supplementary materials The result above is not a vacuous truth – we now provide two possible determinization rules that achieves it. One rule is straightforward from the theorem, using b = 0, choosing arg min C(a) − f (a). Another rule follows the probability ratio between the stochastic policy and the empiaricaCl dis- tribution: arg max f(a) . We denote this determinization LLL, because when generalized counters a C(a) are used instead of counters it becomes arg max logf (s, a) − loglog E (s, a). a 1−α Now we can replace the visit counters C (s, a) with the generalized counters log (E (s, a)) to 1−α create Directed Outreaching Reinforcement Action-Selection – DORA the explorer. By this, we can transform any stochastic or counter-based action-selection rule into a deterministic rule in which exploration propagates over the states and the expected trajectories to follow. Input: Stochastic action-selection rule f, learning rate α, Exploration discount factor γ E initialize Q (s, a) = 0, E (s, a) = 1; foreach episode do init s; while not terminated do Choose a = arg max log f (x|s) − log log E (s, x); x Q 1−α Observe transitions (s, a, r, s(cid:48), a(cid:48)); Q (s, a) ← (1 − α) Q (s, a) + α (r + γ max Q (s(cid:48), x)); x E (s, a) ← (1 − α) E (s, a) + αγ E (s(cid:48), a(cid:48)); E end end Algorithm 1: DORA algorithm using LLL determinization for stochastic policy f 3.3 RESULTS – FINITE MDPS To test this algorithm, the ﬁrst set of experiments were done on Bridge environments of various lengths k (Figure 2). We considered the following agents: (cid:15)(cid:113)-greedy, Softmax and their respective LLL determinizations (as described in ) using both counters and E-values. In addition, we com- pared a more standard counter-based agent in the form of a UCB-like algorithm (Auer et al., 2002) following an action-selection rule with exploration bonus of logt. We tested two variants of this C algorithm, using ordinary visit counters and E-values. Each agent’s hyperparameters ((cid:15) and tem- perature) were ﬁtted separately to optimize learning. For stochastic agents, we averaged the results over 50 trials for each execution. Unless stated otherwise, γ = 0.9. E We also used a normalized version of the bridge environment, where all rewards are between 0 and 1, to compare DORA with the Delayed Q-Learning algorithm (Strehl et al., 2006). Our results (Figure 4) demonstrate that E-value based agents outperform both their counter-based and their stochastic equivalents on the bridge problem. As shown in Figure 4, Stochastic and counter-based (cid:15)-greedy agents, as well as the standard UCB fail to converge. E-value agents are the ﬁrst to reach low error values, indicating that they learn faster. Similar results were achieved 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Published

text: as

text: a

text: conference

text: paper

text: at

text: ICLR

text: We

text: prove

text: that

text: given

text: a

text: stochastic

text: action-selection

text: rule

text: f

text: (a|s),

text: every

text: deterministic

text: policy

text: that

text: does

text: not

text: choose

text: an

text: action

text: that

text: was

text: visited

text: too

text: many

text: times

text: until

text: now

text: (with

text: respect

text: to

text: the

text: expected

text: number

text: according

text: to

text: the

text: probability

text: distribution)

text: is

text: a

text: determinization

text: of

text: f.

text: Formally,

text: lets

text: assume

text: that

text: given

text: a

text: certain

text: Q

text: function

text: and

text: state

text: s

text: we

text: wish

text: a

text: certain

text: ratio

text: choices

text: of

text: actions

text: a

text: A

text: to

text: hold.

text: We

text: denote

text: the

text: frequency

text: of

text: this

text: ratio

text: f

text: (a|s).

text: For

text: brevity

text: we

text: assume

text: s

text: and

text: Q

text: are

text: constants

text: Q

text: and

text: denote

text: f

text: (a|s)

text: f

text: (a).

text: We

text: also

text: assume

text: a

text: counter

text: C

text: (s,

text: a)

text: is

text: kept

text: denoting

text: the

text: number

text: of

text: Q

text: choices

text: of

text: a

text: in

text: s.

text: For

text: brevity

text: we

text: denote

text: C

text: (s,

text: a)

text: C

text: (a)

text: and

text: C

text: (s,

text: a)

text: C.

text: When

text: we

text: look

text: at

text: a

text: the

text: counters

text: after

text: T

text: steps

text: we

text: use

text: subscript

text: C

text: (a).

text: Following

text: this

text: notation,

text: note

text: that

text: C

text: T.

text: T

text: T

text: Theorem

text: For

text: any

text: sub-linear

text: function

text: b

text: (t)

text: and

text: for

text: any

text: deterministic

text: policy

text: which

text: chooses

text: at

text: step

text: T

text: an

text: action

text: a

text: such

text: that

text: CT(a)

text: f

text: (a)

text: b

text: (t)

text: it

text: holds

text: that

text: ∀a

text: A

text: T

text: C

text: (a)

text: T

text: lim

text: f

text: (a)

text: T→∞

text: T

text: Proof.

text: For

text: a

text: full

text: proof

text: of

text: the

text: theorem

text: see

text: Appendix

text: A

text: in

text: the

text: supplementary

text: materials

text: The

text: result

text: above

text: is

text: not

text: a

text: vacuous

text: truth

text: we

text: now

text: provide

text: two

text: possible

text: determinization

text: rules

text: that

text: achieves

text: it.

text: One

text: rule

text: is

text: straightforward

text: from

text: the

text: theorem,

text: using

text: b

text: choosing

text: arg

text: min

text: C(a)

text: f

text: (a).

text: Another

text: rule

text: follows

text: the

text: probability

text: ratio

text: between

text: the

text: stochastic

text: policy

text: and

text: the

text: empiaricaCl

text: dis-

text: tribution:

text: arg

text: max

text: f(a)

text: We

text: denote

text: this

text: determinization

text: LLL,

text: because

text: when

text: generalized

text: counters

text: a

text: C(a)

text: are

text: used

text: instead

text: of

text: counters

text: it

text: becomes

text: arg

text: max

text: logf

text: (s,

text: a)

text: loglog

text: E

text: (s,

text: a).

text: a

text: 1−α

text: Now

text: we

text: can

text: replace

text: the

text: visit

text: counters

text: C

text: (s,

text: a)

text: with

text: the

text: generalized

text: counters

text: log

text: (E

text: (s,

text: a))

text: to

text: 1−α

text: create

text: Directed

text: Outreaching

text: Reinforcement

text: Action-Selection

text: DORA

text: the

text: explorer.

text: By

text: this,

text: we

text: can

text: transform

text: any

text: stochastic

text: or

text: counter-based

text: action-selection

text: rule

text: into

text: a

text: deterministic

text: rule

text: in

text: which

text: exploration

text: propagates

text: over

text: the

text: states

text: and

text: the

text: expected

text: trajectories

text: to

text: follow.

text: Input:

text: Stochastic

text: action-selection

text: rule

text: f,

text: learning

text: rate

text: α,

text: Exploration

text: discount

text: factor

text: γ

text: E

text: initialize

text: Q

text: (s,

text: a)

text: E

text: (s,

text: a)

text: foreach

text: episode

text: do

text: init

text: s;

text: while

text: not

text: terminated

text: do

text: Choose

text: a

text: arg

text: max

text: log

text: f

text: (x|s)

text: log

text: log

text: E

text: (s,

text: x);

text: x

text: Q

text: 1−α

text: Observe

text: transitions

text: (s,

text: a,

text: r,

text: Q

text: (s,

text: a)

text: α)

text: Q

text: (s,

text: a)

text: α

text: (r

text: γ

text: max

text: Q

text: x));

text: x

text: E

text: (s,

text: a)

text: α)

text: E

text: (s,

text: a)

text: αγ

text: E

text: E

text: end

text: end

text: Algorithm

text: DORA

text: algorithm

text: using

text: LLL

text: determinization

text: for

text: stochastic

text: policy

text: f

text: RESULTS

text: FINITE

text: MDPS

text: To

text: test

text: this

text: algorithm,

text: the

text: ﬁrst

text: set

text: of

text: experiments

text: were

text: done

text: on

text: Bridge

text: environments

text: of

text: various

text: lengths

text: k

text: (Figure

text: We

text: considered

text: the

text: following

text: agents:

text: Softmax

text: and

text: their

text: respective

text: LLL

text: determinizations

text: (as

text: described

text: in

text: using

text: both

text: counters

text: and

text: E-values.

text: In

text: addition,

text: we

text: com-

text: pared

text: a

text: more

text: standard

text: counter-based

text: agent

text: in

text: the

text: form

text: of

text: a

text: UCB-like

text: algorithm

text: (Auer

text: et

text: al.,

text: following

text: an

text: action-selection

text: rule

text: with

text: exploration

text: bonus

text: of

text: logt.

text: We

text: tested

text: two

text: variants

text: of

text: this

text: C

text: algorithm,

text: using

text: ordinary

text: visit

text: counters

text: and

text: E-values.

text: Each

text: agent’s

text: hyperparameters

text: and

text: tem-

text: perature)

text: were

text: ﬁtted

text: separately

text: to

text: optimize

text: learning.

text: For

text: stochastic

text: agents,

text: we

text: averaged

text: the

text: results

text: over

text: trials

text: for

text: each

text: execution.

text: Unless

text: stated

text: otherwise,

text: γ

text: E

text: We

text: also

text: used

text: a

text: normalized

text: version

text: of

text: the

text: bridge

text: environment,

text: where

text: all

text: rewards

text: are

text: between

text: and

text: to

text: compare

text: DORA

text: with

text: the

text: Delayed

text: Q-Learning

text: algorithm

text: (Strehl

text: et

text: al.,

text: Our

text: results

text: (Figure

text: demonstrate

text: that

text: E-value

text: based

text: agents

text: outperform

text: both

text: their

text: counter-based

text: and

text: their

text: stochastic

text: equivalents

text: on

text: the

text: bridge

text: problem.

text: As

text: shown

text: in

text: Figure

text: Stochastic

text: and

text: counter-based

text: agents,

text: as

text: well

text: as

text: the

text: standard

text: UCB

text: fail

text: to

text: converge.

text: E-value

text: agents

text: are

text: the

text: ﬁrst

text: to

text: reach

text: low

text: error

text: values,

text: indicating

text: that

text: they

text: learn

text: faster.

text: Similar

text: results

text: were

text: achieved

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##