text: 6 Now set ω = ω + n ω (22) β β Σ − ω = ω + n ω , (23) σ σ Σ − 1 = n +n , use 1/(x i(cid:15)) = 1/x+iπδ(x), and expand in powers of ω . To zero-order all non-resonant contributions β σ Σ cancel and we obtain (cid:88) − 2πe4 ∂2ω − ( iωΣ)3(cid:15)0χ3i1 = 6(cid:126)3V ∂kd∂nkma fmnrnbmrmc nδ(ωnm ω), (24) nmk which contributes to ι . There is also a contribution from χ which we now consider. After symmetrization and 3 3i2 integration by parts χ has 8 terms 3i2 (cid:88) (−iωΣ)3 e(cid:15)40/χ(cid:126)33iV2 = 8 χ3i(cid:88)2,l (cid:18) (cid:19) (cid:88) (cid:18) (cid:19) l=1 − − −iω (va va )rc f rb − iω (va va )rb f rc · · · = Σ n −m m−n mn n−m Σ n −m m−n mn n−m + 6 ω ω ω ω ω 6 ω ω ω ω ω nmk nm β ∆ nm β ;d nmk nm σ ∆ nm σ ;d (25) In these two terms the imaginary parts of the frequency play an important role as the reals parts vanish. After an expansion in powers of ω we have to leading order Σ (cid:88) 2π ∂ ∂ω − χ + χ = ( nm rc rb )f δ(ω ω). (26) 3i2,1 3i2,2 6 ∂kd ∂ka mn nm mn nm nmk This term also contributes to ι . Adding Eq. 24 and Eq. 26 gives ι in Eq. 5 in the main text. 3 3 Jerk current in transition metal dichalcogenides The eﬀective Hamiltonian for transition metal dichalcogenides (TMDs) near the band edges is (cid:126) (cid:126) (cid:126) − H = λvsσ + γ(vk σ + k σ ) + (∆ λvs)σ . (27) vs 2 0 x x y y 2 z It is easy to calculate its eigenvalues and eigenfunctions for each valley (v) and spin (s) and with those compute − the corresponding dipole matrix elements ra and the jerk curren(cid:18)t. Deﬁning ∆ = (∆ (cid:19)λvs)/2 the results are nm vs − − − e4γ2 (cid:18) − 116∆2 48∆(cid:19)4 ιx3vxsxx(0, ω, ω, 0) = θ (ω 2∆vs) 96(cid:126)3ω2 10 ω2 vs + ω4vs , (28) − − − e4γ2 (cid:18) − 92∆2 (cid:19) 16∆4 ιx3vysyx(0, ω, ω, 0) = θ (ω 2∆vs) 96(cid:126)3ω2 6 ω2vs + ω4vs , (29) − − − e4γ2 (cid:18)− 16∆4 (cid:19) ιx3vxsyy(0, ω, ω, 0) = θ (ω 2∆vs) 48(cid:126)3ω2 1 ω4vs , (30) − − − e4γ2∆ − 28∆2 − 32∆4 ιx3vxsyx(0, ω, ω, 0) = θ (ω 2∆vs) iv 12(cid:126)3ωv3s 1 ω2vs ω4vs , (31) ##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: Now

text: set

text: ω

text: ω

text: n

text: ω

text: β

text: β

text: Σ

text: ω

text: ω

text: n

text: ω

text: σ

text: σ

text: Σ

text: n

text: +n

text: use

text: and

text: expand

text: in

text: powers

text: of

text: ω

text: To

text: zero-order

text: all

text: non-resonant

text: contributions

text: β

text: σ

text: Σ

text: cancel

text: and

text: we

text: obtain

text: ∂2ω

text: ∂kd∂nkma

text: fmnrnbmrmc

text: nδ(ωnm

text: ω),

text: nmk

text: which

text: contributes

text: to

text: ι

text: There

text: is

text: also

text: a

text: contribution

text: from

text: χ

text: which

text: we

text: now

text: consider.

text: After

text: symmetrization

text: and

text: integration

text: by

text: parts

text: χ

text: has

text: terms

text: −iω

text: (va

text: va

text: )rc

text: f

text: rb

text: iω

text: (va

text: va

text: )rb

text: f

text: rc

text: Σ

text: n

text: −m

text: m−n

text: mn

text: n−m

text: Σ

text: n

text: −m

text: m−n

text: mn

text: n−m

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: ω

text: nmk

text: nm

text: β

text: nm

text: β

text: ;d

text: nmk

text: nm

text: σ

text: nm

text: σ

text: ;d

text: In

text: these

text: two

text: terms

text: the

text: imaginary

text: parts

text: of

text: the

text: frequency

text: play

text: an

text: important

text: role

text: as

text: the

text: reals

text: parts

text: vanish.

text: After

text: an

text: expansion

text: in

text: powers

text: of

text: ω

text: we

text: have

text: to

text: leading

text: order

text: Σ

text: 2π

text: ∂ω

text: χ

text: χ

text: nm

text: rc

text: rb

text: )f

text: δ(ω

text: ω).

text: ∂kd

text: ∂ka

text: mn

text: nm

text: mn

text: nm

text: nmk

text: This

text: term

text: also

text: contributes

text: to

text: ι

text: Adding

text: Eq.

text: and

text: Eq.

text: gives

text: ι

text: in

text: Eq.

text: in

text: the

text: main

text: text.

text: Jerk

text: current

text: in

text: transition

text: metal

text: dichalcogenides

text: The

text: eﬀective

text: Hamiltonian

text: for

text: transition

text: metal

text: dichalcogenides

text: (TMDs)

text: near

text: the

text: band

text: edges

text: is

text: H

text: λvsσ

text: γ(vk

text: σ

text: k

text: σ

text: λvs)σ

text: vs

text: x

text: x

text: y

text: y

text: z

text: It

text: is

text: easy

text: to

text: calculate

text: its

text: eigenvalues

text: and

text: eigenfunctions

text: for

text: each

text: valley

text: (v)

text: and

text: spin

text: (s)

text: and

text: with

text: those

text: compute

text: the

text: corresponding

text: dipole

text: matrix

text: elements

text: ra

text: and

text: the

text: jerk

text: Deﬁning

text: the

text: results

text: are

text: nm

text: vs

text: ω,

text: ω,

text: θ

text: (ω

text: ω2

text: vs

text: ω,

text: ω,

text: θ

text: (ω

text: ω,

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text: (ω

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text: Achieving Heisenberg-limited metrology with spin cat states via interaction-based ∗ readout † Jiahao Huang1, Min Zhuang1,2, Bo Lu1, Yongguan Ke1,2, and Chaohong Lee1,2,3 ‡ § and 11 (Dated: May 31, 2020) Spin cat states are promising candidates for quantum enhanced measurement. Here, we analyt- ically show that, the ultimate measurement precisions of spin cat states approach the Heisenberg limit, where the uncertainty is inversely proportional to the total particle number. In order to fully exploit their metrological ability, we propose to use the interaction-based readout for implementing phase estimation. It is demonstrated that, the interaction-based readout enables spin cat states to saturate their ultimate precision bounds. The interaction-based readout comprises a one-axis twist- ing, two π pulses, and a population measurement, which can be realized via current experimental 2 techniques. Compared with the twisting echo scheme on spin squeezed states, our scheme with spin cat states is more robust against detection noise. Our scheme may pave an experimentally feasible way to achieve Heisenberg-limited metrology with non-Gaussian entangled states. I. INTRODUCTION nonlinear detection with spin squeezed states [19] and two-mode squeezed vacuum states [20] have been respec- Quantum metrology aims to enhance the measurement tively realized in experiments. More recently, schemes precision and develop optimal schemes for estimating an on interaction-ba(cid:54) sed r†eadout that relax the time-reversal unknown parameter by means of quantum strategies [1– condition (UˆR = Uˆ ) are proposed [21, 22]. These 3]. In comparison to classical strategies, quantum en- pointed out a new direction of utilizing entangled states hanced measurement oﬀers signiﬁcant advantages, where for quantum metrology [23–29]. a dramatic improvement for the achievable precision can Spin cat states, a kind of non-Gaussian entangled be obtained due to the use of quantum entanglement [4– states as a superposition of distinct spin coherent states 6]. Preparing and detecting entangled quantum states (SCS’s), are considered as promising candidates for quan- are two main challenges to achieve the Heisenberg-limited tum enhanced measurement [30–35]. It has been shown metrology. A lot of endeavors had been made to cre- that, spin cat states with modest entanglement can per- ate various kinds of entangled input states, such as spin form high-precision phase measurement beyond the stan- squeezed states [7–10], twin Fock states [11–13], maxi- dard quantum limit (SQL) even under dissipation [35]. mally entangled states [14, 15], and so on. However, to However, to perform the interferometry with spin cat detect the output states, single-particle resolved detec- states in practice, parity measurement is required so that tion is assumed to be necessary, which has so far been single-particle resolution should be accessed [35–39]. The the bottleneck in the practical performances. Moreover, requirement of single-particle detection limits the exper- imperfect detection is one of the key obstacles that ham- imental feasibility of quantum metrology with spin cat per the improvement of measurement precision via many- states. Thus, to overcome this barrier, is it possible to body entangled states. replace the parity measurement with interaction-based In recent, an echo protocol was proposed to perform readout? Compared with the interaction-based readout phase estimation near the Heisenberg limit [16, 17], which scheme with spin squeezed states, will the spin cat states does not require single-particle resolved detection. The oﬀer better robustness against detection noise? input state is generated by the time-evolution under a − In this article, we propose to perform the phase estima- one-axis twisting Hamilton†ian Uˆ = e iHOATt [18], and a tion with spin cat states via interaction-based readout. reversal evolution UˆR = Uˆ is performed on the output We ﬁnd that, spin cat states have the ability to perform state prior to the ﬁnal population measurement. The Heisenberg-limited phase measurement and interaction- nonlinear dyna∝mics−UˆR enables Heisenberg-limit(cid:46)ed√pre- based readout is an optimal method to fully exploit this cision scaling ( N 1) under detection noise σ N, potential ability. In Sec. , we give a general framework where N is the total particle number. This kind of of many-body quantum interferometry and the phase es- timation. In Sec. , we analytically obtain the ultimate precision bound for spin cat states. The ultimate bound ∗ is always inversely proportional to the total particle num- † ber with a constant depending on the separation of the Corresponding author. two superposition SCS’s, which approaches the Heisen- Email: lichaoh2@mail.sysu.edu.cn, chleecn@gmail.com ‡ berg limit. In Sec. , we describe the procedure of quan- § tum interferometry via interaction-based readout with ##LTLine## ##LTLine## ##LTLine##

text: Achieving

label: title

text: Heisenberg-limited

text: metrology

text: with

text: spin

text: cat

text: states

text: via

text: interaction-based

label: paragraph

text: readout

text: Jiahao

label: author

text: Min

text: Bo

text: Yongguan

text: and

text: Chaohong

text: and

text: (Dated:

text: May

label: date

text: Spin

label: abstract

text: cat

text: states

text: are

text: promising

text: candidates

text: for

text: quantum

text: enhanced

text: measurement.

text: Here,

text: we

text: analyt-

text: ically

text: show

text: that,

text: the

text: ultimate

text: measurement

text: precisions

text: of

text: spin

text: cat

text: states

text: approach

text: the

text: Heisenberg

text: limit,

text: where

text: the

text: uncertainty

text: is

text: inversely

text: proportional

text: to

text: the

text: total

text: particle

text: number.

text: In

text: order

text: to

text: fully

text: exploit

text: their

text: metrological

text: ability,

text: we

text: propose

text: to

text: use

text: the

text: interaction-based

text: readout

text: for

text: implementing

text: phase

text: estimation.

text: It

text: is

text: demonstrated

text: that,

text: the

text: interaction-based

text: readout

text: enables

text: spin

text: cat

text: states

text: to

text: saturate

text: their

text: ultimate

text: precision

text: bounds.

text: The

text: interaction-based

text: readout

text: comprises

text: a

text: one-axis

text: twist-

text: ing,

text: two

text: π

text: pulses,

text: and

text: a

text: population

text: measurement,

text: which

text: can

text: be

text: realized

text: via

text: current

text: experimental

text: techniques.

text: Compared

text: with

text: the

text: twisting

text: echo

text: scheme

text: on

text: spin

text: squeezed

text: states,

text: our

text: scheme

text: with

text: spin

text: cat

text: states

text: is

text: more

text: robust

text: against

text: detection

text: noise.

text: Our

text: scheme

text: may

text: pave

text: an

text: experimentally

text: feasible

text: way

text: to

text: achieve

text: Heisenberg-limited

text: metrology

text: with

text: non-Gaussian

text: entangled

text: states.

text: I.

text: INTRODUCTION

text: nonlinear

text: detection

text: with

text: spin

text: squeezed

text: states

text: and

text: two-mode

text: squeezed

text: vacuum

text: states

text: have

text: been

text: respec-

text: Quantum

text: metrology

text: aims

text: to

text: enhance

text: the

text: measurement

text: tively

text: realized

text: in

text: experiments.

text: More

text: recently,

text: schemes

text: precision

text: and

text: develop

text: optimal

text: schemes

text: for

text: estimating

text: an

text: on

text: sed

text: r†eadout

text: that

text: relax

text: the

text: time-reversal

text: unknown

text: parameter

text: by

text: means

text: of

text: quantum

text: strategies

text: condition

text: (UˆR

text: Uˆ

text: are

text: proposed

text: These

text: In

text: comparison

text: to

text: classical

text: strategies,

text: quantum

text: en-

text: pointed

text: out

text: a

text: new

text: direction

text: of

text: utilizing

text: entangled

text: states

text: hanced

text: measurement

text: oﬀers

text: signiﬁcant

text: advantages,

text: where

text: for

text: quantum

text: metrology

text: a

text: dramatic

text: improvement

text: for

text: the

text: achievable

text: precision

text: can

text: Spin

text: cat

text: states,

text: a

text: kind

text: of

text: non-Gaussian

text: entangled

text: be

text: obtained

text: due

text: to

text: the

text: use

text: of

text: quantum

text: entanglement

text: states

text: as

text: a

text: superposition

text: of

text: distinct

text: spin

text: coherent

text: states

text: Preparing

text: and

text: detecting

text: entangled

text: quantum

text: states

text: (SCS’s),

text: are

text: considered

text: as

text: promising

text: candidates

text: for

text: quan-

text: are

text: two

text: main

text: challenges

text: to

text: achieve

text: the

text: Heisenberg-limited

text: tum

text: enhanced

text: measurement

text: It

text: has

text: been

text: shown

text: metrology.

text: A

text: lot

text: of

text: endeavors

text: had

text: been

text: made

text: to

text: cre-

text: that,

text: spin

text: cat

text: states

text: with

text: modest

text: entanglement

text: can

text: per-

text: ate

text: various

text: kinds

text: of

text: entangled

text: input

text: states,

text: such

text: as

text: spin

text: form

text: high-precision

text: phase

text: measurement

text: beyond

text: the

text: stan-

text: squeezed

text: states

text: twin

text: Fock

text: states

text: maxi-

text: dard

text: quantum

text: limit

text: (SQL)

text: even

text: under

text: dissipation

text: mally

text: entangled

text: states

text: and

text: so

text: on.

text: However,

text: to

text: However,

text: to

text: perform

text: the

text: interferometry

text: with

text: spin

text: cat

text: detect

text: the

text: output

text: states,

text: single-particle

text: resolved

text: detec-

text: states

text: in

text: practice,

text: parity

text: measurement

text: is

text: required

text: so

text: that

text: tion

text: is

text: assumed

text: to

text: be

text: necessary,

text: which

text: has

text: so

text: far

text: been

text: single-particle

text: resolution

text: should

text: be

text: accessed

text: The

text: the

text: bottleneck

text: in

text: the

text: practical

text: performances.

text: Moreover,

text: requirement

text: of

text: single-particle

text: detection

text: limits

text: the

text: exper-

text: imperfect

text: detection

text: is

text: one

text: of

text: the

text: key

text: obstacles

text: that

text: ham-

text: imental

text: feasibility

text: of

text: quantum

text: metrology

text: with

text: spin

text: cat

text: per

text: the

text: improvement

text: of

text: measurement

text: precision

text: via

text: many-

text: states.

text: Thus,

text: to

text: overcome

text: this

text: barrier,

text: is

text: it

text: possible

text: to

text: body

text: entangled

text: states.

text: replace

text: the

text: parity

text: measurement

text: with

text: interaction-based

text: In

text: recent,

text: an

text: echo

text: protocol

text: was

text: proposed

text: to

text: perform

text: readout?

text: Compared

text: with

text: the

text: interaction-based

text: readout

text: phase

text: estimation

text: near

text: the

text: Heisenberg

text: limit

text: which

text: scheme

text: with

text: spin

text: squeezed

text: states,

text: will

text: the

text: spin

text: cat

text: states

text: does

text: not

text: require

text: single-particle

text: resolved

text: detection.

text: The

text: oﬀer

text: better

text: robustness

text: against

text: detection

text: noise?

text: input

text: state

text: is

text: generated

text: by

text: the

text: time-evolution

text: under

text: a

text: In

text: this

text: article,

text: we

text: propose

text: to

text: perform

text: the

text: phase

text: estima-

text: one-axis

text: twisting

text: Hamilton†ian

text: Uˆ

text: e

text: iHOATt

text: and

text: a

text: tion

text: with

text: spin

text: cat

text: states

text: via

text: interaction-based

text: readout.

text: reversal

text: evolution

text: UˆR

text: Uˆ

text: is

text: performed

text: on

text: the

text: output

text: We

text: ﬁnd

text: that,

text: spin

text: cat

text: states

text: have

text: the

text: ability

text: to

text: perform

text: state

text: prior

text: to

text: the

text: ﬁnal

text: population

text: measurement.

text: The

text: Heisenberg-limited

text: phase

text: measurement

text: and

text: interaction-

text: nonlinear

text: dyna∝mics−UˆR

text: enables

text: based

text: readout

text: is

text: an

text: optimal

text: method

text: to

text: fully

text: exploit

text: this

text: cision

text: scaling

text: N

text: under

text: detection

text: noise

text: σ

text: N,

text: potential

text: ability.

text: In

text: Sec.

text: we

text: give

text: a

text: general

text: framework

text: where

text: N

text: is

text: the

text: total

text: particle

text: number.

text: This

text: kind

text: of

text: of

text: many-body

text: quantum

text: interferometry

text: and

text: the

text: phase

text: es-

text: timation.

text: In

text: Sec.

text: we

text: analytically

text: obtain

text: the

text: ultimate

text: precision

text: bound

text: for

text: spin

text: cat

text: states.

text: The

text: ultimate

text: bound

text: is

text: always

text: inversely

text: proportional

text: to

text: the

text: total

text: particle

text: num-

text: ber

text: with

text: a

text: constant

text: depending

text: on

text: the

text: separation

text: of

text: the

text: Corresponding

text: author.

text: two

text: superposition

text: SCS’s,

text: which

text: approaches

text: the

text: Heisen-

text: Email:

text: chleecn@gmail.com

text: berg

text: limit.

text: In

text: Sec.

text: we

text: describe

text: the

text: procedure

text: of

text: quan-

text: tum

text: interferometry

text: via

text: interaction-based

text: readout

text: with

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 11 (cid:18) (cid:19) eiπ2 Jˆz2eiπ2 Jˆxe−iφJˆz ((cid:89)|J, k(cid:105) + |J, −k(cid:105)) Finally, we obtain that − 2k √1 | ↓(cid:105) √i | ↑(cid:105) = [co(cid:89)s(kφ(cid:18)) sin(kφ)] (cid:19) l + l 2 2 2J √1 | ↓(cid:105) √l1=1| ↑(cid:105) e−iπ2 Jˆxei−π2 Jˆz2eiπ2 Jˆxe−iφJˆ−z (|J|, k(cid:105)−+(cid:105)|J, −k(cid:105)) l+ (cid:89) (cid:18) l + (cid:19) = [cos(kφ) sin(kφ)] (i)(J k)2 J, k 2 2 l=2k+1 + [cos(kφ)+sin(kφ)] (i)(J−k)2|J, k(cid:105). 2k √i | ↓(cid:105) √1 | ↑(cid:105) [co(cid:89)s(kφ(cid:18))+sin(kφ)] (cid:19) l + l (A18) 2 2 l=1 2J √1 | ↓(cid:105) √1 | ↑(cid:105) + . l l 2 2 l=2k+1 Combining Eq. (A13) and (A18), we can unify as (A17) Eq. (21) in the main text. [1] V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330[2(82]00J4.)H. uang, M. Zhuang, and C. Lee, Phys. Rev. A 97, 032116 (2018). [2] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 9[269,]0S1.04C0h1oi(,2N00.6Y).. Yao, and M. D. Lukin, arXiv: 1801.00042. [3] V. Giovannetti, S. Lloyd, and L. 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Klempt, arXiv: 1711.02658. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 6 FIG. 3. (Color online) The phase precision obtained with spin cat states via interaction-based readout for the estimated phase ∼ ∼ in the vicinity of (a) φ 0 and (b) φ π/2. The standard derivations of the phase ∆φ versus the nonlinear evolution χt for spin | (cid:105) | (cid:105) | (cid:105) | (cid:105) cat states Ψ(0) (blue), Ψ(π/8) (green), Ψ(π/4) (cyan) and Ψ(7π/20) (magenta) are shown. The minimum CAT CAT CAT CAT phase precision ∆φ is obtained by optimizing the nonlinear evolution χt. In the insets, ∆φ and the corresponding optimal min | (cid:105) min evolution time t for diﬀerent MSSCS Ψ(θ) are given. Here, the total particle number N = 100. (c) The log-log scaling opt M | (cid:105) | (cid:105) of the minimum phase precision versus the total particle number for spin cat states Ψ(0) (blue), Ψ(π/8) (green), | (cid:105) | (cid:105) CAT CAT Ψ(π/4) (cyan) and Ψ(7π/20) (magenta). The dotted lines with inverted triangles and the dashed lines with squares CAT ∼ CAT∼ correspond to the cases of φ 0 and φ π/2, respectively. (cid:114) (cid:16) (cid:17) (cid:16) (cid:17) When φ = 0, sin(2kφ) = 0 and cos(2kφ) = 1, the phase ∆Jˆ = (cid:88)(cid:104)Ψ(φ)|Jˆ2|Ψ(cid:2)(φ)(cid:105) − (cid:104)Ψ(φ)|Jˆ |Ψ(cid:3)(φ)(cid:105) 2 measurement precision b(cid:113)ec(cid:80)omes z f z f f z f f (cid:12) (cid:12) { J ∗ − − (cid:12)(cid:80) ∗ (cid:12) = (cid:34) (cid:88) k2akak 1 + ( 1)J k sin(cid:35)(2kφ) | (cid:12) Jk=−J k2akak (cid:12) − ∆φ = . (27) k= J φ=0 J − 2k2a∗a (−1)J−k −(cid:118)(cid:117)(cid:117)(cid:117)k=(cid:88)J−J ka∗kak(−1(cid:34))J(cid:88)−k sin(2kφ) 2}1/2 (cid:35) When φ = π/2, sin(2kk=φ)J= 0 ankdk(−1)J−k cos(2kφ) = (cid:116) 1, the phase measurement(cid:113)precision can be simpliﬁed as (cid:80) 2 J ∗ − J ∗ − − = k2akak kakak( 1)J k sin(2kφ) . (cid:12)(cid:12)(cid:80) ∗ (cid:12)(cid:12) k=−J k=−J | (cid:12) Jk=−J k2akak(cid:12) ∆φ = . (28) (25) φ=π/2 J − 2k2a∗a k= J k k Finally, we can obtain the phase measurement preci- For a spin cat state (7), according to Eqs. (8), (10), (12), (27) sion vi(cid:114)a Eqs. (24) and (25(cid:104)), (cid:105) and (28), we get the phase measurement precision (cid:80) (cid:80) (cid:12) (cid:12) (cid:12)(cid:80) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) J −(cid:12)(cid:12)(cid:80)k2a∗a − J − ka∗a (−1)J−k s(cid:12)(cid:12)in(2kφ) 2 ∆φ| = M , ∆φ = k= J J k−k 2k2a∗ka= (J−1)kJ−kk cos(2kφ) . φ=0 Nm/=2−N/2 2k2cm(θ)∗cm(θ)(−1)N/2−m k= J k k (29) (26) ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: (Color

label: caption

text: online)

text: The

text: phase

text: precision

text: obtained

text: with

text: spin

text: cat

text: states

text: via

text: interaction-based

text: readout

text: for

text: the

text: estimated

text: phase

text: in

text: the

text: vicinity

text: of

text: (a)

text: φ

text: and

text: (b)

text: φ

text: The

text: standard

text: derivations

text: of

text: the

text: phase

text: ∆φ

text: versus

text: the

text: nonlinear

text: evolution

text: χt

text: for

text: spin

text: cat

text: states

text: (blue),

text: (green),

text: (cyan)

text: and

text: (magenta)

text: are

text: shown.

text: The

text: minimum

text: CAT

text: CAT

text: CAT

text: CAT

text: phase

text: precision

text: ∆φ

text: is

text: obtained

text: by

text: optimizing

text: the

text: nonlinear

text: evolution

text: χt.

text: In

text: the

text: insets,

text: ∆φ

text: and

text: the

text: corresponding

text: optimal

text: min

text: min

text: evolution

text: time

text: t

text: for

text: diﬀerent

text: MSSCS

text: Ψ(θ)

text: are

text: given.

text: Here,

text: the

text: total

text: particle

text: number

text: N

text: (c)

text: The

text: log-log

text: scaling

text: opt

text: M

text: of

text: the

text: minimum

text: phase

text: precision

text: versus

text: the

text: total

text: particle

text: number

text: for

text: spin

text: cat

text: states

text: (blue),

text: (green),

text: CAT

text: CAT

text: (cyan)

text: and

text: (magenta).

text: The

text: dotted

text: lines

text: with

text: inverted

text: triangles

text: and

text: the

text: dashed

text: lines

text: with

text: squares

text: CAT

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text: correspond

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text: φ

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text: φ

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text: When

text: φ

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text: the

text: phase

text: ∆Jˆ

text: measurement

text: precision

text: z

text: f

text: z

text: f

text: f

text: z

text: f

text: f

text: J

text: k

text: Jk=−J

text: ∆φ

text: k=

text: J

text: J

text: When

text: φ

text: the

text: phase

text: can

text: be

text: simpliﬁed

text: as

text: J

text: J

text: kakak(

text: k

text: k=−J

text: k=−J

text: Jk=−J

text: ∆φ

text: J

text: k=

text: J

text: k

text: k

text: Finally,

text: we

text: can

text: obtain

text: the

text: phase

text: measurement

text: preci-

text: For

text: a

text: spin

text: cat

text: state

text: according

text: to

text: Eqs.

text: sion

text: Eqs.

text: and

text: and

text: we

text: get

text: the

text: phase

text: measurement

text: precision

text: J

text: J

text: ka∗a

text: ∆φ|

text: M

text: ∆φ

text: k=

text: J

text: J

text: k−k

text: k=

text: J

text: k

text: k

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ (cid:48) Claim 1. The number of corners lying in A with d W is at least (m (α) + | | o(1))N2 W . ∈ Note that for a corner, the condition d W is equivalent to requiring the three vertices of the corner to be in the same outer box. By the pigeonhole ∈ \ { } principle, it follows from the above claim that there is some d W 0 with | | ≥ (cid:48) | |− | | S (m (α) + o(1) + O( W 1))N2. Note that, for a ﬁxed (cid:15), W tends to d | |− inﬁnity with N, so the O( W 1) term can be made as small as we like. Therefore ≥ (cid:48) M(α) m (α), which is Theorem 4. Now we set about proving Claim 1. For each outer box V , we deﬁne α(V ) | ∩ | | ∩ | to be the density of A inside V , α(V ) = V A / V P . The average value of α(V ) over all V is at least α. We will show that if V is a regular outer | | box, then the number of corners in V is at least (m(α(V ) + O((cid:15))) + o(1)) W 3 (this is the content of Claim 2 below). Since m(α) is uniformly continuous, | | | | ≥ (cid:48) | | (m(α(V )+O((cid:15)))+o(1)) W 3 = (m(α(V ))+o(1)) W 3 (m (α(V ))+o(1)) W 3. (cid:48) Then, we can use the convexity of m to take the sum over all V , and so the ∈ (cid:48) | | number of corners with d W is at least (m (α)+o(1))N2 W . Note that here we use the fact that the proportion of irregular boxes is o(1), so the error that comes from those boxes is o(1). So, it is suﬃcient to prove the following claim: | | Claim 2. If V is a regular outer box, then there are at least (T(f) + o(1)) W 3 corners in V , where f is some function of independent random variables X, Y and Z, where X, Y and Z each take ﬁnitely many values, f takes values in [0, 1] E and (f) = α(V ) + O((cid:15)). By Lemma 1, in the above claim, we may equivalently have assumed f was → piecewise constant : [0, 1]3 [0, 1] and X, Y and Z where U[0, 1]. Proof of Claim 2. Let V be a regular outer box. Without loss of generality, ∩ we may assume V = W3. The corners whose vertices lie in V P are in bijection with the elements of V as follows: (x, y, z) correponds to the corner { } − K(x, y, z) := (x+d, y, z), (x, y +d, z), (x, y, z +d) where d = (x+y +z). For ≤ ≤ | || || | ∈ each 1 i, j, k m, there are B C D corners K(x, y, z) with (x, y, z) × × i j k B C D . The three vertices of these corners will lie in the inner boxes i × j k × × (C D ) , (B D ) and (B C ) respectively. We may form a tripartite j k P i k P i j P ∈ graph with vertex sets B , C and D , by adding an edge between x B and ∈ − − ∈i j k i y C if (x, y, x y) A, and similarly for the other two pairs. Triangles j in this graph correspond to corners lying in A. Now, if the three inner boxes × × × (B C ) , (B D ) and (C D ) are all (cid:15)-quasirandom, then we may apply i j P i k P j k P the counting lemma from the proof of the triangle removal lemma to conclude | || || | that the number of such triangles is B C D (δ δ δ +o(1)) (The i j k Bi,Cj Bi,Dk Cj,Dk statement of this counting lemma is Proposition 1.2 in [4]. The triangle removal lemma, or rather a weaker form of it, was ﬁrst proved by Ruzsa and Szemer´edi in 1976 [7]). Without quasirandomness, we still know the number of corners is | || || | at most B C D . Because V is a regular box, the total size of all the non- i j k | | | || || | quasirandom inner boxes is at most 3(cid:15) W 2, so the total of B C D over all i j | k| i, j, k for which some inner box is not (cid:15)-quasirandom is at most 3(cid:15) W 3 11

text: Claim

text: The

text: number

text: of

text: corners

text: lying

text: in

text: A

text: with

text: d

text: W

text: is

text: at

text: least

text: (m

text: (α)

text: W

text: Note

text: that

text: for

text: a

text: corner,

text: the

text: condition

text: d

text: W

text: is

text: equivalent

text: to

text: requiring

text: the

text: three

text: vertices

text: of

text: the

text: corner

text: to

text: be

text: in

text: the

text: same

text: outer

text: box.

text: By

text: the

text: pigeonhole

text: principle,

text: it

text: follows

text: from

text: the

text: above

text: claim

text: that

text: there

text: is

text: some

text: d

text: W

text: with

text: S

text: (m

text: (α)

text: O(

text: W

text: Note

text: that,

text: for

text: a

text: ﬁxed

text: W

text: tends

text: to

text: d

text: inﬁnity

text: with

text: N,

text: so

text: the

text: O(

text: W

text: term

text: can

text: be

text: made

text: as

text: small

text: as

text: we

text: like.

text: Therefore

text: M(α)

text: m

text: (α),

text: which

text: is

text: Theorem

text: Now

text: we

text: set

text: about

text: proving

text: Claim

text: For

text: each

text: outer

text: box

text: V

text: we

text: deﬁne

text: α(V

text: to

text: be

text: the

text: density

text: of

text: A

text: inside

text: V

text: α(V

text: V

text: A

text: V

text: P

text: The

text: average

text: value

text: of

text: α(V

text: over

text: all

text: V

text: is

text: at

text: least

text: α.

text: We

text: will

text: show

text: that

text: if

text: V

text: is

text: a

text: regular

text: outer

text: box,

text: then

text: the

text: number

text: of

text: corners

text: in

text: V

text: is

text: at

text: least

text: (m(α(V

text: W

text: (this

text: is

text: the

text: content

text: of

text: Claim

text: below).

text: Since

text: m(α)

text: is

text: uniformly

text: continuous,

text: (m(α(V

text: W

text: (m(α(V

text: W

text: (m

text: (α(V

text: W

text: Then,

text: we

text: can

text: use

text: the

text: convexity

text: of

text: m

text: to

text: take

text: the

text: sum

text: over

text: all

text: V

text: and

text: so

text: the

text: number

text: of

text: corners

text: with

text: d

text: W

text: is

text: at

text: least

text: (m

text: W

text: Note

text: that

text: here

text: we

text: use

text: the

text: fact

text: that

text: the

text: proportion

text: of

text: irregular

text: boxes

text: is

text: so

text: the

text: error

text: that

text: comes

text: from

text: those

text: boxes

text: is

text: So,

text: it

text: is

text: suﬃcient

text: to

text: prove

text: the

text: following

text: claim:

text: Claim

text: If

text: V

text: is

text: a

text: regular

text: outer

text: box,

text: then

text: there

text: are

text: at

text: least

text: (T(f)

text: W

text: corners

text: in

text: V

text: where

text: f

text: is

text: some

text: function

text: of

text: independent

text: random

text: variables

text: X,

text: Y

text: and

text: Z,

text: where

text: X,

text: Y

text: and

text: Z

text: each

text: take

text: ﬁnitely

text: many

text: values,

text: f

text: takes

text: values

text: in

text: E

text: and

text: (f)

text: α(V

text: By

text: Lemma

text: in

text: the

text: above

text: claim,

text: we

text: may

text: equivalently

text: have

text: assumed

text: f

text: was

text: piecewise

text: constant

text: and

text: X,

text: Y

text: and

text: Z

text: where

text: Proof

text: of

text: Claim

text: Let

text: V

text: be

text: a

text: regular

text: outer

text: box.

text: Without

text: loss

text: of

text: generality,

text: we

text: may

text: assume

text: V

text: The

text: corners

text: whose

text: vertices

text: lie

text: in

text: V

text: P

text: are

text: in

text: bijection

text: with

text: the

text: elements

text: of

text: V

text: as

text: follows:

text: (x,

text: y,

text: z)

text: correponds

text: to

text: the

text: corner

text: K(x,

text: y,

text: z)

text: (x+d,

text: y,

text: z),

text: (x,

text: y

text: +d,

text: z),

text: (x,

text: y,

text: z

text: +d)

text: where

text: d

text: (x+y

text: +z).

text: For

text: each

text: i,

text: j,

text: k

text: m,

text: there

text: are

text: B

text: C

text: D

text: corners

text: K(x,

text: y,

text: z)

text: with

text: (x,

text: y,

text: z)

text: i

text: j

text: k

text: B

text: C

text: D

text: The

text: three

text: vertices

text: of

text: these

text: corners

text: will

text: lie

text: in

text: the

text: inner

text: boxes

text: i

text: j

text: k

text: (C

text: D

text: (B

text: D

text: and

text: (B

text: C

text: respectively.

text: We

text: may

text: form

text: a

text: tripartite

text: j

text: k

text: P

text: i

text: k

text: P

text: i

text: j

text: P

text: graph

text: with

text: vertex

text: sets

text: B

text: C

text: and

text: D

text: by

text: adding

text: an

text: edge

text: between

text: x

text: B

text: and

text: ∈i

text: j

text: k

text: i

text: y

text: C

text: if

text: (x,

text: y,

text: x

text: y)

text: A,

text: and

text: similarly

text: for

text: the

text: other

text: two

text: pairs.

text: Triangles

text: j

text: in

text: this

text: graph

text: correspond

text: to

text: corners

text: lying

text: in

text: A.

text: Now,

text: if

text: the

text: three

text: inner

text: boxes

text: (B

text: C

text: (B

text: D

text: and

text: (C

text: D

text: are

text: all

text: then

text: we

text: may

text: apply

text: i

text: j

text: P

text: i

text: k

text: P

text: j

text: k

text: P

text: the

text: counting

text: lemma

text: from

text: the

text: proof

text: of

text: the

text: triangle

text: removal

text: lemma

text: to

text: conclude

text: that

text: the

text: number

text: of

text: such

text: triangles

text: is

text: B

text: C

text: D

text: (δ

text: δ

text: δ

text: (The

text: i

text: j

text: k

text: Bi,Cj

text: Bi,Dk

text: Cj,Dk

text: statement

text: of

text: this

text: counting

text: lemma

text: is

text: Proposition

text: in

text: The

text: triangle

text: removal

text: lemma,

text: or

text: rather

text: a

text: weaker

text: form

text: of

text: it,

text: was

text: ﬁrst

text: proved

text: by

text: Ruzsa

text: and

text: Szemer´edi

text: in

text: Without

text: quasirandomness,

text: we

text: still

text: know

text: the

text: number

text: of

text: corners

text: is

text: at

text: most

text: B

text: C

text: D

text: Because

text: V

text: is

text: a

text: regular

text: box,

text: the

text: total

text: size

text: of

text: all

text: the

text: non-

text: i

text: j

text: k

text: quasirandom

text: inner

text: boxes

text: is

text: at

text: most

text: W

text: so

text: the

text: total

text: of

text: B

text: C

text: D

text: over

text: all

text: i

text: j

text: k|

text: i,

text: j,

text: k

text: for

text: which

text: some

text: inner

text: box

text: is

text: not

text: is

text: at

text: most

text: W


text: (cid:88) Therefore, the number of corners of V with vertices lying in A is | || || | | | B C D δ δ δ + o(1) W 3. (1) ≤ ≤ i j k Bi,Cj Bi,Dk Cj,Dk 1 i,j,k m Now we deﬁne the random variables X, Y, Z and the function f. For any ∈ (x, y, z) W3, deﬁne B((x, y, z)) to be the B that contains x, C((x, y, z)) to (cid:51) (cid:51) i ∈ be the C y and D((x, y, z)) the D z. Let p W3 be chosen uniformly at j k random, and let X = B(p), Y = C(p), Z = D(p). X, Y and Z are independent random variables, each taking only ﬁnitely many values. Deﬁne | ∩ × × || | (cid:48) A X Y Z W f = | || || | X Y Z and (cid:48) f = min(f , 1). E (cid:48) E (cid:48)| | ∩ × | | || | One can easily check (f ) = α(V ). (f X, Y ) = A (X Y ) / X Y = E (cid:48)| E (cid:48)| P δ Similarly (f X, Z) = δ and (f Y, Z) = δ . Putting this together, X,Y X,Z Y,Z (cid:88) we get | || || | (cid:48) B C D δ δ δ T(f ) = i j k |Bi,C|j Bi,Dk Cj,Dk . ≤ ≤ W 3 1 i,j,k m Comparing this with equation (1), we see that the number of corners is (cid:48) | | (cid:48) (T(f ) + o(1)) W 3. The problem with f , though, is that it may take values (cid:48) greater than 1. We need to bound f . Below, we will be using convolutions of ∗ E − functions on W, which are deﬁned as follows: g h(y) := ∈ g(x)h(y x). x W | ∩ × × | ≤ | ∩ × × | A B C D P B C D i j k i j k |{ ∈ × × | }| = (x, y(cid:88), z) Bi Cj Dk x + y + z = 0 | | ∗ ∗ = W 2B C D (0) i j k | | (cid:88) ∗ ∗ ∧ = W 2 (B C D ) (r) i j k ∈ ∗ r W | | = W 2 Bˆ (r)Cˆ (r)Dˆ (r) i j k ∈ ∗ r W We split the above sum into a “main” term, and error terms, which can be bounded using the uniformity assumption, plus applications of the Cauchy- Schw(cid:88)arz inequality and Parseval’s identity. (cid:88) Bˆ (r)Cˆ (r)Dˆ (r) = δ(B )δ(C )δ(D ) + Bˆ (r)Cˆ (r)Dˆ (r) i j k i j k i j k ∈ ∗ ∈ ∗\{ } r W r W 0 12 ##LTLine## ##LTLine##

text: Therefore,

text: the

text: number

text: of

text: corners

text: of

text: V

text: with

text: vertices

text: lying

text: in

text: A

text: is

text: B

text: C

text: D

text: δ

text: δ

text: δ

text: W

text: i

text: j

text: k

text: Bi,Cj

text: Bi,Dk

text: Cj,Dk

text: i,j,k

text: m

text: Now

text: we

text: deﬁne

text: the

text: random

text: variables

text: X,

text: Y,

text: Z

text: and

text: the

text: function

text: f.

text: For

text: any

text: (x,

text: y,

text: z)

text: deﬁne

text: B((x,

text: y,

text: z))

text: to

text: be

text: the

text: B

text: that

text: contains

text: x,

text: C((x,

text: y,

text: z))

text: to

text: i

text: be

text: the

text: C

text: y

text: and

text: D((x,

text: y,

text: z))

text: the

text: D

text: z.

text: Let

text: p

text: be

text: chosen

text: uniformly

text: at

text: j

text: k

text: random,

text: and

text: let

text: X

text: B(p),

text: Y

text: C(p),

text: Z

text: D(p).

text: X,

text: Y

text: and

text: Z

text: are

text: independent

text: random

text: variables,

text: each

text: taking

text: only

text: ﬁnitely

text: many

text: values.

text: Deﬁne

text: A

text: X

text: Y

text: Z

text: W

text: f

text: X

text: Y

text: Z

text: and

text: f

text: min(f

text: E

text: E

text: One

text: can

text: easily

text: check

text: (f

text: α(V

text: (f

text: X,

text: Y

text: A

text: (X

text: Y

text: X

text: Y

text: E

text: E

text: P

text: δ

text: Similarly

text: (f

text: X,

text: Z)

text: δ

text: and

text: (f

text: Y,

text: Z)

text: δ

text: Putting

text: this

text: together,

text: X,Y

text: X,Z

text: Y,Z

text: we

text: get

text: B

text: C

text: D

text: δ

text: δ

text: δ

text: T(f

text: i

text: j

text: k

text: |Bi,C|j

text: Bi,Dk

text: Cj,Dk

text: W

text: i,j,k

text: m

text: Comparing

text: this

text: with

text: equation

text: we

text: see

text: that

text: the

text: number

text: of

text: corners

text: is

text: (T(f

text: W

text: The

text: problem

text: with

text: f

text: though,

text: is

text: that

text: it

text: may

text: take

text: values

text: greater

text: than

text: We

text: need

text: to

text: bound

text: f

text: Below,

text: we

text: will

text: be

text: using

text: convolutions

text: of

text: E

text: functions

text: on

text: W,

text: which

text: are

text: deﬁned

text: as

text: follows:

text: g

text: h(y)

text: g(x)h(y

text: x).

text: x

text: W

text: A

text: B

text: C

text: D

text: P

text: B

text: C

text: D

text: i

text: j

text: k

text: i

text: j

text: k

text: (x,

text: z)

text: Bi

text: Cj

text: Dk

text: x

text: y

text: z

text: W

text: C

text: D

text: i

text: j

text: k

text: W

text: (B

text: C

text: D

text: (r)

text: i

text: j

text: k

text: r

text: W

text: W

text: Bˆ

text: (r)Cˆ

text: (r)Dˆ

text: (r)

text: i

text: j

text: k

text: r

text: W

text: We

text: split

text: the

text: above

text: sum

text: into

text: a

text: “main”

text: term,

text: and

text: error

text: terms,

text: which

text: can

text: be

text: bounded

text: using

text: the

text: uniformity

text: assumption,

text: plus

text: applications

text: of

text: the

text: Cauchy-

text: inequality

text: and

text: Parseval’s

text: identity.

text: Bˆ

text: (r)Cˆ

text: (r)Dˆ

text: (r)

text: δ(B

text: )δ(C

text: )δ(D

text: Bˆ

text: (r)Cˆ

text: (r)Dˆ

text: (r)

text: i

text: j

text: k

text: i

text: j

text: k

text: i

text: j

text: k

text: r

text: W

text: r

text: W

text: ##LTLine##

text: ##LTLine##


text: 4.2 Lower bound ≥ In this subsection, we will prove the lower bound m(α) α4. For conve- E | E | E | nience, we will write f = (f X, Y ), f = (f X, Z) and f = (f Y, Z). E3 2≤ 1 Firstly, note that (f1(f < α/4)) α/4 for i = 1, 2, 3 (where 1 denotes i the indicator function of an event), because by the law of total expectation E E E | (f1(f < α/4)) = ( (f1(f < α/4) Y, Z)) 1 1 E E | = ( (f Y, Z)1(f < α/4))) 1 E = (f 1(f < α/4)) 1 1 ≤ α/4 And similarly for the others. Therefore (cid:88) E ≥ ≥ ≥ ≥ E − 3 E ≥ (f1(f α/4)1(f α/4)1(f α/4)) (f) (f1(f < α/4)) α/4 1 2 3 i i=1 ≥ ∧ ≥ ∧ ≥ Therefore the event A = (f α/4) (f α/4) (f α/4) has probability 1 2 3 at least α/4. So we have, for any f E ≥ E | P ≥ α4 T(f) = (f f f ) (f f f A) (A) (α/4)3(α/4) = . 1 2 3 1 2 3 256 E Now suppose for contradiction some function h with (h) = α has T(h) < α4. So T(h) = α4/C, some C > 1. Pick an integer k such that Ck > 256. E ⊗ ⊗ Then (h k) = αk but T(h k) = α4k/Ck < α4k/256, contradicting the above ⊗ ≥ inequality for f = h k. Therefore m(α) α4. 4.3 Conclusion Now that we have bounds on m(α), we can combine them with Theorem 3 and Theoram 4 to get bounds on M(α). ≤ For the family of all abelian groups, we have M(α) (27/26)α3+c, where c = 0.131 . . . . Meanwhile, since α4 is a convex function of α, we can deduce F ≥ that for the family G = n, M(α) α4. This last bound also holds for the F n 2 family G = n for any prime p. n p It would be natural to ask what is the correct exponent log(m(α))/ log(α)). So far we only know that this exponent lies somewhere between 3.131 and 4. The author is continuing to study this question and hopes to report on further advances in due course. Acknowledgements. I am grateful to Ben Green for plenty of helpful com- ments and discussion. References [1] M. Ajtai and E. Szemer´edi. Sets of lattice points that form no squares. Stud. Sci. Math. Hungar., 9:9–11 (1975), 1974. 15 ##LTLine##

label: section

text: Lower

text: bound

text: In

text: this

text: subsection,

text: we

text: will

text: prove

text: the

text: lower

text: bound

text: m(α)

text: For

text: conve-

text: E

text: E

text: E

text: nience,

text: we

text: will

text: write

text: f

text: (f

text: X,

text: Y

text: f

text: (f

text: X,

text: Z)

text: and

text: f

text: (f

text: Y,

text: Z).

text: 2≤

text: Firstly,

text: note

text: that

text: for

text: i

text: (where

text: denotes

text: i

text: the

text: indicator

text: function

text: of

text: an

text: event),

text: because

text: by

text: the

text: law

text: of

text: total

text: expectation

text: E

text: E

text: E

text: Y,

text: Z))

text: E

text: E

text: (f

text: Y,

text: E

text: (f

text: And

text: similarly

text: for

text: the

text: others.

text: Therefore

text: E

text: E

text: E

text: (f)

text: i

text: Therefore

text: the

text: event

text: A

text: (f

text: (f

text: (f

text: has

text: probability

text: at

text: least

text: So

text: we

text: have,

text: for

text: any

text: f

text: E

text: E

text: P

text: α4

text: T(f)

text: (f

text: f

text: f

text: (f

text: f

text: f

text: A)

text: (A)

text: E

text: Now

text: suppose

text: for

text: contradiction

text: some

text: function

text: h

text: with

text: (h)

text: α

text: has

text: T(h)

text: So

text: T(h)

text: some

text: C

text: Pick

text: an

text: integer

text: k

text: such

text: that

text: Ck

text: E

text: Then

text: (h

text: k)

text: αk

text: but

text: T(h

text: k)

text: contradicting

text: the

text: above

text: inequality

text: for

text: f

text: h

text: k.

text: Therefore

text: m(α)

text: Conclusion

text: Now

text: that

text: we

text: have

text: bounds

text: on

text: m(α),

text: we

text: can

text: combine

text: them

text: with

text: Theorem

text: and

text: Theoram

text: to

text: get

text: bounds

text: on

text: M(α).

text: For

text: the

text: family

text: of

text: all

text: abelian

text: groups,

text: we

text: have

text: M(α)

text: where

text: c

text: Meanwhile,

text: since

text: α4

text: is

text: a

text: convex

text: function

text: of

text: α,

text: we

text: can

text: deduce

text: F

text: that

text: for

text: the

text: family

text: G

text: n,

text: M(α)

text: This

text: last

text: bound

text: also

text: holds

text: for

text: the

text: F

text: n

text: family

text: G

text: n

text: for

text: any

text: prime

text: p.

text: n

text: p

text: It

text: would

text: be

text: natural

text: to

text: ask

text: what

text: is

text: the

text: correct

text: exponent

text: log(m(α))/

text: log(α)).

text: So

text: far

text: we

text: only

text: know

text: that

text: this

text: exponent

text: lies

text: somewhere

text: between

text: and

text: The

text: author

text: is

text: continuing

text: to

text: study

text: this

text: question

text: and

text: hopes

text: to

text: report

text: on

text: further

text: advances

text: in

text: due

text: course.

text: Acknowledgements.

text: I

text: am

text: grateful

text: to

text: Ben

text: Green

text: for

text: plenty

text: of

text: helpful

text: com-

text: ments

text: and

text: discussion.

text: References

text: M.

text: Ajtai

text: and

text: E.

text: Szemer´edi.

text: Sets

text: of

text: lattice

text: points

text: that

text: form

text: no

text: squares.

text: Stud.

text: Sci.

text: Math.

text: Hungar.,

text: ##LTLine##


text: Proof. First part: Suppose we are given such a function f. Let m, a positive ≤ ≤ · · · ≤ ≤ ≤ · · · ≤ integer, and 0 = x x x = 1, 0 = y y y = 1 and ≤ ≤ · · · ≤0 1 m 0 1 ×m × 0 = z z z = 1 be such that f is constant on [x , x ) [y , y ) 0 1 ≤m ≤ − ∼ i i+1 j j+1 [z , z ) for each 0 i, j, k m 1. Given X, Y, Z U[0, 1] independent we k k+1 (cid:48) (cid:48) (cid:48) (cid:48) deﬁne new random variables X , Y and Z taking values in [m] by setting X to ∈ (cid:48) ∈ (cid:48) be the i such that X [x − , x ), Y to be the j such that Y [y − , y ) and Z ∈i 1 i (cid:48) (cid:48)j 1 j (cid:48) to be the k such that Z [z − , z ). The independence of X , Y and Z follows k 1 k from the indepence of X, Y and Z. By the piecewise constant assumption, the (cid:48) (cid:48) (cid:48) value of the function f is determined by the values of X , Y and Z , so there (cid:48) → (cid:48) (cid:48) (cid:48) (cid:48) exists a function f : [m]3 [0, 1] such that f(X, Y, Z) = f (X , Y , Z ). The ﬁrst part follows easily. (cid:48) (cid:48) Second part: Now suppose we are given indepent random variables X , Y (cid:48) and Z , each taking only ﬁnitely many values. Without loss of generality, we may (cid:48) (cid:48) (cid:48) assume that X , Y and Z take values in [m] for some positive integer m. This (cid:48) (cid:48) (cid:48) is because the actual values taken by X , Y and Z are unimportant, so we can ≤ ≤ · · · ≤ simply relable them. We then deﬁne real numbers 0 = x x x = 1, ≤ ≤ · · · ≤ ≤ ≤ · · ·0 ≤ 1 m 0 = y y y = 1 and 0 = z z z = 1 by setting 0P (cid:48) 1≤ Pm (cid:48) ≤ 0 P 1 (cid:48) ≤ m P (cid:48) x = (X i), y = (Y i) and z = (Z i). Then (X = i) = i − P ∈i i (cid:48) x x − = (X [x − , x )) so we may couple X with X in the same way i i 1 i 1 i (cid:48) as we did in the ﬁrst part. The same applies, of course, for Y with Y and Z (cid:48) → with Z . We can then construct a piecewise constant f : [0, 1]3 [0, 1] with (cid:48) (cid:48) (cid:48) (cid:48) f(X, Y, Z) = f (X , Y , Z ), and so the second part follows. 2 Construction In what follows, we will need some properties of the functions m and M, which we will establish now. (cid:48) → (cid:48) ≤ ∈ Lemma 2. Let m : [0, 1] [0, 1] be convex with m (α) m(α) for all α [0, 1]. (cid:48) M, m and m are all increasing functions of α, and m is uniformly continuous. (cid:48) Proof. M and m are trivially increasing. m(0) = 0 so m (0) = 0, which implies (cid:48) m is increasing by convexity. So we focus on showing that m is uniformly → continuous. Given any piecewise constant function f : [0, 1]3 [0, 1], we can − form a new function f by setting f = (cid:15) + (1 (cid:15))f. This function will have E − E (cid:15) E (cid:15)− E (f ) = (cid:15) + (1 (cid:15)) (f) = (f) + (cid:15)(1 (f)), and (cid:15) E E | E | E | T(f ) = ( (f X, Y ) (f X, Z) (f Y, Z)) (cid:15) (cid:15) (cid:15) (cid:15) ≤ E E | E | E | (( (f X, Y ) + (cid:15))( (f X, Z) + (cid:15))( (f Y, Z) + (cid:15))) ≤ T(f) + 3(cid:15) + 3(cid:15)2 + (cid:15)3. The last inequality above is obtained by expanding and using the trivial bound ≤ E f 1. By taking f to have (f) = α and T(f) approaching m(α), we deduce ≤ − ≤ m(α) m(α + (cid:15)(1 α)) m(α) + 3(cid:15) + 3(cid:15)2 + (cid:15)3, which implies continuity of m at all points except from 1. To show continuity at 1, we use the following 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: First

text: part:

text: Suppose

text: we

text: are

text: given

text: such

text: a

text: function

text: f.

text: Let

text: m,

text: a

text: positive

text: integer,

text: and

text: x

text: x

text: x

text: y

text: y

text: y

text: and

text: ≤0

text: m

text: ×m

text: z

text: z

text: z

text: be

text: such

text: that

text: f

text: is

text: constant

text: on

text: [x

text: x

text: [y

text: y

text: ≤m

text: i

text: j

text: [z

text: z

text: for

text: each

text: i,

text: j,

text: k

text: m

text: Given

text: X,

text: Y,

text: Z

text: independent

text: we

text: k

text: deﬁne

text: new

text: random

text: variables

text: X

text: Y

text: and

text: Z

text: taking

text: values

text: in

text: [m]

text: by

text: setting

text: X

text: to

text: be

text: the

text: i

text: such

text: that

text: X

text: [x

text: x

text: Y

text: to

text: be

text: the

text: j

text: such

text: that

text: Y

text: [y

text: y

text: and

text: Z

text: ∈i

text: i

text: j

text: to

text: be

text: the

text: k

text: such

text: that

text: Z

text: [z

text: z

text: The

text: independence

text: of

text: X

text: Y

text: and

text: Z

text: follows

text: k

text: k

text: from

text: the

text: indepence

text: of

text: X,

text: Y

text: and

text: Z.

text: By

text: the

text: piecewise

text: constant

text: assumption,

text: the

text: value

text: of

text: the

text: function

text: f

text: is

text: determined

text: by

text: the

text: values

text: of

text: X

text: Y

text: and

text: Z

text: so

text: there

text: exists

text: a

text: function

text: f

text: such

text: that

text: f(X,

text: Y,

text: Z)

text: f

text: (X

text: Y

text: Z

text: The

text: ﬁrst

text: part

text: follows

text: easily.

text: Second

text: part:

text: Now

text: suppose

text: we

text: are

text: given

text: indepent

text: random

text: variables

text: X

text: Y

text: and

text: Z

text: each

text: taking

text: only

text: ﬁnitely

text: many

text: values.

text: Without

text: loss

text: of

text: generality,

text: we

text: may

text: assume

text: that

text: X

text: Y

text: and

text: Z

text: take

text: values

text: in

text: [m]

text: for

text: some

text: positive

text: integer

text: m.

text: This

text: is

text: because

text: the

text: actual

text: values

text: taken

text: by

text: X

text: Y

text: and

text: Z

text: are

text: unimportant,

text: so

text: we

text: can

text: simply

text: relable

text: them.

text: We

text: then

text: deﬁne

text: real

text: numbers

text: x

text: x

text: x

text: ·0

text: m

text: y

text: y

text: y

text: and

text: z

text: z

text: z

text: by

text: setting

text: 1≤

text: Pm

text: P

text: m

text: P

text: x

text: (X

text: i),

text: y

text: (Y

text: i)

text: and

text: z

text: (Z

text: i).

text: Then

text: (X

text: i)

text: i

text: P

text: ∈i

text: i

text: x

text: x

text: (X

text: [x

text: x

text: so

text: we

text: may

text: couple

text: X

text: with

text: X

text: in

text: the

text: same

text: way

text: i

text: i

text: i

text: i

text: as

text: we

text: did

text: in

text: the

text: ﬁrst

text: part.

text: The

text: same

text: applies,

text: of

text: course,

text: for

text: Y

text: with

text: Y

text: and

text: Z

text: with

text: Z

text: We

text: can

text: then

text: construct

text: a

text: piecewise

text: constant

text: f

text: with

text: f(X,

text: Y,

text: Z)

text: f

text: (X

text: Y

text: Z

text: and

text: so

text: the

text: second

text: part

text: follows.

text: Construction

text: In

text: what

text: follows,

text: we

text: will

text: need

text: some

text: properties

text: of

text: the

text: functions

text: m

text: and

text: M,

text: which

text: we

text: will

text: establish

text: now.

text: Lemma

text: Let

text: m

text: be

text: convex

text: with

text: m

text: (α)

text: m(α)

text: for

text: all

text: α

text: M,

text: m

text: and

text: m

text: are

text: all

text: increasing

text: functions

text: of

text: α,

text: and

text: m

text: is

text: uniformly

text: continuous.

text: Proof.

text: M

text: and

text: m

text: are

text: trivially

text: increasing.

text: so

text: m

text: which

text: implies

text: m

text: is

text: increasing

text: by

text: convexity.

text: So

text: we

text: focus

text: on

text: showing

text: that

text: m

text: is

text: uniformly

text: continuous.

text: Given

text: any

text: piecewise

text: constant

text: function

text: f

text: we

text: can

text: form

text: a

text: new

text: function

text: f

text: by

text: setting

text: f

text: This

text: function

text: will

text: have

text: E

text: E

text: E

text: E

text: (f

text: (f)

text: (f)

text: (f)),

text: and

text: E

text: E

text: E

text: E

text: T(f

text: (f

text: X,

text: Y

text: (f

text: X,

text: Z)

text: (f

text: Y,

text: Z))

text: E

text: E

text: E

text: E

text: (f

text: X,

text: Y

text: (f

text: X,

text: Z)

text: (f

text: Y,

text: Z)

text: T(f)

text: The

text: last

text: inequality

text: above

text: is

text: obtained

text: by

text: expanding

text: and

text: using

text: the

text: trivial

text: bound

text: E

text: f

text: By

text: taking

text: f

text: to

text: have

text: (f)

text: α

text: and

text: T(f)

text: approaching

text: m(α),

text: we

text: deduce

text: m(α)

text: m(α

text: α))

text: m(α)

text: which

text: implies

text: continuity

text: of

text: m

text: at

text: all

text: points

text: except

text: from

text: To

text: show

text: continuity

text: at

text: we

text: use

text: the

text: following

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ ∈ A C-coloring of G is a function φ that assigns each vertex v V (G) a color φ(v) L(v), such ∈ that for every e = uv E(G), the vertices (u, φ(u)) and (v, φ(v)) are not adjacent in C . We say e that G is C-colorable if such a C-coloring exists. The correspondence chromatic number χ (G) of G is the smallest integer k such that G is DP C-colorable for every k-correspondence assignment (L, C). Let (L, C) be a k-correspondence assignment for a graph G, and let W = v v . . . v with 1 2 m v = v be a closed walk of length m in G. We say that the assignment (L, C) is inconsistent m 1 ∈ ∈ on W if there exist c L(v ) for i [m] such that (v , c )(v , c ) is an edge of C for ∈ − (cid:54) i i i i i+1 i+1 vivi+i i [m 1], and c = c . The (L, C) is consistent if (L, C) is inconsistent on none of the closed 1 m walks in G. Recently, DP-coloring is attractively widely. On the one hand, some results of list coloring were generalized to DP-coloring. For this literature, the readers can see [13, 14, 15, 16]. On the other hand, some new results of DP-coloring are obtained, can be found in [1, 2, 3]. We here pay more attention on the result of Dvoˇr´ak and Postle [9], who solved a conjecture of Borodin in 2007 by the technique developed in DP-coloring, as follows. Theorem 1.1 ([9]) Every planar graph G without cycles of lengths 4 to 8 is 3-choosable. The well-known Four Color Theorem states that all planar graphs can be colored with four colors such that no adjacent vertices share a common color. There was much eﬀort to ﬁnd suﬃcient conditions for a graph to be 3-colorable. The classic theorem by Gr¨otzch([11], 1959) shows that planar graphs without 3-cycles are 3-colorable. In 1969, Havel ([12]) posed a problem: Does there exists a constant C such that every planar graph with the minimum distance between triangles at least C is 3-colorable? Recently, Dvoˇr´ak, Kr´al’ and Thomas [8] says that the distance is at least 10100. In 1976, Steinberg ([17]) conjectured that every planar graph without 4-cycles and 5-cycles is 3-colorable. Erd˝os relaxed Steinberg’s conjecture and suggested to determine the smallest integer k, if it exists, such that every planar graphs without cycles of length from 4 to k is 3-colorable. The best bound for such k is 7, and it is proved by Borodin, Glebov, Raspaud, and Salavatipour [4]. In 2003, Strong Bordeaux Conjecture ([5]) by Borodin and Raspaud was posed as follows: Every planar graphs without 5-cycle and − K is 3-colorable. 4 Recently, Cohen-Addad, Hebdige, Kral, Li and Salgado [7] presented a counterexample to the Steinberg’s Conjecture, as well as to the Strong Bordeaux Conjecture. Borodin, Montassier and Raspaud [6] asked to determine the smallest integer k such that every planar graphs without adjacent cycles of length at most k is 3-colorable and proved the following result. Theorem 1.2 Every planar graph without adjacent cycles of length at most 7 is 3-colorable. Motivated by Theorems 1.1 and 1.2, we present the following result in this paper. Theorem 1.3 Planar graphs without adjacent cycles of length at most 8 are 3-choosable. In the end of this section, we introduce some notations used in the paper. Graphs mentioned in this paper are all simple. Let K be a cycle of a plane graph G. We use int(K) and ext(K) 2

text: A

text: C-coloring

text: of

text: G

text: is

text: a

text: function

text: φ

text: that

text: assigns

text: each

text: vertex

text: v

text: V

text: (G)

text: a

text: color

text: φ(v)

text: L(v),

text: such

text: that

text: for

text: every

text: e

text: uv

text: E(G),

text: the

text: vertices

text: (u,

text: φ(u))

text: and

text: (v,

text: φ(v))

text: are

text: not

text: adjacent

text: in

text: C

text: We

text: say

text: e

text: that

text: G

text: is

text: C-colorable

text: if

text: such

text: a

text: C-coloring

text: exists.

text: The

text: correspondence

text: chromatic

text: number

text: χ

text: (G)

text: of

text: G

text: is

text: the

text: smallest

text: integer

text: k

text: such

text: that

text: G

text: is

text: DP

text: C-colorable

text: for

text: every

text: k-correspondence

text: assignment

text: (L,

text: C).

text: Let

text: (L,

text: C)

text: be

text: a

text: k-correspondence

text: assignment

text: for

text: a

text: graph

text: G,

text: and

text: let

text: W

text: v

text: v

text: v

text: with

text: m

text: v

text: v

text: be

text: a

text: closed

text: walk

text: of

text: length

text: m

text: in

text: G.

text: We

text: say

text: that

text: the

text: assignment

text: (L,

text: C)

text: is

text: inconsistent

text: m

text: on

text: W

text: if

text: there

text: exist

text: c

text: L(v

text: for

text: i

text: [m]

text: such

text: that

text: (v

text: c

text: )(v

text: c

text: is

text: an

text: edge

text: of

text: C

text: for

text: i

text: i

text: i

text: i

text: vivi+i

text: i

text: [m

text: and

text: c

text: c

text: The

text: (L,

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text: measure among artifacts based on skin decorations. To approach this problem, we con- sider a novel extension of the Local Binary Pattern description to surface tessellations based on the evolution of the color over concentric circles around a vertex. To deter- mine these circles we adopt a sphere - edge intersection strategy and for this reason we name our approach edgeLBP. As application of the edgeLBP description, we propose the retrieval and classiﬁcation of color patterns over surfaces. The remainder of the paper is organized as follows. Section brieﬂy reviews the literature on the retrieval of textured images and surfaces. Section introduces the elements of our method, i.e. the edgeLBP operator and how we store it in a descrip- tor. Section presents and analyses the retrieval and classiﬁcation performances of the method over two datasets, while conclusive remarks end the paper, Section . 2 State of art A typical strategy to detect textures on images is to consider local patches that describe the behavior of the texture around a group of pixels. Examples of these de- scriptions are the Local Binary Patterns (LBP) [21, 22], the Scale Invariant Feature Transform (SIFT) [18] and the Histogram of Oriented Gradients (HOG) [11].The gen- eralization of these descriptions to (even textured) surfaces has been explored in sev- eral works, such as the PANORAMA views of the 3D objects [23], the meshHOG [37] and the meshLBP [34, 35]. In general, the methods for matching textured 3D shapes adopt a combination of geometric and colorimetric descriptors. Possible choices of the colorimetric descriptors are: feature-vectors, where the color is treated as a gen- eral property of the shape, [30], or its subparts in [12]; local or global views of the objects [24,36]; point-to-point correspondences among sets of feature points (e.g., the CSHOT descriptor [31]); the evolution of the sub-level sets according to the persistent homology settings [5]. These methods mainly address the shape matching problem without focusing on the surface details and local colorimetric variations. On the con- trary, when looking for patterns, locality and scale are the two key aspects. A detailed evaluation and comparison of methods for 3D texture retrieval and comparison can be found in [7] and several SHREC contests [6,9,13]. However, all these contests focused on the joint comparison of geometry and texture, without considering the comparison of the purely colorimetric information that characterizes the surface decorations. At the best of our knowledge, the Mesh Local Binary Pattern (meshLBP) approach [33–35] is the unique approach that explicitly addresses pattern analysis over surfaces. The meshLBP extends the LBP [21] to triangle meshes. The main idea behind the meshLBP is that triangles play the role of pixels and the 8-neighbor connectivity in an image is ideally substituted by a 6-neighbor connectivity around triangles. Rings on the mesh are computed using a uniform, region growing, triangle-based expansion. From the practical point of view, the meshLBP encodes a pattern efﬁciently, providing a compact representation of it. 3 The edgeLBP We extend the LBP to surfaces using rings deﬁned on the basis of a sphere-mesh intersection. In Section we brieﬂy sum up the deﬁnition of the LBP deﬁnition. Our extension to surface tessellations is described in Section , while Section details the 2

text: measure

text: among

text: artifacts

text: based

text: on

text: skin

text: decorations.

text: To

text: approach

text: this

text: problem,

text: we

text: con-

text: sider

text: a

text: novel

text: extension

text: of

text: the

text: Local

text: Binary

text: Pattern

text: description

text: to

text: surface

text: tessellations

text: based

text: on

text: the

text: evolution

text: of

text: the

text: color

text: over

text: concentric

text: circles

text: around

text: a

text: vertex.

text: To

text: deter-

text: mine

text: these

text: circles

text: we

text: adopt

text: a

text: sphere

text: edge

text: intersection

text: strategy

text: and

text: for

text: this

text: reason

text: we

text: name

text: our

text: approach

text: edgeLBP.

text: As

text: application

text: of

text: the

text: edgeLBP

text: description,

text: we

text: propose

text: the

text: retrieval

text: and

text: classiﬁcation

text: of

text: color

text: patterns

text: over

text: surfaces.

text: The

text: remainder

text: of

text: the

text: paper

text: is

text: organized

text: as

text: follows.

text: Section

text: brieﬂy

text: reviews

text: the

text: literature

text: on

text: the

text: retrieval

text: of

text: textured

text: images

text: and

text: surfaces.

text: Section

text: introduces

text: the

text: elements

text: of

text: our

text: method,

text: i.e.

text: the

text: edgeLBP

text: operator

text: and

text: how

text: we

text: store

text: it

text: in

text: a

text: descrip-

text: tor.

text: Section

text: presents

text: and

text: analyses

text: the

text: retrieval

text: and

text: classiﬁcation

text: performances

text: of

text: the

text: method

text: over

text: two

text: datasets,

text: while

text: conclusive

text: remarks

text: end

text: the

text: paper,

text: Section

text: State

text: of

text: art

text: A

text: typical

text: strategy

text: to

text: detect

text: textures

text: on

text: images

text: is

text: to

text: consider

text: local

text: patches

text: that

text: describe

text: the

text: behavior

text: of

text: the

text: texture

text: around

text: a

text: group

text: of

text: pixels.

text: Examples

text: of

text: these

text: de-

text: scriptions

text: are

text: the

text: Local

text: Binary

text: Patterns

text: (LBP)

text: the

text: Scale

text: Invariant

text: Feature

text: Transform

text: (SIFT)

text: and

text: the

text: Histogram

text: of

text: Oriented

text: Gradients

text: (HOG)

text: gen-

text: eralization

text: of

text: these

text: descriptions

text: to

text: (even

text: textured)

text: surfaces

text: has

text: been

text: explored

text: in

text: sev-

text: eral

text: works,

text: such

text: as

text: the

text: PANORAMA

text: views

text: of

text: the

text: objects

text: the

text: meshHOG

text: and

text: the

text: meshLBP

text: In

text: general,

text: the

text: methods

text: for

text: matching

text: textured

text: shapes

text: adopt

text: a

text: combination

text: of

text: geometric

text: and

text: colorimetric

text: descriptors.

text: Possible

text: choices

text: of

text: the

text: colorimetric

text: descriptors

text: are:

text: feature-vectors,

text: where

text: the

text: color

text: is

text: treated

text: as

text: a

text: gen-

text: eral

text: property

text: of

text: the

text: shape,

text: or

text: its

text: subparts

text: in

text: local

text: or

text: global

text: views

text: of

text: the

text: objects

text: point-to-point

text: correspondences

text: among

text: sets

text: of

text: feature

text: points

text: (e.g.,

text: the

text: CSHOT

text: descriptor

text: the

text: evolution

text: of

text: the

text: sub-level

text: sets

text: according

text: to

text: the

text: persistent

text: homology

text: settings

text: These

text: methods

text: mainly

text: address

text: the

text: shape

text: matching

text: problem

text: without

text: focusing

text: on

text: the

text: surface

text: details

text: and

text: local

text: colorimetric

text: variations.

text: On

text: the

text: con-

text: trary,

text: when

text: looking

text: for

text: patterns,

text: locality

text: and

text: scale

text: are

text: the

text: two

text: key

text: aspects.

text: A

text: detailed

text: evaluation

text: and

text: comparison

text: of

text: methods

text: for

text: texture

text: retrieval

text: and

text: comparison

text: can

text: be

text: found

text: in

text: and

text: several

text: SHREC

text: contests

text: However,

text: all

text: these

text: contests

text: focused

text: on

text: the

text: joint

text: comparison

text: of

text: geometry

text: and

text: texture,

text: without

text: considering

text: the

text: comparison

text: of

text: the

text: purely

text: colorimetric

text: information

text: that

text: characterizes

text: the

text: surface

text: decorations.

text: At

text: the

text: best

text: of

text: our

text: knowledge,

text: the

text: Mesh

text: Local

text: Binary

text: Pattern

text: (meshLBP)

text: approach

text: is

text: the

text: unique

text: approach

text: that

text: explicitly

text: addresses

text: pattern

text: analysis

text: over

text: surfaces.

text: The

text: meshLBP

text: extends

text: the

text: LBP

text: to

text: triangle

text: meshes.

text: The

text: main

text: idea

text: behind

text: the

text: meshLBP

text: is

text: that

text: triangles

text: play

text: the

text: role

text: of

text: pixels

text: and

text: the

text: connectivity

text: in

text: an

text: image

text: is

text: ideally

text: substituted

text: by

text: a

text: connectivity

text: around

text: triangles.

text: Rings

text: on

text: the

text: mesh

text: are

text: computed

text: using

text: a

text: uniform,

text: region

text: growing,

text: triangle-based

text: expansion.

text: From

text: the

text: practical

text: point

text: of

text: view,

text: the

text: meshLBP

text: encodes

text: a

text: pattern

text: efﬁciently,

text: providing

text: a

text: compact

text: representation

text: of

text: it.

text: The

text: edgeLBP

text: We

text: extend

text: the

text: LBP

text: to

text: surfaces

text: using

text: rings

text: deﬁned

text: on

text: the

text: basis

text: of

text: a

text: sphere-mesh

text: intersection.

text: In

text: Section

text: we

text: brieﬂy

text: sum

text: up

text: the

text: deﬁnition

text: of

text: the

text: LBP

text: deﬁnition.

text: Our

text: extension

text: to

text: surface

text: tessellations

text: is

text: described

text: in

text: Section

text: while

text: Section

text: details

text: the


text: P : 15, N : 5R : 0, 2 P : 15, N : 5R : 0, 5 r max r max Figure 8: Confusion matrices and tier images for two of the best runs of the edgeLBP on the Artifacts dataset. In the tier images, the black dots represent the NN element, the red dots correspond to points in FT while blue ones are the ST. Table 1: The NN, FT and ST scores for some runs of the edgeLBP on the Artifacts ∗ dataset. The in the fourth row means that in these settings we adopt h3 instead of h (here, h corresponds to the L-channel). R is expressed in mm. Parameter Settings NN FT ST P : 15, N : 5, R : 0, 2 0.775 0.789 1 r max P : 15, N : 5, R : 0, 3 0.75 0.811 0.989 r max P : 15, N : 5, R : 0, 5 0.75 0.711 0.889 r max ∗ P : 15, N : 5, R : 0, 7 0.725 0.667 0.756 r max P : 12, N : 7, R : 0, 5 0.75 0.789 0.9 r max P : 12, N : 7, R : 0, 2 0.775 0.856 0.978 r max P : 18, N : 5, R : 0, 7 0.7 0.667 0.744 r max 12 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P

text: N

text: P

text: N

text: r

text: max

text: r

text: max

text: Figure

text: Confusion

text: matrices

text: and

text: tier

text: images

text: for

text: two

text: of

text: the

text: best

text: runs

text: of

text: the

text: edgeLBP

text: on

text: the

text: Artifacts

text: dataset.

text: In

text: the

text: tier

text: images,

text: the

text: black

text: dots

text: represent

text: the

text: NN

text: element,

text: the

text: red

text: dots

text: correspond

text: to

text: points

text: in

text: FT

text: while

text: blue

text: ones

text: are

text: the

text: ST.

text: Table

text: The

text: NN,

text: FT

text: and

text: ST

text: scores

text: for

text: some

text: runs

text: of

text: the

text: edgeLBP

text: on

text: the

text: Artifacts

text: dataset.

text: The

text: in

text: the

text: fourth

text: row

text: means

text: that

text: in

text: these

text: settings

text: we

text: adopt

text: instead

text: of

text: h

text: (here,

text: h

text: corresponds

text: to

text: the

text: L-channel).

text: R

text: is

text: expressed

text: in

text: mm.

text: Parameter

text: Settings

text: NN

text: FT

text: ST

text: P

text: N

text: R

text: r

text: max

text: P

text: N

text: R

text: r

text: max

text: P

text: N

text: R

text: r

text: max

text: P

text: N

text: R

text: r

text: max

text: P

text: N

text: R

text: r

text: max

text: P

text: N

text: R

text: r

text: max

text: P

text: N

text: R

text: r

text: max

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: 40K 24K 8K Figure 9: The degradation of one of the model used to test the robustness of the de- scriptor of both edgeLBP and meshLBP. The number on each image is the respective vertex resolution. Table 2: Evaluation measures of the performances on the CPP dataset resampled with 40K vertices. NN FT ST e mAP nDCG edgeLBP 0.95 0.688 0.857 0.59 0.761 0.9 meshLBP 0.77 0.517 0.703 0.47 0.58 0.79 4.4 Robustness over different surface tessellations The strength of the edgeLBP is its ring deﬁnition, which is robust to different sur- face tessellations: in this Section we experimentally discuss this robustness. To this aim we re-sample the triangles meshes with a decreasing number of vertices. The triangle mesh re-sampling with x vertices is done with the MeshLAB tool [10] that approxi- mates the original mesh preserving its geometry as much as possible with the given number of vertices (for instance, x = 40K vertices). This process generally modiﬁes the mesh connectivity and the area of the triangles, discards the smallest details and keeps the overall shape, unless the number of vertices drastically diminishes and the new vertices are too few to preserve it. First, we re-sampled the meshes in the CCP dataset with 40K vertices. On this dataset, we compare the outcome of the edgeLBP with the default settings with the meshLBP, see Table 2. If compared with the performances on the original CPP dataset in Figure 7, the edgeLBP degrades less than the meshLBP, demonstrating of being more robust to mesh degradation and re-sampling. Second, we selected 3 patches from the Artifacts dataset and sub-sampled them with 32K, 24K, 16K and 8K vertices (see Figure 9).These four meshes are compared against the original patch (that has 40K vertices).These four distance values provide an estimate of the error the descriptors do when working with the simpliﬁed meshes. We performed two runs for both the edgeLBP and meshLBP: • Run1: P = 12, N = 7. These settings are the setting used by the meshLBP r as default. Both meshLBP and edgeLBP are run with these settings. For the edgeLBP we set R = 0.5mm. max • Run2: P = 15, N = 5. These settings are those that we consider default for r the edgeLBP. Both the algorithms are run with these settings. As in run1, we set 13 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

text: The

text: degradation

text: of

text: one

text: of

text: the

text: model

text: used

text: to

text: test

text: the

text: robustness

text: of

text: the

text: de-

text: scriptor

text: of

text: both

text: edgeLBP

text: and

text: meshLBP.

text: The

text: number

text: on

text: each

text: image

text: is

text: the

text: respective

text: vertex

text: resolution.

text: Table

text: Evaluation

text: measures

text: of

text: the

text: performances

text: on

text: the

text: CPP

text: dataset

text: resampled

text: with

text: vertices.

text: NN

label: table

text: FT

text: ST

text: e

text: mAP

text: nDCG

text: edgeLBP

text: meshLBP

text: Robustness

text: over

text: different

text: surface

text: tessellations

text: The

text: strength

text: of

text: the

text: edgeLBP

text: is

text: its

text: ring

text: deﬁnition,

text: which

text: is

text: robust

text: to

text: different

text: sur-

text: face

text: tessellations:

text: in

text: this

text: Section

text: we

text: experimentally

text: discuss

text: this

text: robustness.

text: To

text: this

text: aim

text: we

text: re-sample

text: the

text: triangles

text: meshes

text: with

text: a

text: decreasing

text: number

text: of

text: vertices.

text: The

text: triangle

text: mesh

text: re-sampling

text: with

text: x

text: vertices

text: is

text: done

text: with

text: the

text: MeshLAB

text: tool

text: that

text: approxi-

text: mates

text: the

text: original

text: mesh

text: preserving

text: its

text: geometry

text: as

text: much

text: as

text: possible

text: with

text: the

text: given

text: number

text: of

text: vertices

text: (for

text: instance,

text: x

text: vertices).

text: This

text: process

text: generally

text: modiﬁes

text: the

text: mesh

text: connectivity

text: and

text: the

text: area

text: of

text: the

text: triangles,

text: discards

text: the

text: smallest

text: details

text: and

text: keeps

text: the

text: overall

text: shape,

text: unless

text: the

text: number

text: of

text: vertices

text: drastically

text: diminishes

text: and

text: the

text: new

text: vertices

text: are

text: too

text: few

text: to

text: preserve

text: it.

text: First,

text: we

text: re-sampled

text: the

text: meshes

text: in

text: the

text: CCP

text: dataset

text: with

text: vertices.

text: On

text: this

text: dataset,

text: we

text: compare

text: the

text: outcome

text: of

text: the

text: edgeLBP

text: with

text: the

text: default

text: settings

text: with

text: the

text: meshLBP,

text: see

text: Table

text: If

text: compared

text: with

text: the

text: performances

text: on

text: the

text: original

text: CPP

text: dataset

text: in

text: Figure

text: the

text: edgeLBP

text: degrades

text: less

text: than

text: the

text: meshLBP,

text: demonstrating

text: of

text: being

text: more

text: robust

text: to

text: mesh

text: degradation

text: and

text: re-sampling.

text: Second,

text: we

text: selected

text: patches

text: from

text: the

text: Artifacts

text: dataset

text: and

text: sub-sampled

text: them

text: with

text: and

text: vertices

text: (see

text: Figure

text: four

text: meshes

text: are

text: compared

text: against

text: the

text: original

text: patch

text: (that

text: has

text: vertices).These

text: four

text: distance

text: values

text: provide

text: an

text: estimate

text: of

text: the

text: error

text: the

text: descriptors

text: do

text: when

text: working

text: with

text: the

text: simpliﬁed

text: meshes.

text: We

text: performed

text: two

text: runs

text: for

text: both

text: the

text: edgeLBP

text: and

text: meshLBP:

text: P

text: N

text: These

text: settings

text: are

text: the

text: setting

text: used

text: by

text: the

text: meshLBP

text: r

text: as

text: default.

text: Both

text: meshLBP

text: and

text: edgeLBP

text: are

text: run

text: with

text: these

text: settings.

text: For

text: the

text: edgeLBP

text: we

text: set

text: R

text: max

text: P

text: N

text: These

text: settings

text: are

text: those

text: that

text: we

text: consider

text: default

text: for

text: r

text: the

text: edgeLBP.

text: Both

text: the

text: algorithms

text: are

text: run

text: with

text: these

text: settings.

text: As

text: in

text: we

text: set

text: ##LTFigure##

text: ##LTFigure##

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: ##LTLine##


text: [36] C. Wu, B. Clipp, X. Li, J.-M. Frahm, and M. Pollefeys. 3D model matching with Viewpoint-Invariant Patches (VIP). In CVPR, pages 1–8, 2008. [37] A. Zaharescu, E. Boyer, and R. Horaud. Keypoints and local descriptors of scalar functions on 2D manifolds. Int. J. Comput. Vision, 100(1):78–98, 2012. 18

label: reference

text: C.

text: Wu,

text: B.

text: Clipp,

text: X.

text: Li,

text: J.-M.

text: Frahm,

text: and

text: M.

text: Pollefeys.

text: model

text: matching

text: with

text: Viewpoint-Invariant

text: Patches

text: (VIP).

text: In

text: CVPR,

text: pages

text: A.

text: Zaharescu,

text: E.

text: Boyer,

text: and

text: R.

text: Horaud.

text: Keypoints

text: and

text: local

text: descriptors

text: of

text: scalar

text: functions

text: on

text: manifolds.

text: Int.

text: J.

text: Comput.

text: Vision,