text: 15 − ∗ − ( ) = log( ) + + ( 2) ( ) ( ) = + ( 2) (23) h x x x x O x , h x p h p Kx O x . b b b → → Theorem 2: Location of the solution points of (14), when 0 and 0, is as follows: d1 d2 d2 ≈ eK(K(K12−−KK)) d1KK21−−KK . (24) − Proof: Let = 1 + ( ) and = ( ) + ( ) ( ) be sum-rate and distortion R0 h p D0 h p1 h p2 h p b b b b → at ( ) = (0 0). According to (23), after ( ) (0 0), then we have: d1, d2 , d1, d2 , − ∗ − − − = ( ) ( ) ( ) ( ) (25) R R0 h p d h p h d1 h d2 , b b b b − − { } = + log + log + (max 2 2 2 ) Kd d1 d1 d1 d2 d2 d2 O d , d1, d2 − { } = ( 1)( + ) + log + log + (max 2 2 ) K d1 d2 d1 d1 d2 d2 O d1, d2 , − ∗ − ∗ − − ∗ = ( ) ( ) + ( ) ( ) ( ) + ( ) (26) D D0 h p1 d1 h p1 h p2 d2 h p2 h p d h p b b b b b b − { } = + + (max 2 2 2 ) K1d1 K2d2 Kd O d , d1, d2 − − { } = ( ) + ( ) + (max 2 2 ) K1 K d1 K2 K d2 O d1, d2 . By ignoring the high-order terms, the following convex optimization problem is obtained: − min ( 1)( + ) + log + log 0≤d1,d2≤0.5 K d1 d2 d1 d1 d2 d2 (27) − − − s.t. ( ) + ( ) = K1 K d1 K2 K d2 D D0. The Lagrangian of the above minimization problem for the Lagrangian multiplier is L λ given by: May 30, 2020 DRAFT ##LTLine## ##LTLine##

text: log(

text: h

text: x

text: x

text: x

text: x

text: O

text: x

text: h

text: x

text: p

text: h

text: p

text: Kx

text: O

text: x

text: b

text: b

text: b

text: Theorem

text: Location

text: of

text: the

text: solution

text: points

text: of

text: when

text: and

text: is

text: as

text: follows:

text: Proof:

text: Let

text: and

text: be

text: sum-rate

text: and

text: distortion

text: h

text: p

text: h

text: h

text: h

text: p

text: b

text: b

text: b

text: b

text: at

text: According

text: to

text: after

text: then

text: we

text: have:

text: R

text: h

text: p

text: d

text: h

text: p

text: h

text: h

text: b

text: b

text: b

text: b

text: log

text: log

text: (max

text: Kd

text: O

text: d

text: log

text: log

text: (max

text: K

text: O

text: D

text: h

text: h

text: h

text: h

text: h

text: p

text: d

text: h

text: p

text: b

text: b

text: b

text: b

text: b

text: b

text: (max

text: Kd

text: O

text: d

text: (max

text: K

text: K

text: O

text: By

text: ignoring

text: the

text: high-order

text: terms,

text: the

text: following

text: convex

text: optimization

text: problem

text: is

text: obtained:

text: min

text: log

text: log

text: K

text: s.t.

text: K

text: K

text: D

text: The

text: Lagrangian

text: of

text: the

text: above

text: minimization

text: problem

text: for

text: the

text: Lagrangian

text: multiplier

text: is

text: L

text: λ

text: given

text: by:

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##


text: 16 − − − − = ( 1)( + ) + log + log + [( ) + ( ) + ] (28) L K d1 d2 d1 d1 d2 d2 λ K1 K d1 K2 K d2 D D0 . The gradient equation implies: ∂L − = + log + ( ) = 0 for = 1 2 (29) K d λ K K , i , . i i ∂d i Finally, by canceling in the above two equations, it is concluded that: λ K(K2 − K1) + (K2 − K) log d1 − (K1 − K) log d2 = 0 ⇒ d2 = eK(K(K12−−KK)) d1KK12−−KK . (30) Corollary 1: In general, without assuming that the test-channels are BSCs, one can − − still show that can be approximated by a convex function while can R R0 D D0 be approximated by a linear function, in the high-resolution regime. As a consequence, computation of the rate-distortion function can be approximately formulated as a convex optimization problem. A direct implication of this convex optimization formulation is that the binary symmetric test-channel is asymptotically optimal in the high-resolution regime. Corollary 2: The slope of the tangent lines to the curve of location of optimum points in ( ∗ ∗) = (0 0) are, respectively, 1, ∞, or 0 when = , , or . d1, d2 , p1 p2 p1 < p2 p1 > p2 In the following ﬁgure, location of the optimum points in a high-resolution regime 1 and the curve (24) for Fig. 2(b) are depicted. As it is obvious, these curves are approximately the same. By the following lemma, the asymptotic analysis of the problem is investigated around ( ∗ ∗) = (0 5 0 5). d1, d2 . , . Lemma 2: The maximum value of the parameter occurs in ( ) = (0 1) when µ R, D , ( ∗ ∗) = (0 5 0 5) and it equals: d1, d2 . , . 1 and are (10−4) d1 d2 O DRAFT May 30, 2020 ##LTLine## ##LTLine## ##LTLine##

label: paragraph

label: equation

text: log

text: log

text: L

text: K

text: λ

text: K

text: K

text: D

text: The

text: gradient

text: equation

text: implies:

text: ∂L

text: log

text: for

text: K

text: d

text: λ

text: K

text: K

text: i

text: i

text: i

text: ∂d

text: i

text: Finally,

text: by

text: canceling

text: in

text: the

text: above

text: two

text: equations,

text: it

text: is

text: concluded

text: that:

text: λ

text: K)

text: log

text: K)

text: log

text: Corollary

text: In

text: general,

text: without

text: assuming

text: that

text: the

text: test-channels

text: are

text: BSCs,

text: one

text: can

text: still

text: show

text: that

text: can

text: be

text: approximated

text: by

text: a

text: convex

text: function

text: while

text: can

text: R

text: D

text: be

text: approximated

text: by

text: a

text: linear

text: function,

text: in

text: the

text: high-resolution

text: regime.

text: As

text: a

text: consequence,

text: computation

text: of

text: the

text: rate-distortion

text: function

text: can

text: be

text: approximately

text: formulated

text: as

text: a

text: convex

text: optimization

text: problem.

text: A

text: direct

text: implication

text: of

text: this

text: convex

text: optimization

text: formulation

text: is

text: that

text: the

text: binary

text: symmetric

text: test-channel

text: is

text: asymptotically

text: optimal

text: in

text: the

text: high-resolution

text: regime.

text: Corollary

text: The

text: slope

text: of

text: the

text: tangent

text: lines

text: to

text: the

text: curve

text: of

text: location

text: of

text: optimum

text: points

text: in

text: are,

text: respectively,

text: or

text: when

text: or

text: In

text: the

text: following

text: ﬁgure,

text: location

text: of

text: the

text: optimum

text: points

text: in

text: a

text: high-resolution

text: regime

text: and

text: the

text: curve

text: for

text: Fig.

text: are

text: depicted.

text: As

text: it

text: is

text: obvious,

text: these

text: curves

text: are

text: approximately

text: the

text: same.

text: By

text: the

text: following

text: lemma,

text: the

text: asymptotic

text: analysis

text: of

text: the

text: problem

text: is

text: investigated

text: around

text: Lemma

text: The

text: maximum

text: value

text: of

text: the

text: parameter

text: occurs

text: in

text: when

text: µ

text: R,

text: D

text: and

text: it

text: equals:

text: and

text: are

text: O

text: DRAFT

text: May

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 21 (=a) ( | ) − ( | ) = ( ∗ ) − ( ) for = 1 2 H U U3− H U Y h d p h d , i , , i i i i b b i ≥ + ( ; ) R1 R2 I Y1, Y2 U1, U2 = ( ) − ( | ) (=a) 1 + ( | ) − ( | ) − ( | ) H U1, U2 H U1, U2 Y1, Y2 H U1 U2 H U1 Y1 H U2 U2 ∗ − − = 1 + ( ) ( ) ( ) h p d h d1 h d2 , b b b ↔ ↔ ↔ ↔ where (a) follows from the Markov chain rule in the form of d1 p1 p2 d2 . U1 Y1 X Y2 U2 Since a conventional rate-distortion quantizer can asymptotically achieve the compression − rate 1 ( ) for distortion being assumed , it is impossible to get close to (39) by only h d d b i i using the rate-distortion quantizer. Hence, another lossless source encoder should be utilized after the conventional rate-distortion quantizer for achieving the rate (39) in the -th link. i We use an LDGM quantizer concatenated with a Syndrome-Generator (SG), inspired by the “quantize-and-bin” idea in the context of information theory. On the other side, the dominant face of the rate region is a line segment connecting two end points ( 0 0 ) and R1, R2 ( 00 00), where R1, R2 (cid:16) (cid:17) ( 0 0 ) = ( ∗ ) − ( ) 1 − ( ) (40) R1, R2 hb p d hb d1 , hb d2 , and (cid:16) (cid:17) ( 00 00) = 1 − ( ) ( ∗ ) − ( ) (41) R1, R2 hb d1 , hb p d hb d2 . We consider a coding scheme for achieving ( 0 0 ). A similar method can be applied for R1, R2 achieving ( 00 00). Encoder 1 quantizes to using an LDGM code of rate = , R1, R2 y1n un1 R1,1 mn1 then it computes the syndrome sk11 = un1H1T , where H1 is the parity-check matrix of an LDPC code of rate = − . We do this process of quantize and bin by employing a R1 2 m1 k1 , n compound LDGM-LDPC code. It is notable that the total length of the obtained syndrome May 30, 2020 DRAFT ##LTLine## ##LTLine##

text: (=a)

text: for

text: H

text: U

text: H

text: U

text: Y

text: h

text: d

text: p

text: h

text: d

text: i

text: i

text: i

text: i

text: i

text: b

text: b

text: i

text: I

text: (=a)

text: H

text: H

text: H

text: H

text: H

text: h

text: p

text: d

text: h

text: h

text: b

text: b

text: b

text: where

text: (a)

text: follows

text: from

text: the

text: Markov

text: chain

text: rule

text: in

text: the

text: form

text: of

text: X

text: Since

text: a

text: conventional

text: rate-distortion

text: quantizer

text: can

text: asymptotically

text: achieve

text: the

text: compression

text: rate

text: for

text: distortion

text: being

text: assumed

text: it

text: is

text: impossible

text: to

text: get

text: close

text: to

text: by

text: only

text: h

text: d

text: d

text: b

text: i

text: i

text: using

text: the

text: rate-distortion

text: quantizer.

text: Hence,

text: another

text: lossless

text: source

text: encoder

text: should

text: be

text: utilized

text: after

text: the

text: conventional

text: rate-distortion

text: quantizer

text: for

text: achieving

text: the

text: rate

text: in

text: the

text: -th

text: link.

text: i

text: We

text: use

text: an

text: LDGM

text: quantizer

text: concatenated

text: with

text: a

text: Syndrome-Generator

text: (SG),

text: inspired

text: by

text: the

text: “quantize-and-bin”

text: idea

text: in

text: the

text: context

text: of

text: information

text: theory.

text: On

text: the

text: other

text: side,

text: the

text: dominant

text: face

text: of

text: the

text: rate

text: region

text: is

text: a

text: line

text: segment

text: connecting

text: two

text: end

text: points

text: and

text: where

text: hb

text: p

text: d

text: hb

text: hb

text: and

text: hb

text: hb

text: p

text: d

text: hb

text: We

text: consider

text: a

text: coding

text: scheme

text: for

text: achieving

text: A

text: similar

text: method

text: can

text: be

text: applied

text: for

text: achieving

text: Encoder

text: quantizes

text: to

text: using

text: an

text: LDGM

text: code

text: of

text: rate

text: then

text: it

text: computes

text: the

text: syndrome

text: where

text: is

text: the

text: parity-check

text: matrix

text: of

text: an

text: LDPC

text: code

text: of

text: rate

text: We

text: do

text: this

text: process

text: of

text: quantize

text: and

text: bin

text: by

text: employing

text: a

text: n

text: compound

text: LDGM-LDPC

text: code.

text: It

text: is

text: notable

text: that

text: the

text: total

text: length

text: of

text: the

text: obtained

text: syndrome

text: May

text: DRAFT

text: ##LTLine##

text: ##LTLine##


text: 11 [36] M. Gluck, A.R. Kolovsky, H.J.Korsch, and N. Moiseyev, Eur[.4P6]hyJs..YJ.. VDai4s,h2n3a9v (a1n9d98C). W. Clark, Phys. Rev. Lett. 100, 153002 (2008). [37] R. Grimm, M. Weidenmuller, and Y. B. Ovchinnikov, Adv. A[4t7.]MMol..MOeprtk.l,PFhy. sE..4Z2i,m9m5 e(r2,0G00.)J.uzeliunas, and P. Ohberg, EPL 83 54002 (2008). [38] D. S. Barker, N. C. Pisenti, B. J. Reschovsky, and G. K. Cam[4p8b] eCll., QPhuy,sC. .RHeva.mAne9r3, ,M0.5G34o1n7g,(2C0.16Z)h.ang, and P. Engels, Phys. Rev. A 88, 021604(R) (2013). [39] V. D. Ovsyannikov, V. G. Pal’chikov, H. Katori, and M. Tak[a4m9]otLo.,JQ. uLaenbtluanmc,EMle.ctCr.onB.e3el6e,r,3K(.20J0im6)e.nez-Garcia, A. R. Perry, S. Sugawa, R. A. Williams, and I. B. Spielman, New J. Phys. [40] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, New15J.0P7h3y01s.11(22,01036)5.010 (2010). [41] A. Kitaev, AIP Conference Proceedings 1134, 22 (2009). [50] M. Fox, Quantum Optics - An Introduction (Oxford University Press, 2006). [42] S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 77002 (2002). [51] We assume that the other states of atom are energetically much separated from the three states. Thus, the eﬀective [43] Similar result in L. Li, Z. Xu, and S. Chen, Phys. Rev. B 89, 085H11a1m(i2lt0o1n4i)a.n becomes simple form including the only threes states. [44] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). [45] D. J. Gross and A. Neveu, Phys. Rev. D 10, 3235 (1974).

label: reference

text: M.

text: Gluck,

text: A.R.

text: Kolovsky,

text: H.J.Korsch,

text: and

text: N.

text: Moiseyev,

text: W.

text: Clark,

text: Phys.

text: Rev.

text: Lett.

text: R.

text: Grimm,

text: M.

text: Weidenmuller,

text: and

text: Y.

text: B.

text: Ovchinnikov,

text: Adv.

text: and

text: P.

text: Ohberg,

text: EPL

text: D.

text: S.

text: Barker,

text: N.

text: C.

text: Pisenti,

text: B.

text: J.

text: Reschovsky,

text: and

text: G.

text: K.

text: eCll.,

text: QPhuy,sC.

text: and

text: P.

text: Engels,

text: Phys.

text: Rev.

text: A

text: V.

text: D.

text: Ovsyannikov,

text: V.

text: G.

text: Pal’chikov,

text: H.

text: Katori,

text: and

text: M.

text: A.

text: R.

text: Perry,

text: S.

text: Sugawa,

text: R.

text: A.

text: Williams,

text: and

text: I.

text: B.

text: Spielman,

text: New

text: J.

text: Phys.

text: S.

text: Ryu,

text: A.

text: P.

text: Schnyder,

text: A.

text: Furusaki,

text: and

text: A.

text: W.

text: W.

text: Ludwig,

text: A.

text: Kitaev,

text: AIP

text: Conference

text: Proceedings

text: M.

text: Fox,

text: Quantum

text: Optics

text: An

text: Introduction

text: (Oxford

text: University

text: Press,

text: S.

text: Ryu

text: and

text: Y.

text: Hatsugai,

text: Phys.

text: Rev.

text: Lett.

text: We

text: assume

text: that

text: the

text: other

text: states

text: of

text: atom

text: are

text: energetically

text: much

text: separated

text: from

text: the

text: three

text: states.

text: Thus,

text: the

text: eﬀective

text: Similar

text: result

text: in

text: L.

text: Li,

text: Z.

text: Xu,

text: and

text: S.

text: Chen,

text: Phys.

text: Rev.

text: B

text: becomes

text: simple

text: form

text: including

text: the

text: only

text: threes

text: states.

text: F.

text: D.

text: M.

text: Haldane,

text: Phys.

text: Rev.

text: Lett.

text: D.

text: J.

text: Gross

text: and

text: A.

text: Neveu,

text: Phys.

text: Rev.

text: D


text: 4 For example, the label on Λaj means the NN hopping aj+1 between a and a on the a-lattice, which corresponds j j+1 to the hopping term of Eq.(8) as shown in Fig.1 (a) and (c). The other labels have the similar meaning. Next, to establish Λaj , Λbj , Λaj and Λaj with- aj+1 bj+1 bj+1 bj−1 out interfering with each other, suitable tunings of the on-site energy-oﬀset, the lattice tilt and the excitation laser frequencies used in the laser-assisted hoppings are required. One can directly create the hoppings Λaj and aj+1 Λbj by choosing an appropriate excited state for each bj+1 state. However, Λaj and Λaj have to be carefully pre- bj+1 bj−1 pared because we need to prohibit or highly suppress the † † Rabi coupling a b +a b on the same site. Furthermore, j j j j FIG. 2. Energy condition to realize the laser-assisted hop- in the case that the two component fermionic atom comes pings Λabjj+1 and Λabjj−1 from the diﬀerent z-component of internal spin, we must select appropriate polarization of the excitation lasers in (cid:88) creating Λabjj+1 and Λabjj−1. To satisfy these requirements, we exhibit the general tuning condition as the energy di- | (cid:105) | (cid:105) | (cid:105) | (cid:105) agram shown in Fig.2. There, a , b , b and b − † j j j+1 j 1 H2nd(g) = (J˜ g g + h.c.), (13) are two diﬀerent internal states of an atom on lattice sites j,j+1 j+1 j − |j(cid:90)(cid:48) || (cid:48) | j, j + 1 an| d (cid:105)j 1, r|esp(cid:105) ectively. The energy splitting ∆ab J˜j,j+1 = ΩAE4δΩBE boﬀetswetee2n∆ a=j ∆aanbdasbejxpilsainreeldatbeedfotroe.thωea+bo=n-s∆itaeb +en∆ertgyis- the frequency diﬀerence of the two excitation lasers used × ∗ − · − dr W (r rj+1)eδk rW(r rj), in the la−ser-assisted hoppin−g Λabjj+1. (−This corresponds to ω ω in Sec..) ω = ∆ ∆ is also that (14) AE BE ab ab t of Λaj . To realize the hoppings Λaj and Λaj in- bj−1 bj+1 bj−1 dependently, we need to impose the following condition: † (cid:29) − − where g (g ) is a creation (annihilation) operator of ∆ > ∆ δ+( ), where δ+( ) is two photon detuning atom on ljattjice site j, and J˜j,j+1 is a complex hop- usaebd in tthe lasber−-assisted hobpping Λabjj+1 (Λabjj−1). Actu- ping p≡arameter determined by a localized wave function ally, we take δb+( ) zero to create the resonance between W(r) wws(x)w(y)w(z). Here wws(x) is the Wannier- NN sites, and then in order to suﬃciently separate the Stark state [35, 36], determined by the tilted optical lat- resonance conditions of Λaj and Λaj , the diﬀerence tice and, w(y) and w(z) are the Wannier states, deter- between the two resonancebjd+1enoted bbyj−ω1 + − ω− = 2∆ mined by the y- and z-direction optical lattice, which ab ab t should be taken large compared with two-photon Rabi creates a strong conﬁnement potential creating a 1D sys- − frequency in Eq.(14). If these above conditions are sat- tem. δk is deﬁned as δk = k k . By tuning appro- A B isﬁed in the two parallel optical lattices, the transition priately incident angles of the excitation lasers, δk can probability of the same site is negligibly small (at least be uniform along the one-dimensional lattice. (See ap- strongly suppressed ). Then, the hoppings Λaj and pendix B.) Here, it should be noticed that in Eq.(14), bj+1 − ∼ Λaj are dominant. if we set ωBE ωAE ∆t, the tilt energy diﬀerence bj−1 ∆ between NN sites does not appear due to the RWA As explained in the above, by applying the four kinds t with the rotating ﬂame of ω [28]. The hopping terms of laser-assisted hoppings Λaj , Λbj , Λaj and Λaj AB aj+1 bj+1 bj+1 bj−1 of Eq.(13) is a basic ingredient in creating the hopping to the two parallel optical lattices with the same tilt, we terms of Eqs.(8)-(11). can design the hopping terms of Eqs.(8)-(11). There- C. Four kinds of laser-assisted hopping fore, we can design the quantum simulator of the 1D GWDM. Furthermore, controlling parameters of the ex- For the realization of the hopping terms of Eqs.(8)- citation lasers−enables us to set the uniform phases θa, (11), we apply four kinds of laser-assisted hopping to the θb, θ+, and θ and uniform hopping amplitudes rather two parallel optical lattice system as explained in Sec.. freely. (See appendix. B.) By using the two parallel lattices, which are tilted with D. Concrete example by 171Yb the same amount by certain experimental techniques, the tilted energy diﬀerence ∆t between NN lattice can be In general, the above theoretical proposal can be per- introduced. Since the laser-assisted hopping is created formed by using some atomic species, e.g., alkali atoms. by the Λ-like scheme explained in Sec.IIIB, we denote the In this section, as one of candidates, we consider 171Yb four kinds of hoppings by Λaajj+1, Λbbjj+1, Λabjj+1 and Λabjj−1. atoms. In particular, we employ the two internal states ##LTFigure## ##LTFigure## ##LTLine##

text: For

text: example,

text: the

text: label

text: on

text: Λaj

text: means

text: the

text: NN

text: hopping

text: between

text: a

text: and

text: a

text: on

text: the

text: a-lattice,

text: which

text: corresponds

text: j

text: to

text: the

text: hopping

text: term

text: of

text: as

text: shown

text: in

text: (a)

text: and

text: (c).

text: The

text: other

text: labels

text: have

text: the

text: similar

text: meaning.

text: Next,

text: to

text: establish

text: Λaj

text: Λbj

text: Λaj

text: and

text: Λaj

text: with-

text: out

text: interfering

text: with

text: each

text: other,

text: suitable

text: tunings

text: of

text: the

text: on-site

text: energy-oﬀset,

text: the

text: lattice

text: tilt

text: and

text: the

text: excitation

text: laser

text: frequencies

text: used

text: in

text: the

text: laser-assisted

text: hoppings

text: are

text: required.

text: One

text: can

text: directly

text: create

text: the

text: hoppings

text: Λaj

text: and

text: Λbj

text: by

text: choosing

text: an

text: appropriate

text: excited

text: state

text: for

text: each

text: state.

text: However,

text: Λaj

text: and

text: Λaj

text: have

text: to

text: be

text: carefully

text: pre-

text: pared

text: because

text: we

text: need

text: to

text: prohibit

text: or

text: highly

text: suppress

text: the

text: Rabi

text: coupling

text: a

text: b

text: +a

text: b

text: on

text: the

text: same

text: site.

text: Furthermore,

text: j

text: j

text: j

text: j

text: FIG.

text: Energy

text: condition

text: to

text: realize

text: the

text: laser-assisted

text: hop-

text: in

text: the

text: case

text: that

text: the

text: two

text: component

text: fermionic

text: atom

text: comes

text: pings

text: and

text: from

text: the

text: diﬀerent

text: z-component

text: of

text: internal

text: spin,

text: we

text: must

text: select

text: appropriate

text: polarization

text: of

text: the

text: excitation

text: lasers

text: in

text: creating

text: and

text: To

text: satisfy

text: these

text: requirements,

text: we

text: exhibit

text: the

text: general

text: tuning

text: condition

text: as

text: the

text: energy

text: di-

text: agram

text: shown

text: in

text: There,

text: a

text: b

text: b

text: and

text: b

text: j

text: j

text: j

text: (J˜

text: g

text: g

text: h.c.),

text: are

text: two

text: diﬀerent

text: internal

text: states

text: of

text: an

text: atom

text: on

text: lattice

text: sites

text: j

text: j,

text: j

text: an|

text: d

text: ectively.

text: The

text: energy

text: splitting

text: ∆ab

text: a=j

text: +en∆ertgyis-

text: the

text: frequency

text: diﬀerence

text: of

text: the

text: two

text: excitation

text: lasers

text: used

text: dr

text: W

text: (r

text: rW(r

text: rj),

text: in

text: the

text: la−ser-assisted

text: hoppin−g

text: (−This

text: corresponds

text: to

text: ω

text: ω

text: in

text: Sec..)

text: ω

text: is

text: also

text: that

text: AE

text: BE

text: ab

text: ab

text: t

text: of

text: Λaj

text: To

text: realize

text: the

text: hoppings

text: Λaj

text: and

text: Λaj

text: in-

text: dependently,

text: we

text: need

text: to

text: impose

text: the

text: following

text: condition:

text: where

text: g

text: (g

text: is

text: a

text: creation

text: (annihilation)

text: operator

text: of

text: δ+(

text: where

text: δ+(

text: is

text: two

text: photon

text: detuning

text: atom

text: on

text: ljattjice

text: site

text: j,

text: and

text: is

text: a

text: complex

text: hop-

text: usaebd

text: in

text: tthe

text: lasber−-assisted

text: hobpping

text: Actu-

text: ping

text: p≡arameter

text: determined

text: by

text: a

text: localized

text: wave

text: function

text: ally,

text: we

text: take

text: δb+(

text: zero

text: to

text: create

text: the

text: resonance

text: between

text: W(r)

text: wws(x)w(y)w(z).

text: Here

text: wws(x)

text: is

text: the

text: Wannier-

text: NN

text: sites,

text: and

text: then

text: in

text: order

text: to

text: suﬃciently

text: separate

text: the

text: Stark

text: state

text: determined

text: by

text: the

text: tilted

text: optical

text: lat-

text: resonance

text: conditions

text: of

text: Λaj

text: and

text: Λaj

text: the

text: diﬀerence

text: tice

text: and,

text: w(y)

text: and

text: w(z)

text: are

text: the

text: Wannier

text: states,

text: deter-

text: between

text: the

text: two

text: ω−

text: 2∆

text: mined

text: by

text: the

text: y-

text: and

text: z-direction

text: optical

text: lattice,

text: which

text: ab

text: ab

text: t

text: should

text: be

text: taken

text: large

text: compared

text: with

text: two-photon

text: Rabi

text: creates

text: a

text: strong

text: conﬁnement

text: potential

text: creating

text: a

text: sys-

text: frequency

text: in

text: If

text: these

text: above

text: conditions

text: are

text: sat-

text: tem.

text: δk

text: is

text: deﬁned

text: as

text: δk

text: k

text: k

text: By

text: tuning

text: appro-

text: A

text: B

text: isﬁed

text: in

text: the

text: two

text: parallel

text: optical

text: lattices,

text: the

text: transition

text: priately

text: incident

text: angles

text: of

text: the

text: excitation

text: lasers,

text: δk

text: can

text: probability

text: of

text: the

text: same

text: site

text: is

text: negligibly

text: small

text: (at

text: least

text: be

text: uniform

text: along

text: the

text: one-dimensional

text: lattice.

text: (See

text: ap-

text: strongly

text: suppressed

text: Then,

text: the

text: hoppings

text: Λaj

text: and

text: pendix

text: B.)

text: Here,

text: it

text: should

text: be

text: noticed

text: that

text: in

text: Λaj

text: are

text: dominant.

text: if

text: we

text: set

text: ωBE

text: ωAE

text: ∆t,

text: the

text: tilt

text: energy

text: diﬀerence

text: between

text: NN

text: sites

text: does

text: not

text: appear

text: due

text: to

text: the

text: RWA

text: As

text: explained

text: in

text: the

text: above,

text: by

text: applying

text: the

text: four

text: kinds

text: t

text: with

text: the

text: rotating

text: ﬂame

text: of

text: ω

text: The

text: hopping

text: terms

text: of

text: laser-assisted

text: hoppings

text: Λaj

text: Λbj

text: Λaj

text: and

text: Λaj

text: AB

text: of

text: is

text: a

text: basic

text: ingredient

text: in

text: creating

text: the

text: hopping

text: to

text: the

text: two

text: parallel

text: optical

text: lattices

text: with

text: the

text: same

text: tilt,

text: we

text: terms

text: of

text: can

text: design

text: the

text: hopping

text: terms

text: of

text: There-

text: C.

text: Four

text: kinds

text: of

text: laser-assisted

text: hopping

text: fore,

text: we

text: can

text: design

text: the

text: quantum

text: simulator

text: of

text: the

text: GWDM.

text: Furthermore,

text: controlling

text: parameters

text: of

text: the

text: ex-

text: For

text: the

text: realization

text: of

text: the

text: hopping

text: terms

text: of

text: citation

text: lasers−enables

text: us

text: to

text: set

text: the

text: uniform

text: phases

text: θa,

text: we

text: apply

text: four

text: kinds

text: of

text: laser-assisted

text: hopping

text: to

text: the

text: θb,

text: θ+,

text: and

text: θ

text: and

text: uniform

text: hopping

text: amplitudes

text: rather

text: two

text: parallel

text: optical

text: lattice

text: system

text: as

text: explained

text: in

text: Sec..

text: freely.

text: (See

text: appendix.

text: B.)

text: By

text: using

text: the

text: two

text: parallel

text: lattices,

text: which

text: are

text: tilted

text: with

text: D.

text: Concrete

text: example

text: by

text: the

text: same

text: amount

text: by

text: certain

text: experimental

text: techniques,

text: the

text: tilted

text: energy

text: diﬀerence

text: ∆t

text: between

text: NN

text: lattice

text: can

text: be

text: In

text: general,

text: the

text: above

text: theoretical

text: proposal

text: can

text: be

text: per-

text: introduced.

text: Since

text: the

text: laser-assisted

text: hopping

text: is

text: created

text: formed

text: by

text: using

text: some

text: atomic

text: species,

text: e.g.,

text: alkali

text: atoms.

text: by

text: the

text: Λ-like

text: scheme

text: explained

text: in

text: Sec.IIIB,

text: we

text: denote

text: the

text: In

text: this

text: section,

text: as

text: one

text: of

text: candidates,

text: we

text: consider

text: four

text: kinds

text: of

text: hoppings

text: by

text: and

text: atoms.

text: In

text: particular,

text: we

text: employ

text: the

text: two

text: internal

text: states

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 6 To discuss the above problem, we ﬁrst study symme- tries of the GWDM by using the symmetry-classiﬁcation scheme in Refs. [40, 41]. Symmetries of the system de- pend on the external phases, θ’s. We shall also obtain the energy spectrum of the GWDM on a ﬁnite lattice with the sharp boundary. Then, the spectrum is expected to exhibit zero-energy edge states in some parameter regime of the phases, θ’s. The existence of the zero-energy edge states is a direct signal of nontrivial topological phase in the bulk system by the bulk-edge correspondence. FIG. 4. (a) Energy spectra for θa = 3π/4, θb = 0. (b) Energy Hereafter for simplicity, we impose a condition for the spectra for ∆ = 0, θb = 0 | | | | hopping amplitudes of Eqs.(8)-(11) such as J = J = | | | −| a b J+ = J = 1 as it does not change physical results. ab ab the existence of degenerate zero-energy edge states by The above condition again can be realized in certain ex- diagonalizing the Hamiltonian of the 1D lattice system perimental setups [28]. with system size L = 100 (generally, we take L even). A. Bulk-momentum Hamiltonian and winding number The zero-energy edge state is a direct signal of existence of nontrivial topological phases. B. Symmetries, topological phases and zero-energy First, we consider the system under the periodic edge states in 1D GWDM boundary condition, which preserves the discrete trans- lation symmetry. We focus on symmetries of the bulk In this section, we show how to construct topologically Hamiltonian and also the bulk topological properties of nontrivial Hamiltonians in the 1D GWDM, and give ex- the GWDM. The bulk-momentum Hamiltonian H (k) bulk is obtained fr(cid:34)om Eq. (1) as, (cid:35) perimental conditions on laser setups to realize them. As classiﬁcation theory indicates [40, 41], 1D model, which has nontrivial topological phases, belongs to the BDI − − − ∆ 2 cos(k + θ ) e ik+iθ+ + eik+iθ class or AIII class. This means that the 1D model must at Hbulk(k) = e−ik−iθ− + eik−iaθ+ −∆ − 2 cos(k + θb) , least possess chiral symmetryT[42]. Relevant symmetries are time-reversal symmetry ( ) and charge-conjugation C (15) symmetry ( ) for the classiﬁcation scheme. The system has time-reversal symmetry if and only if the Hamilto- nian H satisﬁes the following condition, where we have taken the lattice spacing as the length unity. Under the Dirac condition: θ = 0, θ = π, and − − a b θ+ = θ = π/2, H (k) is just the bulk-momentum T † K K † ∗ bulk : U H U = U H U = H, (18) Hamiltonian of the ordinary Wilson-Dirac fermion, T T T T − ≡ · K Hbulk(k) = [∆ 2 cos k]σz + [2 sin k]σx d(k) σ, where UT is a unitary operator and is the complex-conjugation operator. Similarly for the charge- (16) conjugation symmetry (the particle-hole symmetry), − where d(k) = (dx, dz) = (2 sin k, ∆ 2 cos k). This is C † K K † ∗ − the base model of 1D topological insulator [5] and be- : UC H UC = UCH UC = H, (19) longs to BDI class Hamiltonian. [See later discussion]. Then, the nontrivial topological phase can be character- where U is again a unitary operator. The BDI class ized by calculating the winding number Nw [6], which T CC T C has and symmetries with 2 = +1 and 2 = +1, is obtained by integrating the vector trajectory of the S ≡ T · C bulk-momentum(cid:90)Hamiltonian deﬁne(cid:18)d as, (cid:19) whereas the AIII class has only K symmetry, which is called chiral symmetry. Under , k and θ’s (cid:48) → − (cid:48) transform as (k, θ s) (k, θ s). Then, it is seen that π dk d(k) × d d(k) the Hamiltonian of the ordinary Wilson-Dirac fermion N = | | | | . (17) w − 2π d(k) dk d(k) [Eq. (16)] has both time-reversal and charge-conjugation π symmetries and in this case U = σ and U = σ , − T z C x For nontrivial topological phase, N = +1 or 1, respectively. From the time-reversal and charge conjuga- w whereas for trivial insulating phase N = 0. In the pa- tion operator, the chiral operator is directly obtained as − ≤ ≤ w rameter regime 2 ∆ 2, a nontrivial topological U = σ . S y phase with N = +1 is known to exist [5, 6]. To search the parameter regime of chiral symmetric w We shall also consider the ﬁnite lattice system of the Hamiltonian in the 1D GWDM, we ﬁrst assume the Dirac − − 1D GWDM with sharp boundaries later on and show condition, i.e., θ+ = π/2 and θ = π/2, but relax θ a ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: To

text: discuss

text: the

text: above

text: problem,

text: we

text: ﬁrst

text: study

text: symme-

text: tries

text: of

text: the

text: GWDM

text: by

text: using

text: the

text: symmetry-classiﬁcation

text: scheme

text: in

text: Refs.

text: Symmetries

text: of

text: the

text: system

text: de-

text: pend

text: on

text: the

text: external

text: phases,

text: θ’s.

text: We

text: shall

text: also

text: obtain

text: the

text: energy

text: spectrum

text: of

text: the

text: GWDM

text: on

text: a

text: ﬁnite

text: lattice

text: with

text: the

text: sharp

text: boundary.

text: Then,

text: the

text: spectrum

text: is

text: expected

text: to

text: exhibit

text: zero-energy

text: edge

text: states

text: in

text: some

text: parameter

text: regime

text: of

text: the

text: phases,

text: θ’s.

text: The

text: existence

text: of

text: the

text: zero-energy

text: edge

text: states

text: is

text: a

text: direct

text: signal

text: of

text: nontrivial

text: topological

text: phase

text: in

text: the

text: bulk

text: system

text: by

text: the

text: bulk-edge

text: correspondence.

text: FIG.

text: (a)

label: caption

text: Energy

text: spectra

text: for

text: θa

text: θb

text: (b)

text: Energy

text: Hereafter

text: for

text: simplicity,

text: we

text: impose

text: a

text: condition

text: for

text: the

text: spectra

text: for

text: θb

text: hopping

text: amplitudes

text: of

text: such

text: as

text: J

text: J

text: a

text: b

text: J+

text: J

text: as

text: it

text: does

text: not

text: change

text: physical

text: results.

text: ab

text: ab

text: the

text: existence

text: of

text: degenerate

text: zero-energy

text: edge

text: states

text: by

text: The

text: above

text: condition

text: again

text: can

text: be

text: realized

text: in

text: certain

text: ex-

text: diagonalizing

text: the

text: Hamiltonian

text: of

text: the

text: lattice

text: system

text: perimental

text: setups

text: with

text: system

text: size

text: L

text: (generally,

text: we

text: take

text: L

text: even).

text: A.

text: Bulk-momentum

text: Hamiltonian

text: and

text: winding

text: number

text: The

text: zero-energy

text: edge

text: state

text: is

text: a

text: direct

text: signal

text: of

text: existence

text: of

text: nontrivial

text: topological

text: phases.

text: B.

text: Symmetries,

text: topological

text: phases

text: and

text: zero-energy

text: First,

text: we

text: consider

text: the

text: system

text: under

text: the

text: periodic

text: edge

text: states

text: in

text: GWDM

text: boundary

text: condition,

text: which

text: preserves

text: the

text: discrete

text: trans-

text: lation

text: symmetry.

text: We

text: focus

text: on

text: symmetries

text: of

text: the

text: bulk

text: In

text: this

text: section,

text: we

text: show

text: how

text: to

text: construct

text: topologically

text: Hamiltonian

text: and

text: also

text: the

text: bulk

text: topological

text: properties

text: of

text: nontrivial

text: Hamiltonians

text: in

text: the

text: GWDM,

text: and

text: give

text: ex-

text: the

text: GWDM.

text: The

text: bulk-momentum

text: Hamiltonian

text: H

text: (k)

text: bulk

text: is

text: obtained

text: Eq.

text: as,

text: perimental

text: conditions

text: on

text: laser

text: setups

text: to

text: realize

text: them.

text: As

text: classiﬁcation

text: theory

text: indicates

text: model,

text: which

text: has

text: nontrivial

text: topological

text: phases,

text: belongs

text: to

text: the

text: BDI

text: cos(k

text: θ

text: e

text: ik+iθ+

text: eik+iθ

text: class

text: or

text: AIII

text: class.

text: This

text: means

text: that

text: the

text: model

text: must

text: at

text: Hbulk(k)

text: e−ik−iθ−

text: eik−iaθ+

text: cos(k

text: θb)

text: least

text: possess

text: chiral

text: Relevant

text: symmetries

text: are

text: time-reversal

text: symmetry

text: and

text: charge-conjugation

text: C

text: symmetry

text: for

text: the

text: classiﬁcation

text: scheme.

text: The

text: system

text: has

text: time-reversal

text: symmetry

text: if

text: and

text: only

text: if

text: the

text: Hamilto-

text: nian

text: H

text: satisﬁes

text: the

text: following

text: condition,

text: where

text: we

text: have

text: taken

text: the

text: lattice

text: spacing

text: as

text: the

text: length

text: unity.

text: Under

text: the

text: Dirac

text: condition:

text: θ

text: θ

text: π,

text: and

text: a

text: b

text: θ+

text: θ

text: H

text: (k)

text: is

text: just

text: the

text: bulk-momentum

text: T

text: K

text: K

text: bulk

text: U

text: H

text: U

text: U

text: H

text: U

text: H,

text: Hamiltonian

text: of

text: the

text: ordinary

text: Wilson-Dirac

text: fermion,

text: T

text: T

text: T

text: T

text: K

text: Hbulk(k)

text: cos

text: k]σz

text: sin

text: k]σx

text: d(k)

text: σ,

text: where

text: UT

text: is

text: a

text: unitary

text: operator

text: and

text: is

text: the

text: complex-conjugation

text: operator.

text: Similarly

text: for

text: the

text: charge-

text: conjugation

text: symmetry

text: (the

text: particle-hole

text: symmetry),

text: where

text: d(k)

text: (dx,

text: dz)

text: sin

text: k,

text: cos

text: k).

text: This

text: is

text: C

text: K

text: K

text: the

text: base

text: model

text: of

text: topological

text: insulator

text: and

text: be-

text: UC

text: H

text: UC

text: UCH

text: UC

text: H,

text: longs

text: to

text: BDI

text: class

text: Hamiltonian.

text: [See

text: later

text: discussion].

text: Then,

text: the

text: nontrivial

text: topological

text: phase

text: can

text: be

text: character-

text: where

text: U

text: is

text: again

text: a

text: unitary

text: operator.

text: The

text: BDI

text: class

text: ized

text: by

text: calculating

text: the

text: winding

text: number

text: Nw

text: which

text: T

text: CC

text: T

text: C

text: has

text: and

text: symmetries

text: with

text: and

text: is

text: obtained

text: by

text: integrating

text: the

text: vector

text: trajectory

text: of

text: the

text: S

text: T

text: C

text: as,

text: whereas

text: the

text: AIII

text: class

text: has

text: only

text: K

text: symmetry,

text: which

text: is

text: called

text: chiral

text: symmetry.

text: Under

text: k

text: and

text: θ’s

text: transform

text: as

text: (k,

text: θ

text: s)

text: (k,

text: θ

text: s).

text: Then,

text: it

text: is

text: seen

text: that

text: π

text: dk

text: d(k)

text: d

text: d(k)

text: the

text: Hamiltonian

text: of

text: the

text: ordinary

text: Wilson-Dirac

text: fermion

text: N

text: w

text: 2π

text: d(k)

text: dk

text: d(k)

text: [Eq.

text: has

text: both

text: time-reversal

text: and

text: charge-conjugation

text: π

text: symmetries

text: and

text: in

text: this

text: case

text: U

text: σ

text: and

text: U

text: σ

text: T

text: z

text: C

text: x

text: For

text: nontrivial

text: topological

text: phase,

text: N

text: or

text: respectively.

text: From

text: the

text: time-reversal

text: and

text: charge

text: conjuga-

text: w

text: whereas

text: for

text: trivial

text: insulating

text: phase

text: N

text: In

text: the

text: pa-

text: tion

text: operator,

text: the

text: chiral

text: operator

text: is

text: directly

text: obtained

text: as

text: w

text: rameter

text: regime

text: a

text: nontrivial

text: topological

text: U

text: σ

text: S

text: y

text: phase

text: with

text: N

text: is

text: known

text: to

text: exist

text: To

text: search

text: the

text: parameter

text: regime

text: of

text: chiral

text: symmetric

text: w

text: We

text: shall

text: also

text: consider

text: the

text: ﬁnite

text: lattice

text: system

text: of

text: the

text: Hamiltonian

text: in

text: the

text: GWDM,

text: we

text: ﬁrst

text: assume

text: the

text: Dirac

text: GWDM

text: with

text: sharp

text: boundaries

text: later

text: on

text: and

text: show

text: condition,

text: i.e.,

text: θ+

text: and

text: θ

text: but

text: relax

text: θ

text: a

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Our guide in the development of such new conceptual forms is the orthodox quantum formalism itself assisted by some general physical considerations. Let us begin by the latter. The notion of invariance has always played a major role within the development of theories. As remarked by Max Born [11]: “the idea of invariant is the clue to a rational concept of reality, not only in physics but in every aspect of the world.” The notion of invariance allows us to determine what is to be considered the same within a mathematical formalism regardless of any particular reference frame. In physics, invariants are quantities which can be translated —through mathematical transformations— from one reference frame to another. Even though the values of physical magnitudes might vary from one frame to another —due to the dynamics between reference frames—, in both classical physics and relativity theory there is a consistent translation between the values of magnitudes of diﬀerent frames secured by the speciﬁc laws of transformation —Galilean transformations in the case of classical physics and Lorentzian in the case of relativity. As a consequence, within these theories it is always possible to provide a GBV of the properties that pertain to systems. For example, in classical physics, the position of a rabbit running through the ﬁelds and observed by a distant passenger of a high speed train can be translated to the position of that same rabbit taken from the perspective of another passenger waiting in the platform of the station. The fact that the values of observables (position, momentum, etc.) can be consistently translated from one reference frame to the other allows us to assume that such physical observables also bear an objective real existence completely independent of the speciﬁc choice of the reference frame of the observers. The rabbit has a set of dynamical properties (position, a momentum, etc.) independently of his observers in the train and on the platform. This deﬁnition of invariant observables allows us to argue that the physical system (the rabbit) possesses preexistent properties independently of the observers, their choice of the reference frame or even the measurements they might perform. Consequently we are allowed to claim that such properties are dynamical variations that pertain to the same physical system. It is this type of analysis which allows us to understand what can be considered as an objective representation (see for a detailed analysis [21, 23]). Invariance is our thread of Ariadne in the quantum labyrinth of conceptual representation. But since this quantum labyrinth is clearly not classical, we should not expect classical concepts to help us ﬁnding the way out. And indeed, for almost a century now, no researcher entering this labyrinth has been able to escape with classical ideas. It seems we might need to think diﬀerently. Maybe we should leave aside our classical metaphysical prejudices and commitments and simply follow the thread. Taking into account our analysis about the importance of invariance within physical theories, let us consider the following simple question within QM: given a Ψ, what is independent of the relative observational choice of individual subjects according to the formalism? Or in other words, what is invariant with respect to mathematical contexts (or bases)? The answer for any quantum physicist who knows the orthodox formalism is obviously straightforward. The invariants in QM are the average value of observables considered in relation to that particular vector. The average value of an observable is independent of the particular context (or basis) that one might chose in order to calculate the computation. This is known by quantum physicists as the Born rule. Born Rule: Given a vector Ψ in a Hilbert space, the following rule allows us to predict the average value of (any) observable P . (cid:104) | | (cid:105) (cid:104) (cid:105) Ψ P Ψ = P This prediction is independent of the choice of any particular basis. From the previous considerations, we believe that a good starting point in order to derive an ob- jective representation for QM must rely on the invariants of the formalism. Also, recalling Einstein’s 15

text: Our

text: guide

text: in

text: the

text: development

text: of

text: such

text: new

text: conceptual

text: forms

text: is

text: the

text: orthodox

text: quantum

text: formalism

text: itself

text: assisted

text: by

text: some

text: general

text: physical

text: considerations.

text: Let

text: us

text: begin

text: by

text: the

text: latter.

text: The

text: notion

text: of

text: invariance

text: has

text: always

text: played

text: a

text: major

text: role

text: within

text: the

text: development

text: of

text: theories.

text: As

text: remarked

text: by

text: Max

text: Born

text: “the

text: idea

text: of

text: invariant

text: is

text: the

text: clue

text: to

text: a

text: rational

text: concept

text: of

text: reality,

text: not

text: only

text: in

text: physics

text: but

text: in

text: every

text: aspect

text: of

text: the

text: world.”

text: The

text: notion

text: of

text: invariance

text: allows

text: us

text: to

text: determine

text: what

text: is

text: to

text: be

text: considered

text: the

text: same

text: within

text: a

text: mathematical

text: formalism

text: regardless

text: of

text: any

text: particular

text: reference

text: frame.

text: In

text: physics,

text: invariants

text: are

text: quantities

text: which

text: can

text: be

text: translated

text: —through

text: mathematical

text: transformations—

text: from

text: one

text: reference

text: frame

text: to

text: another.

text: Even

text: though

text: the

text: values

text: of

text: physical

text: magnitudes

text: might

text: vary

text: from

text: one

text: frame

text: to

text: another

text: —due

text: to

text: the

text: dynamics

text: between

text: reference

text: frames—,

text: in

text: both

text: classical

text: physics

text: and

text: relativity

text: theory

text: there

text: is

text: a

text: consistent

text: translation

text: between

text: the

text: values

text: of

text: magnitudes

text: of

text: diﬀerent

text: frames

text: secured

text: by

text: the

text: speciﬁc

text: laws

text: of

text: transformation

text: —Galilean

text: transformations

text: in

text: the

text: case

text: of

text: classical

text: physics

text: and

text: Lorentzian

text: in

text: the

text: case

text: of

text: relativity.

text: As

text: a

text: consequence,

text: within

text: these

text: theories

text: it

text: is

text: always

text: possible

text: to

text: provide

text: a

text: GBV

text: of

text: the

text: properties

text: that

text: pertain

text: to

text: systems.

text: For

text: example,

text: in

text: classical

text: physics,

text: the

text: position

text: of

text: a

text: rabbit

text: running

text: through

text: the

text: ﬁelds

text: and

text: observed

text: by

text: a

text: distant

text: passenger

text: of

text: a

text: high

text: speed

text: train

text: can

text: be

text: translated

text: to

text: the

text: position

text: of

text: that

text: same

text: rabbit

text: taken

text: from

text: the

text: perspective

text: of

text: another

text: passenger

text: waiting

text: in

text: the

text: platform

text: of

text: the

text: station.

text: The

text: fact

text: that

text: the

text: values

text: of

text: observables

text: (position,

text: momentum,

text: etc.)

text: can

text: be

text: consistently

text: translated

text: from

text: one

text: reference

text: frame

text: to

text: the

text: other

text: allows

text: us

text: to

text: assume

text: that

text: such

text: physical

text: observables

text: also

text: bear

text: an

text: objective

text: real

text: existence

text: completely

text: independent

text: of

text: the

text: speciﬁc

text: choice

text: of

text: the

text: reference

text: frame

text: of

text: the

text: observers.

text: The

text: rabbit

text: has

text: a

text: set

text: of

text: dynamical

text: properties

text: (position,

text: a

text: momentum,

text: etc.)

text: independently

text: of

text: his

text: observers

text: in

text: the

text: train

text: and

text: on

text: the

text: platform.

text: This

text: deﬁnition

text: of

text: invariant

text: observables

text: allows

text: us

text: to

text: argue

text: that

text: the

text: physical

text: system

text: (the

text: rabbit)

text: possesses

text: preexistent

text: properties

text: independently

text: of

text: the

text: observers,

text: their

text: choice

text: of

text: the

text: reference

text: frame

text: or

text: even

text: the

text: measurements

text: they

text: might

text: perform.

text: Consequently

text: we

text: are

text: allowed

text: to

text: claim

text: that

text: such

text: properties

text: are

text: dynamical

text: variations

text: that

text: pertain

text: to

text: the

text: same

text: physical

text: system.

text: It

text: is

text: this

text: type

text: of

text: analysis

text: which

text: allows

text: us

text: to

text: understand

text: what

text: can

text: be

text: considered

text: as

text: an

text: objective

text: representation

text: (see

text: for

text: a

text: detailed

text: analysis

text: Invariance

text: is

text: our

text: thread

text: of

text: Ariadne

text: in

text: the

text: quantum

text: labyrinth

text: of

text: conceptual

text: representation.

text: But

text: since

text: this

text: quantum

text: labyrinth

text: is

text: clearly

text: not

text: classical,

text: we

text: should

text: not

text: expect

text: classical

text: concepts

text: to

text: help

text: us

text: ﬁnding

text: the

text: way

text: out.

text: And

text: indeed,

text: for

text: almost

text: a

text: century

text: now,

text: no

text: researcher

text: entering

text: this

text: labyrinth

text: has

text: been

text: able

text: to

text: escape

text: with

text: classical

text: ideas.

text: It

text: seems

text: we

text: might

text: need

text: to

text: think

text: diﬀerently.

text: Maybe

text: we

text: should

text: leave

text: aside

text: our

text: classical

text: metaphysical

text: prejudices

text: and

text: commitments

text: and

text: simply

text: follow

text: the

text: thread.

text: Taking

text: into

text: account

text: our

text: analysis

text: about

text: the

text: importance

text: of

text: invariance

text: within

text: physical

text: theories,

text: let

text: us

text: consider

text: the

text: following

text: simple

text: question

text: within

text: QM:

text: given

text: a

text: Ψ,

text: what

text: is

text: independent

text: of

text: the

text: relative

text: observational

text: choice

text: of

text: individual

text: subjects

text: according

text: to

text: the

text: formalism?

text: Or

text: in

text: other

text: words,

text: what

text: is

text: invariant

text: with

text: respect

text: to

text: mathematical

text: contexts

text: (or

text: bases)?

text: The

text: answer

text: for

text: any

text: quantum

text: physicist

text: who

text: knows

text: the

text: orthodox

text: formalism

text: is

text: obviously

text: straightforward.

text: The

text: invariants

text: in

text: QM

text: are

text: the

text: average

text: value

text: of

text: observables

text: considered

text: in

text: relation

text: to

text: that

text: particular

text: vector.

text: The

text: average

text: value

text: of

text: an

text: observable

text: is

text: independent

text: of

text: the

text: particular

text: context

text: (or

text: basis)

text: that

text: one

text: might

text: chose

text: in

text: order

text: to

text: calculate

text: the

text: computation.

text: This

text: is

text: known

text: by

text: quantum

text: physicists

text: as

text: the

text: Born

text: rule.

text: Born

text: Rule:

text: Given

text: a

text: vector

text: Ψ

text: in

text: a

text: Hilbert

text: space,

text: the

text: following

text: rule

text: allows

text: us

text: to

text: predict

text: the

text: average

text: value

text: of

text: (any)

text: observable

text: P

text: Ψ

text: P

text: Ψ

text: P

text: This

text: prediction

text: is

text: independent

text: of

text: the

text: choice

text: of

text: any

text: particular

text: basis.

text: From

text: the

text: previous

text: considerations,

text: we

text: believe

text: that

text: a

text: good

text: starting

text: point

text: in

text: order

text: to

text: derive

text: an

text: ob-

text: jective

text: representation

text: for

text: QM

text: must

text: rely

text: on

text: the

text: invariants

text: of

text: the

text: formalism.

text: Also,

text: recalling

text: Einstein’s


text: measured. Indeed, given a Ψ8 the KS-impossibility to consider the state of aﬀairs in terms of deﬁnite valued properties, 0 or 1, seemingly precludes an objective (non-relative) representation. However, if we go back to the KS proof discussed in section 4, our global intensive valuation is able to provide a non-contextual account of projection operators which can be easily comprehended through the following graph. The intensities drawn are 1/4, 1/3 or 1/6. By replacing (classical) binary valuations by intensive valuations we are able to relate Ψ to a well deﬁned state of aﬀairs. We are able to now state the following non-contextuality theorem for QM: H Theorem 7.1 (Intensive Non-Contextuality) If is a Hilbert space, then a PSA is possible. (cid:3) Proof: Straightforward, apply the Born rule to obtain a PSA, which is a function to [0, 1]. Notice that escaping the metaphysical constraints imposed by the actual realm, our representa- tion and redeﬁnition of how to valuate a quantum wave function, Ψ, restores an objective (non- contextual) account of the (intensive) values of all projection operators related to the decomposition of Ψ. As we have seen above, our approach can be derived directly from physical considerations (section 6) and the formalism itself in a very natural manner. This also provides a guide regarding the mathematical elements that should be considered in the creation of an objective conceptual representation. Corollary 7.2 The non-contextuality intensive theorem restores the possibility of an objective phys- ical representation of any quantum wave function Ψ. Contrary to the orthodox interpretation of QM in terms of systems with properties (which imply a binary valuation), our conceptual representation of quantum physical reality is not relative to any particular context, it is global. G (cid:3) Proof: Straightforward, for any Ψ there always exists a PSA deﬁned over all . To sum up, we have been able to escape the contextual character of KS theorem, not by chang- ing the formalism in order to restore our classical way of thinking about what there is, but on the 8We avoid using the term “quantum state” for reasons that will become clear in the following. 20 ##LTFigure## ##LTLine##

text: measured.

text: Indeed,

text: given

text: a

text: Ψ8

text: the

text: KS-impossibility

text: to

text: consider

text: the

text: state

text: of

text: aﬀairs

text: in

text: terms

text: of

text: deﬁnite

text: valued

text: properties,

text: or

text: seemingly

text: precludes

text: an

text: objective

text: (non-relative)

text: representation.

text: However,

text: if

text: we

text: go

text: back

text: to

text: the

text: KS

text: proof

text: discussed

text: in

text: section

text: our

text: global

text: intensive

text: valuation

text: is

text: able

text: to

text: provide

text: a

text: non-contextual

text: account

text: of

text: projection

text: operators

text: which

text: can

text: be

text: easily

text: comprehended

text: through

text: the

text: following

text: graph.

text: The

text: intensities

text: drawn

text: are

text: or

text: By

text: replacing

text: (classical)

text: binary

text: valuations

text: by

text: intensive

text: valuations

text: we

text: are

text: able

text: to

text: relate

text: Ψ

text: to

text: a

text: well

text: deﬁned

text: state

text: of

text: aﬀairs.

text: We

text: are

text: able

text: to

text: now

text: state

text: the

text: following

text: non-contextuality

text: theorem

text: for

text: QM:

text: H

text: Theorem

text: (Intensive

text: Non-Contextuality)

text: If

text: is

text: a

text: Hilbert

text: space,

text: then

text: a

text: PSA

text: is

text: possible.

text: Proof:

text: Straightforward,

text: apply

text: the

text: Born

text: rule

text: to

text: obtain

text: a

text: PSA,

text: which

text: is

text: a

text: function

text: to

text: Notice

text: that

text: escaping

text: the

text: metaphysical

text: constraints

text: imposed

text: by

text: the

text: actual

text: realm,

text: our

text: representa-

text: tion

text: and

text: redeﬁnition

text: of

text: how

text: to

text: valuate

text: a

text: quantum

text: wave

text: function,

text: Ψ,

text: restores

text: an

text: objective

text: (non-

text: contextual)

text: account

text: of

text: the

text: (intensive)

text: values

text: of

text: all

text: projection

text: operators

text: related

text: to

text: the

text: decomposition

text: of

text: Ψ.

text: As

text: we

text: have

text: seen

text: above,

text: our

text: approach

text: can

text: be

text: derived

text: directly

text: from

text: physical

text: considerations

text: (section

text: and

text: the

text: formalism

text: itself

text: in

text: a

text: very

text: natural

text: manner.

text: This

text: also

text: provides

text: a

text: guide

text: regarding

text: the

text: mathematical

text: elements

text: that

text: should

text: be

text: considered

text: in

text: the

text: creation

text: of

text: an

text: objective

text: conceptual

text: representation.

text: Corollary

text: The

text: non-contextuality

text: intensive

text: theorem

text: restores

text: the

text: possibility

text: of

text: an

text: objective

text: phys-

text: ical

text: representation

text: of

text: any

text: quantum

text: wave

text: function

text: Ψ.

text: Contrary

text: to

text: the

text: orthodox

text: interpretation

text: of

text: QM

text: in

text: terms

text: of

text: systems

text: with

text: properties

text: (which

text: imply

text: a

text: binary

text: valuation),

text: our

text: conceptual

text: representation

text: of

text: quantum

text: physical

text: reality

text: is

text: not

text: relative

text: to

text: any

text: particular

text: context,

text: it

text: is

text: global.

text: G

text: Proof:

text: Straightforward,

text: for

text: any

text: Ψ

text: there

text: always

text: exists

text: a

text: PSA

text: deﬁned

text: over

text: all

text: To

text: sum

text: up,

text: we

text: have

text: been

text: able

text: to

text: escape

text: the

text: contextual

text: character

text: of

text: KS

text: theorem,

text: not

text: by

text: chang-

text: ing

text: the

text: formalism

text: in

text: order

text: to

text: restore

text: our

text: classical

text: way

text: of

text: thinking

text: about

text: what

text: there

text: is,

text: but

text: on

text: the

text: avoid

text: using

text: the

text: term

text: “quantum

text: state”

text: for

text: reasons

text: that

text: will

text: become

text: clear

text: in

text: the

text: following.

text: ##LTFigure##

text: ##LTLine##


text: The topos approach was originally proposed by Chirs Isham [42], Jeremy Butterﬁeld and Andreas D¨oring [27, 28, 29]. At its origin, in a series of four papers [43, 44, 45, 46], Isham and Butterﬁeld investigated in depth the KS theorem attempting to provide a “neo-realist” solution that would restore a classical understanding about what QM is really talking about. The main ideas put forward by this neo-realist topos approach were summarized by D¨oring and Isham in the following passage: “When dealing with a closed system, what is needed is a realist interpretation of the theory, not one that is instrumentalist. The exact meaning of “realist” is inﬁnitely debatable, but, when used by physicists, it typically means the following: (1) The idea of “a property of the system” (i.e. “the value of a physical quantity”) is mean- ingful, and representable in the theory. (2) Propositions about the system are handled using Boolean logic. This requirement is com- pelling in so far as we humans think in a Boolean way. (3) There is a space of “microstates” such that specifying a microstate leads to unequivocal truth values for all propositions about the system. The existence of such a state space is a natural way of ensuring that the ﬁrst two requirements are satisﬁed. The standard interpretation of classical physics satisﬁes these requirements, and provides the paradigmatic example of a realist philosophy in science. On the other hand, the existence of such an interpretation in quantum theory is foiled by the famous Kochen-Specker theorem.” [27] These presuppositions are, in great measure, analogous to those found within Bohr’s approach to QM —namely, the need of a classical representation to account for phenomena and the existence of a reductionistic limit between QM and classical physics. Indeed, the ﬁrst point makes explicit the necessity of retaining the classical language and representation of physics, in terms of “systems” and “properties”, also for QM. The second presupposition goes in the same direction and points to the requirement that classical logic must be considered as a necessary precondition for thinking about any physical theory —including, obviously, QM. (1) and (2) might thus seem to assume implicitly also the Bohrian claim according to which “it would be a misconception to believe that the diﬃculties of the atomic theory may be evaded by eventually replacing the concepts of classical physics by new conceptual forms.” (3) makes explicit the assumption of an atomist metaphysical picture. The reductionistic understanding of QM with respect to classical physics is also captured by the presupposition that (classical) macrosistems necessarily emerge from (quantum) microsystems (i.e., the quantum to classical limit). As they explain in [43, p. 1], the topos approach is based on the introduction of “a new type of valuation which is deﬁned on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves.” Before addressing this speciﬁc proposal for a “proper valuation” in QM, let us brieﬂy introduce the mathematical scheme in which the topos approach is grounded. C ∈ C Let us start by deﬁning the notion of a sieve. Let be a category and let A . A sieve on A → is a collections S of maps over A closed under precompositions. Speciﬁcally, if the map f : B A → ∈ S is in S and g : C B is an arbitrary map, then gf S. Let us denote by ie(A) to the set of S S C → sieves on A. It is a fact that ie(A) is a Heyting algebra and that ie : Hey is a functor, where Hey is the category of Heyting algebras. O In order to deﬁne a sieve valued valuation as in [43] we need to deﬁne the category and the B O coarse-graining presheaf ool. Objects in are bounded self-adjoint operators on a Hilbert space 6

text: The

text: topos

text: approach

text: was

text: originally

text: proposed

text: by

text: Chirs

text: Isham

text: Jeremy

text: Butterﬁeld

text: and

text: Andreas

text: D¨oring

text: At

text: its

text: origin,

text: in

text: a

text: series

text: of

text: four

text: papers

text: Isham

text: and

text: Butterﬁeld

text: investigated

text: in

text: depth

text: the

text: KS

text: theorem

text: attempting

text: to

text: provide

text: a

text: “neo-realist”

text: solution

text: that

text: would

text: restore

text: a

text: classical

text: understanding

text: about

text: what

text: QM

text: is

text: really

text: talking

text: about.

text: The

text: main

text: ideas

text: put

text: forward

text: by

text: this

text: neo-realist

text: topos

text: approach

text: were

text: summarized

text: by

text: D¨oring

text: and

text: Isham

text: in

text: the

text: following

text: passage:

text: “When

text: dealing

text: with

text: a

text: closed

text: system,

text: what

text: is

text: needed

text: is

text: a

text: realist

text: interpretation

text: of

text: the

text: theory,

text: not

text: one

text: that

text: is

text: instrumentalist.

text: The

text: exact

text: meaning

text: of

text: “realist”

text: is

text: inﬁnitely

text: debatable,

text: but,

text: when

text: used

text: by

text: physicists,

text: it

text: typically

text: means

text: the

text: following:

text: The

text: idea

text: of

text: “a

text: property

text: of

text: the

text: system”

text: (i.e.

text: “the

text: value

text: of

text: a

text: physical

text: quantity”)

text: is

text: mean-

text: ingful,

text: and

text: representable

text: in

text: the

text: theory.

text: Propositions

text: about

text: the

text: system

text: are

text: handled

text: using

text: Boolean

text: logic.

text: This

text: requirement

text: is

text: com-

text: pelling

text: in

text: so

text: far

text: as

text: we

text: humans

text: think

text: in

text: a

text: Boolean

text: way.

text: There

text: is

text: a

text: space

text: of

text: “microstates”

text: such

text: that

text: specifying

text: a

text: microstate

text: leads

text: to

text: unequivocal

text: truth

text: values

text: for

text: all

text: propositions

text: about

text: the

text: system.

text: The

text: existence

text: of

text: such

text: a

text: state

text: space

text: is

text: a

text: natural

text: way

text: of

text: ensuring

text: that

text: the

text: ﬁrst

text: two

text: requirements

text: are

text: satisﬁed.

text: The

text: standard

text: interpretation

text: of

text: classical

text: physics

text: satisﬁes

text: these

text: requirements,

text: and

text: provides

text: the

text: paradigmatic

text: example

text: of

text: a

text: realist

text: philosophy

text: in

text: science.

text: On

text: the

text: other

text: hand,

text: the

text: existence

text: of

text: such

text: an

text: interpretation

text: in

text: quantum

text: theory

text: is

text: foiled

text: by

text: the

text: famous

text: Kochen-Specker

text: theorem.”

text: These

text: presuppositions

text: are,

text: in

text: great

text: measure,

text: analogous

text: to

text: those

text: found

text: within

text: Bohr’s

text: approach

text: to

text: QM

text: —namely,

text: the

text: need

text: of

text: a

text: classical

text: representation

text: to

text: account

text: for

text: phenomena

text: and

text: the

text: existence

text: of

text: a

text: reductionistic

text: limit

text: between

text: QM

text: and

text: classical

text: physics.

text: Indeed,

text: the

text: ﬁrst

text: point

text: makes

text: explicit

text: the

text: necessity

text: of

text: retaining

text: the

text: classical

text: language

text: and

text: representation

text: of

text: physics,

text: in

text: terms

text: of

text: “systems”

text: and

text: “properties”,

text: also

text: for

text: QM.

text: The

text: second

text: presupposition

text: goes

text: in

text: the

text: same

text: direction

text: and

text: points

text: to

text: the

text: requirement

text: that

text: classical

text: logic

text: must

text: be

text: considered

text: as

text: a

text: necessary

text: precondition

text: for

text: thinking

text: about

text: any

text: physical

text: theory

text: —including,

text: obviously,

text: QM.

text: and

text: might

text: thus

text: seem

text: to

text: assume

text: implicitly

text: also

text: the

text: Bohrian

text: claim

text: according

text: to

text: which

text: “it

text: would

text: be

text: a

text: misconception

text: to

text: believe

text: that

text: the

text: diﬃculties

text: of

text: the

text: atomic

text: theory

text: may

text: be

text: evaded

text: by

text: eventually

text: replacing

text: the

text: concepts

text: of

text: classical

text: physics

text: by

text: new

text: conceptual

text: forms.”

text: makes

text: explicit

text: the

text: assumption

text: of

text: an

text: atomist

text: metaphysical

text: picture.

text: The

text: reductionistic

text: understanding

text: of

text: QM

text: with

text: respect

text: to

text: classical

text: physics

text: is

text: also

text: captured

text: by

text: the

text: presupposition

text: that

text: (classical)

text: macrosistems

text: necessarily

text: emerge

text: from

text: (quantum)

text: microsystems

text: (i.e.,

text: the

text: quantum

text: to

text: classical

text: limit).

text: As

text: they

text: explain

text: in

text: p.

text: the

text: topos

text: approach

text: is

text: based

text: on

text: the

text: introduction

text: of

text: “a

text: new

text: type

text: of

text: valuation

text: which

text: is

text: deﬁned

text: on

text: all

text: operators,

text: and

text: which

text: respects

text: an

text: appropriate

text: version

text: of

text: the

text: functional

text: composition

text: principle.

text: The

text: truth-values

text: assigned

text: to

text: propositions

text: are

text: (i)

text: contextual;

text: and

text: (ii)

text: multi-valued,

text: where

text: the

text: space

text: of

text: contexts

text: and

text: the

text: multi-valued

text: logic

text: for

text: each

text: context

text: come

text: naturally

text: from

text: the

text: topos

text: theory

text: of

text: presheaves.”

text: Before

text: addressing

text: this

text: speciﬁc

text: proposal

text: for

text: a

text: “proper

text: valuation”

text: in

text: QM,

text: let

text: us

text: brieﬂy

text: introduce

text: the

text: mathematical

text: scheme

text: in

text: which

text: the

text: topos

text: approach

text: is

text: grounded.

text: C

text: C

text: Let

text: us

text: start

text: by

text: deﬁning

text: the

text: notion

text: of

text: a

text: sieve.

text: Let

text: be

text: a

text: category

text: and

text: let

text: A

text: A

text: sieve

text: on

text: A

text: is

text: a

text: collections

text: S

text: of

text: maps

text: over

text: A

text: closed

text: under

text: precompositions.

text: Speciﬁcally,

text: if

text: the

text: map

text: f

text: B

text: A

text: S

text: is

text: in

text: S

text: and

text: g

text: C

text: B

text: is

text: an

text: arbitrary

text: map,

text: then

text: gf

text: S.

text: Let

text: us

text: denote

text: by

text: ie(A)

text: to

text: the

text: set

text: of

text: S

text: S

text: C

text: sieves

text: on

text: A.

text: It

text: is

text: a

text: fact

text: that

text: ie(A)

text: is

text: a

text: Heyting

text: algebra

text: and

text: that

text: ie

text: Hey

text: is

text: a

text: functor,

text: where

text: Hey

text: is

text: the

text: category

text: of

text: Heyting

text: algebras.

text: O

text: In

text: order

text: to

text: deﬁne

text: a

text: sieve

text: valued

text: valuation

text: as

text: in

text: we

text: need

text: to

text: deﬁne

text: the

text: category

text: and

text: the

text: B

text: O

text: coarse-graining

text: presheaf

text: ool.

text: Objects

text: in

text: are

text: bounded

text: self-adjoint

text: operators

text: on

text: a

text: Hilbert

text: space


text: Facial Emotion Recognition using Min-Max Similarity Classiﬁer Olga Krestinskaya and Alex Pappachen James School of Engineering, Nazarbayev University, Astana www.biomicrosystems.info/alex Email: apj@ieee.org Abstract— features because imaging matrices become sparse and feature Recognition of human emotions from the imaging templates is computation process is not trivial. As facial expressions cause useful in a wide variety of human-computer interaction and intel- a mismatch of intra-class features due to their orientation ligent systems applications. However, the automatic recognition of variability, it is difﬁcult to map them between the imaging facial expressions using image template matching techniques suf- fer from the natural variability with facial features and recording templates. conditions. In spite of the progress achieved in facial emotion Facial emotion recognition accuracy depends on the robust- recognition in recent years, the effective and computationally ness of a feature extraction method to intra-class variations and simple feature selection and classiﬁcation technique for emotion classiﬁer performance under noisy conditions and with various recognition is still an open problem. In this paper, we propose types of occlusions [10]. Even thought a variety of approaches an efﬁcient and straightforward facial emotion recognition algo- rithm to reduce the problem of inter-class pixel mismatch during for the automated recognition of human expressions from classiﬁcation. The proposed method includes the application the face images using template matching methods have been of pixel normalization to remove intensity offsets followed-up investigated and proposed over the last few years [14], the with a Min-Max metric in a nearest neighbor classiﬁer that is emotion recognition method with robust feature extraction capable of suppressing feature outliers. The results indicate an and effective classiﬁcation techniques accompanied by low improvement of recognition performance from 92.85% to 98.57% for the proposed Min-Max classiﬁcation method when tested on computational complexity is still an open research problem JAFFE database. The proposed emotion recognition technique [21]. Therefore, in this paper, we address the issues of outperforms the existing template matching methods. matching templates through pixel normalization followed by Index Terms—Face emotions, Classiﬁer, Emotion recognition, the removal of inter-image feature outliers using a Min-Max spatial ﬁlters, gradients similarity metric. We apply Gaussian normalization method with local mean and standard deviation to normalize the I. INTRODUCTION pixels and extract relevant face features followed by Min- In recent years, the human-computer interaction challenge Max classiﬁcation method to determine an emotion class. The has led to the demand to introduce efﬁcient facial and speech simulation is performed in Matlab for the Japanese Female recognition systems [1]–[5]. Facial emotion recognition is Facial Expression (JAFFE) database [22] and the emotion the identiﬁcation of a human emotion based on the facial recognition accuracy is calculated using leave-one-out cross- expression and mimics [6]. The facial emotion recognition has validation method. a wide range of appliction prospects in different areas, such as The main contributions of this work are the following: medicine [7], robotics [8], [9], computer vision, surveillance • We develop a simpliﬁed approach for facial emotion systems [1], machine learning [10], artiﬁcial intelligence, recognition with template matching method using Gaus- communication [11], [12], psychological studies [4], smart sian normalization, mean and standard deviation based vehicles [9], security and embedded systems [13]. feature extraction and Min-Max classiﬁcation approach. There are two main approaches for facial expression recog- • We present simple and effective facial emotion recog- nition: geometry-based and appearance-based methods [2]. nition algorithm having low computational complexity The geometry-based methods extract main feature points and and ability to suppress the outliers and remove intensity their shapes from the face image and calculate the distances offsets. between them. While, appearance-based methods focus on • We conduct the experiments and simulations on JAFFE the face texture using different classiﬁcation and template database to demonstrate the efﬁciency of the proposed matching methods [14], [15]. In this paper, we focus on facial approach and highlight its advantages, comparing to the emotion recognition based on template matching techniques other existing methods. that remains a challenging task [16]–[18]. Since the orientation The paper is organized as follows. Section presents the of pixel features are sensitive to the changes in illumina- overview of the existing methods for facial emotion recogni- tion, pose, scale and other natural imaging variabilities, the tion, their drawbacks and reasons to propose a new method. In matching errors tend to be high [4], [19], [20]. Pixel matching Section , we show normalization, feature extraction and clas- methods are known to be useful when the images has missing siﬁcation parts of the proposed method, present the algorithm

text: Facial

label: title

text: Emotion

text: Recognition

text: using

text: Min-Max

text: Similarity

text: Classiﬁer

text: Olga

label: author

text: Krestinskaya

text: and

text: Alex

text: Pappachen

text: James

text: School

text: of

text: Engineering,

text: Nazarbayev

text: University,

text: Astana

text: Email:

text: apj@ieee.org

text: Abstract—

text: features

text: because

text: imaging

text: matrices

text: become

text: sparse

text: and

text: feature

text: Recognition

label: abstract

text: of

text: human

text: emotions

text: from

text: the

text: imaging

text: templates

text: is

text: computation

text: process

text: is

text: not

text: trivial.

text: As

text: facial

text: expressions

text: cause

text: useful

text: in

text: a

text: wide

text: variety

text: of

text: human-computer

text: interaction

text: and

text: intel-

text: a

text: mismatch

text: of

text: intra-class

text: features

text: due

text: to

text: their

text: orientation

text: ligent

text: systems

text: applications.

text: However,

text: the

text: automatic

text: recognition

text: of

text: variability,

text: it

text: is

text: difﬁcult

text: to

text: map

text: them

text: between

text: the

text: imaging

text: facial

text: expressions

text: using

text: image

text: template

text: matching

text: techniques

text: suf-

text: fer

text: from

text: the

text: natural

text: variability

text: with

text: facial

text: features

text: and

text: recording

text: templates.

text: conditions.

text: In

text: spite

text: of

text: the

text: progress

text: achieved

text: in

text: facial

text: emotion

text: Facial

text: emotion

text: recognition

text: accuracy

text: depends

text: on

text: the

text: robust-

text: recognition

text: in

text: recent

text: years,

text: the

text: effective

text: and

text: computationally

text: ness

text: of

text: a

text: feature

text: extraction

text: method

text: to

text: intra-class

text: variations

text: and

text: simple

text: feature

text: selection

text: and

text: classiﬁcation

text: technique

text: for

text: emotion

text: classiﬁer

text: performance

text: under

text: noisy

text: conditions

text: and

text: with

text: various

text: recognition

text: is

text: still

text: an

text: open

text: problem.

text: In

text: this

text: paper,

text: we

text: propose

text: types

text: of

text: occlusions

text: Even

text: thought

text: a

text: variety

text: of

text: approaches

text: an

text: efﬁcient

text: and

text: straightforward

text: facial

text: emotion

text: recognition

text: algo-

text: rithm

text: to

text: reduce

text: the

text: problem

text: of

text: inter-class

text: pixel

text: mismatch

text: during

text: for

text: the

text: automated

text: recognition

text: of

text: human

text: expressions

text: from

text: classiﬁcation.

text: The

text: proposed

text: method

text: includes

text: the

text: application

text: the

text: face

text: images

text: using

text: template

text: matching

text: methods

text: have

text: been

text: of

text: pixel

text: normalization

text: to

text: remove

text: intensity

text: offsets

text: followed-up

text: investigated

text: and

text: proposed

text: over

text: the

text: last

text: few

text: years

text: the

text: with

text: a

text: Min-Max

text: metric

text: in

text: a

text: nearest

text: neighbor

text: classiﬁer

text: that

text: is

text: emotion

text: recognition

text: method

text: with

text: robust

text: feature

text: extraction

text: capable

text: of

text: suppressing

text: feature

text: outliers.

text: The

text: results

text: indicate

text: an

text: and

text: effective

text: classiﬁcation

text: techniques

text: accompanied

text: by

text: low

text: improvement

text: of

text: recognition

text: performance

text: from

text: to

text: for

text: the

text: proposed

text: Min-Max

text: classiﬁcation

text: method

text: when

text: tested

text: on

text: computational

text: complexity

text: is

text: still

text: an

text: open

text: research

text: problem

text: JAFFE

text: database.

text: The

text: proposed

text: emotion

text: recognition

text: technique

text: Therefore,

text: in

text: this

text: paper,

text: we

text: address

text: the

text: issues

text: of

text: outperforms

text: the

text: existing

text: template

text: matching

text: methods.

text: matching

text: templates

text: through

text: pixel

text: normalization

text: followed

text: by

text: Index

text: Terms—Face

text: emotions,

text: Classiﬁer,

text: Emotion

text: recognition,

text: the

text: removal

text: of

text: inter-image

text: feature

text: outliers

text: using

text: a

text: Min-Max

text: spatial

text: ﬁlters,

text: gradients

text: similarity

text: metric.

text: We

text: apply

text: Gaussian

text: normalization

text: method

text: with

text: local

text: mean

text: and

text: standard

text: deviation

text: to

text: normalize

text: the

text: I.

text: INTRODUCTION

text: pixels

text: and

text: extract

text: relevant

text: face

text: features

text: followed

text: by

text: Min-

text: In

text: recent

text: years,

text: the

text: human-computer

text: interaction

text: challenge

text: Max

text: classiﬁcation

text: method

text: to

text: determine

text: an

text: emotion

text: class.

text: The

text: has

text: led

text: to

text: the

text: demand

text: to

text: introduce

text: efﬁcient

text: facial

text: and

text: speech

text: simulation

text: is

text: performed

text: in

text: Matlab

text: for

text: the

text: Japanese

text: Female

text: recognition

text: systems

text: Facial

text: emotion

text: recognition

text: is

text: Facial

text: Expression

text: (JAFFE)

text: database

text: and

text: the

text: emotion

text: the

text: identiﬁcation

text: of

text: a

text: human

text: emotion

text: based

text: on

text: the

text: facial

text: recognition

text: accuracy

text: is

text: calculated

text: using

text: leave-one-out

text: cross-

text: expression

text: and

text: mimics

text: The

text: facial

text: emotion

text: recognition

text: has

text: validation

text: method.

text: a

text: wide

text: range

text: of

text: appliction

text: prospects

text: in

text: different

text: areas,

text: such

text: as

text: The

text: main

text: contributions

text: of

text: this

text: work

text: are

text: the

text: following:

text: medicine

text: robotics

text: computer

text: vision,

text: surveillance

text: We

text: develop

text: a

text: simpliﬁed

text: approach

text: for

text: facial

text: emotion

text: systems

text: machine

text: learning

text: artiﬁcial

text: intelligence,

text: recognition

text: with

text: template

text: matching

text: method

text: using

text: Gaus-

text: communication

text: psychological

text: studies

text: smart

text: sian

text: normalization,

text: mean

text: and

text: standard

text: deviation

text: based

text: vehicles

text: security

text: and

text: embedded

text: systems

text: feature

text: extraction

text: and

text: Min-Max

text: classiﬁcation

text: approach.

text: There

text: are

text: two

text: main

text: approaches

text: for

text: facial

text: expression

text: recog-

text: We

text: present

text: simple

text: and

text: effective

text: facial

text: emotion

text: recog-

text: nition:

text: geometry-based

text: and

text: appearance-based

text: methods

text: nition

text: algorithm

text: having

text: low

text: computational

text: complexity

text: The

text: geometry-based

text: methods

text: extract

text: main

text: feature

text: points

text: and

text: and

text: ability

text: to

text: suppress

text: the

text: outliers

text: and

text: remove

text: intensity

text: their

text: shapes

text: from

text: the

text: face

text: image

text: and

text: calculate

text: the

text: distances

text: offsets.

text: between

text: them.

text: While,

text: appearance-based

text: methods

text: focus

text: on

text: We

text: conduct

text: the

text: experiments

text: and

text: simulations

text: on

text: JAFFE

text: the

text: face

text: texture

text: using

text: different

text: classiﬁcation

text: and

text: template

text: database

text: to

text: demonstrate

text: the

text: efﬁciency

text: of

text: the

text: proposed

text: matching

text: methods

text: In

text: this

text: paper,

text: we

text: focus

text: on

text: facial

text: approach

text: and

text: highlight

text: its

text: advantages,

text: comparing

text: to

text: the

text: emotion

text: recognition

text: based

text: on

text: template

text: matching

text: techniques

text: other

text: existing

text: methods.

text: that

text: remains

text: a

text: challenging

text: task

text: Since

text: the

text: orientation

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: Section

text: presents

text: the

text: of

text: pixel

text: features

text: are

text: sensitive

text: to

text: the

text: changes

text: in

text: illumina-

text: overview

text: of

text: the

text: existing

text: methods

text: for

text: facial

text: emotion

text: recogni-

text: tion,

text: pose,

text: scale

text: and

text: other

text: natural

text: imaging

text: variabilities,

text: the

text: tion,

text: their

text: drawbacks

text: and

text: reasons

text: to

text: propose

text: a

text: new

text: method.

text: In

text: matching

text: errors

text: tend

text: to

text: be

text: high

text: Pixel

text: matching

text: Section

text: we

text: show

text: normalization,

text: feature

text: extraction

text: and

text: clas-

text: methods

text: are

text: known

text: to

text: be

text: useful

text: when

text: the

text: images

text: has

text: missing

text: siﬁcation

text: parts

text: of

text: the

text: proposed

text: method,

text: present

text: the

text: algorithm