text: [30, 31] protocols which would require end users (customers) to have quantum resources which are costly. A ﬁnal goal should be to implement unconditionally secure and device independent quantum schemes [32,33]. 5 Conclusions Most of the Indian banks are found to support insecure SSL ciphers. Some of the previous studies were restricted to the banks of speciﬁc region or country (e.g., investigations performed in Refs [4,10,11,15–17,19] were restricted to the banks of India, Hongkong and Malyasia, respectively), whereas others were not restricted. Present study is also focused on Indian banks, but the methodology adopted and the conclusions obtained are valid in general. To emphasis on this point we analyze the security of 11 non-Indian banks, which are selected from diﬀerent continents. In what follows, we brieﬂy note the outcome of such an investigation in Table 5. S. Name of Website Server (Grade) Issues that reduced the grade No the bank (Country) 1 Bank of https://secure.bank 171.161.203.200 (A-) This server does not support Forward America ofamerica.com/ Secrecy with the reference browsers. (USA) Grade will be capped to B from March 2018. This server’s certiﬁcate will be distrusted by Google and Mozilla from September 2018. HTTP Strict Transport Security (HSTS) with long duration deployed on this server. 2 jpmorgan www.chase.com 159.53.224.21 (A-) This server does not support Forward chase Secrecy with the reference browsers. (USA) Grade will be capped to B from March 2018. HTTP Strict Transport Security (HSTS) with long duration deployed on this server. 3 Bank of www.bb 52.84.237.182 (A) Brazil americas.com 52.84.237.72 (A) (Brazil) 52.84.237.12 (A) 52.84.237.180 (A) 52.84.237.121 (A) 52.84.237.77 (A) 52.84.237.80 (A) 52.84.237.203 (A) 4 Steward onlinebanking. 41.216.125.233 (B) This server supports weak Bank (Zim- stewardbank.co.zw Diﬃe-Hellman (DH) key exchange babwe) parameters. Grade capped to B. This site works only in browsers with SNI support. 5 Avtovaz server29.cey- 12.191.20.25 (C) The server supports only older bank ebanking.com protocols, but not the current best (Russia) TLS 1.2. Grade capped to C. This server does not support Forward Secrecy with the reference browsers. Grade will be capped to B from March 2018. This server does not support Authenticated encryption (AEAD) cipher suites. Grade will be capped to B from March 2018. 6 Pingan bank.pingan.com 115.231.227.16 (A) Bank 27.148.164.23(A) (China) 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: protocols

text: which

text: would

text: require

text: end

text: users

text: (customers)

text: to

text: have

text: quantum

text: resources

text: which

text: are

text: costly.

text: A

text: ﬁnal

text: goal

text: should

text: be

text: to

text: implement

text: unconditionally

text: secure

text: and

text: device

text: independent

text: quantum

text: schemes

label: section

text: Conclusions

text: Most

text: of

text: the

text: Indian

text: banks

text: are

text: found

text: to

text: support

text: insecure

text: SSL

text: ciphers.

text: Some

text: of

text: the

text: previous

text: studies

text: were

text: restricted

text: to

text: the

text: banks

text: of

text: speciﬁc

text: region

text: or

text: country

text: (e.g.,

text: investigations

text: performed

text: in

text: Refs

text: were

text: restricted

text: to

text: the

text: banks

text: of

text: India,

text: Hongkong

text: and

text: Malyasia,

text: respectively),

text: whereas

text: others

text: were

text: not

text: restricted.

text: Present

text: study

text: is

text: also

text: focused

text: on

text: Indian

text: banks,

text: but

text: the

text: methodology

text: adopted

text: and

text: the

text: conclusions

text: obtained

text: are

text: valid

text: in

text: general.

text: To

text: emphasis

text: on

text: this

text: point

text: we

text: analyze

text: the

text: security

text: of

text: non-Indian

text: banks,

text: which

text: are

text: selected

text: from

text: diﬀerent

text: continents.

text: In

text: what

text: follows,

text: we

text: brieﬂy

text: note

text: the

text: outcome

text: of

text: such

text: an

text: investigation

text: in

text: Table

text: S.

text: Name

text: of

text: Website

text: Server

text: (Grade)

text: Issues

text: that

text: reduced

text: the

text: grade

text: No

text: the

text: bank

text: (Country)

text: Bank

text: of

text: (A-)

text: This

text: server

text: does

text: not

text: support

text: Forward

text: America

text: ofamerica.com/

text: Secrecy

text: with

text: the

text: reference

text: browsers.

text: (USA)

text: Grade

text: will

text: be

text: capped

text: to

text: B

text: from

text: March

text: This

text: server’s

text: certiﬁcate

text: will

text: be

text: distrusted

text: by

text: Google

text: and

text: Mozilla

text: from

text: September

text: HTTP

text: Strict

text: Transport

text: Security

text: (HSTS)

text: with

text: long

text: duration

text: deployed

text: on

text: this

text: server.

text: jpmorgan

text: (A-)

text: This

text: server

text: does

text: not

text: support

text: Forward

text: chase

text: Secrecy

text: with

text: the

text: reference

text: browsers.

text: (USA)

text: Grade

text: will

text: be

text: capped

text: to

text: B

text: from

text: March

text: HTTP

text: Strict

text: Transport

text: Security

text: (HSTS)

text: with

text: long

text: duration

text: deployed

text: on

text: this

text: server.

text: Bank

text: of

text: (A)

text: Brazil

text: americas.com

text: (A)

text: (Brazil)

text: (A)

text: (A)

text: (A)

text: (A)

text: (A)

text: (A)

text: Steward

text: onlinebanking.

text: (B)

text: This

text: server

text: supports

text: weak

text: Bank

text: (Zim-

text: stewardbank.co.zw

text: Diﬃe-Hellman

text: (DH)

text: key

text: exchange

text: babwe)

text: parameters.

text: Grade

text: capped

text: to

text: B.

text: This

text: site

text: works

text: only

text: in

text: browsers

text: with

text: SNI

text: support.

text: Avtovaz

text: (C)

text: The

text: server

text: supports

text: only

text: older

text: bank

text: ebanking.com

text: protocols,

text: but

text: not

text: the

text: current

text: best

text: (Russia)

text: TLS

text: Grade

text: capped

text: to

text: C.

text: This

text: server

text: does

text: not

text: support

text: Forward

text: Secrecy

text: with

text: the

text: reference

text: browsers.

text: Grade

text: will

text: be

text: capped

text: to

text: B

text: from

text: March

text: This

text: server

text: does

text: not

text: support

text: Authenticated

text: encryption

text: (AEAD)

text: cipher

text: suites.

text: Grade

text: will

text: be

text: capped

text: to

text: B

text: from

text: March

text: Pingan

text: bank.pingan.com

text: (A)

text: Bank

text: (China)

text: ##LTLine##

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text: Figure 12: Plots of Tll (left) and Tkl (right) as a function of v for u = 999, 1000, 1001. Tkk Tkk 4.3.3 Choice of on − ψ I As explained in Section , choosing = 0 leads to a violation of the NEC on all = cst ψ v slices as soon as begins to decrease ( 1000 in the Hayward-like model). In order to m v > avoid this violation of the NEC on −, one has to carefully study the sign of the following I components of the EMT:  h i  Tkk = −−e−2πψ((vu−,vu))∂uψ , (43) Tll = 32eπψ((vu−,v)u) F 2eψ(u,v)∂uψ + ∂vF . ≤ → −∞ First of all, choosing a function such that 0 for ensures that ψ ∂ ψ u T is non negative on −. As concerns , the term iunside the brackets must be negative fokkr I T → −∞ kk . Let us study the sign of , assuming a form for boiling down to an ingoing u ∂ F F Vaidya metric near −:  v I  −  F (v, R) = 1 2M(v,R) , lim ( ) = R ( ) (44) M v, R m v . =cst v→+∞ R Since = ( − ) 2, the leading term of on − is R v u / ∂ F I v 4 ( ) ’ ∂ m v v (45) ∂ F . v u Therefore, as long as m(v) is increasing, T → 0+ on I− assuming that we keep on with ≤ ll such that 0. ψ ∂ ψ u However, as soon as m(v) becomes decreasing, this leads to ∂ F → 0+ on I−. Since ≤ 0, we have to study carefully the sign of . On −, →v 1 and we require that ∂ ψ T I F u → ll ( ) cst so that we recover Minkowski’s metric there. Hence, the study of the sign of eψ u,v T ’ ll comes down to the comparison of the dominant terms of ∂ ψ and ∂ F 4∂vm(v). Ultimately, ﬁnding ( ) boils down to satisfying simultaneously theu three fovllowing cuonditions: ψ u, v 21 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

label: paragraph

text: Plots

label: caption

text: of

text: Tll

text: (left)

text: and

text: Tkl

text: (right)

text: as

text: a

text: function

text: of

text: v

text: for

text: u

text: Tkk

text: Tkk

text: Choice

text: of

text: on

text: ψ

text: I

text: As

text: explained

text: in

text: Section

text: choosing

text: leads

text: to

text: a

text: violation

text: of

text: the

text: NEC

text: on

text: all

text: cst

text: ψ

text: v

text: slices

text: as

text: soon

text: as

text: begins

text: to

text: decrease

text: in

text: the

text: Hayward-like

text: model).

text: In

text: order

text: to

text: m

text: v

text: avoid

text: this

text: violation

text: of

text: the

text: NEC

text: on

text: one

text: has

text: to

text: carefully

text: study

text: the

text: sign

text: of

text: the

text: following

text: I

text: components

text: of

text: the

text: EMT:

label: equation

text: h

text: i

text: Tkk

text: Tll

text: F

text: ∂vF

text: First

text: of

text: all,

text: choosing

text: a

text: function

text: such

text: that

text: for

text: ensures

text: that

text: ψ

text: ψ

text: u

text: T

text: is

text: non

text: negative

text: on

text: As

text: concerns

text: the

text: term

text: iunside

text: the

text: brackets

text: must

text: be

text: negative

text: fokkr

text: I

text: T

text: kk

text: Let

text: us

text: study

text: the

text: sign

text: of

text: assuming

text: a

text: form

text: for

text: boiling

text: down

text: to

text: an

text: ingoing

text: u

text: F

text: F

text: Vaidya

text: metric

text: near

text: v

text: I

text: F

text: (v,

text: R)

text: lim

text: R

text: M

text: v,

text: R

text: m

text: v

text: =cst

text: v→+∞

text: R

text: Since

text: the

text: leading

text: term

text: of

text: on

text: is

text: R

text: v

text: u

text: F

text: I

text: v

text: m

text: v

text: v

text: F

text: v

text: u

text: Therefore,

text: as

text: long

text: as

text: m(v)

text: is

text: increasing,

text: T

text: on

text: I−

text: assuming

text: that

text: we

text: keep

text: on

text: with

text: ll

text: such

text: that

text: ψ

text: ψ

text: u

text: However,

text: as

text: soon

text: as

text: m(v)

text: becomes

text: decreasing,

text: this

text: leads

text: to

text: F

text: on

text: I−.

text: Since

text: we

text: have

text: to

text: study

text: carefully

text: the

text: sign

text: of

text: On

text: →v

text: and

text: we

text: require

text: that

text: ψ

text: T

text: I

text: F

text: u

text: ll

text: cst

text: so

text: that

text: we

text: recover

text: Minkowski’s

text: metric

text: there.

text: Hence,

text: the

text: study

text: of

text: the

text: sign

text: of

text: eψ

text: u,v

text: T

text: ll

text: comes

text: down

text: to

text: the

text: comparison

text: of

text: the

text: dominant

text: terms

text: of

text: ψ

text: and

text: F

text: Ultimately,

text: ﬁnding

text: boils

text: down

text: to

text: satisfying

text: simultaneously

text: theu

text: three

text: fovllowing

text: cuonditions:

text: ψ

text: u,

text: v

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − 2 = ( ) 2 + 2 + 2 Ω2 (1) ds F v, R dv dvdR R d , where is the areal radius, Ω2 = 2 + sin2 2, and is a function of and not R d dθ θdφ F v R speciﬁed yet. This is not the most general spherically symmetric metric, whose expression will be used later (Eq.(27)). However, as Eq.(39) will illustrate, the additional degree of freedom of the general metric does not aﬀect the shape of the horizons given by the metric (1) used in the present section. We will assume that can be written in the following way F 2 ( ) −1 + ( ) −2 + · · · + ( ) + ( ) F (v, R) ≡ 1− M Rv, R = 1−2m(v)RRα α + baα−−1(2vv)RRαα−1 + · · · + ba1(1vv)RR+ b0a(0v)v , b0(v) 6= 0 . α (2) − Assuming that 1 can be written as a ratio of polynomial functions, this is the most F → ∞ general form one can use to recover Schwarzschild’s limit when + . The function R ( ) plays the role of the Misner-Sharp mass for an observer at inﬁnity. m v 2.3.1 Conditions for the existence of closed trapping horizons The existence of closed trapping horizons requires the presence of two horizons, i.e. of one marginally outer and one marginally inner trapped surfaces, whose coordinates ( ) R1 v and ( ) match for at least two diﬀerent values of . R2 v v As mentioned in the Introduction, the locus of the marginally trapped surfaces is deﬁned via the expansion of null outgoing geodesic congruence ≡ ∇ = 0 (3) θ+ hab k , a b where is the induced metric on the 2-spheres of symmetry and an outgoing radial h k ab b null vector. For the metric (1), the expansion is = and thus the locations of the horizons ( ) θ+ F R v are deﬁned by R ⇔ = 0 ( ( )) = 0 (4) θ+ F v, R v . Since the existence of a closed trapped region requires the presence of two horizons, the equation ( ( )) = 0 should thus be at least of degree 2 in . In this minimal case of F v, R v R degree 2, one has + ( ) ( ) = 1 − 2 ( ) R a0 v (5) F v, R m v , 2 + ( ) + ( ) R b1 v R b0 v and 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Ω2

text: ds

text: F

text: v,

text: R

text: dv

text: dvdR

text: R

text: d

text: where

text: is

text: the

text: areal

text: radius,

text: Ω2

text: and

text: is

text: a

text: function

text: of

text: and

text: not

text: R

text: d

text: dθ

text: θdφ

text: F

text: v

text: R

text: speciﬁed

text: yet.

text: This

text: is

text: not

text: the

text: most

text: general

text: spherically

text: symmetric

text: metric,

text: whose

text: expression

text: will

text: be

text: used

text: later

text: However,

text: as

text: will

text: illustrate,

text: the

text: additional

text: degree

text: of

text: freedom

text: of

text: the

text: general

text: metric

text: does

text: not

text: aﬀect

text: the

text: shape

text: of

text: the

text: horizons

text: given

text: by

text: the

text: metric

text: used

text: in

text: the

text: present

text: section.

text: We

text: will

text: assume

text: that

text: can

text: be

text: written

text: in

text: the

text: following

text: way

text: F

text: −1

text: −2

text: F

text: (v,

text: R)

text: 1−

text: M

text: Rv,

text: R

text: α

text: α

text: Assuming

text: that

text: can

text: be

text: written

text: as

text: a

text: ratio

text: of

text: polynomial

text: functions,

text: this

text: is

text: the

text: most

text: F

text: general

text: form

text: one

text: can

text: use

text: to

text: recover

text: Schwarzschild’s

text: limit

text: when

text: The

text: function

text: R

text: plays

text: the

text: role

text: of

text: the

text: Misner-Sharp

text: mass

text: for

text: an

text: observer

text: at

text: inﬁnity.

text: m

text: v

text: Conditions

text: for

text: the

text: existence

text: of

text: closed

text: trapping

text: horizons

text: The

text: existence

text: of

text: closed

text: trapping

text: horizons

text: requires

text: the

text: presence

text: of

text: two

text: horizons,

text: i.e.

text: of

text: one

text: marginally

text: outer

text: and

text: one

text: marginally

text: inner

text: trapped

text: surfaces,

text: whose

text: coordinates

text: v

text: and

text: match

text: for

text: at

text: least

text: two

text: diﬀerent

text: values

text: of

text: v

text: v

text: As

text: mentioned

text: in

text: the

text: Introduction,

text: the

text: locus

text: of

text: the

text: marginally

text: trapped

text: surfaces

text: is

text: deﬁned

text: via

text: the

text: expansion

text: of

text: outgoing

text: geodesic

text: congruence

text: θ+

text: hab

text: k

text: a

text: b

text: where

text: is

text: the

text: induced

text: metric

text: on

text: the

text: of

text: symmetry

text: and

text: an

text: outgoing

text: radial

text: h

text: k

text: ab

text: b

text: vector.

text: For

text: the

text: metric

text: the

text: expansion

text: is

text: and

text: thus

text: the

text: locations

text: of

text: the

text: horizons

text: θ+

text: F

text: R

text: v

text: are

text: deﬁned

text: by

text: R

text: θ+

text: F

text: v,

text: R

text: v

text: Since

text: the

text: existence

text: of

text: a

text: closed

text: trapped

text: region

text: requires

text: the

text: presence

text: of

text: two

text: horizons,

text: the

text: equation

text: should

text: thus

text: be

text: at

text: least

text: of

text: degree

text: in

text: In

text: this

text: minimal

text: case

text: of

text: F

text: v,

text: R

text: v

text: R

text: degree

text: one

text: has

text: R

text: v

text: F

text: v,

text: R

text: m

text: v

text: R

text: v

text: R

text: v

text: and

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 form by further performing the so called rotating wave approximation (RWA) where all summands on the right hand side of (cid:54) (cid:48) Eq. (2) with ω = ω are neglected. The RWA approximation can be performed if the individual levels are well spaced such that | − (cid:48)|− the time-scales given by differences in the transition frequencies, τRWA = ω ω 1, are much shorter than the relaxation time of the system, τ . S Another frequent approximation lies in neglecting the Lamb and Stark shift terms S(ω). This approximation is reasonable − since the Lamb and Stark shift terms are usually very small (for example for hydrogen the Lamb shift is of the order 109 s 1 [28] − − while typical optical transition frequencies are of the order of 1014 1015 s 1) and must be taken into account only in higher orders in the system-reservoir coupling strength. One example where the calculation is performed without dropping the Lamb and Stark shift terms is Ref. [29], where the authors investigate transport through a degenerate quantum dot. Different from − ≡ the Planck’s spectral density ω3/[exp(βh¯ω) 1] used here (β 1/kBT), the authors of the study [29] use the Ohmic spectral density ω/(exp[βh¯ω]+1) which leads to weaker divergences in the Lamb and Stark shift terms. | − (cid:48)|− ≈ If the system contains close energy levels resulting in close transition frequencies ω ω 1 = 1/∆ τS, t|he−abov(cid:48)|e−ap(cid:28)prox- imations must be taken with care. First, the RWA approximation should be performed only in terms where ω ω 1 τ . S This approximation is usually called the secular approximation [29] and it leaves on the right hand side of Eq. (2) the slowly ± (cid:54) rotating terms proportional to exp( i∆t). In case of a non-perfect degeneracy (∆ = 0), these terms cause several mathematical problems. First, the resulting master equation is still of the Redﬁeld type and thus it does not preserve positivity of the system density matrix [16]. Second, the resulting master equation in the Schro¨dinger picture contains terms proportional to ∆. These terms can have similar magnitude as the Lamb and Stark shift terms and also the higher-order terms in the system-reservoir coupling. In order to account for a non-zero ∆ correctly, one thus should not neglect the Lamb and Stark shift terms and should go beyond the week coupling Markov approximation of Eq. (2) at the same time. To sum up, Eq. (2) cannot be expected to give reasonable results for systems with close, but non-degenerate energy levels. In our analysis, we will thus always assume that the close energy levels are perfectly degenerate in which case there are no problems present. B. Summary of assumptions To sum up, for the derivation of the master equation for the V-type system with degenerate energy levels in the next subsection, we assume that the following separation of timescales occurs in the compound system composed of the system and reservoir: (cid:29) τS τR, (5) (cid:29) τS τRWA. (6) Here τ denotes the relaxation time of the system which is determined by the inverse maximum transition rate in the rate S equation, τR denotes the relaxation time of the reservoir which is roughly given by τR = h¯/(kBT) [26, 27, 30], and τRWA is determined by the smallest nonzero transition frequency in the system. The condition (5) is necessary for the validity of the Markov approximation and the condition (6) justiﬁes usage of RWA approximation as described in Sec. . The inequalities (5) and (6) reduce the space of allowed parameters for the models based on the resulting master equation. The timescales τ and τ depend only on reservoir temperature and on the structure of energy levels of the system. Thus R RWA they can be determined a priori before the master equation is derived. On the other hand, the system relaxation time τ must S be calculated a posteriori from the transition rates of the resulting master equation. The validity of inequalities (5) and (6) thus should be checked for every set of parameters separately2. In addition to the described timescale separation, we assume that the system-bath coupling is weak (we use the second order approximation in the coupling strength), we neglect the Lamb and Stark shift terms and we assume that the closely-lying energy levels are actually degenerate. C. Master equation for the V-type system with degenerate levels Using these assumptions, the Redﬁeld equation for the V-type model of Fig. 1 with degenerate upper doublet (∆ = 0) results ≡ ≡ (cid:104) | | (cid:105) in the following set of coupled differential equations for the individual elements ρij ρij(t) i ρ(t) j of the system density matrix in the Schro¨dinger picture: 2 Let us note that Dorfman et al. [9] used the rotating wave approximation while taking ∆ of the same order as standard transition frequencies between the energy levels in their numerical examples (see Table 1 in Ref. [9]). ##LTLine##

text: form

text: by

text: further

text: performing

text: the

text: so

text: called

text: rotating

text: wave

text: approximation

text: (RWA)

text: where

text: all

text: summands

text: on

text: the

text: right

text: hand

text: side

text: of

font: default

text: Eq.

text: with

text: ω

text: ω

text: are

text: neglected.

text: The

text: RWA

text: approximation

text: can

text: be

text: performed

text: if

text: the

text: individual

text: levels

text: are

text: well

text: spaced

text: such

text: that

text: the

text: time-scales

text: given

text: by

text: differences

text: in

text: the

text: transition

text: frequencies,

text: τRWA

text: ω

text: ω

text: are

text: much

text: shorter

text: than

text: the

text: relaxation

text: time

text: of

text: the

text: system,

text: τ

text: S

text: Another

text: frequent

text: approximation

text: lies

text: in

text: neglecting

text: the

text: Lamb

text: and

text: Stark

text: shift

text: terms

text: S(ω).

text: This

text: approximation

text: is

text: reasonable

text: since

text: the

text: Lamb

text: and

text: Stark

text: shift

text: terms

text: are

text: usually

text: very

text: small

text: (for

text: example

text: for

text: hydrogen

text: the

text: Lamb

text: shift

text: is

text: of

text: the

text: order

text: s

text: while

text: typical

text: optical

text: transition

text: frequencies

text: are

text: of

text: the

text: order

text: of

text: s

text: and

text: must

text: be

text: taken

text: into

text: account

text: only

text: in

text: higher

text: orders

text: in

text: the

text: system-reservoir

text: coupling

text: strength.

text: One

text: example

text: where

text: the

text: calculation

text: is

text: performed

text: without

text: dropping

text: the

text: Lamb

text: and

text: Stark

text: shift

text: terms

text: is

text: Ref.

text: where

text: the

text: authors

text: investigate

text: transport

text: through

text: a

text: degenerate

text: quantum

text: dot.

text: Different

text: from

text: the

text: Planck’s

text: spectral

text: density

text: used

text: here

text: (β

text: the

text: authors

text: of

text: the

text: study

text: use

text: the

text: Ohmic

text: spectral

text: density

text: which

text: leads

text: to

text: weaker

text: divergences

text: in

text: the

text: Lamb

text: and

text: Stark

text: shift

text: terms.

text: If

text: the

text: system

text: contains

text: close

text: energy

text: levels

text: resulting

text: in

text: close

text: transition

text: frequencies

text: ω

text: ω

text: τS,

text: imations

text: must

text: be

text: taken

text: with

text: care.

text: First,

text: the

text: RWA

text: approximation

text: should

text: be

text: performed

text: only

text: in

text: terms

text: where

text: ω

text: ω

text: τ

text: S

text: This

text: approximation

text: is

text: usually

text: called

text: the

text: secular

text: approximation

text: and

text: it

text: leaves

text: on

text: the

text: right

text: hand

text: side

text: of

text: Eq.

text: the

text: slowly

text: rotating

text: terms

text: proportional

text: to

text: exp(

text: i∆t).

text: In

text: case

text: of

text: a

text: non-perfect

text: degeneracy

text: these

text: terms

text: cause

text: several

text: mathematical

text: problems.

text: First,

text: the

text: resulting

text: master

text: equation

text: is

text: still

text: of

text: the

text: Redﬁeld

text: type

text: and

text: thus

text: it

text: does

text: not

text: preserve

text: positivity

text: of

text: the

text: system

text: density

text: matrix

text: Second,

text: the

text: resulting

text: master

text: equation

text: in

text: the

text: Schro¨dinger

text: picture

text: contains

text: terms

text: proportional

text: to

text: These

text: terms

text: can

text: have

text: similar

text: magnitude

text: as

text: the

text: Lamb

text: and

text: Stark

text: shift

text: terms

text: and

text: also

text: the

text: higher-order

text: terms

text: in

text: the

text: system-reservoir

text: coupling.

text: In

text: order

text: to

text: account

text: for

text: a

text: non-zero

text: correctly,

text: one

text: thus

text: should

text: not

text: neglect

text: the

text: Lamb

text: and

text: Stark

text: shift

text: terms

text: and

text: should

text: go

text: beyond

text: the

text: week

text: coupling

text: Markov

text: approximation

text: of

text: Eq.

text: at

text: the

text: same

text: time.

text: To

text: sum

text: up,

text: Eq.

text: cannot

text: be

text: expected

text: to

text: give

text: reasonable

text: results

text: for

text: systems

text: with

text: close,

text: but

text: non-degenerate

text: energy

text: levels.

text: In

text: our

text: analysis,

text: we

text: will

text: thus

text: always

text: assume

text: that

text: the

text: close

text: energy

text: levels

text: are

text: perfectly

text: degenerate

text: in

text: which

text: case

text: there

text: are

text: no

text: problems

text: present.

text: B.

text: Summary

text: of

text: assumptions

text: To

text: sum

text: up,

text: for

text: the

text: derivation

text: of

text: the

text: master

text: equation

text: for

text: the

text: V-type

text: system

text: with

text: degenerate

text: energy

text: levels

text: in

text: the

text: next

text: subsection,

text: we

text: assume

text: that

text: the

text: following

text: separation

text: of

text: timescales

text: occurs

text: in

text: the

text: compound

text: system

text: composed

text: of

text: the

text: system

text: and

text: reservoir:

text: τS

text: τR,

text: τS

text: τRWA.

text: Here

text: τ

text: denotes

text: the

text: relaxation

text: time

text: of

text: the

text: system

text: which

text: is

text: determined

text: by

text: the

text: inverse

text: maximum

text: transition

text: rate

text: in

text: the

text: rate

text: S

text: equation,

text: τR

text: denotes

text: the

text: relaxation

text: time

text: of

text: the

text: reservoir

text: which

text: is

text: roughly

text: given

text: by

text: τR

text: h¯/(kBT)

text: and

text: τRWA

text: is

text: determined

text: by

text: the

text: smallest

text: nonzero

text: transition

text: frequency

text: in

text: the

text: system.

text: The

text: condition

text: is

text: necessary

text: for

text: the

text: validity

text: of

text: the

text: Markov

text: approximation

text: and

text: the

text: condition

text: justiﬁes

text: usage

text: of

text: RWA

text: approximation

text: as

text: described

text: in

text: Sec.

text: The

text: inequalities

text: and

text: reduce

text: the

text: space

text: of

text: allowed

text: parameters

text: for

text: the

text: models

text: based

text: on

text: the

text: resulting

text: master

text: equation.

text: The

text: timescales

text: τ

text: and

text: τ

text: depend

text: only

text: on

text: reservoir

text: temperature

text: and

text: on

text: the

text: structure

text: of

text: energy

text: levels

text: of

text: the

text: system.

text: Thus

text: R

text: RWA

text: they

text: can

text: be

text: determined

text: a

text: priori

text: before

text: the

text: master

text: equation

text: is

text: derived.

text: On

text: the

text: other

text: hand,

text: the

text: system

text: relaxation

text: time

text: τ

text: must

text: S

text: be

text: calculated

text: a

text: posteriori

text: from

text: the

text: transition

text: rates

text: of

text: the

text: resulting

text: master

text: equation.

text: The

text: validity

text: of

text: inequalities

text: and

text: thus

text: should

text: be

text: checked

text: for

text: every

text: set

text: of

text: parameters

text: In

text: addition

text: to

text: the

text: described

text: timescale

text: separation,

text: we

text: assume

text: that

text: the

text: system-bath

text: coupling

text: is

text: weak

text: (we

text: use

text: the

text: second

text: order

text: approximation

text: in

text: the

text: coupling

text: strength),

text: we

text: neglect

text: the

text: Lamb

text: and

text: Stark

text: shift

text: terms

text: and

text: we

text: assume

text: that

text: the

text: closely-lying

text: energy

text: levels

text: are

text: actually

text: degenerate.

text: C.

text: Master

text: equation

text: for

text: the

text: V-type

text: system

text: with

text: degenerate

text: levels

text: Using

text: these

text: assumptions,

text: the

text: Redﬁeld

text: equation

text: for

text: the

text: V-type

text: model

text: of

text: Fig.

text: with

text: degenerate

text: upper

text: doublet

text: results

text: in

text: the

text: following

text: set

text: of

text: coupled

text: differential

text: equations

text: for

text: the

text: individual

text: elements

text: ρij

text: ρij(t)

text: i

text: ρ(t)

text: j

text: of

text: the

text: system

text: density

text: matrix

text: in

text: the

text: Schro¨dinger

text: picture:

text: Let

text: us

text: note

text: that

text: Dorfman

text: et

text: al.

text: used

text: the

text: rotating

text: wave

text: approximation

text: while

text: taking

text: of

text: the

text: same

text: order

text: as

text: standard

text: transition

text: frequencies

text: between

text: the

text: energy

text: levels

text: in

text: their

text: numerical

text: examples

text: (see

text: Table

text: in

text: Ref.

text: ##LTLine##


text: 4 − − ρ˙11 = γ1(f13ρ33 f31ρ11) f32Re(γ1c2ρ21), (7) − − ρ˙22 = γ2(f23ρ33 f32ρ22) f31Re(γ1c2ρ21), (8) − − ρ˙33 = γ1(f31ρ11 f13ρ33)+γ2(f32ρ22 f23ρ33)+(f31 + f32)Re(γ1c2ρ21), (9) γc − − − 1 ρ˙12 = 212 [(f13 + f23)ρ33 f32ρ11 f31ρ22] 2(γ1 f31 +γ2 f32)ρ12, (10) ∗ γc − − − 1 ρ˙21 = 122 [(f13 + f23)ρ33 f32ρ11 f31ρ22] 2(γ1 f31 +γ2 f32)ρ21. (11) ≡ | | · ∗ ≡ − Here γi gi 2, γ1c2 = g1 g2 , and fij 2F((Ei Ej)/h¯). T∗he condition E1 = E2 implies f13 = f23 and f31 = f≡32. The structure of equations does guarantee the necessary condition ρ12 = ρ21. The magnitude γ12 of the coupling parameter γ1c2 γ12eiφ controls how strongly the coherence ρ couples to populations. Using it we can get rid of the superﬂuous phase φ and simplify the above 12 system into a more compact form − − ρ˙11 = γ1(f13ρ33 f31ρ11) γ12 f32ρR, (12) − − ρ˙22 = γ2(f23ρ33 f32ρ22) γ12 f31ρR, (13) − − ρ˙33 = γ1(f31ρ11 f13ρ33)+γ2(f32ρ22 f23ρ33)+γ12(f31 + f32)ρR, (14) ρ˙R = γ212 [(f13 + f23)ρ33 − f32ρ11 − f31ρ22]− 12(γ1 f31 +γ2 f32)ρR, (15) √ ≡ − with real ρR ρ12e iφ = ρ21eiφ. The coupling γ12 can attain any value from the interval [0, γ1γ2] depending on the angle between the vectors g and g . In accord with the no√menclature used in the studies [6–9], we will from now on call the case of 1 2 the maximum value of the coupling parameter γ12 = γ1γ2 as the limit of maximum coherence and the limit of the zero value of the coupling parameter γ12 = 0 as the classical or no-coherence limit. Equations (12)–(15) are sufﬁcient to describe the dynamics of populations of the system and the energy currents throughout the system. The dynamics of the elements of the density matrix which are not contained in the above equations is trivial. Due to the linear nature of the Redﬁeld equation (2), using the formulas (12)–(15) one can write analogous master equation for an arbitrary system containing pairs of degenerate levels where the individual transitions are coupled to an arbitrary number of independent reservoirs at different temperatures given that the assumptions described in Sec. are fulﬁlled. It is enough to write the dynamical equations for all V-type, Λ-type and two-level subsystems (dynamical equations for a two level system can be obtained from Eqs. (12)–(15) if one forbids one of the transitions, i.e. after setting g1 = 0 or g2 = 0) composing the whole system in question and then sum up these sub-equations3. While the long-time solution of the resulting dynamical equation for a system coupled to a set of reservoirs at the same temperature is always given by the Boltzmann distribution where coherences vanish, the system attains a nontrivial steady state with nonzero coherences in case it is coupled at least to two reservoirs at different temperatures [6–9, 29]. In the studies [6–9] it was suggested that the coherence induced by the temperature gradient, the so called noise-induced coherence, can be utilized to enhance output power of quantum heat engines. In what follows, we introduce two models of absorption refrigerators whose performance may be also enhanced by the noise-induced coherence. After these models are introduced, we ﬁrst investigate whether the effect of the noise-induced coherence in these systems is a true quantum effect which can not be emulated by a classical stochastic system. Then, we discuss whether the cooling ﬂux in the refrigerators can be enhanced by the noise-induced coherence similarly as the output power of heat engines in Refs. [6–9]. III. REFRIGERATORS: DEFINITIONS OF TWO MODELS Consider the two types of absorption refrigerators depicted by general thermodynamic diagrams in Figs. 2b) and 2c). For both types of the refrigerators we assume the structure of energy levels of the working medium (system) depicted in Fig. 2a). Both refrige−rators are coupled to three reservoirs at the temperatures Tc < Tm < Th (cold-medium-hot) and use the temperature gradient T T to cool the coldest bath. The reservoir at T thus serves as an energy source of the fridge and the reservoir at T h m h m is an entropy sink which collects the heat coming both from the hot and cold baths (see the arrows in Figs. 2b) and 2c)). The allowed transitions between the individual levels of the system are in Fig. 2a) depicted by the colored arrows. For the A-type refrigerator, we assume that the transitions depicted by the individual colors are caused by the photons coming from 3 The linearity of system (12)–(15) only holds if all γc ’s of various bath couplings to a given energy doublet posses the same phase, which we assume here. If 12 not, one has to resort back to the original full set of Eqs. (7)–(11). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ρ˙11

text: ρ˙22

text: ρ˙33

text: γc

text: ρ˙12

text: γc

text: ρ˙21

text: Here

text: γi

text: gi

text: and

text: fij

text: Ej)/h¯).

text: T∗he

text: condition

text: implies

text: and

text: The

text: structure

text: of

text: equations

text: does

text: guarantee

text: the

text: necessary

text: condition

text: The

text: magnitude

text: of

text: the

text: coupling

text: parameter

text: controls

text: how

text: strongly

text: the

text: coherence

text: ρ

text: couples

text: to

text: populations.

text: Using

text: it

text: we

text: can

text: get

text: rid

text: of

text: the

text: superﬂuous

text: phase

text: φ

text: and

text: simplify

text: the

text: above

text: system

text: into

text: a

text: more

text: compact

text: form

text: ρ˙11

text: ρ˙22

text: ρ˙33

text: ρ˙R

text: with

text: real

text: ρR

text: iφ

text: The

text: coupling

text: can

text: attain

text: any

text: value

text: from

text: the

text: interval

text: depending

text: on

text: the

text: angle

text: between

text: the

text: vectors

text: g

text: and

text: g

text: In

text: accord

text: with

text: the

text: no√menclature

text: used

text: in

text: the

text: studies

text: we

text: will

text: from

text: now

text: on

text: call

text: the

text: case

text: of

text: the

text: maximum

text: value

text: of

text: the

text: coupling

text: parameter

text: γ1γ2

text: as

text: the

text: limit

text: of

text: maximum

text: coherence

text: and

text: the

text: limit

text: of

text: the

text: zero

text: value

text: of

text: the

text: coupling

text: parameter

text: as

text: the

text: classical

text: or

text: no-coherence

text: limit.

text: Equations

text: are

text: sufﬁcient

text: to

text: describe

text: the

text: dynamics

text: of

text: populations

text: of

text: the

text: system

text: and

text: the

text: energy

text: currents

text: throughout

text: the

text: system.

text: The

text: dynamics

text: of

text: the

text: elements

text: of

text: the

text: density

text: matrix

text: which

text: are

text: not

text: contained

text: in

text: the

text: above

text: equations

text: is

text: trivial.

text: Due

text: to

text: the

text: linear

text: nature

text: of

text: the

text: Redﬁeld

text: equation

text: using

text: the

text: formulas

text: one

text: can

text: write

text: analogous

text: master

text: equation

text: for

text: an

text: arbitrary

text: system

text: containing

text: pairs

text: of

text: degenerate

text: levels

text: where

text: the

text: individual

text: transitions

text: are

text: coupled

text: to

text: an

text: arbitrary

text: number

text: of

text: independent

text: reservoirs

text: at

text: different

text: temperatures

text: given

text: that

text: the

text: assumptions

text: described

text: in

text: Sec.

text: are

text: fulﬁlled.

text: It

text: is

text: enough

text: to

text: write

text: the

text: dynamical

text: equations

text: for

text: all

text: V-type,

text: Λ-type

text: and

text: two-level

text: subsystems

text: (dynamical

text: equations

text: for

text: a

text: two

text: level

text: system

text: can

text: be

text: obtained

text: from

text: Eqs.

text: if

text: one

text: forbids

text: one

text: of

text: the

text: transitions,

text: i.e.

text: after

text: setting

text: or

text: composing

text: the

text: whole

text: system

text: in

text: question

text: and

text: then

text: sum

text: up

text: these

text: While

text: the

text: long-time

text: solution

text: of

text: the

text: resulting

text: dynamical

text: equation

text: for

text: a

text: system

text: coupled

text: to

text: a

text: set

text: of

text: reservoirs

text: at

text: the

text: same

text: temperature

text: is

text: always

text: given

text: by

text: the

text: Boltzmann

text: distribution

text: where

text: coherences

text: vanish,

text: the

text: system

text: attains

text: a

text: nontrivial

text: steady

text: state

text: with

text: nonzero

text: coherences

text: in

text: case

text: it

text: is

text: coupled

text: at

text: least

text: to

text: two

text: reservoirs

text: at

text: different

text: temperatures

text: In

text: the

text: studies

text: it

text: was

text: suggested

text: that

text: the

text: coherence

text: induced

text: by

text: the

text: temperature

text: gradient,

text: the

text: so

text: called

text: noise-induced

text: coherence,

text: can

text: be

text: utilized

text: to

text: enhance

text: output

text: power

text: of

text: quantum

text: heat

text: engines.

text: In

text: what

text: follows,

text: we

text: introduce

text: two

text: models

text: of

text: absorption

text: refrigerators

text: whose

text: performance

text: may

text: be

text: also

text: enhanced

text: by

text: the

text: noise-induced

text: coherence.

text: After

text: these

text: models

text: are

text: introduced,

text: we

text: ﬁrst

text: investigate

text: whether

text: the

text: effect

text: of

text: the

text: noise-induced

text: coherence

text: in

text: these

text: systems

text: is

text: a

text: true

text: quantum

text: effect

text: which

text: can

text: not

text: be

text: emulated

text: by

text: a

text: classical

text: stochastic

text: system.

text: Then,

text: we

text: discuss

text: whether

text: the

text: cooling

text: ﬂux

text: in

text: the

text: refrigerators

text: can

text: be

text: enhanced

text: by

text: the

text: noise-induced

text: coherence

text: similarly

text: as

text: the

text: output

text: power

text: of

text: heat

text: engines

text: in

text: Refs.

text: III.

text: REFRIGERATORS:

text: DEFINITIONS

text: OF

text: TWO

text: MODELS

text: Consider

text: the

text: two

text: types

text: of

text: absorption

text: refrigerators

text: depicted

text: by

text: general

text: thermodynamic

text: diagrams

text: in

text: Figs.

text: and

text: For

text: both

text: types

text: of

text: the

text: refrigerators

text: we

text: assume

text: the

text: structure

text: of

text: energy

text: levels

text: of

text: the

text: working

text: medium

text: (system)

text: depicted

text: in

text: Fig.

text: Both

text: refrige−rators

text: are

text: coupled

text: to

text: three

text: reservoirs

text: at

text: the

text: temperatures

text: Tc

text: Tm

text: Th

text: (cold-medium-hot)

text: and

text: use

text: the

text: temperature

text: gradient

text: T

text: T

text: to

text: cool

text: the

text: coldest

text: bath.

text: The

text: reservoir

text: at

text: T

text: thus

text: serves

text: as

text: an

text: energy

text: source

text: of

text: the

text: fridge

text: and

text: the

text: reservoir

text: at

text: T

text: h

text: m

text: h

text: m

text: is

text: an

text: entropy

text: sink

text: which

text: collects

text: the

text: heat

text: coming

text: both

text: from

text: the

text: hot

text: and

text: cold

text: baths

text: (see

text: the

text: arrows

text: in

text: Figs.

text: and

text: The

text: allowed

text: transitions

text: between

text: the

text: individual

text: levels

text: of

text: the

text: system

text: are

text: in

text: Fig.

text: depicted

text: by

text: the

text: colored

text: arrows.

text: For

text: the

text: A-type

text: refrigerator,

text: we

text: assume

text: that

text: the

text: transitions

text: depicted

text: by

text: the

text: individual

text: colors

text: are

text: caused

text: by

text: the

text: photons

text: coming

text: from

text: The

text: linearity

text: of

text: system

text: only

text: holds

text: if

text: all

text: γc

text: ’s

text: of

text: various

text: bath

text: couplings

text: to

text: a

text: given

text: energy

text: doublet

text: posses

text: the

text: same

text: phase,

text: which

text: we

text: assume

text: here.

text: If

text: not,

text: one

text: has

text: to

text: resort

text: back

text: to

text: the

text: original

text: full

text: set

text: of

text: Eqs.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ⊕ ⊕ of H onto L2(I) L2(K) L2(K). To this end we show that the range of U is ⊕ − ⊕ − dense.We have U(1(An(AT )m+(AT )nAm)) = rn+m rn+m cos(n m)θ rn+m cos(n 2 − ∈ m)θ, which is the element corresponding to rn+m cos(n m)θ L2(σ(A)) and − ⊕ − ⊕ − − U(1(An(AT )m (AT )nAm)) = rn+m rn+m sin(n m)θ rn+m sin(n m)θ, which 2 − ∈ is the element corresponding to rn+m(sin(n m)θ L2(σ(A)). We already know from Lemma 4.14 that this collection is total. Thus we have proved that U can be extended uniquely as an orthogonal transformation on H, we use the notation U for denoting this extended operator also . Further, a direct computation using ⊕ trigonometric identities shows that UAUT = S on vectors of the form rn+m − ⊕ − ⊕ − ⊕− − rn+m cos(n m)θ rn+m cos(n m)θ and rn+m rn+m sin(n m)θ rn+m sin(n m)θ. This proves the lemma. 4.16 Theorem. Let A be a bounded normal operator on a real Hilbert space H. Then A is orthogonally equivalent to a countable direct sum of operators of the form (4.10). ⊆ C ( ie.there exists a countable family of compact sets E , positive measures µ B i →i symmetric to the real axis, on (E ) and a real orthogonal transformation U : H ⊕ ⊕ i L2(E , µ ) such that UAUT = S , where S on L2(E , µ ) is as in (4.10) ) i i i i i i i i Proof. Use the previous lemma and apply Zorn’s lemma. (cid:0) (cid:1) (cid:76) (cid:0) 4.17 Corollary. If A is a skew symmetric operator on a real Hilbert space H, then (cid:1) ⊂ ∞ there exists a countable family of compact subsets F [0, ) and positive measures B j → ⊕ ⊕ µ on (F ) and a real orthogonal transformation V : H L2(µ ) j j (cid:20) (cid:21) j j j L2(µ ) such that j ⊕ − 0 Mj A = V T ⊕ j λ V (4.15) Mj 0 j λ → (cid:55)→ ∀ ∈ where Mj : L2(µ ) L2(µ ) is such that f λf with (λf)(x) = xf(x), x F . λ j j j Proof. Since A is skew-symmetric σ(A) is contained in the imaginary axis. Use Theorem 4.16 to get L2(E ), where E subset of the positive imaginary axis. Take − j j F = iE . Use the obvious transformation to transfer measure to F . j j j (cid:76) 4.18 Corollary. If A is a skew symmetric invertible operator on a real Hilbert space H, then there exists a real Hilbert space(cid:20) K,→a p(cid:21)ositive invertible operator P on K and a real orthogonal transformation V : H K K such that − 0 P A = V T V (4.16) P 0 We give two proofs for this result the second proof is very elementary and does not use the spectral measure. Proof. (1) If A is invertible then each Mj in Corollary 4.17 is invertible. λ Proof. (2) Assume ﬁrst that A has a cyclic vector x. Note that if A is skew- ∈ N symmetric, A2k+1 is skew-symmetric and A2k is symmetric for k . Therefore (cid:104) (cid:105) ∀ ≥ A2nx, A2m+1x = 0, n, m 0, (4.17) 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: H

text: onto

text: To

text: this

text: end

text: we

text: show

text: that

text: the

text: range

text: of

text: U

text: is

text: dense.We

text: have

text: )m+(AT

text: )nAm))

text: rn+m

text: rn+m

text: cos(n

text: m)θ

text: rn+m

text: cos(n

text: m)θ,

text: which

text: is

text: the

text: element

text: corresponding

text: to

text: rn+m

text: cos(n

text: m)θ

text: and

text: )m

text: (AT

text: )nAm))

text: rn+m

text: rn+m

text: sin(n

text: m)θ

text: rn+m

text: sin(n

text: m)θ,

text: which

text: is

text: the

text: element

text: corresponding

text: to

text: rn+m(sin(n

text: m)θ

text: We

text: already

text: know

text: from

text: Lemma

text: that

text: this

text: collection

text: is

text: total.

text: Thus

text: we

text: have

text: proved

text: that

text: U

text: can

text: be

text: extended

text: uniquely

text: as

text: an

text: orthogonal

text: transformation

text: on

text: H,

text: we

text: use

text: the

text: notation

text: U

text: for

text: denoting

text: this

text: extended

text: operator

text: also

text: Further,

text: a

text: direct

text: computation

text: using

text: trigonometric

text: identities

text: shows

text: that

text: UAUT

text: S

text: on

text: vectors

text: of

text: the

text: form

text: rn+m

text: rn+m

text: cos(n

text: m)θ

text: rn+m

text: cos(n

text: m)θ

text: and

text: rn+m

text: rn+m

text: sin(n

text: m)θ

text: rn+m

text: sin(n

text: m)θ.

text: This

text: proves

text: the

text: lemma.

text: Theorem.

text: Let

text: A

text: be

text: a

text: bounded

text: normal

text: operator

text: on

text: a

text: real

text: Hilbert

text: space

text: H.

text: Then

text: A

text: is

text: orthogonally

text: equivalent

text: to

text: a

text: countable

text: direct

text: sum

text: of

text: operators

text: of

text: the

text: form

text: C

text: ie.there

text: exists

text: a

text: countable

text: family

text: of

text: compact

text: sets

text: E

text: positive

text: measures

text: µ

text: B

text: i

text: →i

text: symmetric

text: to

text: the

text: real

text: axis,

text: on

text: (E

text: and

text: a

text: real

text: orthogonal

text: transformation

text: U

text: H

text: i

text: µ

text: such

text: that

text: UAUT

text: S

text: where

text: S

text: on

text: µ

text: is

text: as

text: in

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: Proof.

text: Use

text: the

text: previous

text: lemma

text: and

text: apply

text: Zorn’s

text: lemma.

text: Corollary.

text: If

text: A

text: is

text: a

text: skew

text: symmetric

text: operator

text: on

text: a

text: real

text: Hilbert

text: space

text: H,

text: then

text: there

text: exists

text: a

text: countable

text: family

text: of

text: compact

text: subsets

text: F

text: and

text: positive

text: measures

text: B

text: j

text: µ

text: on

text: (F

text: and

text: a

text: real

text: orthogonal

text: transformation

text: V

text: H

text: j

text: j

text: j

text: j

text: j

text: such

text: that

text: j

text: Mj

text: A

text: V

text: T

text: j

text: λ

text: V

text: Mj

text: j

text: λ

text: where

text: Mj

text: is

text: such

text: that

text: f

text: λf

text: with

text: (λf)(x)

text: xf(x),

text: x

text: F

text: λ

text: j

text: j

text: j

text: Proof.

text: Since

text: A

text: is

text: skew-symmetric

text: σ(A)

text: is

text: contained

text: in

text: the

text: imaginary

text: axis.

text: Use

text: Theorem

text: to

text: get

text: where

text: E

text: subset

text: of

text: the

text: positive

text: imaginary

text: axis.

text: Take

text: j

text: j

text: F

text: iE

text: Use

text: the

text: obvious

text: transformation

text: to

text: transfer

text: measure

text: to

text: F

text: j

text: j

text: j

text: Corollary.

text: If

text: A

text: is

text: a

text: skew

text: symmetric

text: invertible

text: operator

text: on

text: a

text: real

text: Hilbert

text: space

text: H,

text: then

text: there

text: exists

text: a

text: real

text: Hilbert

text: K,→a

text: invertible

text: operator

text: P

text: on

text: K

text: and

text: a

text: real

text: orthogonal

text: transformation

text: V

text: H

text: K

text: K

text: such

text: that

text: P

text: A

text: V

text: T

text: V

text: P

text: We

text: give

text: two

text: proofs

text: for

text: this

text: result

text: the

text: second

text: proof

text: is

text: very

text: elementary

text: and

text: does

text: not

text: use

text: the

text: spectral

text: measure.

text: Proof.

text: If

text: A

text: is

text: invertible

text: then

text: each

text: Mj

text: in

text: Corollary

text: is

text: invertible.

text: λ

text: Proof.

text: Assume

text: ﬁrst

text: that

text: A

text: has

text: a

text: cyclic

text: vector

text: x.

text: Note

text: that

text: if

text: A

text: is

text: skew-

text: N

text: symmetric,

text: is

text: skew-symmetric

text: and

text: is

text: symmetric

text: for

text: k

text: Therefore

text: n,

text: m

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [label=)] µ is symmetric about the real axis. There exists an orthogonal trans- → formation U : H L2(E, µ) such that UAUT = S, where S on L2(E, µ) is given by (4.10), via the identiﬁcation described there. 12.. Proof. Deﬁne µ = µ(x,x) as in Proposition 4.2. We look at the polar coordinates for making the computa1tions simpler. Let E := {An(AT )mx : n, m ∈ N ∪ {0}}. Deﬁne E → ⊕ ⊕ an o(cid:88)perator U : span (cid:88)L2(I) L2(K) (cid:88)L2(K) by (cid:88) ⊕ − − U( a An((AT )mx) = a rn+m a rn+m(cos(n m)θ + sin(n m)θ) nm nm nm ⊕ − − − a rn+m(cos(n m)θ sin(n m)θ) nm (4.11) We set out to prove that U is inner product preserving and can be extended as an onto map from H. Strictly as an element of L2(σ(A)), U(An(AT )mx) is the − − element rn+m(cos(n m)θ + sin(n m)θ). Therefore,we have (cid:104) (cid:105) U(A(cid:90) k(AT )lx), U(An(AT )mx) (cid:104) − − − − (cid:105) = rk+l(cos(k l)θ + sin(k l)θ), rn+m(cos(n m)θ + sin(n m)θ) − − − − = rk+l(cos(k l)θ + sin(k l)θ)rn+m(cos(n m)θ + sin(n m)θ) dµ (4.12) σ(A) (cid:82) − − Note that since sin( θ) = sin θ and µ is symmetric about the real axis we have ∀ ∈ Z sin qθ dµ = 0, q . Therefore, by expanding (4.12) using trignometric iden- σ(A) tities, we have (cid:90) (cid:104) (cid:105) − − − U(Ak(AT )lx), U(An(AT )mx) = rk+l+n+m cos(k l) (n m)θ dµ (4.13) σ(A) So for proving that U is inner (cid:90)product preserving, we just need to prove (cid:104) (cid:105) − − − Ak(AT )lx, An(AT )mx = rk+l+n+m cos(k l) (n m)θ dµ (4.14) σ(A) Write k + m = s and l + n = t. T(cid:68)hen by usin(cid:69)g elementary properties of normal (cid:10) (cid:11) operators and (4.8) we have left hand side of (4.14) is same as (cid:90) Asx, Atx = As(AT )tx, x − = rs+t cos(s t)θ dµ(x,x) 1 σ(A) which is now same as the right hand side of 4.13 and thus U is inner product preserving. Now we will show that U extends as an orthogonal transformation 10

text: [label=)]

label: list

text: µ

text: is

text: symmetric

text: about

text: the

text: real

text: axis.

text: There

text: exists

text: an

text: orthogonal

text: trans-

text: formation

text: U

text: H

text: µ)

text: such

text: that

text: UAUT

text: S,

text: where

text: S

text: on

text: µ)

text: is

text: given

text: by

text: via

text: the

text: identiﬁcation

text: described

text: there.

text: Proof.

text: Deﬁne

text: µ

text: µ(x,x)

text: as

text: in

text: Proposition

text: We

text: look

text: at

text: the

text: polar

text: coordinates

text: for

text: making

text: the

text: simpler.

text: Let

text: E

text: {An(AT

text: )mx

text: n,

text: m

text: N

text: Deﬁne

text: E

text: an

text: U

text: span

text: by

text: U(

text: a

text: An((AT

text: )mx)

text: a

text: rn+m

text: a

text: rn+m(cos(n

text: m)θ

text: sin(n

text: m)θ)

text: nm

text: nm

text: nm

text: a

text: rn+m(cos(n

text: m)θ

text: sin(n

text: m)θ)

text: nm

text: We

text: set

text: out

text: to

text: prove

text: that

text: U

text: is

text: inner

text: product

text: preserving

text: and

text: can

text: be

text: extended

text: as

text: an

text: onto

text: map

text: from

text: H.

text: Strictly

text: as

text: an

text: element

text: of

text: U(An(AT

text: )mx)

text: is

text: the

text: element

text: rn+m(cos(n

text: m)θ

text: sin(n

text: m)θ).

text: Therefore,we

text: have

text: k(AT

text: )lx),

text: U(An(AT

text: )mx)

text: rk+l(cos(k

text: l)θ

text: sin(k

text: l)θ),

text: rn+m(cos(n

text: m)θ

text: sin(n

text: m)θ)

text: rk+l(cos(k

text: l)θ

text: sin(k

text: l)θ)rn+m(cos(n

text: m)θ

text: sin(n

text: m)θ)

text: dµ

text: σ(A)

text: Note

text: that

text: since

text: sin(

text: θ)

text: sin

text: θ

text: and

text: µ

text: is

text: symmetric

text: about

text: the

text: real

text: axis

text: we

text: have

text: Z

text: sin

text: qθ

text: dµ

text: q

text: Therefore,

text: by

text: expanding

text: using

text: trignometric

text: iden-

text: σ(A)

text: tities,

text: we

text: have

text: U(Ak(AT

text: )lx),

text: U(An(AT

text: )mx)

text: rk+l+n+m

text: cos(k

text: l)

text: (n

text: m)θ

text: dµ

text: σ(A)

text: So

text: for

text: proving

text: that

text: U

text: is

text: inner

text: preserving,

text: we

text: just

text: need

text: to

text: prove

text: Ak(AT

text: )lx,

text: An(AT

text: )mx

text: rk+l+n+m

text: cos(k

text: l)

text: (n

text: m)θ

text: dµ

text: σ(A)

text: Write

text: k

text: m

text: s

text: and

text: l

text: n

text: t.

text: by

text: elementary

text: properties

text: of

text: normal

text: operators

text: and

text: we

text: have

text: left

text: hand

text: side

text: of

text: is

text: same

text: as

text: Asx,

text: Atx

text: As(AT

text: )tx,

text: x

text: rs+t

text: cos(s

text: t)θ

text: dµ(x,x)

text: σ(A)

text: which

text: is

text: now

text: same

text: as

text: the

text: right

text: hand

text: side

text: of

text: and

text: thus

text: U

text: is

text: inner

text: product

text: preserving.

text: Now

text: we

text: will

text: show

text: that

text: U

text: extends

text: as

text: an

text: orthogonal

text: transformation


text: 2 Wenda Li, Lawrence C. Paulson theory of the Cauchy index on the complex plane and thus we succeed to approxi- mate how the path winds. When the path is a cycle and comprises linear and part of circular paths, we can now evaluate the winding number by calculating Cauchy indices along those sub-paths. In addition, by further combining the previous formalisation of the argument principle [13] (which associates the winding number with the number of complex roots), we build eﬀective procedures to count complex roots of a polynomial within some domains, such as a rectangle box and a half-plane. In short, the main contributions of this paper are – a novel tactic to enable users to evaluate the winding number through Cauchy indices, – and novel veriﬁed procedures to count complex roots of a polynomial. The Isabelle sources of this paper are available from the Archive of Formal Proofs [11,10]. Formulations in this paper, such as the deﬁnition of the Cauchy index and statements of some key lemmas, mainly follow Rahman and Schmeisser’s book [18, Chapter 11] and Eisermann’s paper [6]. Nevertheless, we were still obliged to devise some proofs on our own as discussed later. § This paper continues as follows: we start with a motivating example ( ) to explain the diﬃculty of formal evaluation of the winding number in Isabelle/HOL. We then present an intuitive description of the link between the winding number § and the Cauchy indices ( ). Formal development of the previous intuition is pre- § sented in . Next, veriﬁed procedures that count the number of complex roots in § § a domain are presented in , following which we discuss some limitations ( ) and § make some general remarks on the formalisation ( ). Finally, we discuss related § § work in and the conclusion in . 2 A Motivating Example In the formalisation of Cauchy’s residue theorem [13], we demonstrated an ap- (cid:90) plication of this theorem to formally evaluate an improper integral in Isabelle/HOL: ∞ dx = π. (1) −∞ x2 + 1 The idea is to embed this integral into the complex plane, and, as illustrated − in Fig. 1, to construct a linear path L from r to r and a semi-circular path C r r centred at 0 with radius r > 1: ∈ C (t) = reiπt for t [0, 1], r − − ∈ L (t) = (1 t)( r) + tr for t [0, 1]. r Next, by letting 1 f(w) = , w2 + 1 → ∞ and r , we can derive Equation (1) through the following steps ##LTLine## ##LTLine## ##LTLine##

text: Wenda

label: author

text: Li,

text: Lawrence

text: C.

text: Paulson

text: theory

text: of

text: the

text: Cauchy

text: index

text: on

text: the

text: complex

text: plane

text: and

text: thus

text: we

text: succeed

text: to

text: approxi-

text: mate

text: how

text: the

text: path

text: winds.

text: When

text: the

text: path

text: is

text: a

text: cycle

text: and

text: comprises

text: linear

text: and

text: part

text: of

text: circular

text: paths,

text: we

text: can

text: now

text: evaluate

text: the

text: winding

text: number

text: by

text: calculating

text: Cauchy

text: indices

text: along

text: those

text: sub-paths.

text: In

text: addition,

text: by

text: further

text: combining

text: the

text: previous

text: formalisation

text: of

text: the

text: argument

text: principle

text: (which

text: associates

text: the

text: winding

text: number

text: with

text: the

text: number

text: of

text: complex

text: roots),

text: we

text: build

text: eﬀective

text: procedures

text: to

text: count

text: complex

text: roots

text: of

text: a

text: polynomial

text: within

text: some

text: domains,

text: such

text: as

text: a

text: rectangle

text: box

text: and

text: a

text: half-plane.

text: In

text: short,

text: the

text: main

text: contributions

text: of

text: this

text: paper

text: are

text: a

text: novel

text: tactic

text: to

text: enable

text: users

text: to

text: evaluate

text: the

text: winding

text: number

text: through

text: Cauchy

text: indices,

text: and

text: novel

text: veriﬁed

text: procedures

text: to

text: count

text: complex

text: roots

text: of

text: a

text: polynomial.

text: The

text: Isabelle

text: sources

text: of

text: this

text: paper

text: are

text: available

text: from

text: the

text: Archive

text: of

text: Formal

text: Proofs

text: Formulations

text: in

text: this

text: paper,

text: such

text: as

text: the

text: deﬁnition

text: of

text: the

text: Cauchy

text: index

text: and

text: statements

text: of

text: some

text: key

text: lemmas,

text: mainly

text: follow

text: Rahman

text: and

text: Schmeisser’s

text: book

text: Chapter

text: and

text: Eisermann’s

text: paper

text: Nevertheless,

text: we

text: were

text: still

text: obliged

text: to

text: devise

text: some

text: proofs

text: on

text: our

text: own

text: as

text: discussed

text: later.

text: This

text: paper

text: continues

text: as

text: follows:

text: we

text: start

text: with

text: a

text: motivating

text: example

text: to

text: explain

text: the

text: diﬃculty

text: of

text: formal

text: evaluation

text: of

text: the

text: winding

text: number

text: in

text: Isabelle/HOL.

text: We

text: then

text: present

text: an

text: intuitive

text: description

text: of

text: the

text: link

text: between

text: the

text: winding

text: number

text: and

text: the

text: Cauchy

text: indices

text: Formal

text: development

text: of

text: the

text: previous

text: intuition

text: is

text: pre-

text: sented

text: in

text: Next,

text: veriﬁed

text: procedures

text: that

text: count

text: the

text: number

text: of

text: complex

text: roots

text: in

text: a

text: domain

text: are

text: presented

text: in

text: following

text: which

text: we

text: discuss

text: some

text: limitations

text: and

text: make

text: some

text: general

text: remarks

text: on

text: the

text: formalisation

text: Finally,

text: we

text: discuss

text: related

text: work

text: in

text: and

text: the

text: conclusion

text: in

text: A

text: Motivating

text: Example

text: In

text: the

text: formalisation

text: of

text: Cauchy’s

text: residue

text: theorem

text: we

text: demonstrated

text: an

text: ap-

text: plication

text: of

text: this

text: theorem

text: to

text: formally

text: evaluate

text: an

text: improper

text: integral

text: in

text: Isabelle/HOL:

text: dx

text: π.

text: The

text: idea

text: is

text: to

text: embed

text: this

text: integral

text: into

text: the

text: complex

text: plane,

text: and,

text: as

text: illustrated

text: in

text: Fig.

text: to

text: construct

text: a

text: linear

text: path

text: L

text: from

text: r

text: to

text: r

text: and

text: a

text: semi-circular

text: path

text: C

text: r

text: r

text: centred

text: at

text: with

text: radius

text: r

text: C

text: (t)

text: reiπt

text: for

text: t

text: r

text: L

text: (t)

text: t)(

text: r)

text: tr

text: for

text: t

text: r

text: Next,

text: by

text: letting

text: f(w)

text: and

text: r

text: we

text: can

text: derive

text: Equation

text: through

text: the

text: following

text: steps

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 Wenda Li, Lawrence C. Paulson propose veriﬁed procedures to count the number of complex roots of a polynomial in some domain such as a rectangle and a half-plane. Does a winding number have anything to do with the number of roots of a polynomial? The answer is yes. Thanks to the argument principle, we can calculate (cid:73) the number of roots by evaluating a contour integral: (cid:48) 1 p (x) dx = N (14) 2πi p(x) γ ∈ C (cid:48) where p [x], p (x) is the ﬁrst derivative of p and N is the number of complex roots of p (counted with multiplicity) inside the loop γ. Also, by the deﬁnition of (cid:73) winding numbers, we have (cid:48) ◦ 1 p (x) n(p γ, 0) = dx. (15) 2πi p(x) γ Combining Equations (14) and (15) gives us the relationship between a winding number and the number of roots of a polynomial: ◦ n(p γ, 0) = N. (16) ◦ And the question becomes: can we evaluate n(p γ, 0) through Cauchy indices? 5.1 Roots in a Rectangle a4 L3 a3 L4 L2 a1 L1 a2 Fig. 7: Complex roots of a polynomial (red dots) and a rectangular path (L + 1 L + L + L ) on the complex plane 2 3 4 Let N be the number of complex roots of a polynomial p inside the rectangle deﬁned by its lower left corner a and upper right corner a . As illustrated in Fig. 1 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Wenda

text: Li,

text: Lawrence

text: C.

text: Paulson

text: propose

text: veriﬁed

text: procedures

text: to

text: count

text: the

text: number

text: of

text: complex

text: roots

text: of

text: a

text: polynomial

text: in

text: some

text: domain

text: such

text: as

text: a

text: rectangle

text: and

text: a

text: half-plane.

text: Does

text: a

text: winding

text: number

text: have

text: anything

text: to

text: do

text: with

text: the

text: number

text: of

text: roots

text: of

text: a

text: polynomial?

text: The

text: answer

text: is

text: yes.

text: Thanks

text: to

text: the

text: argument

text: principle,

text: we

text: can

text: calculate

text: the

text: number

text: of

text: roots

text: by

text: evaluating

text: a

text: contour

text: integral:

text: p

text: (x)

text: dx

text: N

text: p(x)

text: γ

text: C

text: where

text: p

text: [x],

text: p

text: (x)

text: is

text: the

text: ﬁrst

text: derivative

text: of

text: p

text: and

text: N

text: is

text: the

text: number

text: of

text: complex

text: roots

text: of

text: p

text: (counted

text: with

text: multiplicity)

text: inside

text: the

text: loop

text: γ.

text: Also,

text: by

text: the

text: deﬁnition

text: of

text: winding

text: numbers,

text: we

text: have

text: p

text: (x)

text: n(p

text: γ,

text: dx.

text: p(x)

text: γ

text: Combining

text: Equations

text: and

text: gives

text: us

text: the

text: relationship

text: between

text: a

text: winding

text: number

text: and

text: the

text: number

text: of

text: roots

text: of

text: a

text: polynomial:

text: n(p

text: γ,

text: N.

text: And

text: the

text: question

text: becomes:

text: can

text: we

text: evaluate

text: n(p

text: γ,

text: through

text: Cauchy

text: indices?

text: Roots

text: in

text: a

text: Rectangle

text: Fig.

text: Complex

text: roots

text: of

text: a

text: polynomial

text: (red

text: dots)

text: and

text: a

text: rectangular

text: path

text: (L

text: L

text: L

text: L

text: on

text: the

text: complex

text: plane

text: Let

text: N

text: be

text: the

text: number

text: of

text: complex

text: roots

text: of

text: a

text: polynomial

text: p

text: inside

text: the

text: rectangle

text: deﬁned

text: by

text: its

text: lower

text: left

text: corner

text: a

text: and

text: upper

text: right

text: corner

text: a

text: As

text: illustrated

text: in

text: Fig.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 Wenda Li, Lawrence C. Paulson → C Deﬁnition 3 (Cauchy index along a path) Given a path γ : [0, 1] and a ∈ C point z , the Cauchy index along γ about z is deﬁned as 0 0 Indp(γ, z ) = Ind1(f) 0 0 where − Im(γ(t) z ) f(t) = − 0 . Re(γ(t) z ) 0 In particular, it can be checked that the Cauchy index Indp(γ, z ) captures the (cid:12) { | } (cid:12) 0 way that γ crosses the li(cid:12)ne z Re(z) = Re(z ) , he(cid:12)nce leads to an approximation (cid:12) 0 (cid:12) of Re(n(γ, z )): (cid:12) (cid:12) 0 Indp(γ, z ) 1 Re(n(γ, z )) + 0 < . 0 2 2 More interestingly, by further knowing that γ is a loop we can derive Re(n(γ, z )) = ∈ Z ∈ Z 0 n(γ, z ) and Indp(γ, z )/2 , following which we come to the core proposi- 0 0 tion of this paper: → C ∈ C Proposition 1 Given a valid path γ : [0, 1] and a point z , such that γ 0 is a loop and z is not on the image of γ, we have 0 −Indp(γ, z ) n(γ, z ) = 0 . 0 2 That is, under some assumptions, we can evaluate a winding number through Cauchy indices! Cr Cr i − − r r r r Lr Lr − i (a) (b) − Fig. 3: Evaluating n(L +C , i) and n(L +C , i) through the way that the path r r r r L + C crosses the imaginary axis r r § A formal proof of Proposition 1 will be introduced in . Here, given the state- ment of the proposition we can have alternative proofs for n(L + C , i) = 1 and − r r n(L + C , i) = 0. r r ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Wenda

text: Li,

text: Lawrence

text: C.

text: Paulson

text: C

text: Deﬁnition

text: (Cauchy

text: index

text: along

text: a

text: path)

text: Given

text: a

text: path

text: γ

text: and

text: a

text: C

text: point

text: z

text: the

text: Cauchy

text: index

text: along

text: γ

text: about

text: z

text: is

text: deﬁned

text: as

text: Indp(γ,

text: z

text: where

text: Im(γ(t)

text: z

text: f(t)

text: Re(γ(t)

text: z

text: In

text: particular,

text: it

text: can

text: be

text: checked

text: that

text: the

text: Cauchy

text: index

text: Indp(γ,

text: z

text: captures

text: the

text: way

text: that

text: γ

text: crosses

text: the

text: z

text: Re(z)

text: Re(z

text: leads

text: to

text: an

text: approximation

text: of

text: Re(n(γ,

text: z

text: Indp(γ,

text: z

text: Re(n(γ,

text: z

text: More

text: interestingly,

text: by

text: further

text: knowing

text: that

text: γ

text: is

text: a

text: loop

text: we

text: can

text: derive

text: Re(n(γ,

text: z

text: Z

text: Z

text: n(γ,

text: z

text: and

text: Indp(γ,

text: z

text: following

text: which

text: we

text: come

text: to

text: the

text: core

text: proposi-

text: tion

text: of

text: this

text: paper:

text: C

text: C

text: Proposition

text: Given

text: a

text: valid

text: path

text: γ

text: and

text: a

text: point

text: z

text: such

text: that

text: γ

text: is

text: a

text: loop

text: and

text: z

text: is

text: not

text: on

text: the

text: image

text: of

text: γ,

text: we

text: have

text: −Indp(γ,

text: z

text: n(γ,

text: z

text: That

text: is,

text: under

text: some

text: assumptions,

text: we

text: can

text: evaluate

text: a

text: winding

text: number

text: through

text: Cauchy

text: indices!

text: Cr

text: Cr

text: i

text: r

text: r

text: r

text: r

text: Lr

text: Lr

text: i

text: (a)

text: (b)

text: Fig.

text: Evaluating

text: n(L

text: +C

text: i)

text: and

text: n(L

text: +C

text: i)

text: through

text: the

text: way

text: that

text: the

text: path

text: r

text: r

text: r

text: r

text: L

text: C

text: crosses

text: the

text: imaginary

text: axis

text: r

text: r

text: A

text: formal

text: proof

text: of

text: Proposition

text: will

text: be

text: introduced

text: in

text: Here,

text: given

text: the

text: state-

text: ment

text: of

text: the

text: proposition

text: we

text: can

text: have

text: alternative

text: proofs

text: for

text: n(L

text: C

text: i)

text: and

text: r

text: r

text: n(L

text: C

text: i)

text: r

text: r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 Wenda Li, Lawrence C. Paulson ∀ ∃ ∀ −→ ≤ (LIM x (at_left x). f x :> at_top) = ( z. b<x. y>b. y < x z f y) ∀ ∃ ∀ −→ ≤ w∞here z. b<x. y>b. y < x z f y is the usual deﬁnition of limu→x− f(u) = in textbooks. We can then encode Indba(f) and Indp(γ, z0) (see Deﬁnitions 2 and 3) as cindexE and cindex pathE respectively: (cid:80) ⇒ ⇒ ⇒ ⇒ deﬁnition(cid:80)cindexE::"real real (real real) real" where "cindexE ∈a b f = (cid:54) ∧ ≤ ∧ ( x {∈x. jumpF f (at_right x) =(cid:54) 0 ∧a x ∧ x<≤b}. jumpF f (at_right x)) - ( x {x. jumpF f (at_left x) =0 a<x x b}. jumpF f (at_left x))" ⇒ ⇒ ⇒ deﬁnition cindex_pathE::"(real complex) complex real" where "cindex_pathE γ z0 = cindexE 0 1 (λt. Im (γ t - z0) / Re (γ t - z0))" (cid:88) Note, in the deﬁnition of Indb(f) we have a term a jump (f, x) + ∈ x [a,b) which actually hides an assumption that only a ﬁnite number of points within the interval [a, b) contribute to the sum. This assumption is made explicit when deﬁning cindexE, where the sum is over the set (cid:80) (cid:54) ∧ ≤ ∧ {x. jumpF f (at_right x) = 0 a x x < b}. In the case that the set above is inﬁnite (i.e., the sum ∈ jump (f, x) is not (cid:80) x [a,b) + mathematically well-deﬁned) we have ∈ (cid:54) ∧ ≤ ∧ ( x {x. jumpF f (at_right x)=0 a x x<b}. jumpF f (at_right x)) = 0. In other words, a default value (i.e., 0) is used in Isabelle/HOL when summing over an inﬁnite set. Due to the issue of well-deﬁned sums, many of our lemmas related cindexE will have an assumption finite jumpFs: ⇒ ⇒ ⇒ ⇒ deﬁnition finite_jumpFs::"(real real) real real(cid:54) bool" where "finite_jumpFs f a b = fin∨ite {x. (jumpF f (at_lef(cid:54) t x) ∧=0≤ ∧ ≤ jumpF f (at_right x) =0) a x x b}" which guarantees the well-deﬁnedness of cindexE. Now, suppose that we know that Indp is well-deﬁned (i.e., there are only ﬁnite number of jumps over the path). What is the strategy we can employ to formally prove Proposition 1? Naturally, we may want to divide the path into a ﬁnite num- ber of segments (subpaths) separated by those jumps, and then perform inductions on these segments. To formalise the ﬁniteness of such segments, we can have: ⇒ ⇒ ⇒ ⇒ inductive finite_Psegments::"(real bool) real real bool" for P wh≥ere ⇒ emptyI: "a b =∈ finite_Pseg∨ments P∀a ∈b"| insertI_1: "[[s {a..<b}; s=a ⇒P s; t {s<..<b}. P t; finite_Pseg∈ments P a s]] =∨ finit∀e_∈Psegments P¬a b"| insertI_2: "[[s {a..<b}; s=a ⇒P s; t {s<..<b}. P t; finite_Psegments P a s]] = finite_Psegments P a b" ⇒ ⇒ ⇒ deﬁnition finite_ReZ_segments::"(real complex) complex bool" where "finite_ReZ_segments γ z0 = finite_Psegments (λt. Re (γ t - z0) = 0) 0 1" ##LTLine## ##LTLine## ##LTLine##

text: Wenda

text: Li,

text: Lawrence

text: C.

text: Paulson

text: (LIM

text: x

text: x).

text: f

text: x

text: z.

text: b<x.

text: y>b.

text: y

text: x

text: z

text: f

text: y)

text: w∞here

text: z.

text: b<x.

text: y>b.

text: y

text: x

text: z

text: f

text: y

text: is

text: the

text: usual

text: deﬁnition

text: of

text: limu→x−

text: f(u)

text: in

text: textbooks.

text: We

text: can

text: then

text: encode

text: Indba(f)

text: and

text: Indp(γ,

text: (see

text: Deﬁnitions

text: and

text: as

text: cindexE

text: and

text: cindex

text: pathE

text: respectively:

text: real

text: (real

text: real)

text: real"

text: where

text: "cindexE

text: ∈a

text: b

text: f

text: x

text: {∈x.

text: jumpF

text: f

text: x)

text: ∧a

text: x

text: x<≤b}.

text: jumpF

text: f

text: x))

text: x

text: {x.

text: jumpF

text: f

text: x)

text: a<x

text: x

text: b}.

text: jumpF

text: f

text: x))"

text: deﬁnition

text: complex)

text: complex

text: real"

text: where

text: γ

text: cindexE

text: (λt.

text: Im

text: (γ

text: t

text: Re

text: (γ

text: t

text: Note,

text: in

text: the

text: deﬁnition

text: of

text: Indb(f)

text: we

text: have

text: a

text: term

text: a

text: jump

text: (f,

text: x)

text: x

text: [a,b)

text: which

text: actually

text: hides

text: an

text: assumption

text: that

text: only

text: a

text: ﬁnite

text: number

text: of

text: points

text: within

text: the

text: interval

text: [a,

text: b)

text: contribute

text: to

text: the

text: sum.

text: This

text: assumption

text: is

text: made

text: explicit

text: when

text: deﬁning

text: cindexE,

text: where

text: the

text: sum

text: is

text: over

text: the

text: set

text: {x.

text: jumpF

text: f

text: x)

text: a

text: x

text: x

text: b}.

text: In

text: the

text: case

text: that

text: the

text: set

text: above

text: is

text: inﬁnite

text: (i.e.,

text: the

text: sum

text: jump

text: (f,

text: x)

text: is

text: not

text: x

text: [a,b)

text: mathematically

text: well-deﬁned)

text: we

text: have

text: x

text: {x.

text: jumpF

text: f

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text: 2 Fixed Path detector Movable deﬁned as the upper bound to the probability with which | (cid:105) | (cid:105) detector detector the states d1 , . . . , dn can| be(cid:105)distinguished without any 1 S 2 e| rr|or [3, 25]. If the state dk occurs with a probability D1 D2 ck 2, the path distinguisha(cid:213)bility has the following form [3, 25] L 2 D L1 Multi slit D ≡ − 1 | || ||(cid:104) | (cid:105)| 1 − c c d d . (3) Coincidence Q n 1 i j i j counter i(cid:44)j FIG. 1: A modiﬁed ghost-interference experiment with entan- The path-distinguishability can take values between 0 gled pairs of particles. A which-path detector can detect which and 1. of the n slits particle 1 went through. Detector D1 is ﬁxed at III. GENERAL ANALYSIS z (cid:3) 0, but D2 is free to move along the z-axis. Next we formulate the n-slit ghost interference, with a path-detector, in a most general way. We assume that the be a consequence of entanglement, and generated lot of particles travel in opposite directions along the x-axis. research activity [10–18]. The entanglement is in the z-direction. The entangled | ( )(cid:105) We look at a modiﬁed ghost interference experiment, two particle state, at the source is given by Ψ 0 . We as- shown in Fig.1, where the double-slit is replaced by a sume that after travelling for a time t , particle 1 reaches 0 n-slit. A n-slit ghost interference experiment has actu- the n-slit (vt0 (cid:3) L2), and particle 2 travels a distance L2 ally been performed by Zeilinger’s group [19]. However, towards detector D2. The two particle state is now given | ( )(cid:105) here we will consider another modiﬁcation of it. En- by Ψ t0 . tangled particle pairs, which could also more generally We take into account the eﬀect of the n-slit on the be something other than photons, emerge from a source entangled state as follows. We assume that the n-slit S. Particle 1 passes through a n-slit and also interacts allows the portions of the wave-function in front of the with a path-detector. We do not assume any form of slits to pass through, and blocks the other portions. We the path-detector, |bu(cid:105)t j|ust(cid:105) a|ssu(cid:105)me th|at(cid:105)it is an n-state assume that what emerge from the n-slit are localised system with states d1 , d2 , d3 , . . . , dn , which get en- states, whose width is approximately the width of a tangled with the n paths of particle 1. This entangle- slit. The states of particle 1, which pass through the | (cid:105) | (cid:105) | (cid:105) ment is bound to happen if the path-detector acquires slits 1, 2, . . . , n, are denoted by φ , φ , . . . , φ , re- 1 2 n the relevant information about which slit particle 1 went spectively. The state representing the situation which | (cid:105) through. Particle 1 then travels further and is detected particle 1 gets blocked is, say, χ . These n+1 states are by a ﬁxed detector D1. Particle 2 travels undisturbed to obviously orthogonal, because they represent mutually detector D2. Since the two particles have to be counted exclusive possibilities, and form a complete set because in coincidence, the paths travelled by both the particles, they exhaust all the possibilities for particle 1. Thus the before reaching their respective detectors, are assumed entangled two-particle state can be expanded in terms of (cid:32) (cid:33) to be equal. Without the path-detector, our experiment these. We can thu(cid:213)s write: would just be a n-slit generalization of the original two- slit ghost interference experiment [9]. It is expected to show a n-slit ghost interference for particle 2. By in- | ( )(cid:105) n | (cid:105)(cid:104) | | (cid:105)(cid:104) | ⊗ | ( )(cid:105) (cid:213) troducing a path-detector, we are probing if acquiring Ψ t0 (cid:3) φk φk + χ χ 12 Ψ t0 path information about particle 1, has any eﬀect on the k(cid:3)1 interference shown by particle 2. n | (cid:105)(cid:104) | ( )(cid:105) | (cid:105)(cid:104) | ( )(cid:105) | (cid:105) | (cid:105) We assume d , . . . , d to be normalized, but not (cid:3) φk φk Ψ t0 + χ χ Ψ t0 , (4) 1 n necessarily orthogonal. The ultimate limit to the knowl- k(cid:3)1 edge we can acquire as to which slit particle 1 went | (cid:105) | (cid:105) where is the unit operator for the space of particle through| , i(cid:105)s set b|y h(cid:105)ow distinct the states d1 , . . . , dn 2. Now, s1in2 ce we are only interested in those instances are. If d , . . . , d are orthogonal, we can in principle 1 n (through coincidence counting) where particle 1 does know with certainty which slit the particle went through. | (cid:105) | (cid:105) pass through the multi-slit, and does not get blocked, Of course in general, d , . . . , d may not be all orthog- | (cid:105) 1 n the χ dependent term can be discarded, and one is left onal to each other. In such a situation, one is left with the with the two-particle sta(cid:213)te problem of unambigiously telliing which is of the states | (cid:105) | (cid:105) d , . . . , d , is the given unknown path-detector state 1 n which the path-detector process throws up. The best bet | ( )(cid:105) n | (cid:105)(cid:104) | ( )(cid:105) to answer this question is using unambigious quantum Ψ t0 (cid:3) φk φk Ψ t0 , (5) state discrimination (UQSD) [20–24]. D k(cid:3)1 Using UQSD, a path-distinguishability can then be Q ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: BISQMA+TeXGyrePagellaX-Regular

text: Fixed

text: Path

text: detector

text: Movable

text: deﬁned

text: as

text: the

text: upper

text: bound

text: to

text: the

text: probability

text: with

text: which

font: OCQTEU+pxsys

text: detector

text: detector

text: the

text: states

font: VXLVVM+URWPalladioL-Ital

font: IZCYQZ+Rpxmi

text: dn

text: can|

text: without

text: any

text: S

text: e|

text: rr|or

text: If

text: the

text: state

text: dk

text: occurs

text: with

text: a

text: probability

text: ck

text: the

text: path

text: has

text: the

text: following

text: form

text: L

text: D

text: Multi

text: slit

text: D

text: c

text: c

text: d

text: d

text: Coincidence

text: Q

text: n

text: i

text: j

text: i

text: j

text: counter

text: FIG.

text: A

text: modiﬁed

text: ghost-interference

text: experiment

text: with

text: entan-

text: The

text: path-distinguishability

text: can

text: take

text: values

text: between

text: gled

text: pairs

text: of

text: particles.

text: A

text: which-path

text: detector

text: can

text: detect

text: which

text: and

text: of

text: the

text: n

text: slits

text: particle

text: went

text: through.

text: Detector

text: is

text: ﬁxed

text: at

text: III.

text: GENERAL

text: ANALYSIS

text: z

text: but

text: is

text: free

text: to

text: move

text: along

text: the

text: z-axis.

text: Next

text: we

text: formulate

text: the

text: n-slit

text: ghost

text: interference,

text: with

text: a

text: path-detector,

text: in

text: a

text: most

text: general

text: way.

text: We

text: assume

text: that

text: the

text: be

text: a

text: consequence

text: of

text: entanglement,

text: and

text: generated

text: lot

text: of

text: particles

text: travel

text: in

text: opposite

text: directions

text: along

text: the

text: x-axis.

text: research

text: activity

text: The

text: entanglement

text: is

text: in

text: the

text: z-direction.

text: The

text: entangled

text: We

text: look

text: at

text: a

text: modiﬁed

text: ghost

text: interference

text: experiment,

text: two

text: particle

text: state,

text: at

text: the

text: source

text: is

text: given

text: by

text: Ψ

text: We

text: as-

text: shown

text: in

text: where

text: the

text: double-slit

text: is

text: replaced

text: by

text: a

text: sume

text: that

text: after

text: travelling

text: for

text: a

text: time

text: t

text: particle

text: reaches

text: n-slit.

text: A

text: n-slit

text: ghost

text: interference

text: experiment

text: has

text: actu-

text: the

text: n-slit

text: and

text: particle

text: travels

text: a

text: distance

text: ally

text: been

text: performed

text: by

text: Zeilinger’s

text: group

text: However,

text: towards

text: detector

text: The

text: two

text: particle

text: state

text: is

text: now

text: given

text: here

text: we

text: will

text: consider

text: another

text: modiﬁcation

text: of

text: it.

text: En-

text: by

text: Ψ

text: tangled

text: particle

text: pairs,

text: which

text: could

text: also

text: more

text: generally

text: We

text: take

text: into

text: account

text: the

text: eﬀect

text: of

text: the

text: n-slit

text: on

text: the

text: be

text: something

text: other

text: than

text: photons,

text: emerge

text: from

text: a

text: source

text: entangled

text: state

text: as

text: follows.

text: We

text: assume

text: that

text: the

text: n-slit

text: S.

text: Particle

text: passes

text: through

text: a

text: n-slit

text: and

text: also

text: interacts

text: allows

text: the

text: portions

text: of

text: the

text: wave-function

text: in

text: front

text: of

text: the

text: with

text: a

text: path-detector.

text: We

text: do

text: not

text: assume

text: any

text: form

text: of

text: slits

text: to

text: pass

text: through,

text: and

text: blocks

text: the

text: other

text: portions.

text: We

text: the

text: path-detector,

text: is

text: an

text: n-state

text: assume

text: that

text: what

text: emerge

text: from

text: the

text: n-slit

text: are

text: localised

text: system

text: with

text: states

text: dn

text: which

text: get

text: en-

text: states,

text: whose

text: width

text: is

text: approximately

text: the

text: width

text: of

text: a

text: tangled

text: with

text: the

text: n

text: paths

text: of

text: particle

text: This

text: entangle-

text: slit.

text: The

text: states

text: of

text: particle

text: which

text: pass

text: through

text: the

text: ment

text: is

text: bound

text: to

text: happen

text: if

text: the

text: path-detector

text: acquires

text: slits

text: n,

text: are

text: denoted

text: by

text: φ

text: φ

text: φ

text: re-

text: n

text: the

text: relevant

text: information

text: about

text: which

text: slit

text: particle

text: went

text: spectively.

text: The

text: state

text: representing

text: the

text: situation

text: which

text: through.

text: Particle

text: then

text: travels

text: further

text: and

text: is

text: detected

text: particle

text: gets

text: blocked

text: is,

text: say,

text: χ

text: These

text: states

text: are

text: by

text: a

text: ﬁxed

text: detector

text: Particle

text: travels

text: undisturbed

text: to

text: obviously

text: orthogonal,

text: because

text: they

text: represent

text: mutually

text: detector

text: Since

text: the

text: two

text: particles

text: have

text: to

text: be

text: counted

text: exclusive

text: possibilities,

text: and

text: form

text: a

text: complete

text: set

text: because

text: in

text: coincidence,

text: the

text: paths

text: travelled

text: by

text: both

text: the

text: particles,

text: they

text: exhaust

text: all

text: the

text: possibilities

text: for

text: particle

text: Thus

text: the

text: before

text: reaching

text: their

text: respective

text: detectors,

text: are

text: assumed

text: entangled

text: two-particle

text: state

text: can

text: be

text: expanded

text: in

text: terms

text: of

text: to

text: be

text: equal.

text: Without

text: the

text: path-detector,

text: our

text: experiment

text: these.

text: We

text: can

text: write:

text: would

text: just

text: be

text: a

text: n-slit

text: generalization

text: of

text: the

text: original

text: two-

text: slit

text: ghost

text: interference

text: experiment

text: It

text: is

text: expected

text: to

text: show

text: a

text: n-slit

text: ghost

text: interference

text: for

text: particle

text: By

text: in-

text: n

text: troducing

text: a

text: path-detector,

text: we

text: are

text: probing

text: if

text: acquiring

text: Ψ

text: φk

text: φk

text: χ

text: χ

text: Ψ

text: path

text: information

text: about

text: particle

text: has

text: any

text: eﬀect

text: on

text: the

text: interference

text: shown

text: by

text: particle

text: n

text: We

text: assume

text: d

text: d

text: to

text: be

text: normalized,

text: but

text: not

text: φk

text: φk

text: Ψ

text: χ

text: χ

text: Ψ

text: n

text: necessarily

text: orthogonal.

text: The

text: ultimate

text: limit

text: to

text: the

text: knowl-

text: edge

text: we

text: can

text: acquire

text: as

text: to

text: which

text: slit

text: particle

text: went

text: where

text: is

text: the

text: unit

text: operator

text: for

text: the

text: space

text: of

text: particle

text: through|

text: set

text: b|y

text: distinct

text: the

text: states

text: dn

text: Now,

text: ce

text: we

text: are

text: only

text: interested

text: in

text: those

text: instances

text: are.

text: If

text: d

text: d

text: are

text: orthogonal,

text: we

text: can

text: in

text: principle

text: n

text: (through

text: coincidence

text: counting)

text: where

text: particle

text: does

text: know

text: with

text: certainty

text: which

text: slit

text: the

text: particle

text: went

text: through.

text: pass

text: through

text: the

text: multi-slit,

text: and

text: does

text: not

text: get

text: blocked,

text: Of

text: course

text: in

text: general,

text: d

text: d

text: may

text: not

text: be

text: all

text: orthog-

text: n

text: the

text: χ

text: dependent

text: term

text: can

text: be

text: discarded,

text: and

text: one

text: is

text: left

text: onal

text: to

text: each

text: other.

text: In

text: such

text: a

text: situation,

text: one

text: is

text: left

text: with

text: the

text: with

text: the

text: two-particle

text: problem

text: of

text: unambigiously

text: telliing

text: which

text: is

text: of

text: the

text: states

text: d

text: d

text: is

text: the

text: given

text: unknown

text: path-detector

text: state

text: n

text: which

text: the

text: path-detector

text: process

text: throws

text: up.

text: The

text: best

text: bet

text: n

text: to

text: answer

text: this

text: question

text: is

text: using

text: unambigious

text: quantum

text: Ψ

text: φk

text: φk

text: Ψ

text: state

text: discrimination

text: (UQSD)

text: D

text: Using

text: UQSD,

text: a

text: path-distinguishability

text: can

text: then

text: be

text: Q

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

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