text: 5 (cid:112) Eq. (B20), whereas for the CV GHZ state it is equal to gives GIE for all symmetric GLEMS satisfying inequality ± ∓ q = r + ln( x−/x )/2 with x± = (e 2r + 2e 2r)/3, (B18),   + (cid:16) (cid:17) where r ≥ 0 is a squeezing parameter. Minimizing there- (cid:113)  fore the quantity g, Eq. (B21), exactly as in Ref. [35] and restricting ourselves only to entangled states satisfying a E↓G ρ(1) = ln . (B26) the necessary and suﬃcient condition for entanglement, AB a2 − k2 1 > (a − k )(a − k ), one ﬁnds, that the quantity pos- p x p sesses three candidates for the minimum which lie on the edges of the volume O and correspond to: Note, that the RHS of the latter formula is nothing 1. Homodyne detection of quadrature p on mode but the mutual information of the joint distribution of → ∞ E − E, i.e., ΓE = Γtp + , where Γtp ≡ diag(e2t, e 2t), or outcomes of measurements of quadratures pA and pB on discarding of mod(cid:32)e E(cid:114), which (cid:33)yields (cid:32) (cid:33) a symmetric Gaussian state with CM (B2). (cid:16) (cid:17) In particular, for the CV G(cid:114)HZ state we have (cid:112) U1 ρ(A1B) = ln a aa ++ kkxp = ln a2a− kx2 . a = √x+x−, kx,p = xx∓± (x− − x+) (B27) (B22) which gives after substitution into the RHS of Eq. (B26) the formula (cid:18) (cid:19) (cid:0) (cid:1) Here, the second equality is a consequence of the equal- ity x− E↓G ρGHZ = ln √ . (B28) AB er x + 1 a + kp = a − k , (B23) The latter expression agrees with the GIE for the CV x (cid:112) GHZ state derived in Ref. [35]. It remains to identify the set of symmetric entangled which follows from the deﬁning equality for symmetric GLEMS for which the GIE is given by formula (B26). GLEMS, ν2 = (a − kx)(a + kp) = 1. This set comprise all the states, for which inequality 2. Heterodyne detection on mode E, i.e., Γ = 11, (B18) is satisﬁed for all CMs Γ . In order to ﬁnd the E E which gives(cid:16) (cid:17) (cid:20) (cid:18) (cid:19)(cid:21) states, we can again use the results of Ref. [35]. There, minimum of the left-hand side (LHS) of the inequality for CV GHZ state was found over all CMs Γ , which E U ρ(1) = ln 1 z z + 1 , (B24) was strictly positive for all values of the squeezing pa- 2 AB 2 A B z z rameter r. Hence, the inequality is satisﬁed for all CV A B GHZ states and therefore GIE is for all the states given by formula (B28). where the parameters z and z are deﬁned below A B Inspired by the approach of Ref. [35], we can also min- Eq. (B7). imize the LHS of inequality (B18). Like in the case of 3. Homodyne detection of quadrature x on mode E, i.e., Γ = Γt→+∞, where Γt ≡ diag(e−2t, eE2t), for which quantity (B21), the LHS depends on variables Vp, Vx E x x and ϕ that we minimize over in the same way, as the the upper bound(cid:32)(B1(cid:115)4) is equ(cid:33)al to   LHS for the CV GHZ state, up to the already mentioned (cid:16) (cid:17) diﬀerence in the form of the parameter q. Therefore, (cid:113)  we can minimize the LHS of inequality (B18) exactly in a − k a the same way as the LHS of the inequality was mini- U ρ(1) = ln a x = ln . 3 AB a − kp a2 − k2 mized in Ref. [35] for a particul(cid:112)ar case of the CV GHZ p state. Hence it follows, that also for symmetric entan- (B25) gled GLEMS both the quantities a˜ and s˜ attain maximal values of a˜ = a and s˜ = a2 − k2 for projection max max x of mode E onto an inﬁnitely hot thermal state which is equivalent with dropping of mode E. Consequently, Comparison of the upper bounds (B22), (B24) and the LHS of inequality (B18) possesses the following lower (B25) shows that U ≥ U and U ≥ U and therefore, 1 3 2 3 the upper bound (B14) is equal to U , i.e., U(ρ(1) ) = bound: (cid:112) 3 AB U (ρ(1) ). Because the bound also represents at homo- 3 AB 1 1 dyne detections of quadratures xA and xB on modes A 2 + − s˜ ≥ 2 + − a2 − k2. (B29) and B, the least mutual information over all Gaussian a˜ a x measurements on mode E, the upper bound (B25) also ##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: Eq.

text: whereas

text: for

text: the

text: CV

text: GHZ

text: state

text: it

text: is

text: equal

text: to

text: gives

text: GIE

text: for

text: all

text: symmetric

text: GLEMS

text: satisfying

text: inequality

text: q

text: r

text: ln(

text: x−/x

text: with

text: x±

text: (e

text: where

text: r

text: is

text: a

text: squeezing

text: parameter.

text: Minimizing

text: there-

text: fore

text: the

text: quantity

text: g,

text: Eq.

text: exactly

text: as

text: in

text: Ref.

text: and

text: restricting

text: ourselves

text: only

text: to

text: entangled

text: states

text: satisfying

text: a

text: E↓G

text: ln

text: the

text: necessary

text: and

text: suﬃcient

text: condition

text: for

text: entanglement,

text: AB

text: (a

text: k

text: )(a

text: k

text: one

text: ﬁnds,

text: that

text: the

text: quantity

text: pos-

text: p

text: x

text: p

text: sesses

text: three

text: candidates

text: for

text: the

text: minimum

text: which

text: lie

text: on

text: the

text: edges

text: of

text: the

text: volume

text: O

text: and

text: correspond

text: to:

text: Note,

text: that

text: the

text: RHS

text: of

text: the

text: latter

text: formula

text: is

text: nothing

text: Homodyne

text: detection

text: of

text: quadrature

text: p

text: on

text: mode

text: but

text: the

text: mutual

text: information

text: of

text: the

text: joint

text: distribution

text: of

text: E

text: E,

text: i.e.,

text: ΓE

text: Γtp

text: where

text: Γtp

text: e

text: or

text: outcomes

text: of

text: measurements

text: of

text: quadratures

text: pA

text: and

text: pB

text: on

text: discarding

text: of

text: which

text: a

text: symmetric

text: Gaussian

text: state

text: with

text: CM

text: In

text: particular,

text: for

text: the

text: CV

text: state

text: we

text: have

text: ln

text: a

text: aa

text: kkxp

text: ln

text: a

text: √x+x−,

text: kx,p

text: xx∓±

text: (x−

text: x+)

text: which

text: gives

text: after

text: substitution

text: into

text: the

text: RHS

text: of

text: Eq.

text: the

text: formula

text: Here,

text: the

text: second

text: equality

text: is

text: a

text: consequence

text: of

text: the

text: equal-

text: ity

text: x−

text: E↓G

text: ρGHZ

text: ln

text: AB

text: er

text: x

text: a

text: kp

text: a

text: k

text: The

text: latter

text: expression

text: agrees

text: with

text: the

text: GIE

text: for

text: the

text: CV

text: x

text: GHZ

text: state

text: derived

text: in

text: Ref.

text: It

text: remains

text: to

text: identify

text: the

text: set

text: of

text: symmetric

text: entangled

text: which

text: follows

text: from

text: the

text: deﬁning

text: equality

text: for

text: symmetric

text: GLEMS

text: for

text: which

text: the

text: GIE

text: is

text: given

text: by

text: formula

text: GLEMS,

text: ν2

text: (a

text: kx)(a

text: kp)

text: This

text: set

text: comprise

text: all

text: the

text: states,

text: for

text: which

text: inequality

text: Heterodyne

text: detection

text: on

text: mode

text: E,

text: i.e.,

text: Γ

text: is

text: satisﬁed

text: for

text: all

text: CMs

text: Γ

text: In

text: order

text: to

text: ﬁnd

text: the

text: E

text: E

text: which

text: states,

text: we

text: can

text: again

text: use

text: the

text: results

text: of

text: Ref.

text: There,

text: minimum

text: of

text: the

text: left-hand

text: side

text: (LHS)

text: of

text: the

text: inequality

text: for

text: CV

text: GHZ

text: state

text: was

text: found

text: over

text: all

text: CMs

text: Γ

text: which

text: E

text: U

text: ln

text: z

text: z

text: was

text: strictly

text: positive

text: for

text: all

text: values

text: of

text: the

text: squeezing

text: pa-

text: AB

text: A

text: B

text: z

text: z

text: rameter

text: r.

text: Hence,

text: the

text: inequality

text: is

text: satisﬁed

text: for

text: all

text: CV

text: A

text: B

text: GHZ

text: states

text: and

text: therefore

text: GIE

text: is

text: for

text: all

text: the

text: states

text: given

text: by

text: formula

text: where

text: the

text: parameters

text: z

text: and

text: z

text: are

text: deﬁned

text: below

text: A

text: B

text: Inspired

text: by

text: the

text: approach

text: of

text: Ref.

text: we

text: can

text: also

text: min-

text: Eq.

text: imize

text: the

text: LHS

text: of

text: inequality

text: Like

text: in

text: the

text: case

text: of

text: Homodyne

text: detection

text: of

text: quadrature

text: x

text: on

text: mode

text: E,

text: i.e.,

text: Γ

text: Γt→+∞,

text: where

text: Γt

text: for

text: which

text: quantity

text: the

text: LHS

text: depends

text: on

text: variables

text: Vp,

text: Vx

text: E

text: x

text: x

text: and

text: ϕ

text: that

text: we

text: minimize

text: over

text: in

text: the

text: same

text: way,

text: as

text: the

text: the

text: upper

text: is

text: to

text: LHS

text: for

text: the

text: CV

text: GHZ

text: state,

text: up

text: to

text: the

text: already

text: mentioned

text: diﬀerence

text: in

text: the

text: form

text: of

text: the

text: parameter

text: q.

text: Therefore,

text: we

text: can

text: minimize

text: the

text: LHS

text: of

text: inequality

text: exactly

text: in

text: a

text: k

text: a

text: the

text: same

text: way

text: as

text: the

text: LHS

text: of

text: the

text: inequality

text: was

text: mini-

text: U

text: ln

text: a

text: x

text: ln

text: AB

text: a

text: kp

text: mized

text: in

text: Ref.

text: for

text: a

text: case

text: of

text: the

text: CV

text: GHZ

text: p

text: state.

text: Hence

text: it

text: follows,

text: that

text: also

text: for

text: symmetric

text: entan-

text: gled

text: GLEMS

text: both

text: the

text: quantities

text: a˜

text: and

text: s˜

text: attain

text: maximal

text: values

text: of

text: a˜

text: a

text: and

text: s˜

text: for

text: projection

text: max

text: max

text: x

text: of

text: mode

text: E

text: onto

text: an

text: inﬁnitely

text: hot

text: thermal

text: state

text: which

text: is

text: equivalent

text: with

text: dropping

text: of

text: mode

text: E.

text: Consequently,

text: Comparison

text: of

text: the

text: upper

text: bounds

text: and

text: the

text: LHS

text: of

text: inequality

text: possesses

text: the

text: following

text: lower

text: shows

text: that

text: U

text: U

text: and

text: U

text: U

text: and

text: therefore,

text: the

text: upper

text: bound

text: is

text: equal

text: to

text: U

text: i.e.,

text: bound:

text: AB

text: U

text: Because

text: the

text: bound

text: also

text: represents

text: at

text: homo-

text: AB

text: dyne

text: detections

text: of

text: quadratures

text: xA

text: and

text: xB

text: on

text: modes

text: A

text: s˜

text: and

text: B,

text: the

text: least

text: mutual

text: information

text: over

text: all

text: Gaussian

text: a˜

text: a

text: x

text: measurements

text: on

text: mode

text: E,

text: the

text: upper

text: bound

text: also

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##

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text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: 13 If we now use in the latter formula equality γ = ν11 where L± = (λ ± λ )/2 and where we have writ- (cid:48) E 4 1 2 together with Eq. (B90), we express CM Γ in the block ten down explicitly the dependence of the matrix Q (cid:18) (cid:19) E form with respect to x|p splitting, on the variables ϕ, λ and λ . From equality Q(ϕ + 1 2 π, λ , λ ) = Q(ϕ, λ , λ ) it further follows that we can as- 1 2 1 2 sume without loss of generality ϕ ∈ [0, π) whereas equal- (cid:48) α˜ γ˜ ΓE = γ˜T β˜ , (B98) ity Q(ϕ + π2 , λ1, λ2) = Q(ϕ, λ2, λ1) allows us to assume λ ≥ λ in what follows. 1 2 Before performing minimization of the RHS of we take into account the formula (B92) and we use the Eq. (B99) over all matrices (B101), we ﬁrst express the blockwise inversion (B61), we ﬁnd after some algebra the quantity (B(cid:34)97) to be (cid:18) (cid:19) (cid:35) inverse matr(cid:18)ices on th(cid:19)e RHS as − Kh = 1(cid:2)+ 2 sinh(v)(cid:104)eA|σx ν111 + Q −1 σ(cid:3)x|eA(cid:105) ν111 + Q 1 = ν111 +dJ˜QJT , − − ν11 + JQJT × 1 − 2 sinh(v)(cid:104)e |(ν11 + Q) 1|e (cid:105) , (B99) (ν11 + Q) 1 = (B102) A A d where where matrix J is(cid:18)deﬁned (cid:19)in(cid:18)Eq. (A1(cid:19)0) and − Q ≡ α˜ − γ˜(β˜ + ν11) 1γ˜T . (B100) 1 1 d˜ ≡ + λ + λ , ν 1 ν 2 Our task is now to minimize the quantity (B99) over d ≡ (ν + λ )(ν + λ ). (B103) 1 2 all CMs (B98). Note ﬁrst, that the RHS of formula (B99) (cid:48) does not depend on the entire CM Γ but only on the E 2 × 2 matrix (B100) which consists of 2 × 2 submatrices Substituting further into the obtained formula for ma- (cid:48) α˜, β˜ and γ˜ of the CM. Since Γ is a CM, the submatrix trix Q from Eq. (B101) and using Eqs. (B93) and (B103) E α˜ has the meaning of a classical correlation matrix of as well as identities a bivariate Gaussian distribution of outcomes of homo- dyne detections of a pair of x-quadratures and therefore z4 − 1 k 2z2 ν it can be an arbitrary positive semideﬁnite matrix. Simi- = , = , (B104) larly, the oﬀ-diagonal submatrix γ˜ describes correlations z4 + 1 a z4 + 1 a between the x- and p-quadratures. Consider now an ad- missible pure-state CM given by the following direct sum one gets (cid:48) − Γ = α˜ ⊕(α˜) 1, where α˜ is an arbitrary positive deﬁnite E − − matrix, i.e., α˜ = P(ϕ)diag(e 2tA, e 2tB)PT (ϕ). Here, the matrix P(ϕ) is deﬁned in Eq. (B9), ϕ ∈ [0, 2π), and K = {E + F [k cos(2ϕ) − ν sin(2ϕ)]} tA,B ∈ (−∞, +∞). Hence, from Eq. (B100) it then fol- h h(λ1) lows that Q = α˜. Up to now we have considered only {E + F [k cos(2ϕ) + ν sin(2ϕ)]} (cid:48) CMs Γ built of the invertible matrices α˜. The remain- × . E h(λ2) ing case of singular matrices α˜, which is also admissible, (B105) is recovered if we take in the latter equality the limit t → +∞ and/or t → +∞. Consequently, the ma- A B trix Q can be an arbitrary positive semideﬁnite matrix Here, and therefore instead of minimizing the RHS of Eq. (B99) (cid:48) over all two-mode CMs Γ we can just carry out the min- E imization over all 2 × 2 real symmetric positive semidef- E ≡ 1 + λ λ + cosh(v)(λ + λ ), (B106) 1 2 1 2 inite matrices Q. sinh(v) As we have already mentioned, such a matrix can be F ≡ (λ − λ ), (B107) expressed as Q = P(ϕ)diag(λ , λ )PT (ϕ), where ϕ ∈ a 1 2 1 2 [0, 2π) and λ ≥ 0. By multiplying the matrices in the 1,2 (cid:18) (cid:19) latter deﬁnition one ﬁnds after some algebra that where we have set cosh(v) = (ν + 1/ν)/2, and L + L− cos(2ϕ) L− sin(2ϕ) Q(ϕ, λ1, λ2) = +L− sin(2ϕ) L − L− cos(2ϕ) , h(λj) ≡ 1 + 2 cosh(v)λj + λ2j, j = 1, 2. (B108) + (B101) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: If

text: we

text: now

text: use

text: in

text: the

text: latter

text: formula

text: equality

text: γ

text: where

text: L±

text: (λ

text: λ

text: and

text: where

text: we

text: have

text: writ-

text: E

text: together

text: with

text: Eq.

text: we

text: express

text: CM

text: Γ

text: in

text: the

text: block

text: ten

text: down

text: explicitly

text: the

text: dependence

text: of

text: the

text: matrix

text: Q

label: equation

text: E

text: form

text: with

text: respect

text: to

text: x|p

text: splitting,

text: on

text: the

text: variables

text: ϕ,

text: λ

text: and

text: λ

text: From

text: equality

text: Q(ϕ

text: π,

text: λ

text: λ

text: Q(ϕ,

text: λ

text: λ

text: it

text: further

text: follows

text: that

text: we

text: can

text: as-

text: sume

text: without

text: loss

text: of

text: generality

text: ϕ

text: π)

text: whereas

text: equal-

text: α˜

text: γ˜

text: ΓE

text: γ˜T

text: β˜

text: ity

text: Q(ϕ

text: π2

text: Q(ϕ,

text: allows

text: us

text: to

text: assume

text: λ

text: λ

text: in

text: what

text: follows.

text: Before

text: performing

text: minimization

text: of

text: the

text: RHS

text: of

text: we

text: take

text: into

text: account

text: the

text: formula

text: and

text: we

text: use

text: the

text: Eq.

text: over

text: all

text: matrices

text: we

text: ﬁrst

text: express

text: the

text: blockwise

text: inversion

text: we

text: ﬁnd

text: after

text: some

text: algebra

text: the

text: quantity

text: to

text: be

text: inverse

text: on

text: RHS

text: as

text: Kh

text: Q

text: −1

text: Q

text: +dJ˜QJT

text: JQJT

text: Q)

text: Q)

text: A

text: A

text: d

text: where

text: where

text: matrix

text: J

text: and

text: Q

text: α˜

text: γ˜(β˜

text: 1γ˜T

text: d˜

text: λ

text: λ

text: ν

text: ν

text: Our

text: task

text: is

text: now

text: to

text: minimize

text: the

text: quantity

text: over

text: d

text: (ν

text: λ

text: )(ν

text: λ

text: all

text: CMs

text: Note

text: ﬁrst,

text: that

text: the

text: RHS

text: of

text: formula

text: does

text: not

text: depend

text: on

text: the

text: entire

text: CM

text: Γ

text: but

text: only

text: on

text: the

text: E

text: matrix

text: which

text: consists

text: of

text: submatrices

text: Substituting

text: further

text: into

text: the

text: obtained

text: formula

text: for

text: ma-

text: α˜,

text: β˜

text: and

text: γ˜

text: of

text: the

text: CM.

text: Since

text: Γ

text: is

text: a

text: CM,

text: the

text: submatrix

text: trix

text: Q

text: from

text: Eq.

text: and

text: using

text: Eqs.

text: and

text: E

text: α˜

text: has

text: the

text: meaning

text: of

text: a

text: classical

text: correlation

text: matrix

text: of

text: as

text: well

text: as

text: identities

text: a

text: bivariate

text: Gaussian

text: distribution

text: of

text: outcomes

text: of

text: homo-

text: dyne

text: detections

text: of

text: a

text: pair

text: of

text: x-quadratures

text: and

text: therefore

text: k

text: ν

text: it

text: can

text: be

text: an

text: arbitrary

text: positive

text: semideﬁnite

text: matrix.

text: Simi-

text: larly,

text: the

text: oﬀ-diagonal

text: submatrix

text: γ˜

text: describes

text: correlations

text: a

text: a

text: between

text: the

text: x-

text: and

text: p-quadratures.

text: Consider

text: now

text: an

text: ad-

text: missible

text: pure-state

text: CM

text: given

text: by

text: the

text: following

text: direct

text: sum

text: one

text: gets

text: Γ

text: α˜

text: ⊕(α˜)

text: where

text: α˜

text: is

text: an

text: arbitrary

text: positive

text: deﬁnite

text: E

text: matrix,

text: i.e.,

text: α˜

text: P(ϕ)diag(e

text: e

text: (ϕ).

text: Here,

text: the

text: matrix

text: P(ϕ)

text: is

text: deﬁned

text: in

text: Eq.

text: ϕ

text: and

text: K

text: {E

text: F

text: [k

text: ν

text: tA,B

text: Hence,

text: from

text: Eq.

text: it

text: then

text: fol-

text: h

text: lows

text: that

text: Q

text: α˜.

text: Up

text: to

text: now

text: we

text: have

text: considered

text: only

text: {E

text: F

text: [k

text: ν

text: CMs

text: Γ

text: built

text: of

text: the

text: invertible

text: matrices

text: α˜.

text: The

text: remain-

text: E

text: ing

text: case

text: of

text: singular

text: matrices

text: α˜,

text: which

text: is

text: also

text: admissible,

text: is

text: recovered

text: if

text: we

text: take

text: in

text: the

text: latter

text: equality

text: the

text: limit

text: t

text: and/or

text: t

text: Consequently,

text: the

text: ma-

text: A

text: B

text: trix

text: Q

text: can

text: be

text: an

text: arbitrary

text: positive

text: semideﬁnite

text: matrix

text: Here,

text: and

text: therefore

text: instead

text: of

text: minimizing

text: the

text: RHS

text: of

text: Eq.

text: over

text: all

text: two-mode

text: CMs

text: Γ

text: we

text: can

text: just

text: carry

text: out

text: the

text: min-

text: E

text: imization

text: over

text: all

text: real

text: symmetric

text: positive

text: semidef-

text: E

text: λ

text: λ

text: cosh(v)(λ

text: λ

text: inite

text: matrices

text: Q.

text: sinh(v)

text: As

text: we

text: have

text: already

text: mentioned,

text: such

text: a

text: matrix

text: can

text: be

text: F

text: (λ

text: λ

text: expressed

text: as

text: Q

text: P(ϕ)diag(λ

text: λ

text: )PT

text: (ϕ),

text: where

text: ϕ

text: a

text: and

text: λ

text: By

text: multiplying

text: the

text: matrices

text: in

text: the

text: latter

text: deﬁnition

text: one

text: ﬁnds

text: after

text: some

text: algebra

text: that

text: where

text: we

text: have

text: set

text: cosh(v)

text: (ν

text: and

text: L

text: L−

text: L−

text: Q(ϕ,

text: +L−

text: L

text: L−

text: h(λj)

text: cosh(v)λj

text: j

text: ##LTLine##

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text: ##LTLine##


text: 17 (cid:115) (cid:115) where we have used equality x2−y2 = 1. Provided that λ = 4 x2V˜x + y2 , λ = 4 y2V˜x + x2 , (C18) inequality (C21) is satisﬁed for any CM ΓE, the upper A x2V˜ + y2 B y2V˜ + x2 bound (B14) can be calculated by minimization of GCMI p p (C22) over all CMs Γ . This amounts to minimization of E does the job and brings CM (C16) to standard form quantity (C24) over all admissible values of the param- (A16), where(cid:113)parameters a, b, kx and kp are replaced with eter V˜x deﬁned in Eq. (C13), which is equivalent with maximization of the function (cid:113) a˜ = (x2V˜ + y2)(x2V˜ + y2), x p V˜ F(V˜ ) ≡ x . (C25) ˜b = (y2V˜x + x2)(y2V˜p + x2), x (V˜x + 1)2 xy(V˜ + 1) k˜x = λ xλ , k˜p = xy(V˜p + 1)λAλB. Equation (C13) reveals, that the parameter V˜x lies A B in the interval V˜ ∈ [1, ν˜] on which function (C25) (C19) x is monotonically decreasing. Thus F is maximized by V˜ = 1, which corresponds to heterodyne detection on x Consequently, parameters (C19) are also parameters of Eve’s mode E, i.e., a projection of the mode onto a co- the standard form of CM γ¯(3) | as needed. herent state. Then we get F(V˜x = 1) = 1/4 which further AB E gives To evaluate GIE for states (≡ ρ¯(3) ) with CM (C8), we AB now use the method of Sec. . In this method, instead of carrying out directly optimization in the deﬁnition of h ≡ h(V˜ = 1) = 1 − 1 = 1 − (a − b + 2)2 , GIE, Eq. (A1), we ﬁrst calculate an easier computable min x 1 + 4x2y2 (a + b)2 upper bound, E(cid:16)q. (B(cid:17)14), (cid:104) (cid:16) (cid:17)(cid:105) (C26) where in the derivation of the second equality we have U ρ¯(A3B) = inf IcG ρ¯(A3B) |E , (C20) used equality 1+4x2y2 = (a+b)2/(a−b+2)2. Insertion ΓE of the minimal value of quantity (C23) given in Eq. (C26) where IG(ρ¯(3) | ) is GCMI (B16) of conditional state into the RHS of E(cid:16)q. (C(cid:17)22) yi(cid:18)elds ﬁnally (cid:19)the upper bound c AB E ρ¯(3) | with CM (C15). For CM γ¯(3) | for which inequal- AB E AB E a + b irtaym(eBte4r6)k˜h,olid.es(cid:115).,, where parameter c˜x is replaced with pa- U ρ¯(A3B) = ln a − b + 2 . (C27) (cid:114)x (cid:113) (cid:112) Viewed from a diﬀerent perspective, the RHS of the a˜ ˜b 1 latter equality represents for homodyne detection of ˜b + a˜ + a˜˜b − a˜˜b − k˜x2 ≥ 0, (C21) quadratures xA and xB on modes A and B the least mu- tual information (A2) over all Gaussian measurements on mode E. Since the mutual information saturates the the GCMI is giv(cid:16)en by (cid:17) upper bound, no Gaussian measurements on modes A and B can give a larger mutual information minimized IG ρ¯(3) | = 1 ln 1 , (C22) over all CMs ΓE, and GIE thus coincides with the upper c AB E 2 1 − h bound (C27). Therefore, for asymmetric GLEMS with CM (C8) for which inequality (C21) is satisﬁed for any where CM ΓE, GIE read(cid:16)s as (cid:17) (cid:18) (cid:19) k˜2 a + b h = x , (C23) E↓G ρ¯(3) = ln . (C28) a˜˜b AB a − b + 2 and it is reached by homodyne detection of quadra- It remains to identify the set of states with CM (C8), tures xA and xB on modes A and B. If we now substitute where a > b, for which inequality (C21) is satisﬁed for to the RHS from Eqs. (C19) one ﬁnds, that any CM Γ and therefore GIE is given by formula (C28). E Unfortunately, inequality (C21) is very complex which x2y2(V˜ + 1)2 1 makes identiﬁcation of the whole set a diﬃcult task. h = (x2V˜x + y2)x(y2V˜x + x2) = 1 + x2y2(V˜V˜xx+1)2 , (C24) Huseerea,minusctheamdoorfedsiismcupslseiningeiqnueaqluitayli(tBy5(1C)2t1h)e, wfuelﬁtlhmereenftoroef ##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: where

text: we

text: have

text: used

text: equality

text: Provided

text: that

text: λ

text: λ

text: inequality

text: is

text: satisﬁed

text: for

text: any

text: CM

text: ΓE,

text: the

text: upper

text: A

text: B

text: bound

text: can

text: be

text: calculated

text: by

text: minimization

text: of

text: GCMI

text: p

text: p

text: over

text: all

text: CMs

text: Γ

text: This

text: amounts

text: to

text: minimization

text: of

text: E

text: does

text: the

text: job

text: and

text: brings

text: CM

text: to

text: standard

text: form

text: quantity

text: over

text: all

text: admissible

text: values

text: of

text: the

text: param-

text: a,

text: b,

text: kx

text: and

text: kp

text: are

text: replaced

text: with

text: eter

text: V˜x

text: deﬁned

text: in

text: Eq.

text: which

text: is

text: equivalent

text: with

text: maximization

text: of

text: the

text: function

text: a˜

text: x

text: p

text: V˜

text: F(V˜

text: x

text: ˜b

text: x

text: (V˜x

text: xy(V˜

text: k˜x

text: λ

text: xλ

text: k˜p

text: xy(V˜p

text: Equation

text: reveals,

text: that

text: the

text: parameter

text: V˜x

text: lies

text: A

text: B

text: in

text: the

text: interval

text: V˜

text: ν˜]

text: on

text: which

text: function

text: x

text: is

text: monotonically

text: decreasing.

text: Thus

text: F

text: is

text: maximized

text: by

text: V˜

text: which

text: corresponds

text: to

text: heterodyne

text: detection

text: on

text: x

text: Consequently,

text: parameters

text: are

text: also

text: parameters

text: of

text: Eve’s

text: mode

text: E,

text: i.e.,

text: a

text: projection

text: of

text: the

text: mode

text: onto

text: a

text: co-

text: the

text: standard

text: form

text: of

text: CM

text: as

text: needed.

text: herent

text: state.

text: Then

text: we

text: get

text: F(V˜x

text: which

text: further

text: AB

text: E

text: gives

text: To

text: evaluate

text: GIE

text: for

text: states

text: with

text: CM

text: we

text: AB

text: now

text: use

text: the

text: method

text: of

text: Sec.

text: In

text: this

text: method,

text: instead

text: of

text: carrying

text: out

text: directly

text: optimization

text: in

text: the

text: deﬁnition

text: of

text: h

text: h(V˜

text: (a

text: b

text: GIE,

text: Eq.

text: we

text: ﬁrst

text: calculate

text: an

text: easier

text: computable

text: min

text: x

text: (a

text: upper

text: bound,

text: where

text: in

text: the

text: derivation

text: of

text: the

text: second

text: equality

text: we

text: have

text: U

text: inf

text: IcG

text: |E

text: used

text: equality

text: Insertion

text: ΓE

text: of

text: the

text: minimal

text: value

text: of

text: quantity

text: given

text: in

text: Eq.

text: where

text: is

text: GCMI

text: of

text: conditional

text: state

text: into

text: the

text: RHS

text: of

text: ﬁnally

text: upper

text: bound

text: c

text: AB

text: E

text: with

text: CM

text: For

text: CM

text: for

text: which

text: inequal-

text: AB

text: E

text: AB

text: E

text: a

text: b

text: where

text: parameter

text: c˜x

text: is

text: replaced

text: with

text: pa-

text: U

text: ln

text: a

text: b

text: Viewed

text: from

text: a

text: diﬀerent

text: perspective,

text: the

text: RHS

text: of

text: the

text: a˜

text: ˜b

text: latter

text: equality

text: represents

text: for

text: homodyne

text: detection

text: of

text: ˜b

text: a˜

text: a˜˜b

text: a˜˜b

text: quadratures

text: xA

text: and

text: xB

text: on

text: modes

text: A

text: and

text: B

text: the

text: least

text: mu-

text: tual

text: information

text: over

text: all

text: Gaussian

text: measurements

text: on

text: mode

text: E.

text: Since

text: the

text: mutual

text: information

text: saturates

text: the

text: the

text: GCMI

text: is

text: by

text: upper

text: bound,

text: no

text: Gaussian

text: measurements

text: on

text: modes

text: A

text: and

text: B

text: can

text: give

text: a

text: larger

text: mutual

text: information

text: minimized

text: IG

text: ln

text: over

text: all

text: CMs

text: ΓE,

text: and

text: GIE

text: thus

text: coincides

text: with

text: the

text: upper

text: c

text: AB

text: E

text: h

text: bound

text: Therefore,

text: for

text: asymmetric

text: GLEMS

text: with

text: CM

text: for

text: which

text: inequality

text: is

text: satisﬁed

text: for

text: any

text: where

text: CM

text: ΓE,

text: GIE

text: as

text: a

text: b

text: h

text: x

text: E↓G

text: ln

text: a˜˜b

text: AB

text: a

text: b

text: and

text: it

text: is

text: reached

text: by

text: homodyne

text: detection

text: of

text: quadra-

text: It

text: remains

text: to

text: identify

text: the

text: set

text: of

text: states

text: with

text: CM

text: tures

text: xA

text: and

text: xB

text: on

text: modes

text: A

text: and

text: B.

text: If

text: we

text: now

text: substitute

text: where

text: a

text: b,

text: for

text: which

text: inequality

text: is

text: satisﬁed

text: for

text: to

text: the

text: RHS

text: from

text: Eqs.

text: one

text: ﬁnds,

text: that

text: any

text: CM

text: Γ

text: and

text: therefore

text: GIE

text: is

text: given

text: by

text: formula

text: E

text: Unfortunately,

text: inequality

text: is

text: very

text: complex

text: which

text: makes

text: identiﬁcation

text: of

text: the

text: whole

text: set

text: a

text: diﬃcult

text: task.

text: h

text: wfuelﬁtlhmereenftoroef

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: 2 where γ(0) denotes the Williamson normal form (A11) in terms of parameters a, b, k and k . This can be done AB x p of CM γ , i.e., γ(0) = diag(ν , ν , ν , ν ). using either a method of Ref. [47] or a method of Ref. [48]. AB AB 1 1 2 2 For a generic two-mode CM (A16) the form of the matrix In order to calculate the GIE we now need to express S is complex and therefore we do not write it here explic- CM (A13) as well as the symplectic matrix S appearing itly. In what follows, we work with particular subclasses in Eq. (A12) in terms of the elements of the CM γ . AB of the class of generic two-mode Gaussian states for which For this purpose it is convenient to express the CM in a block form with respect (cid:18)to the A|B(cid:19) splitting, S attains a simple form which is presented explicitly in the respective subsection. Appendix B: GIE for symmetric states A C γ = . (A14) AB CT B In this section we calculate GIE deﬁned in Eq. (8) of the main text for some subclasses of the class of two-mode Owing to the invariance of GIE (A1) under the Gaus- symmetric Gaussian states. The states are characterized sian local unitary operations [35] we can without loss of by the condition a = b whence their standard-form CMs generality assume CM (A14) to be in the standard form (A15) and (A16) reduce to  [44],         a 0 c 0   x a 0 cx 0 γ = 0 a 0 cp (B1) γ = 0 a 0 cp (A15) AB cx 0 a 0 AB cx 0 b 0 0 cp 0 a 0 c 0 b p and   with c ≥ |c | ≥ 0. Since states with c c ≥ 0 are sep- x p x p arable [44] and thus possess zero GIE [35], in calculations     we can restrict ourself only to CMs satisfying cxcp < 0. a 0 kx 0 Introducing new more convenient parameters kx ≡ cx γ = 0 a 0 −kp , (B2) and kp ≡ |cp| = −cp, we arrive at the following standard- AB kx 0 a 0 form CM which weshall consider in what follows [9]: 0 −k 0 a p     a 0 k 0 respectively. For the sake of further use let us also x 0 a 0 −k recap here that the matrix (B2) describes a CM of a γ = p , (A16) AB k 0 b 0 physical quantum state if and only if a2 − k2 ≥ 1 and x x 0 −k 0 b that the CM corresponds to an entangled state if and p only if 1 > (a − k )(a − k ) [9]. x p From Eq. (A17) it further follows that the symplectic where k ≥ k > 0. x p The symplectic eigenvalues of CM (A16) can be cal- eigenvalues of CM (B2(cid:113)) read explicitly as culated conveniently from the eigenvalues of the matrix iΩγ which are of the form {±ν , ±ν } [6]. In terms of (cid:113) AB 1 2 parameters a, b, k and k (cid:115)they read explicitly as ν = (a + k ) (a − k ), x p 1 x p ν = (a − k ) (a + k ). (B3) √ 2 x p ∆ ± D ν = , (A17) 1,2 2 (cid:0) (cid:1) As for the symplectic matrix S, we calculate it using the method of Ref. [47]. Here, we seek the matrix in the where form of a product S = ⊕2 (cid:18)V ∗ WT(cid:19), where i=1 ∆ = a(cid:0)2 + b2 −(cid:1) 2kxkp, 1 i −i V = √ (B4) D = ∆2 − 4detγAB 2 1 1 = a2 − b2 2 + 4 (ak − bk ) (bk − ak ) . x p x p (A18) and W contains in its columns the eigenvectors of the matrix iΩγ which are chosen such that S is real, it AB satisﬁes the symplectic condition (A9) and it does not Similarly, we can express the symplectic matrix S mix position and momentum quadratures. Thus we ﬁnd which brings CM (A16) to Williamson normal form (A11) the symplectic matrix S that brings the CM (B2) to the ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: denotes

text: the

text: Williamson

text: normal

text: form

text: in

text: terms

text: of

text: parameters

text: a,

text: b,

text: k

text: and

text: k

text: This

text: can

text: be

text: done

text: AB

text: x

text: p

text: of

text: CM

text: γ

text: i.e.,

text: diag(ν

text: ν

text: ν

text: ν

text: using

text: either

text: a

text: method

text: of

text: Ref.

text: or

text: a

text: method

text: of

text: Ref.

text: AB

text: AB

text: For

text: a

text: generic

text: two-mode

text: CM

text: the

text: form

text: of

text: the

text: matrix

text: In

text: order

text: to

text: calculate

text: the

text: GIE

text: we

text: now

text: need

text: to

text: express

text: S

text: is

text: complex

text: and

text: therefore

text: we

text: do

text: not

text: write

text: it

text: here

text: explic-

text: CM

text: as

text: well

text: as

text: the

text: symplectic

text: matrix

text: S

text: appearing

text: itly.

text: In

text: what

text: follows,

text: we

text: work

text: with

text: particular

text: subclasses

text: in

text: Eq.

text: in

text: terms

text: of

text: the

text: elements

text: of

text: the

text: CM

text: γ

text: AB

text: of

text: the

text: class

text: of

text: generic

text: two-mode

text: Gaussian

text: states

text: for

text: which

text: For

text: this

text: purpose

text: it

text: is

text: convenient

text: to

text: express

text: the

text: CM

text: in

text: a

text: block

text: form

text: with

text: respect

text: the

text: splitting,

text: S

text: attains

text: a

text: simple

text: form

text: which

text: is

text: presented

text: explicitly

text: in

text: the

text: respective

text: subsection.

text: Appendix

text: B:

text: GIE

text: for

text: symmetric

text: states

text: A

text: C

text: γ

text: AB

text: CT

text: B

text: In

text: this

text: section

text: we

text: calculate

text: GIE

text: deﬁned

text: in

text: Eq.

text: of

text: the

text: main

text: text

text: for

text: some

text: subclasses

text: of

text: the

text: class

text: of

text: two-mode

text: Owing

text: to

text: the

text: invariance

text: of

text: GIE

text: under

text: the

text: Gaus-

text: symmetric

text: Gaussian

text: states.

text: The

text: states

text: are

text: characterized

text: sian

text: local

text: unitary

text: operations

text: we

text: can

text: without

text: loss

text: of

text: by

text: the

text: condition

text: a

text: b

text: whence

text: their

text: standard-form

text: CMs

text: generality

text: assume

text: CM

text: to

text: be

text: in

text: the

text: standard

text: form

text: and

text: reduce

text: to

text: a

text: c

text: x

text: a

text: cx

text: γ

text: a

text: cp

text: γ

text: a

text: cp

text: AB

text: cx

text: a

text: AB

text: cx

text: b

text: cp

text: a

text: c

text: b

text: p

text: and

text: with

text: c

text: |c

text: Since

text: states

text: with

text: c

text: c

text: are

text: sep-

text: x

text: p

text: x

text: p

text: arable

text: and

text: thus

text: possess

text: zero

text: GIE

text: in

text: calculations

text: we

text: can

text: restrict

text: ourself

text: only

text: to

text: CMs

text: satisfying

text: cxcp

text: a

text: kx

text: Introducing

text: new

text: more

text: convenient

text: parameters

text: kx

text: cx

text: γ

text: a

text: −kp

text: and

text: kp

text: |cp|

text: −cp,

text: we

text: arrive

text: at

text: the

text: following

text: standard-

text: AB

text: kx

text: a

text: form

text: CM

text: which

text: weshall

text: consider

text: in

text: what

text: follows

text: −k

text: a

text: p

text: a

text: k

text: respectively.

text: For

text: the

text: sake

text: of

text: further

text: use

text: let

text: us

text: also

text: x

text: a

text: −k

text: recap

text: here

text: that

text: the

text: matrix

text: describes

text: a

text: CM

text: of

text: a

text: γ

text: p

text: AB

text: k

text: b

text: physical

text: quantum

text: state

text: if

text: and

text: only

text: if

text: and

text: x

text: x

text: −k

text: b

text: that

text: the

text: CM

text: corresponds

text: to

text: an

text: entangled

text: state

text: if

text: and

text: p

text: only

text: if

text: (a

text: k

text: )(a

text: k

text: x

text: p

text: From

text: Eq.

text: it

text: further

text: follows

text: that

text: the

text: symplectic

text: where

text: k

text: k

text: x

text: p

text: The

text: symplectic

text: eigenvalues

text: of

text: CM

text: can

text: be

text: cal-

text: eigenvalues

text: of

text: CM

text: read

text: explicitly

text: as

text: culated

text: conveniently

text: from

text: the

text: eigenvalues

text: of

text: the

text: matrix

text: iΩγ

text: which

text: are

text: of

text: the

text: form

text: {±ν

text: ±ν

text: In

text: terms

text: of

text: AB

text: parameters

text: a,

text: b,

text: k

text: and

text: k

text: read

text: explicitly

text: as

text: ν

text: (a

text: k

text: (a

text: k

text: x

text: p

text: x

text: p

text: ν

text: (a

text: k

text: (a

text: k

text: x

text: p

text: D

text: ν

text: As

text: for

text: the

text: symplectic

text: matrix

text: S,

text: we

text: calculate

text: it

text: using

text: the

text: method

text: of

text: Ref.

text: Here,

text: we

text: seek

text: the

text: matrix

text: in

text: the

text: where

text: form

text: of

text: a

text: product

text: S

text: ⊕2

text: where

text: i

text: −i

text: V

text: D

text: ∆2

text: (ak

text: bk

text: (bk

text: ak

text: x

text: p

text: x

text: p

text: and

text: W

text: contains

text: in

text: its

text: columns

text: the

text: eigenvectors

text: of

text: the

text: matrix

text: iΩγ

text: which

text: are

text: chosen

text: such

text: that

text: S

text: is

text: real,

text: it

text: AB

text: satisﬁes

text: the

text: symplectic

text: condition

text: and

text: it

text: does

text: not

text: Similarly,

text: we

text: can

text: express

text: the

text: symplectic

text: matrix

text: S

text: mix

text: position

text: and

text: momentum

text: quadratures.

text: Thus

text: we

text: ﬁnd

text: which

text: brings

text: CM

text: to

text: Williamson

text: normal

text: form

text: the

text: symplectic

text: matrix

text: S

text: that

text: brings

text: the

text: CM

text: to

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 AUTOMORPHIC BUNDLES ON ZIP-SCHEMES 4. Example 1 : Groups of type An 1 In this section, we study Question 2.1.4 for F -groups G of type An. As a p 1 corollary, we deduce results about Hilbert modular varieties. Z We prove Conjecture 2.1.6 when G is of type An (and of Borel-type). There- 1 fore, in all this section one has X = Y . If G is F -split or if G splits over the p ∈ quadratic extension F , then we show that CY = C holds for all w W, p2 ,w Y,w which gives a complete answer to Question 2.1.4. In the general case, we deﬁne a set of admissible strata for which one has CY = ,w C . However, we conjecture that not all w satisfy this equality of cones. Y,w ≥ · · · ≥ Let n 1 be an integer; let n = n + + n , be a partition of n with n 1 1 r i for all i = 1, ..., r. Consider the F -reductive group G deﬁned by p × × (4.1.1) G(cid:80):= G ... G , G := Res (SL ) 1 r i Fpni/Fp 2,Fpni ≤ ≤ Deﬁne N = m n for all 1 m r and N := 0. Denote again by σ the m { i=1 }i 0 · · · permutation of 1, ..., n deﬁned as a product σ = c c where c is the n -cycle − · · · 1 r i i c = (N (N 1) (N − + 1)) for i = 1, ..., r. There is an isomorphism i i i i 1 (cid:39) (4.1.2) G SLn k 2,k ∈ (cid:39) such that the action of σ Gal(k/F ) on G(k) SL (k)n is given by p 2 (4.1.3) σ(x , ..., x ) := (ϕ(x ), ϕ(x ), ..., ϕ(x )). 1 n σ(1) σ(2) σ(n) (cid:101) ⊂ ∗ ∈ Let T SL(cid:101)2,k be the d−iagon(cid:55)→al torus. We identify X×(T)×= Z⊂by sending m Z to the chara∗cter diag(cid:101)(x, x 1) ⊂ xm. Deﬁne T := T (cid:101)... T Gk and identify similarly X (T) = Zn. Let B SL be the Borel subgroup of lower-triangular × × 2,⊂k matrices, and deﬁne B := B ... B G. D(cid:101)en(cid:101)ot(cid:101)e by(cid:101)B− the opposite Borel. The × × Weyl group of G is W = S ... S . Z 2 2 X Let be the Borel-type zip datum (G, B, T, B−, T, ϕ). Denote by the cor- → X responding stack of G-zips. Fix a map ζ : X satisfying the assumptions of ⊂ Conjecture 2.1.6. Deﬁne a Zariski open subset U SL as the non-vanishing locus 2 of the function (cid:18) (cid:19) → a b (cid:55)→ (4.1.4) h : SL A1, h : a. 2,k k c d ⊂ Denote by Z SL the zero locus of h (note that Z is a reduced subscheme). 2,k ⊂ { } Identify the elements of W with subsets S 1, ..., n by the map → P { } (cid:55)→ { ∈ { } } (4.1.5) W ( 1, ..., n ), τ = (τ , ..., τ ) i 1, ..., n : τ = 1 . 1 n i ⊂ { } For a subset S 1, ..., n , write

text: AUTOMORPHIC

text: BUNDLES

text: ON

text: ZIP-SCHEMES

label: section

text: Example

text: Groups

text: of

text: type

text: An

text: In

text: this

text: section,

text: we

text: study

text: Question

text: for

text: F

text: -groups

text: G

text: of

text: type

text: An.

text: As

text: a

text: p

text: corollary,

text: we

text: deduce

text: results

text: about

text: Hilbert

text: modular

text: varieties.

text: Z

text: We

text: prove

text: Conjecture

text: when

text: G

text: is

text: of

text: type

text: An

text: (and

text: of

text: Borel-type).

text: There-

text: fore,

text: in

text: all

text: this

text: section

text: one

text: has

text: X

text: Y

text: If

text: G

text: is

text: F

text: -split

text: or

text: if

text: G

text: splits

text: over

text: the

text: p

text: quadratic

text: extension

text: F

text: then

text: we

text: show

text: that

text: CY

text: C

text: holds

text: for

text: all

text: w

text: W,

text: ,w

text: Y,w

text: which

text: gives

text: a

text: complete

text: answer

text: to

text: Question

text: In

text: the

text: general

text: case,

text: we

text: deﬁne

text: a

text: set

text: of

text: admissible

text: strata

text: for

text: which

text: one

text: has

text: CY

text: ,w

text: C

text: However,

text: we

text: conjecture

text: that

text: not

text: all

text: w

text: satisfy

text: this

text: equality

text: of

text: cones.

text: Y,w

text: Let

text: n

text: be

text: an

text: integer;

text: let

text: n

text: n

text: n

text: be

text: a

text: partition

text: of

text: n

text: with

text: n

text: r

text: i

text: for

text: all

text: i

text: r.

text: Consider

text: the

text: F

text: -reductive

text: group

text: G

text: deﬁned

text: by

text: p

text: G

text: G

text: G

text: Res

text: (SL

text: r

text: i

text: Fpni/Fp

text: Deﬁne

text: N

text: m

text: n

text: for

text: all

text: m

text: r

text: and

text: N

text: Denote

text: again

text: by

text: σ

text: the

text: m

text: }i

text: permutation

text: of

text: n

text: deﬁned

text: as

text: a

text: product

text: σ

text: c

text: c

text: where

text: c

text: is

text: the

text: n

text: -cycle

text: r

text: i

text: i

text: c

text: (N

text: (N

text: (N

text: for

text: i

text: r.

text: There

text: is

text: an

text: isomorphism

text: i

text: i

text: i

text: i

text: G

text: SLn

text: k

text: such

text: that

text: the

text: action

text: of

text: σ

text: Gal(k/F

text: on

text: G(k)

text: SL

text: (k)n

text: is

text: given

text: by

text: p

text: σ(x

text: x

text: (ϕ(x

text: ϕ(x

text: ϕ(x

text: n

text: σ(n)

text: Let

text: T

text: be

text: the

text: torus.

text: We

text: identify

text: X×(T)×=

text: Z⊂by

text: sending

text: m

text: Z

text: to

text: the

text: chara∗cter

text: x

text: xm.

text: Deﬁne

text: T

text: T

text: T

text: Gk

text: and

text: identify

text: similarly

text: X

text: (T)

text: Zn.

text: Let

text: B

text: SL

text: be

text: the

text: Borel

text: subgroup

text: of

text: lower-triangular

text: matrices,

text: and

text: deﬁne

text: B

text: B

text: B

text: G.

text: the

text: opposite

text: Borel.

text: The

text: Weyl

text: group

text: of

text: G

text: is

text: W

text: S

text: S

text: Z

text: X

text: Let

text: be

text: the

text: Borel-type

text: zip

text: datum

text: (G,

text: B,

text: T,

text: B−,

text: T,

text: ϕ).

text: Denote

text: by

text: the

text: cor-

text: X

text: responding

text: stack

text: of

text: G-zips.

text: Fix

text: a

text: map

text: ζ

text: X

text: satisfying

text: the

text: assumptions

text: of

text: Conjecture

text: Deﬁne

text: a

text: Zariski

text: open

text: subset

text: U

text: SL

text: as

text: the

text: non-vanishing

text: locus

text: of

text: the

text: function

text: a

text: b

text: h

text: SL

text: h

text: a.

text: k

text: c

text: d

text: Denote

text: by

text: Z

text: SL

text: the

text: zero

text: locus

text: of

text: h

text: (note

text: that

text: Z

text: is

text: a

text: reduced

text: subscheme).

text: Identify

text: the

text: elements

text: of

text: W

text: with

text: subsets

text: S

text: n

text: by

text: the

text: map

text: P

text: W

text: n

text: τ

text: (τ

text: τ

text: i

text: n

text: τ

text: n

text: i

text: For

text: a

text: subset

text: S

text: n

text: write


text: 10 AUTOMORPHIC BUNDLES ON ZIP-SCHEMES X → X ˜ . Consider the ﬁber product X˜ ζ˜ (cid:47)(cid:47) X˜ ιX ι (cid:15)(cid:15) (cid:15)(cid:15) ζ X X (cid:47)(cid:47) . − ∗ ∗ → ∗ Put T˜ = ι 1(T), a maximal torus of G˜. Write ι : X (T) X (T˜) for the restriction map. → X X Lemma 3.3.1. Let X be a stack and ζ : X arbitrary (X = allowed). For ∈ ∗ ⇐⇒ all w W, one has ι C = C . In particular, CY = C CY = C . Y,w Y˜,w ,w Y,w ˜,w Y˜,w ∗ ⊂ ∈ Proof. By pullback of sections, ι C C for all w W. The reverse inclusions Y,w Y˜,w (cid:3) follow from the descent lemma [5, 3.2.2]. Consequently, the equality of global sections cones for (X, ζ) depends only on the type of G, not on G itself. ∈ ∈ ∗ Deﬁnition 3.4.1. Let w W and λ X (T). Y∗ ∈ Y∗ (a) A partial Hasse invariant of LY(λ) on is a section s H0( , LY(λ)) w ∗ w which is pulled back from the Schubert stratum Sbt (§). ⊂ w ∈ ∗ (b) The Schubert cone C CY of w is the cone of λ X (T) such that Sbt,w ,w Y ≥ LY(Nλ) admits a partial Hasse invariant on for some N 1. w Y∗ ∗ ∈ Recall that and Y are normal, so we may consider Weil divisors. If s Y∗ w w H0( , LY(λ)) is a partial Hasse invariant, its divisor will be supported on a w Y∗ (possibly empty) union of codimension one strata closures in . If ζ is smooth, ∗ w the multiplicities in div(ζ (s)) equal those of div(s) . Y ∈ { } ∈ Deﬁnition 3.4.2. Let w W and w n the set of lower neighbors of w W. i i=1 Y A separating system of partial Hasse invariants for is a set of partial Hasse { } ∈ Y∗ w Y∗ invariants s n with s H0( , LY(λ )) such that div(s ) = . i i=1 i w i i wi Remark 3.4.3. (a) For groups of type An, there always exists a particularly simple separating 1 system of partial Hasse invariants, see §. (b) In general, many strata do not admit a separating system, as the number of ∈ lower neighbors of w W can exceed the semisimple rank of G. ∈ (c) If Pic(G) = 0, then clearly the element w W admits a separating system. 0 Proposition(cid:84)3.4.4. Let w ∈ W with lower neighbors {w }n . Assume that: i i=1 Y (a) There exists a separating system of partial Hasse invariants for . ⊂ w (b) One has n CY CY . i=1 ,wi ,w ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: AUTOMORPHIC

text: BUNDLES

text: ON

text: ZIP-SCHEMES

text: X

text: X

text: Consider

text: the

text: ﬁber

text: product

text: X˜

text: ζ˜

font: default

text: X˜

text: ιX

text: ι

text: ζ

text: X

text: X

text: Put

text: T˜

text: ι

text: a

text: maximal

text: torus

text: of

text: G˜.

text: Write

text: ι

text: X

text: (T)

text: X

text: (T˜)

text: for

text: the

text: restriction

text: map.

text: X

text: X

text: Lemma

text: Let

text: X

text: be

text: a

text: stack

text: and

text: ζ

text: X

text: arbitrary

text: (X

text: allowed).

text: For

text: all

text: w

text: W,

text: one

text: has

text: ι

text: C

text: C

text: In

text: particular,

text: CY

text: C

text: CY

text: C

text: Y,w

text: Y˜,w

text: ,w

text: Y,w

text: ˜,w

text: Y˜,w

text: Proof.

text: By

text: pullback

text: of

text: sections,

text: ι

text: C

text: C

text: for

text: all

text: w

text: W.

text: The

text: reverse

text: inclusions

text: Y,w

text: Y˜,w

text: follow

text: from

text: the

text: descent

text: lemma

text: Consequently,

text: the

text: equality

text: of

text: global

text: sections

text: cones

text: for

text: (X,

text: ζ)

text: depends

text: only

text: on

text: the

text: type

text: of

text: G,

text: not

text: on

text: G

text: itself.

text: Deﬁnition

text: Let

text: w

text: W

text: and

text: λ

text: X

text: (T).

text: Y∗

text: Y∗

text: (a)

text: A

text: partial

text: Hasse

text: invariant

text: of

text: LY(λ)

text: on

text: is

text: a

text: section

text: s

text: LY(λ))

text: w

text: w

text: which

text: is

text: pulled

text: back

text: from

text: the

text: Schubert

text: stratum

text: Sbt

text: w

text: (b)

text: The

text: Schubert

text: cone

text: C

text: CY

text: of

text: w

text: is

text: the

text: cone

text: of

text: λ

text: X

text: (T)

text: such

text: that

text: Sbt,w

text: ,w

text: Y

text: LY(Nλ)

text: admits

text: a

text: partial

text: Hasse

text: invariant

text: on

text: for

text: some

text: N

text: w

text: Y∗

text: Recall

text: that

text: and

text: Y

text: are

text: normal,

text: so

text: we

text: may

text: consider

text: Weil

text: divisors.

text: If

text: s

text: Y∗

text: w

text: w

text: LY(λ))

text: is

text: a

text: partial

text: Hasse

text: invariant,

text: its

text: divisor

text: will

text: be

text: supported

text: on

text: a

text: w

text: Y∗

text: (possibly

text: empty)

text: union

text: of

text: codimension

text: one

text: strata

text: closures

text: in

text: If

text: ζ

text: is

text: smooth,

text: w

text: the

text: multiplicities

text: in

text: div(ζ

text: (s))

text: equal

text: those

text: of

text: div(s)

text: Y

text: Deﬁnition

text: Let

text: w

text: W

text: and

text: w

text: n

text: the

text: set

text: of

text: lower

text: neighbors

text: of

text: w

text: W.

text: i

text: Y

text: A

text: separating

text: system

text: of

text: partial

text: Hasse

text: invariants

text: for

text: is

text: a

text: set

text: of

text: partial

text: Hasse

text: Y∗

text: w

text: Y∗

text: invariants

text: s

text: n

text: with

text: s

text: LY(λ

text: such

text: that

text: div(s

text: i

text: i

text: w

text: i

text: i

text: wi

text: Remark

text: (a)

text: For

text: groups

text: of

text: type

text: An,

text: there

text: always

text: exists

text: a

text: particularly

text: simple

text: separating

text: system

text: of

text: partial

text: Hasse

text: invariants,

text: see

text: (b)

text: In

text: general,

text: many

text: strata

text: do

text: not

text: admit

text: a

text: separating

text: system,

text: as

text: the

text: number

text: of

text: lower

text: neighbors

text: of

text: w

text: W

text: can

text: exceed

text: the

text: semisimple

text: rank

text: of

text: G.

text: (c)

text: If

text: Pic(G)

text: then

text: clearly

text: the

text: element

text: w

text: W

text: admits

text: a

text: separating

text: system.

text: Let

text: w

text: W

text: with

text: lower

text: neighbors

text: {w

text: }n

text: Assume

text: that:

text: i

text: Y

text: (a)

text: There

text: exists

text: a

text: separating

text: system

text: of

text: partial

text: Hasse

text: invariants

text: for

text: w

text: (b)

text: One

text: has

text: n

text: CY

text: CY

text: ,wi

text: ,w

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ANALYSIS OF A STOCHASTIC MODEL OF REPLICATION IN LARGE DISTRIBUTED STORAGE SYSTEMS: A MEAN-FIELD APPROACH WEN SUN, VE´RONIQUE SIMON, SE´BASTIEN MONNET, PHILIPPE ROBERT, AND PIERRE SENS Abstract. Distributed storage systems such as Hadoop File System or Google File System (GFS) ensure data availability and durability using replication. Persistence is achieved by replicating the same data block on several nodes, and ensuring that a minimum number of copies are available on the system at any time. Whenever the contents of a node are lost, for instance due to a hard disk crash, the system regenerates the data blocks stored before the failure by transferring them from the remaining replicas. This paper is focused on the analysis of the eﬃciency of replication mechanism that determines the location of the copies of a given ﬁle at some server. The variability of the loads of the nodes of the network is investigated for several policies. Three replication mechanisms are tested against simulations in the context of a real implementation of a such a system: Random, Least Loaded and Power of Choice. The simulations show that some of these policies may lead to quite unbal- anced situations: if β is the average number of copies per node it turns out that, at equilibrium, the load of the nodes may exhibit a high variability. It is shown in this paper that a simple variant of a power of choice type algorithm has a striking eﬀect on the loads of the nodes: at equilibrium, the distribution of the load of a node has a bounded support, most of nodes have a load less than 2β which is an interesting property for the design of the storage space of these systems. Mathematical models are introduced and investigated to explain this in- teresting phenomenon. The analysis of these systems turns out to be quite complicated mainly because of the large dimensionality of the state spaces involved. Our study relies on probabilistic methods, mean-ﬁeld analysis, to analyze the asymptotic behavior of an arbitrary node of the network when the total number of nodes gets large. An additional ingredient is the use of stochastic calculus with marked Poisson point processes to establish some of our results. 1. Introduction For scalability, performance or for fault-tolerance concerns in distributed storage systems, the pieces of data are spread among many distributed nodes. Most famous distributed data stores include Google File System (GFS) [7], Hadoop Distributed File System (HDFS) [2], Cassandra [12], Dynamo [5], Bigtable [3], PAST [22] or DHASH [4]. Most systems rely on data redistribution. Large amounts of data have to be stored in a distributed and reliable manner. They use a hash function in The author’s work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX- 0098). 1 ##LTLine##

text: ANALYSIS

text: OF

text: A

text: STOCHASTIC

text: MODEL

text: OF

text: REPLICATION

text: IN

text: LARGE

text: DISTRIBUTED

text: STORAGE

text: SYSTEMS:

text: A

text: MEAN-FIELD

text: APPROACH

text: WEN

text: SUN,

text: VE´RONIQUE

text: SIMON,

text: SE´BASTIEN

text: MONNET,

text: PHILIPPE

text: ROBERT,

text: AND

text: PIERRE

text: SENS

text: Abstract.

text: Distributed

label: abstract

text: storage

text: systems

text: such

text: as

text: Hadoop

text: File

text: System

text: or

text: Google

text: File

text: System

text: (GFS)

text: ensure

text: data

text: availability

text: and

text: durability

text: using

text: replication.

text: Persistence

text: is

text: achieved

text: by

text: replicating

text: the

text: same

text: data

text: block

text: on

text: several

text: nodes,

text: and

text: ensuring

text: that

text: a

text: minimum

text: number

text: of

text: copies

text: are

text: available

text: on

text: the

text: system

text: at

text: any

text: time.

text: Whenever

text: the

text: contents

text: of

text: a

text: node

text: are

text: lost,

text: for

text: instance

text: due

text: to

text: a

text: hard

text: disk

text: crash,

text: the

text: system

text: regenerates

text: the

text: data

text: blocks

text: stored

text: before

text: the

text: failure

text: by

text: transferring

text: them

text: from

text: the

text: remaining

text: replicas.

text: This

text: paper

text: is

text: focused

text: on

text: the

text: analysis

text: of

text: the

text: eﬃciency

text: of

text: replication

text: mechanism

text: that

text: determines

text: the

text: location

text: of

text: the

text: copies

text: of

text: a

text: given

text: ﬁle

text: at

text: some

text: server.

text: The

text: variability

text: of

text: the

text: loads

text: of

text: the

text: nodes

text: of

text: the

text: network

text: is

text: investigated

text: for

text: several

text: policies.

text: Three

text: replication

text: mechanisms

text: are

text: tested

text: against

text: simulations

text: in

text: the

text: context

text: of

text: a

text: real

text: implementation

text: of

text: a

text: such

text: a

text: system:

text: Random,

text: Least

text: Loaded

text: and

text: Power

text: of

text: Choice.

text: The

text: simulations

text: show

text: that

text: some

text: of

text: these

text: policies

text: may

text: lead

text: to

text: quite

text: unbal-

text: anced

text: situations:

text: if

text: β

text: is

text: the

text: average

text: number

text: of

text: copies

text: per

text: node

text: it

text: turns

text: out

text: that,

text: at

text: equilibrium,

text: the

text: load

text: of

text: the

text: nodes

text: may

text: exhibit

text: a

text: high

text: variability.

text: It

text: is

text: shown

text: in

text: this

text: paper

text: that

text: a

text: simple

text: variant

text: of

text: a

text: power

text: of

text: choice

text: type

text: algorithm

text: has

text: a

text: striking

text: eﬀect

text: on

text: the

text: loads

text: of

text: the

text: nodes:

text: at

text: equilibrium,

text: the

text: distribution

text: of

text: the

text: load

text: of

text: a

text: node

text: has

text: a

text: bounded

text: support,

text: most

text: of

text: nodes

text: have

text: a

text: load

text: less

text: than

text: 2β

text: which

text: is

text: an

text: interesting

text: property

text: for

text: the

text: design

text: of

text: the

text: storage

text: space

text: of

text: these

text: systems.

text: Mathematical

text: models

text: are

text: introduced

text: and

text: investigated

text: to

text: explain

text: this

text: in-

text: teresting

text: phenomenon.

text: The

text: analysis

text: of

text: these

text: systems

text: turns

text: out

text: to

text: be

text: quite

text: complicated

text: mainly

text: because

text: of

text: the

text: large

text: dimensionality

text: of

text: the

text: state

text: spaces

text: involved.

text: Our

text: study

text: relies

text: on

text: probabilistic

text: methods,

text: mean-ﬁeld

text: analysis,

text: to

text: analyze

text: the

text: asymptotic

text: behavior

text: of

text: an

text: arbitrary

text: node

text: of

text: the

text: network

text: when

text: the

text: total

text: number

text: of

text: nodes

text: gets

text: large.

text: An

text: additional

text: ingredient

text: is

text: the

text: use

text: of

text: stochastic

text: calculus

text: with

text: marked

text: Poisson

text: point

text: processes

text: to

text: establish

text: some

text: of

text: our

text: results.

text: Introduction

text: For

text: scalability,

text: performance

text: or

text: for

text: fault-tolerance

text: concerns

text: in

text: distributed

text: storage

text: systems,

text: the

text: pieces

text: of

text: data

text: are

text: spread

text: among

text: many

text: distributed

text: nodes.

text: Most

text: famous

text: distributed

text: data

text: stores

text: include

text: Google

text: File

text: System

text: (GFS)

text: Hadoop

text: Distributed

text: File

text: System

text: (HDFS)

text: Cassandra

text: Dynamo

text: Bigtable

text: PAST

text: or

text: DHASH

text: Most

text: systems

text: rely

text: on

text: data

text: redistribution.

text: Large

text: amounts

text: of

text: data

text: have

text: to

text: be

text: stored

text: in

text: a

text: distributed

text: and

text: reliable

text: manner.

text: They

text: use

text: a

text: hash

text: function

text: in

text: The

text: author’s

text: work

text: is

text: supported

text: by

text: a

text: public

text: grant

text: overseen

text: by

text: the

text: French

text: National

text: Research

text: Agency

text: (ANR)

text: as

text: part

text: of

text: the

text: “Investissements

text: d’Avenir”

text: program

text: (reference:

text: ##LTLine##


text: 18 W. SUN, V. SIMON, S. MONNET, PH. ROBERT, AND P. SENS with a+ = max(a, 0), Y is a uniformly distributed random variable on the interval [0, 2]. The family of random variables (Y ) has therefore a unique limiting point β when β goes to inﬁnity. One deduces the convergence in distribution. The theorem is proved. (cid:3) 5. Conclusion Our investigations through simulations and mathematical models have shown that — a simple, random placement strategy may lead to heavily unbalanced situ- ations; — Classical load balancing techniques, like choosing the least loaded nodes are optimal from the point of view of placement. They have the drawback of requiring a detailed information on the state of the network, hence a signiﬁcant cost in terms of complexity and bandwidth. — the power of two random choices policy has the advantage of having good performance with a limited cost in terms of storage space and of complexity. Appendix A. Convergence Results The technical results of this section concern the random allocation scheme. The notations of the corresponding section are used. (cid:90) Propo(cid:10)sition(cid:11) 7. The previsible increasing process of the martingale (MN(t)) is 1 t (cid:90) (cid:20) (cid:21) (12) MN (t) = LN(s)2 ds (cid:88) 1 1 0 − N t 1 N 2 + − LN(s)2 + − LN(s) ds. (N 1)2 m (N 1)2 m m=2 0 Concerning previsible increasing processes of martingales, see Section VI-34 page 377 of Rogers and Williams [21]. Proof. The proof is not diﬃcult, it is included for the sake of completeness for readers not fa(cid:18)m(cid:90)iliar with th(cid:104)e properties of mar(cid:105)t(cid:19)ingale(cid:18)s(cid:90)associated to (cid:19)marked Poisson point processes. The previsible increasing process of the martingale t − N M − t LN(s ) (dt, ) ds is LN(s)2 ds , 1 1 U 1 0 0 see Theorem (2(cid:90)8.1(cid:20))(cid:90)page 50 of [21]. By independence of the Po(cid:21)isson processes, it (cid:0) (cid:1) is enough to calculate the previsible increasing process of the martingale t − N − LN(s) M1N,m(t) d=ef. M z1 LNm(s ), u (cid:88)m(ds,(cid:88)du) Nm−1 ds (cid:90) 0 U − LNm(tmn ) − t LN(s) = tm≤t p=1 1{Upm,n=1} 0 Nm−1 ds, n ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: W.

text: SUN,

text: V.

text: SIMON,

text: S.

text: MONNET,

text: PH.

text: ROBERT,

text: AND

text: P.

text: SENS

text: with

text: a+

text: max(a,

text: Y

text: is

text: a

text: uniformly

text: distributed

text: random

text: variable

text: on

text: the

text: interval

text: The

text: family

text: of

text: random

text: variables

text: (Y

text: has

text: therefore

text: a

text: unique

text: limiting

text: point

text: β

text: when

text: β

text: goes

text: to

text: inﬁnity.

text: One

text: deduces

text: the

text: convergence

text: in

text: distribution.

text: The

text: theorem

text: is

text: proved.

text: Conclusion

text: Our

text: investigations

text: through

text: simulations

text: and

text: mathematical

text: models

text: have

text: shown

text: that

label: list

text: a

text: simple,

text: random

text: placement

text: strategy

text: may

text: lead

text: to

text: heavily

text: unbalanced

text: situ-

text: ations;

text: Classical

text: load

text: balancing

text: techniques,

text: like

text: choosing

text: the

text: least

text: loaded

text: nodes

text: are

text: optimal

text: from

text: the

text: point

text: of

text: view

text: of

text: placement.

text: They

text: have

text: the

text: drawback

text: of

text: requiring

text: a

text: detailed

text: information

text: on

text: the

text: state

text: of

text: the

text: network,

text: hence

text: a

text: signiﬁcant

text: cost

text: in

text: terms

text: of

text: complexity

text: and

text: bandwidth.

text: the

text: power

text: of

text: two

text: random

text: choices

text: policy

text: has

text: the

text: advantage

text: of

text: having

text: good

text: performance

text: with

text: a

text: limited

text: cost

text: in

text: terms

text: of

text: storage

text: space

text: and

text: of

text: complexity.

text: Appendix

text: A.

text: Convergence

text: Results

text: The

text: technical

text: results

text: of

text: this

text: section

text: concern

text: the

text: random

text: allocation

text: scheme.

text: The

text: notations

text: of

text: the

text: corresponding

text: section

text: are

text: used.

text: The

text: previsible

text: increasing

text: process

text: of

text: the

text: martingale

text: (MN(t))

text: is

text: t

text: MN

text: (t)

text: ds

text: N

text: t

text: N

text: LN(s)

text: ds.

text: (N

text: m

text: (N

text: m

text: Concerning

text: previsible

text: increasing

text: processes

text: of

text: martingales,

text: see

text: Section

text: page

text: of

text: Rogers

text: and

text: Williams

text: Proof.

text: The

text: proof

text: is

text: not

text: diﬃcult,

text: it

text: is

text: included

text: for

text: the

text: sake

text: of

text: completeness

text: for

text: readers

text: not

text: with

text: properties

text: of

text: to

text: Poisson

text: point

text: processes.

text: The

text: previsible

text: increasing

text: process

text: of

text: the

text: martingale

text: t

text: N

text: M

text: t

text: LN(s

text: (dt,

text: ds

text: is

text: ds

text: U

text: see

text: Theorem

text: of

text: By

text: independence

text: of

text: the

text: processes,

text: it

text: is

text: enough

text: to

text: calculate

text: the

text: previsible

text: increasing

text: process

text: of

text: the

text: martingale

text: t

text: N

text: LN(s)

text: d=ef.

text: M

text: LNm(s

text: u

text: ds

text: U

text: LNm(tmn

text: t

text: LN(s)

text: tm≤t

text: ds,

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: STOCHASTIC MODEL OF REPLICATION 3 We also consider only the steady state of these systems in our results, mainly for the sake of mathematical tractability. These mathematical models appear never- theless to explain some of the phenomena concerning the load of the nodes observed in the simulations. The Least Loaded policy is, without a surprise, quite optimal, the load of the nodes being almost constant in this case, it varies only within some small set of values. We show that, for a large network with an average load β per node , if R P X , [resp. X ] is the load of a random node at equilibrium for the Random policy β β ≥ (cid:32) (cid:33) [resp. Power of choice policy] then, for x 0, R lim P (cid:32)Xβ ≥ x(cid:33) = e(cid:40)−x, → ∞ β + β P − (1) lim P Xβ ≥ x = 1 x/2 if x ≥< 2, → ∞ β + β 0 if x 2. See Theorems 2 and 3 below. The striking feature is that, for the Power of choice policy, the distribution of the load of a node has a ﬁnite support [0, 2β] for a large average load per node β. This is an important and desirable property for the design of such systems, to dimension the storage of the nodes in particular. Note that this is not the case for the Random policy. Our simulations of a real system exhibit this surprising phenomenon, even for moderately large loads, see Figure 2. Another interesting feature is the fact that, in the limit, the distribution of the load of a node is uniform on [0, 2β]. It should be noted that the ﬁnite support feature is only an asymptotic property, for large N and β, of the distribution of the load of a node. Additionally it does not imply, of course, that the maximum of the loads of the nodes is bounded. Usually Power of choice policies used in computer science and communication networks are associated with log log N loads instead of log N loads, see Mitzen- macher [15]; or with double exponential decay for the tail distribution of the load at equilibrium, instead of an exponential decay, see Vvedenskaya et al. [24]. Here the phenomenon is that the number of ﬁles stored at a node is bounded in the limit, i.e. it has a ﬁnite support, instead of an exponential decay for the tail distribution of this variable. The mathematical analysis of these systems turns out to be quite complicated mainly because of the large dimensionality of the state spaces involved. Our study relies on probabilistic methods to analyze the asymptotic behavior of an arbitrary node of the network when the total number of nodes gets large. An additional ingredient is the use of stochastic calculus with marked Poisson point processes to establish some of our results. The paper is organized as follows. The main placement policies are introduced in Section . Section describes the simulation model and presents the results ob- tained with the simulator. Concerning mathematical models, the Random policy is analyzed in Section and Power of Choice policy in Section . All (quite) technical details of the proofs of the results for the Random policy are included. This is not the case for the Power of choice policy, for sake of simplicity and due to the much more complex framework of general mean-ﬁeld results, convergence results of the sample paths (Proposition 4) and of the invariant distributions (Proposition 6) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: STOCHASTIC

text: MODEL

text: OF

text: REPLICATION

text: We

text: also

text: consider

text: only

text: the

text: steady

text: state

text: of

text: these

text: systems

text: in

text: our

text: results,

text: mainly

text: for

text: the

text: sake

text: of

text: mathematical

text: tractability.

text: These

text: mathematical

text: models

text: appear

text: never-

text: theless

text: to

text: explain

text: some

text: of

text: the

text: phenomena

text: concerning

text: the

text: load

text: of

text: the

text: nodes

text: observed

text: in

text: the

text: simulations.

text: The

text: Least

text: Loaded

text: policy

text: is,

text: without

text: a

text: surprise,

text: quite

text: optimal,

text: the

text: load

text: of

text: the

text: nodes

text: being

text: almost

text: constant

text: in

text: this

text: case,

text: it

text: varies

text: only

text: within

text: some

text: small

text: set

text: of

text: values.

text: We

text: show

text: that,

text: for

text: a

text: large

text: network

text: with

text: an

text: average

text: load

text: β

text: per

text: node

text: if

text: R

text: P

text: X

text: [resp.

text: X

text: is

text: the

text: load

text: of

text: a

text: random

text: node

text: at

text: equilibrium

text: for

text: the

text: Random

text: policy

text: β

text: β

text: [resp.

text: Power

text: of

text: choice

text: policy]

text: then,

text: for

text: x

text: R

text: lim

text: P

text: β

text: β

text: P

text: lim

text: P

text: Xβ

text: x

text: if

text: x

text: β

text: β

text: if

text: x

text: See

text: Theorems

text: and

text: below.

text: The

text: striking

text: feature

text: is

text: that,

text: for

text: the

text: Power

text: of

text: choice

text: policy,

text: the

text: distribution

text: of

text: the

text: load

text: of

text: a

text: node

text: has

text: a

text: ﬁnite

text: support

text: for

text: a

text: large

text: average

text: load

text: per

text: node

text: β.

text: This

text: is

text: an

text: important

text: and

text: desirable

text: property

text: for

text: the

text: design

text: of

text: such

text: systems,

text: to

text: dimension

text: the

text: storage

text: of

text: the

text: nodes

text: in

text: particular.

text: Note

text: that

text: this

text: is

text: not

text: the

text: case

text: for

text: the

text: Random

text: policy.

text: Our

text: simulations

text: of

text: a

text: real

text: system

text: exhibit

text: this

text: surprising

text: phenomenon,

text: even

text: for

text: moderately

text: large

text: loads,

text: see

text: Figure

text: Another

text: interesting

text: feature

text: is

text: the

text: fact

text: that,

text: in

text: the

text: limit,

text: the

text: distribution

text: of

text: the

text: load

text: of

text: a

text: node

text: is

text: uniform

text: on

text: It

text: should

text: be

text: noted

text: that

text: the

text: ﬁnite

text: support

text: feature

text: is

text: only

text: an

text: asymptotic

text: property,

text: for

text: large

text: N

text: and

text: β,

text: of

text: the

text: distribution

text: of

text: the

text: load

text: of

text: a

text: node.

text: Additionally

text: it

text: does

text: not

text: imply,

text: of

text: course,

text: that

text: the

text: maximum

text: of

text: the

text: loads

text: of

text: the

text: nodes

text: is

text: bounded.

text: Usually

text: Power

text: of

text: choice

text: policies

text: used

text: in

text: computer

text: science

text: and

text: communication

text: networks

text: are

text: associated

text: with

text: log

text: log

text: N

text: loads

text: instead

text: of

text: log

text: N

text: loads,

text: see

text: Mitzen-

text: macher

text: or

text: with

text: double

text: exponential

text: decay

text: for

text: the

text: tail

text: distribution

text: of

text: the

text: load

text: at

text: equilibrium,

text: instead

text: of

text: an

text: exponential

text: decay,

text: see

text: Vvedenskaya

text: et

text: al.

text: Here

text: the

text: phenomenon

text: is

text: that

text: the

text: number

text: of

text: ﬁles

text: stored

text: at

text: a

text: node

text: is

text: bounded

text: in

text: the

text: limit,

text: i.e.

text: it

text: has

text: a

text: ﬁnite

text: support,

text: instead

text: of

text: an

text: exponential

text: decay

text: for

text: the

text: tail

text: distribution

text: of

text: this

text: variable.

text: The

text: mathematical

text: analysis

text: of

text: these

text: systems

text: turns

text: out

text: to

text: be

text: quite

text: complicated

text: mainly

text: because

text: of

text: the

text: large

text: dimensionality

text: of

text: the

text: state

text: spaces

text: involved.

text: Our

text: study

text: relies

text: on

text: probabilistic

text: methods

text: to

text: analyze

text: the

text: asymptotic

text: behavior

text: of

text: an

text: arbitrary

text: node

text: of

text: the

text: network

text: when

text: the

text: total

text: number

text: of

text: nodes

text: gets

text: large.

text: An

text: additional

text: ingredient

text: is

text: the

text: use

text: of

text: stochastic

text: calculus

text: with

text: marked

text: Poisson

text: point

text: processes

text: to

text: establish

text: some

text: of

text: our

text: results.

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: The

text: main

text: placement

text: policies

text: are

text: introduced

text: in

text: Section

text: Section

text: describes

text: the

text: simulation

text: model

text: and

text: presents

text: the

text: results

text: ob-

text: tained

text: with

text: the

text: simulator.

text: Concerning

text: mathematical

text: models,

text: the

text: Random

text: policy

text: is

text: analyzed

text: in

text: Section

text: and

text: Power

text: of

text: Choice

text: policy

text: in

text: Section

text: All

text: (quite)

text: technical

text: details

text: of

text: the

text: proofs

text: of

text: the

text: results

text: for

text: the

text: Random

text: policy

text: are

text: included.

text: This

text: is

text: not

text: the

text: case

text: for

text: the

text: Power

text: of

text: choice

text: policy,

text: for

text: sake

text: of

text: simplicity

text: and

text: due

text: to

text: the

text: much

text: more

text: complex

text: framework

text: of

text: general

text: mean-ﬁeld

text: results,

text: convergence

text: results

text: of

text: the

text: sample

text: paths

text: (Proposition

text: and

text: of

text: the

text: invariant

text: distributions

text: (Proposition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 W. SUN, V. SIMON, S. MONNET, PH. ROBERT, AND P. SENS constant. This is not the case in practice but the ﬂuctuations are nevertheless not really signiﬁcant. See [17] and the beginning of Section . Simulation parameters. In the simulations, based on PeerSim, the parameters have been ﬁxed as follows: — The number of nodes N=200, ∗ — the number of data blocks F =10000, N — the block size s=10MB, — the replication degree of data blocks d=3, — the mean time between failures (MTBF) is 7 days. The network latency is ﬁxed to 0.1s and the bandwidth is 5.5Mbps. ∗ At the beginning of each simulation, the F blocks and their copies are placed N using the corresponding policy and the system is simulated for a period of 2 years. We have studied the storage load distribution and its time evolution. With these × ∗ parameters, the average load is β=d F /N=150 blocks per node. The optimal N placement from the point of view of load balancing would consist of having 150 blocks at every node. We will investigate the deviation from this scenario for the three policies. Network simulation. The impact of policies on bandwidth management has been carefully monitored. In case of failure, many data blocks have to be transferred among a subset of nodes to repair the system. Taking into account bandwidth limitation and network contention is crucial since a delayed recovery may lead to the loss of additional blocks because of additional crashes in the meantime. Network simulation. Figure 1 shows the evolution of the average load of a node with respect to the duration of its lifetime within the network. One can conclude that: — For the Least Loaded strategy, the load remains almost constant and equal to the optimal value 150 . By systematically choosing the least loaded node to store a data block copy, the storage load tends to be constant among nodes. As observed in simulations, this policy has however two main drawbacks. First, it requires that nodes maintain an up-to-date knowledge of the load of all the nodes. Second, it is more likely to generate network contention for the following reason: If one of the nodes is really underloaded, it will receive most of the transfer requests of its neighborhood. See [23]. — For the Random strategy, the load increases linearly until the failure of the node.This is an undesired feature since it implies that the failure of “old” nodes will imply in this case a lot of transfers to recover the large number of lost blocks. — The growth of the Power of Choice policy is slow as it can be seen from the ﬁgure. It should be noted that, contrary to the Least Loaded Policy, the required information to allocate data blocks is limited. Indeed, its cost is only of a communication with each of the two nodes chosen. Furthermore, the random choice of nodes for the allocation of copies of ﬁles has the ad- vantage of spreading the load from the point of view of network contention. The distribution function of the storage loads after two simulated years is pre- sented in Figure 2. For clarity, the ﬁgure has been truncated. Each point of each

text: W.

text: SUN,

text: V.

text: SIMON,

text: S.

text: MONNET,

text: PH.

text: ROBERT,

text: AND

text: P.

text: SENS

text: constant.

text: This

text: is

text: not

text: the

text: case

text: in

text: practice

text: but

text: the

text: ﬂuctuations

text: are

text: nevertheless

text: not

text: really

text: signiﬁcant.

text: See

text: and

text: the

text: beginning

text: of

text: Section

text: Simulation

text: parameters.

text: In

text: the

text: simulations,

text: based

text: on

text: PeerSim,

text: the

text: parameters

text: have

text: been

text: ﬁxed

text: as

text: follows:

text: The

text: number

text: of

text: nodes

text: the

text: number

text: of

text: data

text: blocks

text: F

text: N

text: the

text: block

text: size

text: the

text: replication

text: degree

text: of

text: data

text: blocks

text: the

text: mean

text: time

text: between

text: failures

text: (MTBF)

text: is

text: days.

text: The

text: network

text: latency

text: is

text: ﬁxed

text: to

text: and

text: the

text: bandwidth

text: is

text: At

text: the

text: beginning

text: of

text: each

text: simulation,

text: the

text: F

text: blocks

text: and

text: their

text: copies

text: are

text: placed

text: N

text: using

text: the

text: corresponding

text: policy

text: and

text: the

text: system

text: is

text: simulated

text: for

text: a

text: period

text: of

text: years.

text: We

text: have

text: studied

text: the

text: storage

text: load

text: distribution

text: and

text: its

text: time

text: evolution.

text: With

text: these

text: parameters,

text: the

text: average

text: load

text: is

text: β=d

text: F

text: blocks

text: per

text: node.

text: The

text: optimal

text: N

text: placement

text: from

text: the

text: point

text: of

text: view

text: of

text: load

text: balancing

text: would

text: consist

text: of

text: having

text: blocks

text: at

text: every

text: node.

text: We

text: will

text: investigate

text: the

text: deviation

text: from

text: this

text: scenario

text: for

text: the

text: three

text: policies.

text: Network

text: simulation.

text: The

text: impact

text: of

text: policies

text: on

text: bandwidth

text: management

text: has

text: been

text: carefully

text: monitored.

text: In

text: case

text: of

text: failure,

text: many

text: data

text: blocks

text: have

text: to

text: be

text: transferred

text: among

text: a

text: subset

text: of

text: nodes

text: to

text: repair

text: the

text: system.

text: Taking

text: into

text: account

text: bandwidth

text: limitation

text: and

text: network

text: contention

text: is

text: crucial

text: since

text: a

text: delayed

text: recovery

text: may

text: lead

text: to

text: the

text: loss

text: of

text: additional

text: blocks

text: because

text: of

text: additional

text: crashes

text: in

text: the

text: meantime.

text: Network

text: simulation.

text: Figure

text: shows

text: the

text: evolution

text: of

text: the

text: average

text: load

text: of

text: a

text: node

text: with

text: respect

text: to

text: the

text: duration

text: of

text: its

text: lifetime

text: within

text: the

text: network.

text: One

text: can

text: conclude

text: that:

text: For

text: the

text: Least

text: Loaded

text: strategy,

text: the

text: load

text: remains

text: almost

text: constant

text: and

text: equal

text: to

text: the

text: optimal

text: value

text: By

text: systematically

text: choosing

text: the

text: least

text: loaded

text: node

text: to

text: store

text: a

text: data

text: block

text: copy,

text: the

text: storage

text: load

text: tends

text: to

text: be

text: constant

text: among

text: nodes.

text: As

text: observed

text: in

text: simulations,

text: this

text: policy

text: has

text: however

text: two

text: main

text: drawbacks.

text: First,

text: it

text: requires

text: that

text: nodes

text: maintain

text: an

text: up-to-date

text: knowledge

text: of

text: the

text: load

text: of

text: all

text: the

text: nodes.

text: Second,

text: it

text: is

text: more

text: likely

text: to

text: generate

text: network

text: contention

text: for

text: the

text: following

text: reason:

text: If

text: one

text: of

text: the

text: nodes

text: is

text: really

text: underloaded,

text: it

text: will

text: receive

text: most

text: of

text: the

text: transfer

text: requests

text: of

text: its

text: neighborhood.

text: See

text: For

text: the

text: Random

text: strategy,

text: the

text: load

text: increases

text: linearly

text: until

text: the

text: failure

text: of

text: the

text: node.This

text: is

text: an

text: undesired

text: feature

text: since

text: it

text: implies

text: that

text: the

text: failure

text: of

text: “old”

text: nodes

text: will

text: imply

text: in

text: this

text: case

text: a

text: lot

text: of

text: transfers

text: to

text: recover

text: the

text: large

text: number

text: of

text: lost

text: blocks.

text: The

text: growth

text: of

text: the

text: Power

text: of

text: Choice

text: policy

text: is

text: slow

text: as

text: it

text: can

text: be

text: seen

text: from

text: the

text: ﬁgure.

text: It

text: should

text: be

text: noted

text: that,

text: contrary

text: to

text: the

text: Least

text: Loaded

text: Policy,

text: the

text: required

text: information

text: to

text: allocate

text: data

text: blocks

text: is

text: limited.

text: Indeed,

text: its

text: cost

text: is

text: only

text: of

text: a

text: communication

text: with

text: each

text: of

text: the

text: two

text: nodes

text: chosen.

text: Furthermore,

text: the

text: random

text: choice

text: of

text: nodes

text: for

text: the

text: allocation

text: of

text: copies

text: of

text: ﬁles

text: has

text: the

text: ad-

text: vantage

text: of

text: spreading

text: the

text: load

text: from

text: the

text: point

text: of

text: view

text: of

text: network

text: contention.

text: The

text: distribution

text: function

text: of

text: the

text: storage

text: loads

text: after

text: two

text: simulated

text: years

text: is

text: pre-

text: sented

text: in

text: Figure

text: For

text: clarity,

text: the

text: ﬁgure

text: has

text: been

text: truncated.

text: Each

text: point

text: of

text: each


text: STOCHASTIC MODEL OF REPLICATION 7 Figure 1. Evolution of the Average Load of a Node with Respect to its Age in the System. policy has been obtained with 210 runs. At the beginning, the data block copies are placed using the corresponding strategy. After two years of failures and reparations, one gets that: — The Random strategy presents a highly non-uniform distribution proﬁle, note that more 10% of the nodes have a loaded greater than 350. This is consistent with our previous remark on the fact that old nodes are over- loaded. — For the Least Loaded strategy, as expected, the load is highly concentrated around 150. — The striking feature concerning the Power of Choice policy is that the load of a node seems to a uniform distribution between 0 and 300. In particular almost all nodes have a load bounded by 300 which is absolutely remarkable. Table 1 gives the maximum loads that have been observed for each strategy over 132,090 samples: starting from day 100, the maximal load has been measured and recorded every day, this for the 210 runs. We can see that the mean maximum load of the random strategy is already high (more than ﬁve times the average), and furthermore, the load varies a lot, the maximum measured load being 2188 data blocks. This implies that, with the random strategy, the storage device for each node has to be over-sized, recall that the average load is 150 data blocks. Strategy Mean of Max Min Max Least Loaded 153 150 165 Random 864 465 2188 Power of Choice 300 269 328 Table 1. Statistics of Maximal Loads ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: STOCHASTIC

text: MODEL

text: OF

text: REPLICATION

text: Figure

text: Evolution

label: caption

text: of

text: the

text: Average

text: Load

text: of

text: a

text: Node

text: with

text: Respect

text: to

text: its

text: Age

text: in

text: the

text: System.

text: policy

text: has

text: been

text: obtained

text: with

text: runs.

text: At

text: the

text: beginning,

text: the

text: data

text: block

text: copies

text: are

text: placed

text: using

text: the

text: corresponding

text: strategy.

text: After

text: two

text: years

text: of

text: failures

text: and

text: reparations,

text: one

text: gets

text: that:

text: The

text: Random

text: strategy

text: presents

text: a

text: highly

text: non-uniform

text: distribution

text: proﬁle,

text: note

text: that

text: more

text: of

text: the

text: nodes

text: have

text: a

text: loaded

text: greater

text: than

text: This

text: is

text: consistent

text: with

text: our

text: previous

text: remark

text: on

text: the

text: fact

text: that

text: old

text: nodes

text: are

text: over-

text: loaded.

text: For

text: the

text: Least

text: Loaded

text: strategy,

text: as

text: expected,

text: the

text: load

text: is

text: highly

text: concentrated

text: around

text: The

text: striking

text: feature

text: concerning

text: the

text: Power

text: of

text: Choice

text: policy

text: is

text: that

text: the

text: load

text: of

text: a

text: node

text: seems

text: to

text: a

text: uniform

text: distribution

text: between

text: and

text: In

text: particular

text: almost

text: all

text: nodes

text: have

text: a

text: load

text: bounded

text: by

text: which

text: is

text: absolutely

text: remarkable.

text: Table

text: gives

text: the

text: maximum

text: loads

text: that

text: have

text: been

text: observed

text: for

text: each

text: strategy

text: over

text: samples:

text: starting

text: from

text: day

text: the

text: maximal

text: load

text: has

text: been

text: measured

text: and

text: recorded

text: every

text: day,

text: this

text: for

text: the

text: runs.

text: We

text: can

text: see

text: that

text: the

text: mean

text: maximum

text: load

text: of

text: the

text: random

text: strategy

text: is

text: already

text: high

text: (more

text: than

text: ﬁve

text: times

text: the

text: average),

text: and

text: furthermore,

text: the

text: load

text: varies

text: a

text: lot,

text: the

text: maximum

text: measured

text: load

text: being

text: data

text: blocks.

text: This

text: implies

text: that,

text: with

text: the

text: random

text: strategy,

text: the

text: storage

text: device

text: for

text: each

text: node

text: has

text: to

text: be

text: over-sized,

text: recall

text: that

text: the

text: average

text: load

text: is

text: data

text: blocks.

text: Strategy

text: Mean

text: of

text: Max

text: Min

text: Max

text: Least

text: Loaded

text: Random

text: Power

text: of

text: Choice

text: Table

text: Statistics

text: of

text: Maximal

text: Loads

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##