text: 8 with the special case A(t) = exp(tLˆ)A(0). The exponentiated operator exp(tLˆ) may thus be used to evolve any function, or projected function of the system coordinates in time. The ﬁrst step in our derivation is to partition the dynamical evolution into a subspace of ‘fast’ and ‘slow’ variables. We have already seen that the projection operator Pˆ projects any function A(r(t), v(t), Φ(t), Φ˙ (t)) onto the space of functions which only take (r(t), v(t)) as arguments. The projection operator is idempotent, i.e. Pˆ2 = Pˆ, so a function f(r(t), v(t)) that only depends on (r(t), v(t)) is unchanged, i.e. f = Pˆf. We can also deﬁne the complimentary projection operator Qˆ ≡ I − Pˆ, which projects any function A(r(t), v(t), Φ(t), Φ˙ (t)) out of the space of functions − which only take (r(t), v(t)) as arguments. It is simple to show that PˆQˆ = QˆPˆ = Pˆ Pˆ = 0. As a result, a function f(r(t), v(t)) only depending on (r(t), v(t)) vanishes under Qˆ, i.e. Qˆf = QˆPˆf = 0. Just as Pˆ projected out defect coordinates, Qˆ eliminates functions that do not have dependence on the vibrational coordinates. In this way, we can think of the ‘slow’ dynamics being selected by Pˆ and the ‘fast’ dynamics being selected by Qˆ. The two projection operators Pˆ and Qˆ give a natural way to partition the defect equation of motion m˜ v˙ = −∂ V (r(t), v(t), Φ(t), Φ˙ (t)) = −∂ V (t), using Pˆ + Qˆ = I to write r r −(cid:104) (cid:105) − m˜ v˙(t) = ∂ V ; r(t) Qˆ∂ V (t), (24) r r − −(cid:104) (cid:105) separating the force into a force expectation Pˆ∂ V (t) = ∂ V ; r(t) that depends only on the defect coordinate − (cid:104) (cid:105) − r r and a remainder Qˆ∂ V (t) = ∂ V ; r(t) ∂ V (t) which clearly vanishes under Pˆ as PˆQˆ = 0. Using the Liouville r r r operator Lˆ to extract the time evolution, the defect force then becomes − (cid:104) (cid:105) − m˜ v˙(t) = etLˆ ∂ V ; r(0) etLˆQˆ∂ V (0), (25) r r where exp(tLˆ) is used to evolve the system from its conﬁguration at a time t = 0. Following a standard method we now use an integral identity26 (cid:90) t − etLˆ = etQˆLˆ + dse(t s)LˆPˆLˆesQˆLˆ, (26) 0 that is designed to arrive at the desired result as quickly as possible. This result can be checked by noting the − integrand is equal to etLˆ d (e sLˆesQˆLˆ). Using this identity in ((cid:90)25) gives ds −(cid:104) (cid:105) t (cid:104) − (cid:105) m˜ v˙(t) = ∂ V ; r(t) + η (t) + ds Lˆη (t); r(t s) , (27) r Q Q 0 where we have deﬁned a ‘noise’ term η (t) = etQˆLˆQˆ∂ V (0), (28) Q r (cid:104) − (cid:105) and a ‘memory’ term Lˆη (t); r(t s) which by the deﬁnition of the projection operator as a conditional expectation Q may be written (cid:90) (cid:104) − (cid:105) − Lˆη (t); r(t s) = ρ(r(t s), Φ, Φ˙ )Lˆη (t) Q Q × · · δ(∂ U Φ)δ(∂ U Φ˙ )dΦdΦ˙ . (29) r r ##LTLine##

text: with

text: the

text: special

text: case

text: A(t)

text: The

text: exponentiated

text: operator

text: exp(tLˆ)

text: may

text: thus

text: be

text: used

text: to

text: evolve

text: any

text: function,

text: or

text: projected

text: function

text: of

text: the

text: system

text: coordinates

text: in

text: time.

text: The

text: ﬁrst

text: step

text: in

text: our

text: derivation

text: is

text: to

text: partition

text: the

text: dynamical

text: evolution

text: into

text: a

text: subspace

text: of

text: ‘fast’

text: and

text: ‘slow’

text: variables.

text: We

text: have

text: already

text: seen

text: that

text: the

text: projection

text: operator

text: Pˆ

text: projects

text: any

text: function

text: A(r(t),

text: v(t),

text: Φ(t),

text: Φ˙

text: (t))

text: onto

text: the

text: space

text: of

text: functions

text: which

text: only

text: take

text: (r(t),

text: v(t))

text: as

text: arguments.

text: The

text: projection

text: operator

text: is

text: idempotent,

text: i.e.

text: Pˆ,

text: so

text: a

text: function

text: f(r(t),

text: v(t))

text: that

text: only

text: depends

text: on

text: (r(t),

text: v(t))

text: is

text: unchanged,

text: i.e.

text: f

text: Pˆf.

text: We

text: can

text: also

text: deﬁne

text: the

text: complimentary

text: projection

text: operator

text: Qˆ

text: I

text: Pˆ,

text: which

text: projects

text: any

text: function

text: A(r(t),

text: v(t),

text: Φ(t),

text: Φ˙

text: (t))

text: out

text: of

text: the

text: space

text: of

text: functions

text: which

text: only

text: take

text: (r(t),

text: v(t))

text: as

text: arguments.

text: It

text: is

text: simple

text: to

text: show

text: that

text: PˆQˆ

text: QˆPˆ

text: Pˆ

text: Pˆ

text: As

text: a

text: result,

text: a

text: function

text: f(r(t),

text: v(t))

text: only

text: depending

text: on

text: (r(t),

text: v(t))

text: vanishes

text: under

text: Qˆ,

text: i.e.

text: Qˆf

text: QˆPˆf

text: Just

text: as

text: Pˆ

text: projected

text: out

text: defect

text: coordinates,

text: Qˆ

text: eliminates

text: functions

text: that

text: do

text: not

text: have

text: dependence

text: on

text: the

text: vibrational

text: coordinates.

text: In

text: this

text: way,

text: we

text: can

text: think

text: of

text: the

text: ‘slow’

text: dynamics

text: being

text: selected

text: by

text: Pˆ

text: and

text: the

text: ‘fast’

text: dynamics

text: being

text: selected

text: by

text: Qˆ.

text: The

text: two

text: projection

text: operators

text: Pˆ

text: and

text: Qˆ

text: give

text: a

text: natural

text: way

text: to

text: partition

text: the

text: defect

text: equation

text: of

text: motion

text: m˜

text: v˙

text: V

text: (r(t),

text: v(t),

text: Φ(t),

text: Φ˙

text: (t))

text: V

text: (t),

text: using

text: Pˆ

text: Qˆ

text: I

text: to

text: write

text: r

text: r

text: m˜

text: v˙(t)

text: V

text: r(t)

text: Qˆ∂

text: V

text: (t),

text: r

text: r

text: separating

text: the

text: force

text: into

text: a

text: force

text: expectation

text: Pˆ∂

text: V

text: (t)

text: V

text: r(t)

text: that

text: depends

text: only

text: on

text: the

text: defect

text: coordinate

text: r

text: r

text: and

text: a

text: remainder

text: Qˆ∂

text: V

text: (t)

text: V

text: r(t)

text: V

text: (t)

text: which

text: clearly

text: vanishes

text: under

text: Pˆ

text: as

text: PˆQˆ

text: Using

text: the

text: Liouville

text: r

text: r

text: r

text: operator

text: Lˆ

text: to

text: extract

text: the

text: time

text: evolution,

text: the

text: defect

text: force

text: then

text: becomes

text: m˜

text: v˙(t)

text: etLˆ

text: V

text: etLˆQˆ∂

text: V

text: r

text: r

text: where

text: exp(tLˆ)

text: is

text: used

text: to

text: evolve

text: the

text: system

text: from

text: its

text: conﬁguration

text: at

text: a

text: time

text: t

text: Following

text: a

text: standard

text: method

text: we

text: now

text: use

text: an

text: integral

text: t

text: etLˆ

text: etQˆLˆ

text: dse(t

text: s)LˆPˆLˆesQˆLˆ,

text: that

text: is

text: designed

text: to

text: arrive

text: at

text: the

text: desired

text: result

text: as

text: quickly

text: as

text: possible.

text: This

text: result

text: can

text: be

text: checked

text: by

text: noting

text: the

text: integrand

text: is

text: equal

text: to

text: etLˆ

text: d

text: (e

text: sLˆesQˆLˆ).

text: Using

text: this

text: identity

text: in

text: gives

text: ds

text: t

text: m˜

text: v˙(t)

text: V

text: r(t)

text: η

text: (t)

text: ds

text: Lˆη

text: (t);

text: r(t

text: s)

text: r

text: Q

text: Q

text: where

text: we

text: have

text: deﬁned

text: a

text: ‘noise’

text: term

text: η

text: (t)

text: etQˆLˆQˆ∂

text: V

text: Q

text: r

text: and

text: a

text: ‘memory’

text: term

text: Lˆη

text: (t);

text: r(t

text: s)

text: which

text: by

text: the

text: deﬁnition

text: of

text: the

text: projection

text: operator

text: as

text: a

text: conditional

text: expectation

text: Q

text: may

text: be

text: written

text: Lˆη

text: (t);

text: r(t

text: s)

text: ρ(r(t

text: s),

text: Φ,

text: Φ˙

text: )Lˆη

text: (t)

text: Q

text: Q

text: δ(∂

text: U

text: Φ)δ(∂

text: U

text: Φ˙

text: )dΦdΦ˙

text: r

text: r

text: ##LTLine##


text: 4 operators a , a , b and b (creation operators would H V† H V have a dagger, ). Here H and V stand for horizontal ⊥ − | (cid:105) (cid:104) | and vertical polarizations, and a, b denote propagation ΠˆA0 = Iˆa 0, 0 aa 0, 0 , (32) beams, towards Alice and Bob, respectively. The opera- tors for arbitrarily oriented (linear) polarization analyz- where Iˆ is the identity operator in the space, and ers are: | (cid:105) a | (cid:105) 0, 0 is the vacuum of the space satisfying a 0, 0 = | a (cid:105) θ a a ⊥ 0, 0 = 0. θ a cos θa + sin θa = a , (22) As in quantum optics violations of Bell inequalities of H V θ − the type presented in [1] usually decreases with the grow- sin θa + cos θa = a ⊥, (23) H V θ ing number of photons, see e.g. [16], the inequalities in- cos φb + sin φb = b , (24) H V φ troduced here might be less prone to this eﬀect, because − sin φb + cos φb = b ⊥. (25) their form warrants that that terms with higher photon H V φ numbers contribute less to correlation functions K(θ, φ) Q and to the local averages S (α) . Thus we shall ben- J+ Q eﬁt from their use twice. They are not only formulated ⊥ − Let us assume the the operators with denote exit without additional assumptions (which is a kind of built of the local polarizers, and the other ones + output. Un- in loophole), but also we shall gain on the sensitivity of der the chosen intensity model given by the inequalities in detecting non-classicality. This will be shown in the examples of the next Section. But be- fore this we shall derive CHSH-like Bell inequalities for † Iˆ = c c , (26) quantum optics based on similar concepts. J+ α α IˆJ− = c†αcα⊥, (27) B. CHSH-like inequalities The CHSH-like inequalities can be derived using the where c = a, b, α = θ, φ for J = A, B, respectively. the same basic hidden variable objects IA±(θ, λ) and IB±(φ, λ), quantum versions of the functions in the inequality read: and their functions RA+(θ, λ) and RB+(φ, λ), plus addi- tional ones K(θ, φ) = Tr[(cid:37)Rˆ (θ)Rˆ (φ)], (28) Q A+ B+ IA−(θ, λ) RA(cid:80)−(θ, λ) = I (θ, λ) (33) A and where I (θ, λ) = ± I (θ, λ), and A i= Ai S (α) = Tr[(cid:37)Rˆ (α)], (29) J+ Q J+ IA−(φ, λ) RB−(φ, λ) = I (φ, λ) . (34) where (cid:37) is the quantum state, and Rˆ (α) are given A J+ by: Recall the all R variables equal zero for zero total lo- (cid:80) cal intensity. Obviously this can be done by introducing ⊥ Iˆ ⊥ correlation functi(cid:90)ons Rˆ (α) = Πˆ 0 J+ Πˆ 0. (30) J+ J i=± IˆJ± J − The operators Πˆ⊥0 are projectors into the subspace, of F(θ, φ) = dλρ(λ)[RA+(θ, λ) RA−(θ, λ)] J × − the Fock space, of the modes described by annihilation [RB+(φ, λ) RB−(φ, λ)]. (35) operators c and c , which does not contain vacuum. H V In case the notation seems a bit too dense, let us write − ≤ − ≤ an example of such an operator explicitly. E.g., the op- As 1 RJ+(α, λ) RJ−(α, λ) 1, one can proceed erator Rˆ (θ) acts in the joint Fock space of modes a in the usual way to derive: a+ θ and a ⊥, and reads θ | (cid:48) (cid:48) − (cid:48) (cid:48) | ≤ F(θ, φ) + F(θ, φ ) + F(θ , φ) F(θ , φ ) 2. (36) † ⊥ a a ⊥ RˆA+(θ) = ΠˆA0 a†θaθ +θaθ†θ⊥aθ⊥ ΠˆA0 (31) while this is a correct inequality, its usefulness for quantum optics is highly limited. This is because in with most Bell experiments the produced states are such that ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: operators

text: a

text: a

text: b

text: and

text: b

text: (creation

text: operators

text: would

text: H

text: V†

text: H

text: V

text: have

text: a

text: dagger,

text: Here

text: H

text: and

text: V

text: stand

text: for

text: horizontal

label: equation

text: and

text: vertical

text: polarizations,

text: and

text: a,

text: b

text: denote

text: propagation

text: ΠˆA0

text: Iˆa

text: aa

text: beams,

text: towards

text: Alice

text: and

text: Bob,

text: respectively.

text: The

text: opera-

text: tors

text: for

text: arbitrarily

text: oriented

text: (linear)

text: polarization

text: analyz-

text: where

text: Iˆ

text: is

text: the

text: identity

text: operator

text: in

text: the

text: space,

text: and

text: ers

text: are:

text: a

text: is

text: the

text: vacuum

text: of

text: the

text: space

text: satisfying

text: a

text: a

text: θ

text: a

text: a

text: θ

text: a

text: cos

text: θa

text: sin

text: θa

text: a

text: As

text: in

text: quantum

text: optics

text: violations

text: of

text: Bell

text: inequalities

text: of

text: H

text: V

text: θ

text: the

text: type

text: presented

text: in

text: usually

text: decreases

text: with

text: the

text: grow-

text: sin

text: θa

text: cos

text: θa

text: a

text: H

text: V

text: θ

text: ing

text: number

text: of

text: photons,

text: see

text: e.g.

text: the

text: inequalities

text: in-

text: cos

text: φb

text: sin

text: φb

text: b

text: H

text: V

text: φ

text: troduced

text: here

text: might

text: be

text: less

text: prone

text: to

text: this

text: eﬀect,

text: because

text: sin

text: φb

text: cos

text: φb

text: b

text: their

text: form

text: warrants

text: that

text: that

text: terms

text: with

text: higher

text: photon

text: H

text: V

text: φ

text: numbers

text: contribute

text: less

text: to

text: correlation

text: functions

text: K(θ,

text: φ)

text: Q

text: and

text: to

text: the

text: local

text: averages

text: S

text: (α)

text: Thus

text: we

text: shall

text: ben-

text: J+

text: Q

text: eﬁt

text: from

text: their

text: use

text: twice.

text: They

text: are

text: not

text: only

text: formulated

text: Let

text: us

text: assume

text: the

text: the

text: operators

text: with

text: denote

text: exit

text: without

text: additional

text: assumptions

text: (which

text: is

text: a

text: kind

text: of

text: built

text: of

text: the

text: local

text: polarizers,

text: and

text: the

text: other

text: ones

text: output.

text: Un-

text: in

text: loophole),

text: but

text: also

text: we

text: shall

text: gain

text: on

text: the

text: sensitivity

text: of

text: der

text: the

text: chosen

text: intensity

text: model

text: given

text: by

text: the

text: inequalities

text: in

text: detecting

text: non-classicality.

text: This

text: will

text: be

text: shown

text: in

text: the

text: examples

text: of

text: the

text: next

text: Section.

text: But

text: be-

text: fore

text: this

text: we

text: shall

text: derive

text: CHSH-like

text: Bell

text: inequalities

text: for

text: Iˆ

text: c

text: c

text: quantum

text: optics

text: based

text: on

text: similar

text: concepts.

text: J+

text: α

text: α

text: IˆJ−

text: c†αcα⊥,

text: B.

text: CHSH-like

text: inequalities

text: The

text: CHSH-like

text: inequalities

text: can

text: be

text: derived

text: using

text: the

text: where

text: c

text: a,

text: b,

text: α

text: θ,

text: φ

text: for

text: J

text: A,

text: B,

text: respectively.

text: the

text: same

text: basic

text: hidden

text: variable

text: objects

text: IA±(θ,

text: λ)

text: and

text: IB±(φ,

text: λ),

text: quantum

text: versions

text: of

text: the

text: functions

text: in

text: the

text: inequality

text: read:

text: and

text: their

text: functions

text: RA+(θ,

text: λ)

text: and

text: RB+(φ,

text: λ),

text: plus

text: addi-

text: tional

text: ones

text: K(θ,

text: φ)

text: (θ)Rˆ

text: (φ)],

text: Q

text: A+

text: B+

text: IA−(θ,

text: λ)

text: λ)

text: I

text: (θ,

text: λ)

text: A

text: and

text: where

text: I

text: (θ,

text: λ)

text: I

text: (θ,

text: λ),

text: and

text: A

text: i=

text: Ai

text: S

text: (α)

text: (α)],

text: J+

text: Q

text: J+

text: IA−(φ,

text: λ)

text: RB−(φ,

text: λ)

text: I

text: (φ,

text: λ)

text: where

text: is

text: the

text: quantum

text: state,

text: and

text: Rˆ

text: (α)

text: are

text: given

text: A

text: J+

text: by:

text: Recall

text: the

text: all

text: R

text: variables

text: equal

text: zero

text: for

text: zero

text: total

text: lo-

text: cal

text: intensity.

text: Obviously

text: this

text: can

text: be

text: done

text: by

text: introducing

text: Iˆ

text: correlation

text: Rˆ

text: (α)

text: Πˆ

text: J+

text: Πˆ

text: J+

text: J

text: i=±

text: IˆJ±

text: J

text: The

text: operators

text: Πˆ⊥0

text: are

text: projectors

text: into

text: the

text: subspace,

text: of

text: F(θ,

text: φ)

text: dλρ(λ)[RA+(θ,

text: λ)

text: RA−(θ,

text: λ)]

text: J

text: the

text: Fock

text: space,

text: of

text: the

text: modes

text: described

text: by

text: annihilation

text: [RB+(φ,

text: λ)

text: RB−(φ,

text: λ)].

text: operators

text: c

text: and

text: c

text: which

text: does

text: not

text: contain

text: vacuum.

text: H

text: V

text: In

text: case

text: the

text: notation

text: seems

text: a

text: bit

text: too

text: dense,

text: let

text: us

text: write

text: an

text: example

text: of

text: such

text: an

text: operator

text: explicitly.

text: E.g.,

text: the

text: op-

text: As

text: RJ+(α,

text: λ)

text: RJ−(α,

text: λ)

text: one

text: can

text: proceed

text: erator

text: Rˆ

text: (θ)

text: acts

text: in

text: the

text: joint

text: Fock

text: space

text: of

text: modes

text: a

text: in

text: the

text: usual

text: way

text: to

text: derive:

text: a+

text: θ

text: and

text: a

text: and

text: reads

text: θ

text: F(θ,

text: φ)

text: F(θ,

text: φ

text: F(θ

text: φ)

text: F(θ

text: φ

text: a

text: a

text: RˆA+(θ)

text: ΠˆA0

text: a†θaθ

text: +θaθ†θ⊥aθ⊥

text: ΠˆA0

text: while

text: this

text: is

text: a

text: correct

text: inequality,

text: its

text: usefulness

text: for

text: quantum

text: optics

text: is

text: highly

text: limited.

text: This

text: is

text: because

text: in

text: with

text: most

text: Bell

text: experiments

text: the

text: produced

text: states

text: are

text: such

text: that

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 L Γcrit Γcrit will be shown that with the new approach we can get Chained Chained, ratios 2 0.915 1.123 better entanglement witnesses (conditions). This is done by replacing the Stokes parameters in old entanglement 3 1.053 1.367 conditions of the type presented in Ref. [13–15] by 4 1.165 1.482 5 1.260 1.586 † − † 67 11..344257 11..678975 SˆA(cid:48) (θ) = Πˆ⊥A0 aθ†aθ aθ†⊥aθ⊥ Πˆ⊥A0. (53) aθaθ + aθ⊥aθ⊥ TABLE II. Critical values of gain Γcrit and Γcrit Chained Chained, ratios above which chained inequalities for moduli of intensity dif- As a matter of fact all entanglement witnesses for ferences and and similar ones for ratios cannot violated in states of two-qubit systems can be directly mapped into function of the number of local settings L entanglement indicators, for optical four mode ﬁelds (of the kind studied here), involving polarization measure- ments. one needs only 4 local settings, whereas it takes 7 settings VI. ACKNOWLEDGMENTS to violate the original inequality. V. CLOSING REMARKS The work is a part of EU grant BRISQ2. It is sub- sidized form funds for science for years 2012-2015 ap- One can generalize the approach to situations involving proved for international co-ﬁnanced project BRISQ2 more observation stations, compare Ref [16], and Bell by Polish Ministry of Science and Higher Education experiments involving generalizations of Bell inequalities (MNiSW). MZ and WL are supported by TEAM project for many parties, like [17]. of FNP. MW acknowledges the ﬁnancial support of The new approach in fact leads to a possibility of a new the Polish National Science Centre (NCN), grant DEC- look at Stokes parameters for non-classical light. This 2013/11/D/ST2/02638. We thank prof. M. Chekhova will be presented in a forthcoming paper [12], where it for discussions. [1] M. D. Reid and D. F. Walls, Phys. Rev. A, 34, 1260 (1986)[,1D0]. MF.. WZuaklloswasnkdi, GA..JZ. eMiliinlbguerrn, ,MQ.Aua. nHtourmneO, Apt.icKs.(ESkperrint,gePrh,ys. Rev. Lett. 71, 4287 (1993). Berlin, New York, 1994) [11] K. Rosol(cid:32)ek, M. Stobin´ska, M. Wie´sniak, M. Z˙ ukowski, Phys. Rev. Lett., in print [2] T.W. Marshall, E. Santos, Phys. Rev. A 39, 6271 (1989). [12] M. Zukowski el al., to be submitted. [3] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phy[1s3.]RDev..BLoeutwt.m2e3e,st8e8r0, C(1.9S6i9m).on, Phys. Rev. Lett., 91, 053601 (2003). [4] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, S. Wehner[,14R]eTv.imMuordS.hP.hIsyksh. a8k6o,v4,1I9va(n20N1.4A).gafonov, Maria V. Chekhova, and Gerd Leuchs, Physical Review Letters, 109, 150502 (2012). [5] K. Banaszek, K. Wo´dkiewicz, Phys. Rev. A 58, 4345 (1998);[1P5h]yMs.aRgedva.leLneattS.t8o2b,in2s0k0a9, F(1a9lk99T)o. ppel, Pavel Sekatski, and Maria V. Chekhova, Phys. Rev. A 86, 022323 (2012) [6] M. Zukowski, Phys. Lett. A Volume: 134 Issue: 6 Pages: 351[-1365]3LPauskboliwshskedi,:WJAieNs(cid:32)la1w6;1W98i9es.niak, Marcin; Zukowski, Marek; Bourennane, Mohamed; Weinfurter, Harald Journal of Physics [7] J. F. Clauser, M. A. Horne, Phys. Rev. D, (1974). B: Atomic, Molecular, and Optical Physics, Volume 42, Issue 11, article id. 114004, 12 pp. (2009). [8] W. Laskowski, M. Wiesniak, M. Zukowski, M. Bourennane, H[1.7W] Mei.nfZuurkteorw,sJkoi,urCn.aBl oruf kPnheyrs,iPcshyBs:.ARteovm. iLce,tMt.,ol(e2c0u0l2a)r, and Optical Physics 42, 114004, (2009). [9] J.-W. Pan, et al., Rev. Mod. Phys. 84, 777 (2012). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: L

label: table

text: Γcrit

text: Γcrit

text: will

text: be

text: shown

text: that

text: with

text: the

text: new

text: approach

text: we

text: can

text: get

text: Chained

text: Chained,

text: ratios

text: better

text: entanglement

text: witnesses

text: (conditions).

text: This

text: is

text: done

text: by

text: replacing

text: the

text: Stokes

text: parameters

text: in

text: old

text: entanglement

text: conditions

text: of

text: the

text: type

text: presented

text: in

text: Ref.

text: by

text: (θ)

text: Πˆ⊥A0

text: aθ†aθ

text: aθ†⊥aθ⊥

text: Πˆ⊥A0.

text: aθaθ

text: aθ⊥aθ⊥

text: TABLE

text: II.

text: Critical

label: caption

text: values

text: of

text: gain

text: Γcrit

text: and

text: Γcrit

text: Chained

text: Chained,

text: ratios

text: above

text: which

text: chained

text: inequalities

text: for

text: moduli

text: of

text: intensity

text: dif-

text: As

text: a

text: matter

text: of

text: fact

text: all

text: entanglement

text: witnesses

text: for

text: ferences

text: and

text: and

text: similar

text: ones

text: for

text: ratios

text: cannot

text: violated

text: in

text: states

text: of

text: two-qubit

text: systems

text: can

text: be

text: directly

text: mapped

text: into

text: function

text: of

text: the

text: number

text: of

text: local

text: settings

text: L

text: entanglement

text: indicators,

text: for

text: optical

text: four

text: mode

text: ﬁelds

text: (of

text: the

text: kind

text: studied

text: here),

text: involving

text: polarization

text: measure-

text: ments.

text: one

text: needs

text: only

text: local

text: settings,

text: whereas

text: it

text: takes

text: settings

text: VI.

text: ACKNOWLEDGMENTS

text: to

text: violate

text: the

text: original

text: inequality.

text: V.

text: CLOSING

text: REMARKS

text: The

text: work

text: is

text: a

text: part

text: of

text: EU

text: grant

text: It

text: is

text: sub-

text: sidized

text: form

text: funds

text: for

text: science

text: for

text: years

text: ap-

text: One

text: can

text: generalize

text: the

text: approach

text: to

text: situations

text: involving

text: proved

text: for

text: international

text: co-ﬁnanced

text: project

text: more

text: observation

text: stations,

text: compare

text: Ref

text: and

text: Bell

text: by

text: Polish

text: Ministry

text: of

text: Science

text: and

text: Higher

text: Education

text: experiments

text: involving

text: generalizations

text: of

text: Bell

text: inequalities

text: (MNiSW).

text: MZ

text: and

text: WL

text: are

text: supported

text: by

text: TEAM

text: project

text: for

text: many

text: parties,

text: like

text: of

text: FNP.

text: MW

text: acknowledges

text: the

text: ﬁnancial

text: support

text: of

text: The

text: new

text: approach

text: in

text: fact

text: leads

text: to

text: a

text: possibility

text: of

text: a

text: new

text: the

text: Polish

text: National

text: Science

text: Centre

text: (NCN),

text: grant

text: DEC-

text: look

text: at

text: Stokes

text: parameters

text: for

text: non-classical

text: light.

text: This

text: We

text: thank

text: prof.

text: M.

text: Chekhova

text: will

text: be

text: presented

text: in

text: a

text: forthcoming

text: paper

text: where

text: it

text: for

text: discussions.

text: M.

text: D.

text: Reid

text: and

text: D.

text: F.

text: Walls,

text: Phys.

text: Rev.

text: A,

text: MF..

text: WZuaklloswasnkdi,

text: GA..JZ.

text: eMiliinlbguerrn,

text: ,MQ.Aua.

text: nHtourmneO,

text: Apt.icKs.(ESkperrint,gePrh,ys.

text: Rev.

text: Lett.

text: Berlin,

text: New

text: York,

text: K.

text: M.

text: Stobin´ska,

text: M.

text: Wie´sniak,

text: M.

text: Z˙

text: ukowski,

text: Phys.

text: Rev.

text: Lett.,

text: in

text: print

text: T.W.

text: Marshall,

text: E.

text: Santos,

text: Phys.

text: Rev.

text: A

text: M.

text: Zukowski

text: el

text: al.,

text: to

text: be

text: submitted.

text: J.

text: F.

text: Clauser,

text: M.

text: A.

text: Horne,

text: A.

text: Shimony,

text: and

text: R.

text: A.

text: Holt,

text: Phys.

text: Rev.

text: Lett.,

text: N.

text: Brunner,

text: D.

text: Cavalcanti,

text: S.

text: Pironio,

text: V.

text: Scarani,

text: S.

text: Maria

text: V.

text: Chekhova,

text: and

text: Gerd

text: Leuchs,

text: Physical

text: Review

text: Letters,

text: K.

text: Banaszek,

text: K.

text: Wo´dkiewicz,

text: Phys.

text: Rev.

text: A

text: ppel,

text: Pavel

text: Sekatski,

text: and

text: Maria

text: V.

text: Chekhova,

text: Phys.

text: Rev.

text: A

text: M.

text: Zukowski,

text: Phys.

text: Lett.

text: A

text: Volume:

text: Issue:

text: Pages:

text: Marcin;

text: Zukowski,

text: Marek;

text: Bourennane,

text: Mohamed;

text: Weinfurter,

text: Harald

text: Journal

text: of

text: Physics

text: J.

text: F.

text: Clauser,

text: M.

text: A.

text: Horne,

text: Phys.

text: Rev.

text: D,

text: B:

text: Atomic,

text: Molecular,

text: and

text: Optical

text: Physics,

text: Volume

text: Issue

text: article

text: id.

text: pp.

text: W.

text: Laskowski,

text: M.

text: Wiesniak,

text: M.

text: Zukowski,

text: M.

text: Bourennane,

text: Mei.nfZuurkteorw,sJkoi,urCn.aBl

text: oruf

text: kPnheyrs,iPcshyBs:.ARteovm.

text: and

text: Optical

text: Physics

text: J.-W.

text: Pan,

text: et

text: al.,

text: Rev.

text: Mod.

text: Phys.

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text: DISCRETE NAHM EQUATIONS FOR SU(N) HYPERBOLIC MONOPOLES 17 β k p1+12 1 β k p1+32 1 ... − β − k 1 pi 12 i βpi+12 ki α = 1 βpi+32 ki ... − βpN−12 kN − βpN+12 kN 0 γ k k1 p1+1 1 ... 0 k1 ... γpi−1 − 0k1 γpi ki 1 0k1+k2 γpi+1 ki α = 2 0k1+k2 γpi+2 ki ... ... − 0−kN γpN kN − 0− k kN N ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DISCRETE

text: NAHM

text: EQUATIONS

text: FOR

text: SU(N)

text: HYPERBOLIC

text: MONOPOLES

text: β

text: k

text: β

text: k

text: β

text: k

text: pi

text: i

text: ki

text: α

text: ki

text: kN

text: kN

text: γ

text: k

text: γpi

text: ki

text: ki

text: α

text: ki

text: γpN

text: kN

text: 0−

text: k

text: kN

text: N

text: ##LTLine##

font: default

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: DISCRETE NAHM EQUATIONS FOR SU(N) HYPERBOLIC MONOPOLES 26 C k1+...+ki (cid:56)(cid:56) pi βpi+12 γpi (cid:38)(cid:38) Ckpi1+...+ki γpi (cid:47)(cid:47) Ckpi1+...+ki−1 βpi−12 (cid:47)(cid:47) C(cid:56)(cid:56)kpi1+...+ki−1 api (cid:38)(cid:38) C bpi pi The cohomology of these small monads are holomorphic line bundles deﬁned ﬁbre- wise C → C C L (x) = ker( 2k1+...+2ki−1+ki+1 k1+...+ki−1)/A ( k1+...+ki) pi x E which are exactly the line bundles in the splitting of . Furthermore, there is a natural interpretation of the maps and , restricted to A B x x each weight space of weight as a pair of maps, p i C → C Bt : k1+...+ki−1 2k1+...+2ki−1+ki+1 x ∼ C → C ⊥ C ⊂ C A : k1+...+ki Bt ( k1+...+ki−1) = k1+...+ki+1 2k1+...+2ki−1+ki+1 x x ⊥ deﬁning a map into the two term partial ﬂag manifold Fl F = (A (H ), B (L ) ) (k + i x pi x pi 1 . . . + k , k + . . . + k + 1, 2k + . . . + 2k − + k + 1). Then each line bundle L and i 1 i 1 i 1 i pi its U connection is the pullback of the invariant line bundle and connection over the (1) two term partial ﬂag manifold. This proves (2) of the theorem. P The map F thus deﬁned is an embedding of 1 into the partial ﬂag manifold, for i the ADHM equations guarantee that the monad is non-degenerate [8], and so im F i has no self-intersections and its derivative is non-zero. Compose with the Plücker F i embedding and then the Segre embedding to get P P → P (cid:32) F : 1(cid:33)(cid:44)(cid:32) k(i) (cid:33) i where k(i) = 2k1+...+2ki−1+ki+1 2k1+...+2ki−1+ki+1 − 1. k1+...+ki k1+...+ki+1 The pullback of the U invariant connection by the embedding P induces (k(i)+1) A F i i P a Kähler form F (the curvature form of A ) on 1. The work of Calabi [13] tells us Ai i ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DISCRETE

text: NAHM

text: EQUATIONS

text: FOR

text: SU(N)

text: HYPERBOLIC

text: MONOPOLES

text: C

text: pi

text: γpi

text: γpi

text: api

text: C

text: bpi

text: pi

text: The

text: cohomology

text: of

text: these

text: small

text: monads

text: are

text: holomorphic

text: line

text: bundles

text: deﬁned

text: ﬁbre-

text: wise

text: C

text: C

text: C

text: L

text: (x)

text: ker(

text: pi

text: x

text: E

text: which

text: are

text: exactly

text: the

text: line

text: bundles

text: in

text: the

text: splitting

text: of

text: Furthermore,

text: there

text: is

text: a

text: natural

text: interpretation

text: of

text: the

text: maps

text: and

text: restricted

text: to

text: A

text: B

text: x

text: x

text: each

text: weight

text: space

text: of

text: weight

text: as

text: a

text: pair

text: of

text: maps,

text: p

text: i

text: C

text: C

text: Bt

text: x

text: C

text: C

text: C

text: C

text: A

text: Bt

text: x

text: x

text: deﬁning

text: a

text: map

text: into

text: the

text: two

text: term

text: partial

text: ﬂag

text: manifold

text: Fl

text: F

text: (A

text: (H

text: B

text: (L

text: (k

text: i

text: x

text: pi

text: x

text: pi

text: k

text: k

text: k

text: k

text: Then

text: each

text: line

text: bundle

text: L

text: and

text: i

text: i

text: i

text: i

text: pi

text: its

text: U

text: connection

text: is

text: the

text: pullback

text: of

text: the

text: invariant

text: line

text: bundle

text: and

text: connection

text: over

text: the

text: two

text: term

text: partial

text: ﬂag

text: manifold.

text: This

text: proves

text: of

text: the

text: theorem.

text: P

text: The

text: map

text: F

text: thus

text: deﬁned

text: is

text: an

text: embedding

text: of

text: into

text: the

text: partial

text: ﬂag

text: manifold,

text: for

text: i

text: the

text: ADHM

text: equations

text: guarantee

text: that

text: the

text: monad

text: is

text: non-degenerate

text: and

text: so

text: im

text: F

text: i

text: has

text: no

text: self-intersections

text: and

text: its

text: derivative

text: is

text: non-zero.

text: Compose

text: with

text: the

text: Plücker

text: F

text: i

text: embedding

text: and

text: then

text: the

text: Segre

text: embedding

text: to

text: get

text: P

text: P

text: P

text: F

text: k(i)

text: i

text: where

text: k(i)

text: The

text: pullback

text: of

text: the

text: U

text: invariant

text: connection

text: by

text: the

text: embedding

text: P

text: induces

text: A

text: F

text: i

text: i

text: P

text: a

text: Kähler

text: form

text: F

text: (the

text: curvature

text: form

text: of

text: A

text: on

text: The

text: work

text: of

text: Calabi

text: tells

text: us

text: Ai

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: DISCRETE NAHM EQUATIONS FOR SU(N) HYPERBOLIC MONOPOLES 27 that any such embedding P is locally rigid, that is, the embedding is determined by F i the Kähler form up to the isometry group of the target space. Hence the boundary values (A , . . . , A − ) descend by the twistor transform to U(1) 1 N 1 connections on S1 and determine the small monad for the weight spaces corresponding − to the weights p ,. . . ,p − . These small monads provide boundary values for the (N 1)- 1 N 1 interval discrete Nahm equations and their propagation uniquely speciﬁes a complete solution up to gauge transformations. Thus the boundary values on S∞1 or equivalently P 1 uniquely determine the monopole. + On the moduli space of SU framed hyperbolic monopoles, the boundary values (N) (cid:55)→ (A , . . . , A − ) are local coordinates. Thus A (A , . . . , A − ) is a local immersion 1 N 1 1 N 1 − of the moduli of monopoles into the moduli of -tuples of U connections on (N 1) (1) (cid:3) S1. 6. Final Remarks I have shown in this paper that (1) There is an equivalence between framed SU hyperbolic monopoles (N) (A, φ) of charge (p , . . . , p − ) and charge (k , . . . , k − ), and solutions 1 N 1 1 N 1 { } { } { } { } − of the -interval discrete Nahm equations of type ( β , γ , a , b ) (N 1) i i pj pi (p , . . . , p − ; k , . . . , k − ); 1 N 1 1 N 1 P → − (2) The rational map 1 SU(N)/U(1)N 1 of a hyperbolic monopole can be writ- { } { } { } { } ten explicitly from a solution of the discrete Nahm equa- ( β , γ , a , b ) j j pi pi tions; and (3) An SU hyperbolic monopole is determined by its boundary value (N) (A, φ) − (N 1)-tuple of U(1) connections (A , . . . , A − ) on the conformal boundary 1 N 1 P sphere 1∞ of H3. − − Note that the -interval discrete Nahm equations are are essentially (N 1) (N 1) copies of the (SU ) discrete Nahm equations linked by an equation of the form (2) − β − γ γ β + b a = 0. pi 12 pi pi pi+12 pi pi − It is interesting to interpret the -interval discrete Nahm equations as a rep- (N 1) resentation of the type A quiver. The A − Dynkin quiver diagram is the directed N 1 graph ◦ (cid:6)(cid:6) ◦ (cid:6)(cid:6) ◦ (cid:6)(cid:6) ◦ (cid:6)(cid:6) ◦ (cid:6)(cid:6) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) . . . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DISCRETE

text: NAHM

text: EQUATIONS

text: FOR

text: SU(N)

text: HYPERBOLIC

text: MONOPOLES

text: that

text: any

text: such

text: embedding

text: P

text: is

text: locally

text: rigid,

text: that

text: is,

text: the

text: embedding

text: is

text: determined

text: by

text: F

text: i

text: the

text: Kähler

text: form

text: up

text: to

text: the

text: isometry

text: group

text: of

text: the

text: target

text: space.

text: Hence

text: the

text: boundary

text: values

text: (A

text: A

text: descend

text: by

text: the

text: twistor

text: transform

text: to

text: N

text: connections

text: on

text: and

text: determine

text: the

text: small

text: monad

text: for

text: the

text: weight

text: spaces

text: corresponding

text: to

text: the

text: weights

text: p

text: ,p

text: These

text: small

text: monads

text: provide

text: boundary

text: values

text: for

text: the

text: (N

text: N

text: interval

text: discrete

text: Nahm

text: equations

text: and

text: their

text: propagation

text: uniquely

text: speciﬁes

text: a

text: complete

text: solution

text: up

text: to

text: gauge

text: transformations.

text: Thus

text: the

text: boundary

text: values

text: on

text: or

text: equivalently

text: P

text: uniquely

text: determine

text: the

text: monopole.

text: On

text: the

text: moduli

text: space

text: of

text: SU

text: framed

text: hyperbolic

text: monopoles,

text: the

text: boundary

text: values

text: (N)

text: (A

text: A

text: are

text: local

text: coordinates.

text: Thus

text: A

text: (A

text: A

text: is

text: a

text: local

text: immersion

text: N

text: N

text: of

text: the

text: moduli

text: of

text: monopoles

text: into

text: the

text: moduli

text: of

text: -tuples

text: of

text: U

text: connections

text: on

text: (N

text: Final

text: Remarks

text: I

text: have

text: shown

text: in

text: this

text: paper

text: that

text: There

text: is

text: an

text: equivalence

text: between

text: framed

text: SU

text: hyperbolic

text: monopoles

text: (N)

text: (A,

text: φ)

text: of

text: charge

text: (p

text: p

text: and

text: charge

text: (k

text: k

text: and

text: solutions

text: N

text: N

text: of

text: the

text: -interval

text: discrete

text: Nahm

text: equations

text: of

text: type

text: β

text: γ

text: a

text: b

text: (N

text: i

text: i

text: pj

text: pi

text: (p

text: p

text: k

text: k

text: N

text: N

text: P

text: The

text: rational

text: map

text: of

text: a

text: hyperbolic

text: monopole

text: can

text: be

text: writ-

text: ten

text: explicitly

text: from

text: a

text: solution

text: of

text: the

text: discrete

text: Nahm

text: equa-

text: β

text: γ

text: a

text: b

text: j

text: j

text: pi

text: pi

text: tions;

text: and

text: An

text: SU

text: hyperbolic

text: monopole

text: is

text: determined

text: by

text: its

text: boundary

text: value

text: (N)

text: (A,

text: φ)

text: (N

text: of

text: connections

text: (A

text: A

text: on

text: the

text: conformal

text: boundary

text: N

text: P

text: sphere

text: 1∞

text: of

text: Note

text: that

text: the

text: -interval

text: discrete

text: Nahm

text: equations

text: are

text: are

text: essentially

text: (N

text: (N

text: copies

text: of

text: the

text: (SU

text: discrete

text: Nahm

text: equations

text: linked

text: by

text: an

text: equation

text: of

text: the

text: form

text: β

text: γ

text: γ

text: β

text: b

text: a

text: pi

text: pi

text: pi

text: pi

text: pi

text: It

text: is

text: interesting

text: to

text: interpret

text: the

text: -interval

text: discrete

text: Nahm

text: equations

text: as

text: a

text: rep-

text: (N

text: resentation

text: of

text: the

text: type

text: A

text: quiver.

text: The

text: A

text: Dynkin

text: quiver

text: diagram

text: is

text: the

text: directed

text: N

text: graph

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DISCRETE NAHM EQUATIONS FOR SU(N) HYPERBOLIC MONOPOLES 6 P P P The ADHM construction can be carried out over either 2 or 3. The 3 construction P can always yield the 2 construction via geometric invariant theory but the converse is not true. By a theorem of Donaldson [8], there is a natural correspondence between framed P ⊂ P instantons and holomorphic bundles on 2 3 (with ﬁrst Chern class c = 0 since 1 P SU(N) has determinant 1) with a ﬁxed holomorphic trivialisation at the ﬁbre 1∞ of inﬁnity via the twistor ﬁbration (2.1). P Such a holomorphic bundle E on 2 can be constructed as the cohomology of monads P [9]. A monad over 2 is the following pair of maps → → H AX K BX L where ⊗ O − ⊗ O ⊗ O (1) , , ; H = H ( 1) K = K L = L (1) C (2) , , are , , dimensional vector spaces over respectively; H K L κ κ + N κ O P (3) (1) is the Hopf bundle over 2 and ∈ P (4) A ,B are linear maps for each [x : y : z] = X 2 and depend linearly on X. X X ≡ The map needs to be injective, the map needs to be surjective and . A B B A 0 X X X X κ ∈ P Since the maps A ,B vary holomorphically with X 2, the holomorphic bundle X X can be deﬁned ﬁbre-wise by the cohomology E im E = ker B / A X X X of the monad. For an instanton, this construction is unique up to an action of GL = HKL × × GL GL GL . (H) (K) (L)   Following Donaldson, the conditions on and imply that a basis can be chosen A B X X such that     x + zα 1 (cid:104)A = y + zα (cid:105) X 2 za − − B = y zα x + zα zb X 2 1 × × × where and are matrices, is a matrix, is a matrix which we α α κ κ a N κ b κ N 1 2 call ADHM matrices; they satisfy the complex ADHM equation (2.2) [α , α ] + ba = 0. 1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DISCRETE

text: NAHM

text: EQUATIONS

text: FOR

text: SU(N)

text: HYPERBOLIC

text: MONOPOLES

text: P

text: P

text: P

text: The

text: ADHM

text: construction

text: can

text: be

text: carried

text: out

text: over

text: either

text: or

text: The

text: construction

text: P

text: can

text: always

text: yield

text: the

text: construction

text: via

text: geometric

text: invariant

text: theory

text: but

text: the

text: converse

text: is

text: not

text: true.

text: By

text: a

text: theorem

text: of

text: Donaldson

text: there

text: is

text: a

text: natural

text: correspondence

text: between

text: framed

text: P

text: P

text: instantons

text: and

text: holomorphic

text: bundles

text: on

text: (with

text: ﬁrst

text: Chern

text: class

text: c

text: since

text: P

text: SU(N)

text: has

text: determinant

text: with

text: a

text: ﬁxed

text: holomorphic

text: trivialisation

text: at

text: the

text: ﬁbre

text: 1∞

text: of

text: inﬁnity

text: via

text: the

text: twistor

text: ﬁbration

text: P

text: Such

text: a

text: holomorphic

text: bundle

text: E

text: on

text: can

text: be

text: constructed

text: as

text: the

text: cohomology

text: of

text: monads

text: P

text: A

text: monad

text: over

text: is

text: the

text: following

text: pair

text: of

text: maps

text: H

text: AX

text: K

text: BX

text: L

text: where

text: O

text: O

text: O

text: H

text: H

text: K

text: K

text: L

text: L

text: C

text: are

text: dimensional

text: vector

text: spaces

text: over

text: respectively;

text: H

text: K

text: L

text: κ

text: κ

text: N

text: κ

text: O

text: P

text: is

text: the

text: Hopf

text: bundle

text: over

text: and

text: P

text: A

text: ,B

text: are

text: linear

text: maps

text: for

text: each

text: [x

text: y

text: z]

text: X

text: and

text: depend

text: linearly

text: on

text: X.

text: X

text: X

text: The

text: map

text: needs

text: to

text: be

text: injective,

text: the

text: map

text: needs

text: to

text: be

text: surjective

text: and

text: A

text: B

text: B

text: A

text: X

text: X

text: X

text: X

text: κ

text: P

text: Since

text: the

text: maps

text: A

text: ,B

text: vary

text: holomorphically

text: with

text: X

text: the

text: holomorphic

text: bundle

text: X

text: X

text: can

text: be

text: deﬁned

text: ﬁbre-wise

text: by

text: the

text: cohomology

text: E

text: im

text: E

text: ker

text: B

text: A

text: X

text: X

text: X

text: of

text: the

text: monad.

text: For

text: an

text: instanton,

text: this

text: construction

text: is

text: unique

text: up

text: to

text: an

text: action

text: of

text: GL

text: HKL

text: GL

text: GL

text: GL

text: (H)

text: (K)

text: (L)

text: Following

text: Donaldson,

text: the

text: conditions

text: on

text: and

text: imply

text: that

text: a

text: basis

text: can

text: be

text: chosen

text: A

text: B

text: X

text: X

text: such

text: that

text: x

text: zα

text: y

text: zα

text: X

text: za

text: B

text: y

text: zα

text: x

text: zα

text: zb

text: X

text: where

text: and

text: are

text: matrices,

text: is

text: a

text: matrix,

text: is

text: a

text: matrix

text: which

text: we

text: α

text: α

text: κ

text: κ

text: a

text: N

text: κ

text: b

text: κ

text: N

text: call

text: ADHM

text: matrices;

text: they

text: satisfy

text: the

text: complex

text: ADHM

text: equation

text: [α

text: α

text: ba

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DISCRETE NAHM EQUATIONS FOR SU(N) HYPERBOLIC MONOPOLES 9 − − No(cid:104)te that x = ρ1xσ 1 implies(cid:105)that ρ(cid:104)1 = σ and cy = ρ2yσ 1 im(cid:105)plies t(cid:0)hat ρ2 = cσ. (cid:1) (cid:48) − Likewise, B = σ (c)B ρ 1(c) in the chosen basis reads as (x,cy,z) (x,y,z) − − (cid:48) − − − − − cy zα x + zα zb = σ y zα x + zα zb diag ρ 1, ρ 1, ρ 1 . 2 1 2 1 1 2 3 − − (cid:48) − (cid:48) (cid:48) − From the ﬁrst two blocks, cy = σ yρ 1 implies that cρ = σ and x = σ xρ 1 implies 1 1 2 (cid:48) that . ρ = σ 2 (cid:48) ∈ C Together, this means and for some GL . σ = P = ρ σ = cP = ρ P (κ, ) c 1 c 2 c Recall that the last basis elements of provide the framing so needs to be the N K ρ 3 representation . Thus, the conditions (1)-(4) of the theorem are exactly the conditions λ c C× (cid:3) for the -equivariance of and . A B X X C× Thus we see that in the case of a circle invariant monopole, the -action on the monad’s bundles is multiplication by (cid:55)→ ∈ × × diag diag GL GL GL c (P , (P , cP , λ ) , cP ) (H) (K) (L). c c c c c C× The homomorphism is a representation of so we can diagonalise it. This means P c C× that , and can be decomposed into weight spaces for the -action. The ADHM H K L data must then preserve these weight spaces. α , α , a, b 1 2 Austin and Braam [3] found the weight space decomposition for the SU case via (2) the equivariant index theorem. In the next section, we will see a calculation of the C× weight spaces for any SU . It is enough to compute the -representation over (N) P c P the ﬁxed line 1 since this is enough to determine the ADHM data (α , α , a, b). + 1 2 3. A Chern Characters Calculation The starting point of the calculation is the following display (which can be found in [9]) for a monad

text: DISCRETE

text: NAHM

text: EQUATIONS

text: FOR

text: SU(N)

text: HYPERBOLIC

text: MONOPOLES

text: that

text: x

text: ρ1xσ

text: σ

text: and

text: cy

text: ρ2yσ

text: ρ2

text: cσ.

text: Likewise,

text: B

text: σ

text: (c)B

text: ρ

text: in

text: the

text: chosen

text: basis

text: reads

text: as

text: (x,cy,z)

text: (x,y,z)

text: cy

text: zα

text: x

text: zα

text: zb

text: σ

text: y

text: zα

text: x

text: zα

text: zb

text: diag

text: ρ

text: ρ

text: ρ

text: From

text: the

text: ﬁrst

text: two

text: blocks,

text: cy

text: σ

text: yρ

text: implies

text: that

text: cρ

text: σ

text: and

text: x

text: σ

text: xρ

text: implies

text: that

text: ρ

text: σ

text: C

text: Together,

text: this

text: means

text: and

text: for

text: some

text: GL

text: σ

text: P

text: ρ

text: σ

text: cP

text: ρ

text: P

text: (κ,

text: c

text: c

text: c

text: Recall

text: that

text: the

text: last

text: basis

text: elements

text: of

text: provide

text: the

text: framing

text: so

text: needs

text: to

text: be

text: the

text: N

text: K

text: ρ

text: representation

text: Thus,

text: the

text: conditions

text: of

text: the

text: theorem

text: are

text: exactly

text: the

text: conditions

text: λ

text: c

text: C×

text: for

text: the

text: -equivariance

text: of

text: and

text: A

text: B

text: X

text: X

text: C×

text: Thus

text: we

text: see

text: that

text: in

text: the

text: case

text: of

text: a

text: circle

text: invariant

text: monopole,

text: the

text: -action

text: on

text: the

text: monad’s

text: bundles

text: is

text: multiplication

text: by

text: diag

text: diag

text: GL

text: GL

text: GL

text: c

text: (P

text: (P

text: cP

text: λ

text: cP

text: (H)

text: (K)

text: (L).

text: c

text: c

text: c

text: c

text: c

text: C×

text: The

text: homomorphism

text: is

text: a

text: representation

text: of

text: so

text: we

text: can

text: diagonalise

text: it.

text: This

text: means

text: P

text: c

text: C×

text: that

text: and

text: can

text: be

text: decomposed

text: into

text: weight

text: spaces

text: for

text: the

text: -action.

text: The

text: ADHM

text: H

text: K

text: L

text: data

text: must

text: then

text: preserve

text: these

text: weight

text: spaces.

text: α

text: α

text: a,

text: b

text: Austin

text: and

text: Braam

text: found

text: the

text: weight

text: space

text: decomposition

text: for

text: the

text: SU

text: case

text: via

text: the

text: equivariant

text: index

text: theorem.

text: In

text: the

text: next

text: section,

text: we

text: will

text: see

text: a

text: calculation

text: of

text: the

text: C×

text: weight

text: spaces

text: for

text: any

text: SU

text: It

text: is

text: enough

text: to

text: compute

text: the

text: -representation

text: over

text: (N)

text: P

text: c

text: P

text: the

text: ﬁxed

text: line

text: since

text: this

text: is

text: enough

text: to

text: determine

text: the

text: ADHM

text: data

text: (α

text: α

text: a,

text: b).

text: A

text: Chern

text: Characters

text: Calculation

text: The

text: starting

text: point

text: of

text: the

text: calculation

text: is

text: the

text: following

text: display

text: (which

text: can

text: be

text: found

text: in

text: for

text: a

text: monad


text: Contents II.ntroduction 2 IBI.ackground 3 IHIIi.lbert-Schmidt Analysis 4 IRVa. ndom Induced Measure Analyses 6 AT.he case K=3 6 BT. he case K=5 7 VB.ures Analysis 7 VTIw. o-Rebit Hilbert-Schmidt Analysis 8 AD. yson-index relations between rebit and qubit r = r formulas 8 A B | | | | | | | | VDIiIv. ision of separability probabilities between ρPT > ρ and ρ > ρPT > 0 9 AH. ilbert-Schmidt (K = 4) case 9 BR. andom Induced K = 3 case 9 CR. andom Induced K = 5 case 10 DT.wo-rebit case 10 EB. ivariate Extension 10 VCIoIIn.cluding Remarks 10 IAXp. pendix A: X-states analyses 13 AX. -states bivariate formulas 13 BC. ertain univariate X-states separability probability conditional distributions 15 CU. se of Fano correlation parameter c = M 16 33 zz DR.andom Induced K = 5 case 16 ER. andom Induced K = 6 case 18 FR. andom Induced K = 7 case 18 GR.andom Induced K = 3 case 18 2

text: Contents

text: II.ntroduction

text: IBI.ackground

text: IHIIi.lbert-Schmidt

text: Analysis

text: IRVa.

text: ndom

text: Induced

text: Measure

text: Analyses

text: AT.he

text: case

text: BT.

text: he

text: case

text: VB.ures

text: Analysis

text: VTIw.

text: o-Rebit

text: Hilbert-Schmidt

text: Analysis

text: AD.

text: yson-index

text: relations

text: between

text: rebit

text: and

text: qubit

text: r

text: r

text: formulas

text: A

text: B

text: VDIiIv.

text: ision

text: of

text: separability

text: probabilities

text: between

text: ρPT

text: ρ

text: and

text: ρ

text: ρPT

text: AH.

text: ilbert-Schmidt

text: (K

text: case

text: BR.

text: andom

text: Induced

text: K

text: case

text: CR.

text: andom

text: Induced

text: K

text: case

text: DT.wo-rebit

text: case

text: EB.

text: ivariate

text: Extension

text: VCIoIIn.cluding

text: Remarks

text: IAXp.

text: pendix

text: A:

text: X-states

text: analyses

text: AX.

text: -states

text: bivariate

text: formulas

text: BC.

text: ertain

text: univariate

text: X-states

text: separability

text: probability

text: conditional

text: distributions

text: CU.

text: se

text: of

text: Fano

text: correlation

text: parameter

text: c

text: M

text: zz

text: DR.andom

text: Induced

text: K

text: case

text: ER.

text: andom

text: Induced

text: K

text: case

text: FR.

text: andom

text: Induced

text: K

text: case

text: GR.andom

text: Induced

text: K

text: case


text: sep. prob 0.7 0.6 0.5 0.4 0.3 0.2 0.1 r 0.2 0.4 0.6 0.8 FIG. 23: Joint plot of the last two ﬁgures sep. prob. 0.4285 0.4280 0.4275 0.4270 0.4265 0.4260 0.4255 0.4250 r 0.2 0.4 0.6 0.8 FIG. 24: Estimated random induced (K = 5) two-qubit separability probabilities over either one of the Bloch radii, along with 95% conﬁdence limits about the conjectured value ≈ of 61 0.426573 143 35 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: sep.

text: prob

text: r

text: FIG.

text: Joint

text: plot

text: of

text: the

text: last

text: two

text: ﬁgures

text: sep.

text: prob.

text: r

text: FIG.

text: Estimated

text: random

text: induced

text: (K

text: two-qubit

text: separability

text: probabilities

text: over

text: either

text: one

text: of

text: the

text: Bloch

text: radii,

text: along

text: with

text: conﬁdence

text: limits

text: about

text: the

text: conjectured

text: value

text: of

text: ##LTFigure##

text: ##LTFigure##

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text: FIG. 26: Histogram of randomly sampled (with respect to the Bures measure) separable two-qubit density matrices 37 ##LTFigure## ##LTFigure##

text: FIG.

text: Histogram

text: of

text: randomly

text: sampled

text: (with

text: respect

text: to

text: the

text: Bures

text: measure)

text: separable

text: two-qubit

text: density

text: matrices

text: ##LTFigure##

label: figure

text: ##LTFigure##


text: II. BACKGROUND Milz and Strunz found numerically-based evidence that the Hilbert-Schmidt volumes of − the two-qubit systems and of their separable subsystems were both proportional to (1 r2 )6 A [11, eqs. (23), (30),(31)], with the consequent constant ratio of the two (simply proportional) volume functions being the aforementioned separability probability of 8 . 33 For the recently much-investigated “toy” model of X-states [31–33] [34, App. B], occu- pying a seven-dimensional subspace of the full ﬁfteen-dimensional space, it was possible for them to formally demonstrate that the counterpart volume functions, somewhat similarly, − were both again proportional, but now to (1 r2 )3 (the square root of the higher-dimensional A result). The corresponding (constant, but at r = 1) separability probability was greater A than 8 , that is 2 [11, Apps. A, B]. This 2 result was also subsequently proven in [15], along 33 5 5 with companion X-states ﬁndings for the broader class of random induced measures [13, 14]. However, the joint distributions over the two radii in which we are expressly interested here in discerning (either in the X-states and/or full model), did not seem readily derivable, even in the analytical frameworks of those two X-states studies (cf. [35]). So recourse to numerical methods seemed indicated. The two marginal univariate distributions of the − desired joint bivariate volume distributions should, of course, be proportional to (1 r2 )6 A − and (1 r2 )3 in the full and X-states models, respectively. (In Appendix A, we do succeed A in constructing the desired bivariate X-states total and separable volume functions and, hence, the associated separability probability function.) III. HILBERT-SCHMIDT ANALYSIS We generated 2,548,000,000 two-qubit density matrices, randomly with respect to Hilbert- Schmidt measure, using the simple (Ginibre ensemble) algorithm outlined in [36, eq. (1)]. (That is: “(a) prepare a square complex random matrix A of size N pertaining to the Ginibre ensemble (with real and imaginary parts of each element being independent normal random variables); (b) compute the matrix H = AA?/(tr AA?), (generically positive deﬁnite)” [13].) For each such matrix, we found the values of r and r , as well as performed the well- A B known Peres-Horodecki (determinantal-based [37]) test for separability [38, 39] on the partial transpose of ρ. (That is, a two-qubit state is separable if and only if this determinant is nonnegative.) Then, we discretized/binned the values of the r ’s and r ’s obtained to lie A B 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: II.

label: section

text: BACKGROUND

text: Milz

text: and

text: Strunz

text: found

text: numerically-based

text: evidence

text: that

text: the

text: Hilbert-Schmidt

text: volumes

text: of

text: the

text: two-qubit

text: systems

text: and

text: of

text: their

text: separable

text: subsystems

text: were

text: both

text: proportional

text: to

text: A

text: eqs.

text: with

text: the

text: consequent

text: constant

text: ratio

text: of

text: the

text: two

text: (simply

text: proportional)

text: volume

text: functions

text: being

text: the

text: aforementioned

text: separability

text: probability

text: of

text: For

text: the

text: recently

text: much-investigated

text: “toy”

text: model

text: of

text: X-states

text: App.

text: B],

text: occu-

text: pying

text: a

text: seven-dimensional

text: subspace

text: of

text: the

text: full

text: ﬁfteen-dimensional

text: space,

text: it

text: was

text: possible

text: for

text: them

text: to

text: formally

text: demonstrate

text: that

text: the

text: counterpart

text: volume

text: functions,

text: somewhat

text: similarly,

text: were

text: both

text: again

text: proportional,

text: but

text: now

text: to

text: (the

text: square

text: root

text: of

text: the

text: higher-dimensional

text: A

text: result).

text: The

text: corresponding

text: (constant,

text: but

text: at

text: r

text: separability

text: probability

text: was

text: greater

text: A

text: than

text: that

text: is

text: Apps.

text: A,

text: B].

text: This

text: result

text: was

text: also

text: subsequently

text: proven

text: in

text: along

text: with

text: companion

text: X-states

text: ﬁndings

text: for

text: the

text: broader

text: class

text: of

text: random

text: induced

text: measures

text: However,

text: the

text: joint

text: distributions

text: over

text: the

text: two

text: radii

text: in

text: which

text: we

text: are

text: expressly

text: interested

text: here

text: in

text: discerning

text: (either

text: in

text: the

text: X-states

text: and/or

text: full

text: model),

text: did

text: not

text: seem

text: readily

text: derivable,

text: even

text: in

text: the

text: analytical

text: frameworks

text: of

text: those

text: two

text: X-states

text: studies

text: (cf.

text: So

text: recourse

text: to

text: numerical

text: methods

text: seemed

text: indicated.

text: The

text: two

text: marginal

text: univariate

text: distributions

text: of

text: the

text: desired

text: joint

text: bivariate

text: volume

text: distributions

text: should,

text: of

text: course,

text: be

text: proportional

text: to

text: A

text: and

text: in

text: the

text: full

text: and

text: X-states

text: models,

text: respectively.

text: (In

text: Appendix

text: A,

text: we

text: do

text: succeed

text: A

text: in

text: constructing

text: the

text: desired

text: bivariate

text: X-states

text: total

text: and

text: separable

text: volume

text: functions

text: and,

text: hence,

text: the

text: associated

text: separability

text: probability

text: function.)

text: III.

text: HILBERT-SCHMIDT

text: ANALYSIS

text: We

text: generated

text: two-qubit

text: density

text: matrices,

text: randomly

text: with

text: respect

text: to

text: Hilbert-

text: Schmidt

text: measure,

text: using

text: the

text: simple

text: (Ginibre

text: ensemble)

text: algorithm

text: outlined

text: in

text: eq.

text: (That

text: is:

text: “(a)

text: prepare

text: a

text: square

text: complex

text: random

text: matrix

text: A

text: of

text: size

text: N

text: pertaining

text: to

text: the

text: Ginibre

text: ensemble

text: (with

text: real

text: and

text: imaginary

text: parts

text: of

text: each

text: element

text: being

text: independent

text: normal

text: random

text: variables);

text: (b)

text: compute

text: the

text: matrix

text: H

text: AA?/(tr

text: AA?),

text: (generically

text: positive

text: deﬁnite)”

text: For

text: each

text: such

text: matrix,

text: we

text: found

text: the

text: values

text: of

text: r

text: and

text: r

text: as

text: well

text: as

text: performed

text: the

text: well-

text: A

text: B

text: known

text: Peres-Horodecki

text: (determinantal-based

text: test

text: for

text: separability

text: on

text: the

text: partial

text: transpose

text: of

text: ρ.

text: (That

text: is,

text: a

text: two-qubit

text: state

text: is

text: separable

text: if

text: and

text: only

text: if

text: this

text: determinant

text: is

text: nonnegative.)

text: Then,

text: we

text: discretized/binned

text: the

text: values

text: of

text: the

text: r

text: ’s

text: and

text: r

text: ’s

text: obtained

text: to

text: lie

text: A

text: B

text: ##LTLine##

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text: sep. prob. 0.1220 0.1215 0.1210 0.1205 r 0.2 0.4 0.6 0.8 FIG. 40: Estimated two-qubit separability probability for the Hilbert-Schmidt case (k = 0) | | | | associated with the determinantal inequality ρPT > ρ , along with 95% conﬁdence limits ≈ about 1( 8 ) = 4 0.121212. 2 33 33 sep. prob. 0.159 0.158 0.157 0.156 r 0.2 0.4 0.6 0.8 FIG. 41: Two-qubit separability probability for the random induced measure case of | | | | K = 5 (k = 1) associated with the inequality ρPT > ρ , along with 95% conﬁdence limits ≈ | | | | about 45 0.157343, with the complementary probability for ρ > ρPT > 0 being 286 − ≈ 61 45 = 7 0.269231. 143 286 26 45 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: sep.

text: prob.

text: r

text: FIG.

text: Estimated

text: two-qubit

text: separability

text: probability

text: for

text: the

text: Hilbert-Schmidt

text: case

text: (k

text: associated

text: with

text: the

text: determinantal

text: inequality

text: ρPT

text: ρ

text: along

text: with

text: conﬁdence

text: limits

text: about

text: sep.

text: prob.

text: r

text: FIG.

text: Two-qubit

text: separability

text: probability

text: for

text: the

text: random

text: induced

text: measure

text: case

text: of

text: K

text: (k

text: associated

text: with

text: the

text: inequality

text: ρPT

text: ρ

text: along

text: with

text: conﬁdence

text: limits

text: about

text: with

text: the

text: complementary

text: probability

text: for

text: ρ

text: ρPT

text: being

text: ##LTFigure##

text: ##LTFigure##

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text: FIG. 44: Bivariate Hilbert-Schmidt volume distribution (7) for the X-state model FIG. 45: Bivariate Hilbert-Schmidt separable volume distribution (8) for the X-state model 47 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: Bivariate

text: Hilbert-Schmidt

text: volume

text: distribution

text: for

text: the

text: X-state

text: model

text: FIG.

text: Bivariate

text: Hilbert-Schmidt

text: separable

text: volume

text: distribution

text: for

text: the

text: X-state

text: model

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: - X states sep. prob 0.6 0.4 0.2 r 0.2 0.4 0.6 0.8 1.0 − FIG. 50: The r = r and (largely dominant) r = 1 r sections of the K = 5 X-states A B A B separability probability function (20) FIG. 51: Numerically-based estimate of bivariate separability probability function for the K = 3 induced measure X-states model 50 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: states

text: sep.

text: prob

text: r

text: FIG.

text: The

text: r

text: r

text: and

text: (largely

text: dominant)

text: r

text: r

text: sections

text: of

text: the

text: K

text: X-states

text: A

text: B

text: A

text: B

text: separability

text: probability

text: function

text: FIG.

text: Numerically-based

text: estimate

text: of

text: bivariate

text: separability

text: probability

text: function

text: for

text: the

text: K

text: induced

text: measure

text: X-states

text: model

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

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text: − where the times τ are the jump times of the process N and ∆N = N N − is the signed volume of the ∈τ τ τ order at time τ. Thus, q > 0 is a measure of market liquidity, ν [0, 1] the proportion of permanent impact, and ρ > 0 the resilience speed of the transient part of the price. In [1], the order ﬂow is modeled by a two-dimensional Hawkes process, which allows self and mutual excitation between buy and sell orders. An interesting feature of this model is that it accounts for herding behavior and meta-orders splitting, see Bacry − and Muzy [7]. Namely, let N+ and N be two nondecreasing càdlàg pure jump processes that describe − − respectively the volumes of incoming buy and sell orders. We have N = N+ N , and we proposed in [1] ± the following model for the respective jump intensities of N (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (cid:88) κ+t = κ∞ + (cid:88) (cid:20)1{∆Nτ>0}ϕs (cid:18)∆mNτ (cid:19)+ 1{∆Nτ<0}ϕc −(cid:18)∆mNτ (cid:19)(cid:21) e−β(t−τ), (2) 1 1 τ<t − −∆N ∆N − − κt = κ∞ + 1{∆Nτ<0}ϕs m τ + 1{∆Nτ>0}ϕc m τ e β(t τ), (3) 1 1 τ<t ≥ − where κ∞ 0 is the common baseline intensity of N+ and N , β is the resilience speed of the intensity and ϕs, ϕc : R+ → R(cid:82) + are measurable positive functions that encode intensity feedback. We assume that the sizes of orders are independent variables distributed according to a square integrable probability law µ on R , and m = ∞ xµ(dx) is the average amplitude of the jumps of N. This price model is called MIH, as + 1 0 Mixed-Impact Hawkes. In this model, we provide a closed-form solution for the optimal liquidation strategy, and determine a set of conditions on ν, ρ, β, ϕs, ϕc that exclude Price Manipulation Strategies (as deﬁned in [20]) from the model. These are referred to as the MIHM (Mixed Impact Hawkes Martingale) conditions. One of the beneﬁts of the framework introduced in [1] is that it is possible to calibrate the model on ﬁnancial data, without eﬀectively trading (which can be costly). One only has to observe the order ﬂow and price process of the market, and to estimate the price impact of trades issued by other participants, which is expected to be similar to the impact that the liquidating trader would have. The aim of the present paper is to conduct such a calibration on real stock data. This enables us to evaluate the realism of the theoretical price model of [1], as well as the performance of the optimal strategy in a practical context. Since our main goal is to confront the model to market data, we test the validity of our calibration protocol on simulations and we leave its mathematical justiﬁcation for further research. Many studies have explored the estimation of Hawkes parameters in various contexts (see for instance Bacry et al. [6], Bouchaud and Hardiman [17], Reynaud-Bouret [26], Lemonnier and Vayatis [23]). The present paper focuses on marked Hawkes processes used to model price jumps triggered by transactions in ﬁnancial markets, where the marks of the jumps are either the traded volumes or the price jumps. As opposed to most Hawkes models in ﬁnance, price moves which do not correspond to trades are treated separately through the propagator function. Propagator price models have been studied extensively in theoretical frameworks such as Gatheral [15], Alfonsi et al. [4] and Gatheral et al. [16], Bouchaud et al. [9] and Farmer et al. [14]. However, to the best of our knowledge, very few empirical studies have described the form of the propagator curve, or only asymptotically. Here, we suggest an estimation protocol for the propagator and discuss the quality of ﬁt of exponential and multi-exponential decays. We also describe the behavior of the curve on the ﬁrst seconds, where it is found to have an increasing part. The paper is structured as follows. First, we present the model in Section . It extends the one considered in [1] to general decay kernels, while preserving most of its properties. Then, in Section we describe our dataset and our calibration method. In particular, we explain how we slightly modify the original model to be in accordance with practical considerations. Section validates our calibration procedure with simulations and discusses the calibration results on real stock data. Eventually, we test in Section the relevance of the optimal execution strategy described in Section and discuss whether it may constitute Price Manipulation 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: the

text: times

text: τ

text: are

text: the

text: jump

text: times

text: of

text: the

text: process

text: N

text: and

text: ∆N

text: N

text: N

text: is

text: the

text: signed

text: volume

text: of

text: the

text: ∈τ

text: τ

text: τ

text: order

text: at

text: time

text: τ.

text: Thus,

text: q

text: is

text: a

text: measure

text: of

text: market

text: liquidity,

text: ν

text: the

text: proportion

text: of

text: permanent

text: impact,

text: and

text: ρ

text: the

text: resilience

text: speed

text: of

text: the

text: transient

text: part

text: of

text: the

text: price.

text: In

text: the

text: order

text: ﬂow

text: is

text: modeled

text: by

text: a

text: two-dimensional

text: Hawkes

text: process,

text: which

text: allows

text: self

text: and

text: mutual

text: excitation

text: between

text: buy

text: and

text: sell

text: orders.

text: An

text: interesting

text: feature

text: of

text: this

text: model

text: is

text: that

text: it

text: accounts

text: for

text: herding

text: behavior

text: and

text: meta-orders

text: splitting,

text: see

text: Bacry

text: and

text: Muzy

text: Namely,

text: let

text: N+

text: and

text: N

text: be

text: two

text: nondecreasing

text: càdlàg

text: pure

text: jump

text: processes

text: that

text: describe

text: respectively

text: the

text: volumes

text: of

text: incoming

text: buy

text: and

text: sell

text: orders.

text: We

text: have

text: N

text: N+

text: N

text: and

text: we

text: proposed

text: in

text: the

text: following

text: model

text: for

text: the

text: respective

text: jump

text: intensities

text: of

text: N

text: κ+t

text: κ∞

text: e−β(t−τ),

text: τ<t

text: −∆N

text: ∆N

text: κt

text: κ∞

text: m

text: τ

text: m

text: τ

text: e

text: β(t

text: τ),

text: τ<t

text: where

text: κ∞

text: is

text: the

text: common

text: baseline

text: intensity

text: of

text: N+

text: and

text: N

text: β

text: is

text: the

text: resilience

text: speed

text: of

text: the

text: intensity

text: and

text: ϕs,

text: ϕc

text: R+

text: are

text: measurable

text: positive

text: functions

text: that

text: encode

text: intensity

text: feedback.

text: We

text: assume

text: that

text: the

text: sizes

text: of

text: orders

text: are

text: independent

text: variables

text: distributed

text: according

text: to

text: a

text: square

text: integrable

text: probability

text: law

text: µ

text: on

text: R

text: and

text: m

text: xµ(dx)

text: is

text: the

text: average

text: amplitude

text: of

text: the

text: jumps

text: of

text: N.

text: This

text: price

text: model

text: is

text: called

text: MIH,

text: as

text: Mixed-Impact

text: Hawkes.

text: In

text: this

text: model,

text: we

text: provide

text: a

text: closed-form

text: solution

text: for

text: the

text: optimal

text: liquidation

text: strategy,

text: and

text: determine

text: a

text: set

text: of

text: conditions

text: on

text: ν,

text: ρ,

text: β,

text: ϕs,

text: ϕc

text: that

text: exclude

text: Price

text: Manipulation

text: Strategies

text: (as

text: deﬁned

text: in

text: from

text: the

text: model.

text: These

text: are

text: referred

text: to

text: as

text: the

text: MIHM

text: (Mixed

text: Impact

text: Hawkes

text: Martingale)

text: conditions.

text: One

text: of

text: the

text: beneﬁts

text: of

text: the

text: framework

text: introduced

text: in

text: is

text: that

text: it

text: is

text: possible

text: to

text: calibrate

text: the

text: model

text: on

text: ﬁnancial

text: data,

text: without

text: eﬀectively

text: trading

text: (which

text: can

text: be

text: costly).

text: One

text: only

text: has

text: to

text: observe

text: the

text: order

text: ﬂow

text: and

text: price

text: process

text: of

text: the

text: market,

text: and

text: to

text: estimate

text: the

text: price

text: impact

text: of

text: trades

text: issued

text: by

text: other

text: participants,

text: which

text: is

text: expected

text: to

text: be

text: similar

text: to

text: the

text: impact

text: that

text: the

text: liquidating

text: trader

text: would

text: have.

text: The

text: aim

text: of

text: the

text: present

text: paper

text: is

text: to

text: conduct

text: such

text: a

text: calibration

text: on

text: real

text: stock

text: data.

text: This

text: enables

text: us

text: to

text: evaluate

text: the

text: realism

text: of

text: the

text: theoretical

text: price

text: model

text: of

text: as

text: well

text: as

text: the

text: performance

text: of

text: the

text: optimal

text: strategy

text: in

text: a

text: practical

text: context.

text: Since

text: our

text: main

text: goal

text: is

text: to

text: confront

text: the

text: model

text: to

text: market

text: data,

text: we

text: test

text: the

text: validity

text: of

text: our

text: calibration

text: protocol

text: on

text: simulations

text: and

text: we

text: leave

text: its

text: mathematical

text: justiﬁcation

text: for

text: further

text: research.

text: Many

text: studies

text: have

text: explored

text: the

text: estimation

text: of

text: Hawkes

text: parameters

text: in

text: various

text: contexts

text: (see

text: for

text: instance

text: Bacry

text: et

text: al.

text: Bouchaud

text: and

text: Hardiman

text: Reynaud-Bouret

text: Lemonnier

text: and

text: Vayatis

text: The

text: present

text: paper

text: focuses

text: on

text: marked

text: Hawkes

text: processes

text: used

text: to

text: model

text: price

text: jumps

text: triggered

text: by

text: transactions

text: in

text: ﬁnancial

text: markets,

text: where

text: the

text: marks

text: of

text: the

text: jumps

text: are

text: either

text: the

text: traded

text: volumes

text: or

text: the

text: price

text: jumps.

text: As

text: opposed

text: to

text: most

text: Hawkes

text: models

text: in

text: ﬁnance,

text: price

text: moves

text: which

text: do

text: not

text: correspond

text: to

text: trades

text: are

text: treated

text: separately

text: through

text: the

text: propagator

text: function.

text: Propagator

text: price

text: models

text: have

text: been

text: studied

text: extensively

text: in

text: theoretical

text: frameworks

text: such

text: as

text: Gatheral

text: Alfonsi

text: et

text: al.

text: and

text: Gatheral

text: et

text: al.

text: Bouchaud

text: et

text: al.

text: and

text: Farmer

text: et

text: al.

text: However,

text: to

text: the

text: best

text: of

text: our

text: knowledge,

text: very

text: few

text: empirical

text: studies

text: have

text: described

text: the

text: form

text: of

text: the

text: propagator

text: curve,

text: or

text: only

text: asymptotically.

text: Here,

text: we

text: suggest

text: an

text: estimation

text: protocol

text: for

text: the

text: propagator

text: and

text: discuss

text: the

text: quality

text: of

text: ﬁt

text: of

text: exponential

text: and

text: multi-exponential

text: decays.

text: We

text: also

text: describe

text: the

text: behavior

text: of

text: the

text: curve

text: on

text: the

text: ﬁrst

text: seconds,

text: where

text: it

text: is

text: found

text: to

text: have

text: an

text: increasing

text: part.

text: The

text: paper

text: is

text: structured

text: as

text: follows.

text: First,

text: we

text: present

text: the

text: model

text: in

text: Section

text: It

text: extends

text: the

text: one

text: considered

text: in

text: to

text: general

text: decay

text: kernels,

text: while

text: preserving

text: most

text: of

text: its

text: properties.

text: Then,

text: in

text: Section

text: we

text: describe

text: our

text: dataset

text: and

text: our

text: calibration

text: method.

text: In

text: particular,

text: we

text: explain

text: how

text: we

text: slightly

text: modify

text: the

text: original

text: model

text: to

text: be

text: in

text: accordance

text: with

text: practical

text: considerations.

text: Section

text: validates

text: our

text: calibration

text: procedure

text: with

text: simulations

text: and

text: discusses

text: the

text: calibration

text: results

text: on

text: real

text: stock

text: data.

text: Eventually,

text: we

text: test

text: in

text: Section

text: the

text: relevance

text: of

text: the

text: optimal

text: execution

text: strategy

text: described

text: in

text: Section

text: and

text: discuss

text: whether

text: it

text: may

text: constitute

text: Price

text: Manipulation

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: • (cid:34) (cid:35) The multi-exponential curve (cid:88) p − − − R(t) = γ 1 λ (1 exp( ρ t)) , (29) i i (cid:80) i=1 (cid:80) · · · · · · ∈ is a generalization of the previous one, with 2p + 1 parameters γ, ρ , , ρ > 0, λ , , λ [0, 1], ≤ ≤ ≤ 1 p 1 p λ 1. For 1 i p, λ is the proportion of transient impact that decays at speed ρ , and i i − i i ν = 1 λ is the proportion of permanent impact. i i For both parameterizations, we est(cid:118)(cid:117)imate(cid:20)a posteriori the volatility σ of the(cid:21)Brownian noise with (cid:117) (cid:80) (cid:80) (cid:117) (cid:116) n − − − 2 Pi Pi ∆Mi G(T τ) T 0 τ σˆ = i=1 0<τ×<T , (30) n T ∈ { · · · } where n is the number of days of the sample, and for i 1, , n , Pi and Mi are respectively the real price and the midprice jump process for day i, and the τ’s are the jump times of Mi. Also, since the prediction model deﬁned by (25) and (26) can be seen as a continuous-time linear regression, where the explained − variables are the price increments P P − and the regressors are the past price jumps ∆M triggered t t ∆RW τ by trades, we can evaluate its quality of ﬁt using(cid:80)a usua(cid:80)l analysis of variance. We deﬁne the r2 value as n − (cid:80) (cid:80) (cid:2) [Pˆi Pi]2 (cid:3) − θ θ r2 = 1 i=1 ∆RW<θ<T , (31) n − − Pi Pi− ∆P 2 θ θ ∆RW i=1 ∆RW<θ<T ∈ { · · · } where for i 1, , n , Pˆi is the predicted price process for day i, and (cid:88) (cid:88) (cid:0) (cid:1) (cid:80) 1 n − ∆P = Pi Pi− n #θ θ θ ∆RW i=1 i i=1 ∆RW<θ<T − is the average price move between θ ∆RW and θ, where the θ’s are the times of price changes with no executed volumes and and #θ is the number of such price changes on day i. Note that since there is no i constant in the regression model (26), the r2 could theoretically be negative, but this is not the case in practice. The r2 constitutes a useful comparison criterion between diﬀerent estimated propagators, and we use it in Section . Now that the global practical framework is set, the estimation protocol for G needs to be detailed. This is the object of the following section. 3.3.2 Estimation protocol E We use a multi-step estimation protocol, that mainly resorts to the minimization of the quadratic error E deﬁned in (27). When G(t) is linear with respect to its parameters, is quadratic and one step of Newton- Raphson’s algorithm is enough to ﬁnd the minimum (see Appendix ). When the dependency in the parameters is non-linear, we ﬁrst use grid minimizations to ﬁnd a suitable starting point for the algorithm. 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: The

text: multi-exponential

text: curve

text: p

text: R(t)

text: γ

text: λ

text: exp(

text: ρ

text: t))

text: i

text: i

text: is

text: a

text: generalization

text: of

text: the

text: previous

text: one,

text: with

text: parameters

text: γ,

text: ρ

text: ρ

text: λ

text: λ

text: p

text: p

text: λ

text: For

text: i

text: p,

text: λ

text: is

text: the

text: proportion

text: of

text: transient

text: impact

text: that

text: decays

text: at

text: speed

text: ρ

text: and

text: i

text: i

text: i

text: i

text: ν

text: λ

text: is

text: the

text: proportion

text: of

text: permanent

text: impact.

text: i

text: i

text: For

text: both

text: parameterizations,

text: we

text: posteriori

text: the

text: volatility

text: σ

text: of

text: noise

text: with

text: n

text: Pi

text: Pi

text: ∆Mi

text: G(T

text: τ)

text: T

text: τ

text: σˆ

text: n

text: T

text: where

text: n

text: is

text: the

text: number

text: of

text: days

text: of

text: the

text: sample,

text: and

text: for

text: i

text: n

text: Pi

text: and

text: Mi

text: are

text: respectively

text: the

text: real

text: price

text: and

text: the

text: midprice

text: jump

text: process

text: for

text: day

text: i,

text: and

text: the

text: τ’s

text: are

text: the

text: jump

text: times

text: of

text: Mi.

text: Also,

text: since

text: the

text: prediction

text: model

text: deﬁned

text: by

text: and

text: can

text: be

text: seen

text: as

text: a

text: continuous-time

text: linear

text: regression,

text: where

text: the

text: explained

text: variables

text: are

text: the

text: price

text: increments

text: P

text: P

text: and

text: the

text: regressors

text: are

text: the

text: past

text: price

text: jumps

text: ∆M

text: triggered

text: t

text: t

text: ∆RW

text: τ

text: by

text: trades,

text: we

text: can

text: evaluate

text: its

text: quality

text: of

text: ﬁt

text: analysis

text: of

text: variance.

text: We

text: deﬁne

text: the

text: value

text: as

text: n

text: [Pˆi

text: θ

text: θ

text: ∆RW<θ<T

text: n

text: Pi

text: Pi−

text: ∆P

text: θ

text: θ

text: ∆RW

text: ∆RW<θ<T

text: where

text: for

text: i

text: n

text: Pˆi

text: is

text: the

text: predicted

text: price

text: process

text: for

text: day

text: i,

text: and

text: n

text: ∆P

text: Pi

text: Pi−

text: n

text: #θ

text: θ

text: θ

text: ∆RW

text: i

text: ∆RW<θ<T

text: is

text: the

text: average

text: price

text: move

text: between

text: θ

text: ∆RW

text: and

text: θ,

text: where

text: the

text: θ’s

text: are

text: the

text: times

text: of

text: price

text: changes

text: with

text: no

text: executed

text: volumes

text: and

text: and

text: #θ

text: is

text: the

text: number

text: of

text: such

text: price

text: changes

text: on

text: day

text: i.

text: Note

text: that

text: since

text: there

text: is

text: no

text: i

text: constant

text: in

text: the

text: regression

text: model

text: the

text: could

text: theoretically

text: be

text: negative,

text: but

text: this

text: is

text: not

text: the

text: case

text: in

text: practice.

text: The

text: constitutes

text: a

text: useful

text: comparison

text: criterion

text: between

text: diﬀerent

text: estimated

text: propagators,

text: and

text: we

text: use

text: it

text: in

text: Section

text: Now

text: that

text: the

text: global

text: practical

text: framework

text: is

text: set,

text: the

text: estimation

text: protocol

text: for

text: G

text: needs

text: to

text: be

text: detailed.

text: This

text: is

text: the

text: object

text: of

text: the

text: following

text: section.

text: Estimation

text: protocol

text: E

text: We

text: use

text: a

text: multi-step

text: estimation

text: protocol,

text: that

text: mainly

text: resorts

text: to

text: the

text: minimization

text: of

text: the

text: quadratic

text: error

text: E

text: deﬁned

text: in

text: When

text: G(t)

text: is

text: linear

text: with

text: respect

text: to

text: its

text: parameters,

text: is

text: quadratic

text: and

text: one

text: step

text: of

text: Newton-

text: Raphson’s

text: algorithm

text: is

text: enough

text: to

text: ﬁnd

text: the

text: minimum

text: (see

text: Appendix

text: When

text: the

text: dependency

text: in

text: the

text: parameters

text: is

text: non-linear,

text: we

text: ﬁrst

text: use

text: grid

text: minimizations

text: to

text: ﬁnd

text: a

text: suitable

text: starting

text: point

text: for

text: the

text: algorithm.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##