text: 12 ered here, we ﬁnd that the parallelogram test with four phase-space points is more powerful than the triangle test with three phase-space points. IV. CONNECTION BETWEEN SINGLE-MODE AND TWO-MODE TESTS Finally, we address the connection between our single- mode nonclassicality test and the BW two-mode nonlo- cality test more directly. We start our discussion with a single-mode state ρ that satisﬁes a condition N | | | | [ρ] > 2e12 max[ Da , Db ]2, (43) N P ≡ N − where [ρ] is the quantity deﬁned in Eq. (41) (s = 0), FIG. 11: (Color online) (a, b) Optimal [ρ] and−Da = S[q1 + ip1] − S[q0 + ip0] and Db = S[q1 + 2e12 max[|Da|,|Db|]2 and B for squeezed stPates as Bfunctions of ip0] S[q0 + ip1] correspond to the diagonal lengths of a sq|u(cid:105)e(cid:104)ez|ing str−engt|h(cid:105)(cid:104)r.| (c, d) Optimal and for states parallelogram with the transformation S[α] = α cosh r + f 0 0 +P(1 f) 2 2 as functions of the vacuum fraction α∗e2iφ sinh r. For the case of taking r = 0, the four points f. Every > 0 in our single-mode test is monBotonically con- N nected to the violation of the nonlocality test > 2 in phase simply form a rectangle; then [ρ] has a classical bound space. 2. In other words, Eq. (43) becomes a nonclassicality test | | | | and the quantity 2e12 max[ Da , Db ]2 > 2 represents a de- gree of nonclassicality. On the other hand, for the case of taking r > 0, Eq. (43) may not be immediately regarded as a nonclassicality test, since a squeezing operation can √1 − α = (2S S + S ), 0 00 10 01 create nonclassicality to the input state. Nevertheless, 2 2 we show below that if a state ρ satisﬁes the condition in √1 − α = (2S + S S ), Eq. (43) for any values of r and φ, it manifests nonlocality 1 11 10 01 2 2 under the BW test √1 − β = (2S + S S ), 0 00 10 01 2 2 √1 − B ≡ π2 { β1 = (2S11 S10 + S01), (46) [ρ ] W (α , β ) + W (α , β ) 2 2 AB 4 ρAB 0 0 ρAB 1 0 − } + W (α , β ) W (α , β ) > 2, (44) ρAB 0 1 ρAB 1 1 and obtain B (cid:48) π { − } − | | by mixing it with a vacuum at a 50:50 beam splitter. [ρAB] = 2 Wρ(S00; 0) Wρ(S11; 0) e 12 Db 2 π { } − | | It has been known that a single-mode nonclassical state + 2 Wρ(S10; 0) + Wρ(S01; 0) e 12 Da 2 can be turned into an entangled state using a beam- ≥ N − | | | | splitter setting [27–31]. By mixing a single-mode state [ρ]e 12 max[ Da , Db ]2 ρ with a vacuum at a 50:50 beam splitter, we obtain a > 2, (47) two-mode Wigner function as (cid:18) (cid:19) (cid:18) (cid:19) with the identity S + S = S + S . We have 00 − 11 10 01 − above used that W (S ; 0) W (S ; 0) and W (S ; 0)+ α√+ β α√+ β ρ 00 ρ N11 ρ 10 Wρ(cid:48) (α, β) = Wρ W|0(cid:105)(cid:104)0| , (45) Wρ(S01; 0) are both pNositive if [ρ] > 2; if the for- AB 2 2 mer is negative, then [ρ] < 2 due to the constraint { } ≤ π W (S ; 0) + W (S ; 0) 2. 2 ρ 10 ρ 01 As a by-product, we now conﬁrm that Eq. (43) rep- ≡ − | | where W| (cid:105)(cid:104) |(γ) 2 e 2 γ 2 is the Wigner function of a resents a nonclassicality test for any values of r and φ. 0 0 π vacuum state. Let us assume that Eq. (43) is satisﬁed for This is because nonclassicality is a necessary condition ≡ the state ρ. Using a short-hand notation S[q +ip ] S , to make a nonlocal resource in the beam-splitter setting. i j ij we choose the phase-space points for the BW test as As an illustration, in Fig. 11 we plot the optimal value of ##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##

text: ered

text: here,

text: we

text: ﬁnd

text: that

text: the

text: parallelogram

text: test

text: with

text: four

text: phase-space

text: points

text: is

text: more

text: powerful

text: than

text: the

text: triangle

text: test

text: with

text: three

text: phase-space

text: points.

text: IV.

text: CONNECTION

text: BETWEEN

text: SINGLE-MODE

text: AND

text: TWO-MODE

text: TESTS

text: Finally,

text: we

text: address

text: the

text: connection

text: between

text: our

text: single-

text: mode

text: nonclassicality

text: test

text: and

text: the

text: BW

text: two-mode

text: nonlo-

text: cality

text: test

text: more

text: directly.

text: We

text: start

text: our

text: discussion

text: with

text: a

text: single-mode

text: state

text: ρ

text: that

text: satisﬁes

text: a

text: condition

text: N

label: equation

text: [ρ]

text: max[

text: Da

text: Db

text: N

text: P

text: N

text: where

text: [ρ]

text: is

text: the

text: quantity

text: deﬁned

text: in

text: Eq.

text: (s

text: FIG.

text: (Color

text: online)

text: (a,

text: b)

text: Optimal

text: [ρ]

text: and−Da

text: and

text: Db

text: and

text: B

text: for

text: squeezed

text: stPates

text: as

text: Bfunctions

text: of

text: correspond

text: to

text: the

text: diagonal

text: lengths

text: of

text: a

text: (c,

text: d)

text: Optimal

text: and

text: for

text: states

text: parallelogram

text: with

text: the

text: transformation

text: S[α]

text: α

text: cosh

text: r

text: f

text: f)

text: as

text: functions

text: of

text: the

text: vacuum

text: fraction

text: α∗e2iφ

text: sinh

text: r.

text: For

text: the

text: case

text: of

text: taking

text: r

text: the

text: four

text: points

text: f.

text: Every

text: in

text: our

text: single-mode

text: test

text: is

text: monBotonically

text: con-

text: N

text: nected

text: to

text: the

text: violation

text: of

text: the

text: nonlocality

text: test

text: in

text: phase

text: simply

text: form

text: a

text: rectangle;

text: then

text: [ρ]

text: has

text: a

text: classical

text: bound

text: space.

text: In

text: other

text: words,

text: Eq.

text: becomes

text: a

text: nonclassicality

text: test

text: and

text: the

text: quantity

text: max[

text: Da

text: Db

text: represents

text: a

text: de-

text: gree

text: of

text: nonclassicality.

text: On

text: the

text: other

text: hand,

text: for

text: the

text: case

text: of

text: taking

text: r

text: Eq.

text: may

text: not

text: be

text: immediately

text: regarded

text: as

text: a

text: nonclassicality

text: test,

text: since

text: a

text: squeezing

text: operation

text: can

text: √1

text: α

text: S

text: S

text: create

text: nonclassicality

text: to

text: the

text: input

text: state.

text: Nevertheless,

text: we

text: show

text: below

text: that

text: if

text: a

text: state

text: ρ

text: satisﬁes

text: the

text: condition

text: in

text: √1

text: α

text: S

text: S

text: Eq.

text: for

text: any

text: values

text: of

text: r

text: and

text: φ,

text: it

text: manifests

text: nonlocality

text: under

text: the

text: BW

text: test

text: √1

text: β

text: S

text: S

text: √1

text: B

text: π2

text: β1

text: [ρ

text: W

text: (α

text: β

text: W

text: (α

text: β

text: AB

text: ρAB

text: ρAB

text: W

text: (α

text: β

text: W

text: (α

text: β

text: ρAB

text: ρAB

text: and

text: obtain

text: B

text: π

text: by

text: mixing

text: it

text: with

text: a

text: vacuum

text: at

text: a

text: beam

text: splitter.

text: [ρAB]

text: e

text: Db

text: π

text: It

text: has

text: been

text: known

text: that

text: a

text: single-mode

text: nonclassical

text: state

text: e

text: Da

text: can

text: be

text: turned

text: into

text: an

text: entangled

text: state

text: using

text: a

text: beam-

text: N

text: splitter

text: setting

text: By

text: mixing

text: a

text: single-mode

text: state

text: [ρ]e

text: max[

text: Da

text: Db

text: ρ

text: with

text: a

text: vacuum

text: at

text: a

text: beam

text: splitter,

text: we

text: obtain

text: a

text: two-mode

text: Wigner

text: function

text: as

text: with

text: the

text: identity

text: S

text: S

text: S

text: S

text: We

text: have

text: above

text: used

text: that

text: W

text: (S

text: W

text: (S

text: and

text: W

text: (S

text: α√+

text: β

text: α√+

text: β

text: ρ

text: ρ

text: ρ

text: (α,

text: β)

text: Wρ

text: are

text: both

text: pNositive

text: if

text: [ρ]

text: if

text: the

text: for-

text: AB

text: mer

text: is

text: negative,

text: then

text: [ρ]

text: due

text: to

text: the

text: constraint

text: π

text: W

text: (S

text: W

text: (S

text: ρ

text: ρ

text: As

text: a

text: by-product,

text: we

text: now

text: conﬁrm

text: that

text: Eq.

text: rep-

text: where

text: W|

text: |(γ)

text: e

text: γ

text: is

text: the

text: Wigner

text: function

text: of

text: a

text: resents

text: a

text: nonclassicality

text: test

text: for

text: any

text: values

text: of

text: r

text: and

text: φ.

text: π

text: vacuum

text: state.

text: Let

text: us

text: assume

text: that

text: Eq.

text: is

text: satisﬁed

text: for

text: This

text: is

text: because

text: nonclassicality

text: is

text: a

text: necessary

text: condition

text: the

text: state

text: ρ.

text: Using

text: a

text: short-hand

text: notation

text: S[q

text: +ip

text: S

text: to

text: make

text: a

text: nonlocal

text: resource

text: in

text: the

text: beam-splitter

text: setting.

text: i

text: j

text: ij

text: we

text: choose

text: the

text: phase-space

text: points

text: for

text: the

text: BW

text: test

text: as

text: As

text: an

text: illustration,

text: in

text: Fig.

text: we

text: plot

text: the

text: optimal

text: value

text: of

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##

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##

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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##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 J J (cid:48) FIG. 2: (Color online) (a, b) Maximum and (test with Wigner function s = 0) for Gaussian states with respect to s=0 s=0 J J (cid:48) squeezing strength κ = 2 tanh 2r. (c, d) Maximum values of and , respectively, among all Gaussian states with respect 0 s s to s (generalized quasidistributions). These maximum values become critical bounds to test genuine non-Gaussianity for each J J (cid:48) s in Eq. (35). (e, f) Contour plots for optimal and with respect to s and κ . Black dashed lines represent the squeezing s s 0 strength κ that yields the maximal values shown in (c) and (d). (g, h) Contour plots of a critical parameter s for a Gaussian 0 c state with purity µ and squeezing strength κ , above which nonclassicality can be detected via rectangle and triangle testing, 0 respectively. The colored regions thus represent the parameter space of Gaussian states in which there exist s-parametrized ∈ distributions s [s , 0] for a successful nonclassicality test. c Equation (15) shows that the overall factor of (cid:80) − † − − ≤ where Sˆ(r, φ) = exp[ r(e2iφaˆ 2 e 2iφaˆ2)] is the W (q, p; s) in Eq. (12) is bounded by 0 < f 1. Its 2 σ s squeezing operator (r: squeezing strength, φ: squeezing maximum f = 1 is achieved by every pure Gaussian ∞ | (cid:105)(cid:104) | s axis), and σth(n¯) = n=0 (n¯+n¯1n)n+1 n n is a thermal state (n¯ = 0) for s = 0, as fs=≡0 = 1+12n¯ represents the state with mean photon number n¯. For a Gaussian state purity of a Gaussian state µ trσ2 = 1 . On the { } 1+2n¯ with parameters α, r, φ, n¯ , its ﬁrst moments are given other hand, only a vacuum state attains the maximum (cid:104) (cid:105) (cid:104) (cid:105) by qˆ = Re[α] and pˆ = Im[α], and the covariance ma- f = 1 for s < 0. In general, with r and s ﬁxed,(cid:101) f s s trix elements by(cid:18) (cid:19) increases with purity (n¯ decreasing). Rewriting Eq. (12) using the rotated quadratures Q = R 1(cid:18) 1(cid:19) − (x, y)T = (θ)(q, p)T , we obtain Γ = n¯ + (cosh 2r sinh 2r cos 2φ), 11 2 2 (cid:20) (cid:21) 1 1 (cid:18) (cid:19) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) Γ22 = 2 n¯ + 2 (cosh 2r + sinh 2r cos 2φ), Wσ(x, y; s) 2f − 1 − (cid:104) (cid:105) − − (cid:104) (cid:105) Γ12 = Γ21 = −12 n¯ + 12 sinh 2r sin 2φ, (14) = π(1 −(cid:101)s s) exp 2(Q Q )T Γs 1(Q Q ) , (16) (cid:101) R R − where Γ = (θ)Γ ( θ) is the covariance matrix in which yields s (cid:104)s (cid:105) − − (cid:112) a rotated frame and Q = (Re[αe iθ], Im[αe iθ]). From − now on, we set α = 0, as the displacement operation 1 s f = − . (15) has no eﬀect on the optimal values (Sec. II B 1). Every s (1 + 2n¯)2 + s2 2(1 + 2n¯)s cosh 2r Gaussian function in the form of Eq. (16) can be recast to ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: J

label: caption

text: J

text: FIG.

text: (Color

text: online)

text: (a,

text: b)

text: Maximum

text: and

text: (test

text: with

text: Wigner

text: function

text: s

text: for

text: Gaussian

text: states

text: with

text: respect

text: to

text: J

text: J

text: squeezing

text: strength

text: κ

text: tanh

text: (c,

text: d)

text: Maximum

text: values

text: of

text: and

text: respectively,

text: among

text: all

text: Gaussian

text: states

text: with

text: respect

text: s

text: s

text: to

text: s

text: (generalized

text: quasidistributions).

text: These

text: maximum

text: values

text: become

text: critical

text: bounds

text: to

text: test

text: genuine

text: non-Gaussianity

text: for

text: each

text: J

text: J

text: s

text: in

text: Eq.

text: (e,

text: f)

text: Contour

text: plots

text: for

text: optimal

text: and

text: with

text: respect

text: to

text: s

text: and

text: κ

text: Black

text: dashed

text: lines

text: represent

text: the

text: squeezing

text: s

text: s

text: strength

text: κ

text: that

text: yields

text: the

text: maximal

text: values

text: shown

text: in

text: (c)

text: and

text: (d).

text: (g,

text: h)

text: Contour

text: plots

text: of

text: a

text: critical

text: parameter

text: s

text: for

text: a

text: Gaussian

text: c

text: state

text: with

text: purity

text: µ

text: and

text: squeezing

text: strength

text: κ

text: above

text: which

text: nonclassicality

text: can

text: be

text: detected

text: via

text: rectangle

text: and

text: triangle

text: testing,

text: respectively.

text: The

text: colored

text: regions

text: thus

text: represent

text: the

text: parameter

text: space

text: of

text: Gaussian

text: states

text: in

text: which

text: there

text: exist

text: s-parametrized

text: distributions

text: s

text: [s

text: for

text: a

text: successful

text: nonclassicality

text: test.

text: c

text: Equation

text: shows

text: that

text: the

text: overall

text: factor

text: of

text: where

text: Sˆ(r,

text: φ)

text: exp[

text: e

text: is

text: the

text: W

text: (q,

text: p;

text: s)

text: in

text: Eq.

text: is

text: bounded

text: by

text: f

text: Its

text: σ

text: s

text: squeezing

text: operator

text: (r:

text: squeezing

text: strength,

text: φ:

text: squeezing

text: maximum

text: f

text: is

text: achieved

text: by

text: every

text: pure

text: Gaussian

text: s

text: axis),

text: and

text: σth(n¯)

text: n

text: n

text: is

text: a

text: thermal

text: state

text: (n¯

text: for

text: s

text: as

text: represents

text: the

text: state

text: with

text: mean

text: photon

text: number

text: n¯.

text: For

text: a

text: Gaussian

text: state

text: purity

text: of

text: a

text: Gaussian

text: state

text: µ

text: On

text: the

text: with

text: parameters

text: α,

text: r,

text: φ,

text: n¯

text: its

text: ﬁrst

text: moments

text: are

text: given

text: other

text: hand,

text: only

text: a

text: vacuum

text: state

text: attains

text: the

text: maximum

text: by

text: qˆ

text: Re[α]

text: and

text: pˆ

text: Im[α],

text: and

text: the

text: covariance

text: ma-

text: f

text: for

text: s

text: In

text: general,

text: with

text: r

text: and

text: s

text: f

text: s

text: s

text: trix

text: elements

text: increases

text: with

text: purity

text: (n¯

text: decreasing).

text: Rewriting

text: Eq.

text: using

text: the

text: rotated

text: quadratures

text: Q

text: R

text: (x,

text: y)T

text: (θ)(q,

text: p)T

text: we

text: obtain

text: Γ

text: n¯

text: (cosh

text: sinh

text: cos

text: n¯

text: (cosh

text: sinh

text: cos

text: Wσ(x,

text: y;

text: s)

text: n¯

text: sinh

text: sin

text: s)

text: exp

text: Q

text: )T

text: Γs

text: Q

text: R

text: R

text: where

text: Γ

text: (θ)Γ

text: θ)

text: is

text: the

text: covariance

text: matrix

text: in

text: which

text: yields

text: s

text: a

text: rotated

text: frame

text: and

text: Q

text: (Re[αe

text: iθ],

text: Im[αe

text: iθ]).

text: From

text: now

text: on,

text: we

text: set

text: α

text: as

text: the

text: displacement

text: operation

text: s

text: f

text: has

text: no

text: eﬀect

text: on

text: the

text: optimal

text: values

text: (Sec.

text: II

text: B

text: Every

text: s

text: cosh

text: Gaussian

text: function

text: in

text: the

text: form

text: of

text: Eq.

text: can

text: be

text: recast

text: to

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 Sˆ. As we illustrate in Fig. 9, under the squeezing trans- formation S[α], a parallelogram in an unsqueezed proﬁle corresponds to a rectangle in a squeezed proﬁle. Impor- tantly, it means that we do not need to implement a squeezing operation on a given state in order to have an enhanced test. Instead, we may simply choose the four points at the vertices of the parallelogram corresponding to the squeezing operation and do our tests for the given initial state. On the other hand, for a nonzero s < 0, the above argument is not directly applicable. This is because the s-parametrized function of a state after squeezing is not simply obtained by reshaping of the original s- parametrized function (squeezing of the proﬁle). Instead, we ﬁnd the following identity (with its proof in the Ap- pendix): (cid:18) (cid:19) − J J (cid:48) π(1 s) π √ α FpoIGin.ts8:of(Caolroerctoannlginlee)((raig,hbt)tri0anagnlde) f0oropSˆt{imf|i0z(cid:105)e(cid:104)d0|o+ver(1th−e 2 Wρ(S[α]; s) = 2 WL(cid:48)[SˆρSˆ†] 1 − s; 0 , (39) | (cid:105)(cid:104) |} † f) 2 2 Sˆ as functions of the vacuum fraction f. Blue solid, L(cid:48) purple dotted, and red dashed lines denote the case of squeez- where [ρ] represents a beam-splitter interaction with { } ianngdarpp=lie{d0t,o1,a2}giv(ebn),straestpe,ecwtiitvhelsyq.ueWezeinﬁgnrd =no 0v,i0o.l5a,t1ion(ao)f tarnadnsamistqtauneeczeedη v=acu1u−1ms Sbˆ|e0t(cid:105)w(cid:104)e0e|Snˆ†.a qTuhaenrteufomre,staatlle sρ- Eq. (36) for an initial state (r = 0), while a state with a suﬃ- parametrized functions taken at the vertices of a par- ciently large squeezing violates Eq. (36) (blue colored region) even when its Wigner function is positive (f > 1). (c, d) alleogram can be understood as the Wigner function Nansglaen)dinNEs(cid:48) qosp.ti(m41iz)edanodve(r42th) efoprofin|0ts(cid:105)(cid:104)o0f| a+p(a1r−allfel)o|22g(cid:105)r(cid:104)a2m| w(tirtih- tsatakteenLa(cid:48)t[SˆtρhSˆe†]v.erAtisceths eowf haolreecptraoncgelses fLor(cid:48)[SˆtρhSeˆ†d] eiscoGhearuesd- respect to f. Black solid, orange dotted, brown dashed lines sian, it does not create non-Gaussianity. We then use { − − } { − − } represent s = 0, 0.01, 0.02 (c) and s = 0, 0.02, 0.04 the Gaussian bounds obtained for s = 0 in order to de- (d), respectively. The Wigner function s = 0 shows the best tect genuine non-Gaussianity under the s-parametrized performance to demonstrate quantum non-Gaussianity. functions. We thus propose enhanced non-Gaussianity tests for an arbitrary s as practical side, it implies that the non-Gaussianity test for a decohered state under a lossy channel can be harder than expected from the analysis only based on the s- − N ≤ 8 1 < [ρ ] , parametrized function of the original state. s MG 39/8 − N(cid:48) ≤ A. Optimizing non-Gaussianity tests 1 < [ρ ] 2, (40) s MG The test of genuine non-Gaussianity in Eq. (35) can be further enhanced by applying squeezing Sˆ on a given with state. Note that a mixture of Gaussian state remains to be a Gaussian mixture under Gaussian operations. Thus, − N π(1 s){ if the state under squeezing shows violation of Gaus- [ρ] = W (S[q + ip ]; s) + W (S[q + ip ]; s) s 2 ρ 0 0 ρ 1 0 sian bounds, the original state must be genuinely non- − } + W (S[q + ip ]; s) W (S[q + ip ]; s) , (41) Gaussian. For example, we show the case of the mixed ρ 0 1 ρ 1 1 | (cid:105)(cid:104) | − | (cid:105)(cid:104) | state f 0 0 + (1 f) 2 2 in Figs. 8(a) and 8(b), which shows that the squeezing operation on the state enhances and the detected region. For the case of a Wigner function, we may address the − problem by using its interesting property [3] N(cid:48) π(1 s){ [ρ] = W (S[q + ip ]; s) + W (S[q + ip ]; s) s 2 ρ 1 0 ρ 0 1 − } WSˆρSˆ†(α; 0) = Wρ(S[α]; 0), (38) Wρ(S[q1 + ip1]; s) , (42) ∗ where S[α] = α cosh r+α e2iφ sinh r is the transforma- which take into account the vertices of parallelo- tion of phase-space points due to the squeezing operation gram and triangle, respectively, with S[α] = α cosh r + ##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##

text: Sˆ.

text: As

text: we

text: illustrate

text: in

text: Fig.

text: under

text: the

text: squeezing

text: trans-

text: formation

text: S[α],

text: a

text: parallelogram

text: in

text: an

text: unsqueezed

text: proﬁle

text: corresponds

text: to

text: a

text: rectangle

text: in

text: a

text: squeezed

text: proﬁle.

text: Impor-

text: tantly,

text: it

text: means

text: that

text: we

text: do

text: not

text: need

text: to

text: implement

text: a

text: squeezing

text: operation

text: on

text: a

text: given

text: state

text: in

text: order

text: to

text: have

text: an

text: enhanced

text: test.

text: Instead,

text: we

text: may

text: simply

text: choose

text: the

text: four

text: points

text: at

text: the

text: vertices

text: of

text: the

text: parallelogram

text: corresponding

text: to

text: the

text: squeezing

text: operation

text: and

text: do

text: our

text: tests

text: for

text: the

text: given

text: initial

text: state.

text: On

text: the

text: other

text: hand,

text: for

text: a

text: nonzero

text: s

text: the

text: above

text: argument

text: is

text: not

text: directly

text: applicable.

text: This

text: is

text: because

text: the

text: s-parametrized

text: function

text: of

text: a

text: state

text: after

text: squeezing

text: is

text: not

text: simply

text: obtained

text: by

text: reshaping

text: of

text: the

text: original

text: s-

text: parametrized

text: function

text: (squeezing

text: of

text: the

text: proﬁle).

text: Instead,

text: we

text: ﬁnd

text: the

text: following

text: identity

text: (with

text: its

text: proof

text: in

text: the

text: Ap-

text: pendix):

text: J

text: J

text: s)

text: π

text: α

text: Wρ(S[α];

text: s)

text: s;

text: f)

text: Sˆ

text: as

text: functions

text: of

text: the

text: vacuum

text: fraction

text: f.

text: Blue

text: solid,

text: purple

text: dotted,

text: and

text: red

text: dashed

text: lines

text: denote

text: the

text: case

text: of

text: squeez-

text: where

text: [ρ]

text: represents

text: a

text: beam-splitter

text: interaction

text: with

text: =no

text: tarnadnsamistqtauneeczeedη

text: qTuhaenrteufomre,staatlle

text: sρ-

text: Eq.

text: for

text: an

text: initial

text: state

text: (r

text: while

text: a

text: state

text: with

text: a

text: suﬃ-

text: parametrized

text: functions

text: taken

text: at

text: the

text: vertices

text: of

text: a

text: par-

text: ciently

text: large

text: squeezing

text: violates

text: Eq.

text: (blue

text: colored

text: region)

text: even

text: when

text: its

text: Wigner

text: function

text: is

text: positive

text: (f

text: (c,

text: d)

text: alleogram

text: can

text: be

text: understood

text: as

text: the

text: Wigner

text: function

text: w(tirtih-

text: eowf

text: haolreecptraoncgelses

text: eiscoGhearuesd-

text: respect

text: to

text: f.

text: Black

text: solid,

text: orange

text: dotted,

text: brown

text: dashed

text: lines

text: sian,

text: it

text: does

text: not

text: create

text: non-Gaussianity.

text: We

text: then

text: use

text: represent

text: s

text: (c)

text: and

text: s

text: the

text: Gaussian

text: bounds

text: obtained

text: for

text: s

text: in

text: order

text: to

text: de-

text: (d),

text: respectively.

text: The

text: Wigner

text: function

text: s

text: shows

text: the

text: best

text: tect

text: genuine

text: non-Gaussianity

text: under

text: the

text: s-parametrized

text: performance

text: to

text: demonstrate

text: quantum

text: non-Gaussianity.

text: functions.

text: We

text: thus

text: propose

text: enhanced

text: non-Gaussianity

text: tests

text: for

text: an

text: arbitrary

text: s

text: as

text: practical

text: side,

text: it

text: implies

text: that

text: the

text: non-Gaussianity

text: test

text: for

text: a

text: decohered

text: state

text: under

text: a

text: lossy

text: channel

text: can

text: be

text: harder

text: than

text: expected

text: from

text: the

text: analysis

text: only

text: based

text: on

text: the

text: s-

text: N

text: [ρ

text: parametrized

text: function

text: of

text: the

text: original

text: state.

text: s

text: MG

text: A.

text: Optimizing

text: non-Gaussianity

text: tests

text: [ρ

text: s

text: MG

text: The

text: test

text: of

text: genuine

text: non-Gaussianity

text: in

text: Eq.

text: can

text: be

text: further

text: enhanced

text: by

text: applying

text: squeezing

text: Sˆ

text: on

text: a

text: given

text: with

text: state.

text: Note

text: that

text: a

text: mixture

text: of

text: Gaussian

text: state

text: remains

text: to

text: be

text: a

text: Gaussian

text: mixture

text: under

text: Gaussian

text: operations.

text: Thus,

text: N

text: s){

text: if

text: the

text: state

text: under

text: squeezing

text: shows

text: violation

text: of

text: Gaus-

text: [ρ]

text: W

text: (S[q

text: ip

text: s)

text: W

text: (S[q

text: ip

text: s)

text: s

text: ρ

text: ρ

text: sian

text: bounds,

text: the

text: original

text: state

text: must

text: be

text: genuinely

text: non-

text: W

text: (S[q

text: ip

text: s)

text: W

text: (S[q

text: ip

text: s)

text: Gaussian.

text: For

text: example,

text: we

text: show

text: the

text: case

text: of

text: the

text: mixed

text: ρ

text: ρ

text: state

text: f

text: f)

text: in

text: Figs.

text: and

text: which

text: shows

text: that

text: the

text: squeezing

text: operation

text: on

text: the

text: state

text: enhances

text: and

text: the

text: detected

text: region.

text: For

text: the

text: case

text: of

text: a

text: Wigner

text: function,

text: we

text: may

text: address

text: the

text: problem

text: by

text: using

text: its

text: interesting

text: property

text: s){

text: [ρ]

text: W

text: (S[q

text: ip

text: s)

text: W

text: (S[q

text: ip

text: s)

text: s

text: ρ

text: ρ

text: WSˆρSˆ†(α;

text: Wρ(S[α];

text: s)

text: where

text: S[α]

text: α

text: cosh

text: r+α

text: sinh

text: r

text: is

text: the

text: transforma-

text: which

text: take

text: into

text: account

text: the

text: vertices

text: of

text: parallelo-

text: tion

text: of

text: phase-space

text: points

text: due

text: to

text: the

text: squeezing

text: operation

text: gram

text: and

text: triangle,

text: respectively,

text: with

text: S[α]

text: α

text: cosh

text: r

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: MNRAS in press, 1–11 (2015) Preprint 28th May 2020 Compiled using MNRAS LATEX style ﬁle v3.0 10C continued: a deeper radio survey at 15.7 GHz I. H. Whittam1,2(cid:63), J. M. Riley2, D. A. Green2, M. L. Davies2, T. M. O. Franzen3, C. Rumsey2, M. P. Schammel and E. M. Waldram2. 1Physics Department, University of the Western Cape, Bellville 7535, South Africa 2Astrophysics Group, Cavendish Laboratory, 19 J. J. Thomson Avenue, Cambridge CB3 0HE 3 International Centre for Radio Astronomy Research (ICRAR), Curtin University, Bentley, WA 6102, Australia Accepted —; received —; in original form — ABSTRACT We present deep 15.7-GHz observations made with the Arcminute Microkelvin Imager Large Array in two ﬁelds previously observed as part of the Tenth Cambridge (10C) survey. These observations allow the source counts to be calculated down to 0.1 mJy, a factor of ﬁve deeper than achieved by the 10C survey. The new source counts are consistent with the extrapolated ﬁt to the 10C source count, and display no evidence for either steepening or ﬂattening of the counts. There is thus no evidence for the emergence of a signiﬁcant new population of sources (e.g. starforming) at 15.7 GHz ﬂux densities above 0.1 mJy, the ﬂux density level at which we expect starforming galaxies to begin to contribute. Comparisons with the de Zotti et al. model and the SKADS Simulated Sky show that they both underestimate the observed number of sources by a factor of two at this ﬂux density level. We suggest that this is due to the ﬂat- spectrum cores of radio galaxies contributing more signiﬁcantly to the counts than predicted by the models. Key words: galaxies: active – radio continuum: galaxies – catalogues – surveys 1 INTRODUCTION 2008) at 15.7 GHz. The observations, mapping and source extrac- tion are described in AMI Consortium: Franzen et al. (2011) (here- The high-frequency radio sky (ν (cid:38) 10 GHz) has been much after Paper F) and the source counts and spectral properties are less widely studied than the population at lower radio frequencies presented in AMI Consortium: Davies et al. (2011) (hereafter Pa- (e.g. 1.4 GHz), mainly due to the increased time required to sur- per D). The 10C survey is complete to 1 mJy in ten diﬀerent ﬁelds vey an area to an equivalent depth at higher frequencies. In re- ≈ ≈ covering a total of 27 deg2; deep areas covering 12 deg2, con- cent years several high-frequency surveys have been conducted, tained within these ﬁelds, are complete to 0.5 mJy, making the 10C albeit with higher ﬂux density limits than surveys at lower frequen- survey the deepest high-frequency radio survey published to date. cies. Waldram et al. (2003, 2010) carried out the Ninth Cambridge (9C) survey at 15 GHz using the Ryle Telescope. This survey cov- Whittam et al. (2013) used data at a range of frequencies to ≈ ers 520 deg2 to a completeness limit of 25 mJy, and was the study the spectral indices of 10C sources in the Lockman Hole. ﬁrst high-frequency survey to cover a signiﬁcant proportion of the They found a signiﬁcant change in spectral index with ﬂux dens- ∝ − sky. A series of deeper regions were also observed (Waldram et al. ity; the median spectral index α (where S ν α for a source with ≈ 2010), with 115 deg2 complete to 10 mJy and 29 deg2 complete ﬂux density S at frequency ν) calculated between 1.4 and 15.7 GHz ≈ to 5.5 mJy. changes from 0.75 for sources with S15.7 GHz > 1.5 mJy to 0.08 for The whole Southern sky has been surveyed by the Australia sources with S15.7 GHz < 0.8 mJy. This shows that there is a popu- Telescope 20 GHz (AT20G) survey (Massardi et al. 2011), which lation of ﬂat-spectrum sources emerging below 1 mJy; Whittam et has a ﬂux density limit of 40 mJy and is 93 per cent complete al. suggest that this may be due to the cores of Fanaroﬀ and Riley above 100 mJy. This survey is complementary to the 9C survey type I (FRI; Fanaroﬀ & Riley 1974) sources becoming dominant at as it covers a larger area at a shallower ﬂux density. More recently, 15.7 GHz. the AT20G-deep pilot survey (Franzen et al. 2014) surveyed 5 deg2 There have been several attempts to model the high-frequency to a completeness level of 2.5 mJy. radio sky, often extrapolating from lower frequencies. Early evolu- The Tenth Cambridge (10C) survey was observed with the tionary models of radio sources (Dunlop & Peacock 1990; Jackson Cambridge Arcminute Microkelvin Imager (AMI; Zwart et al. & Wall 1999; Toﬀolatti et al. 1998) successfully ﬁtted the available data at frequencies (cid:46) 10 GHz down to ﬂux densities of a few mJy. More recently, de Zotti et al. (2005) produced a model of the ra- (cid:63) email: i.whittam@mrao.cam.ac.uk dio source counts at frequencies (cid:38) 5 GHz (up to 30 GHz) which (cid:13) c 2015 The Authors

text: MNRAS

text: in

text: press,

text: Preprint

text: May

text: Compiled

text: using

text: MNRAS

text: LATEX

text: style

text: ﬁle

label: title

text: continued:

text: a

text: deeper

text: radio

text: survey

text: at

text: GHz

text: I.

label: author

text: H.

text: J.

text: M.

text: D.

text: A.

text: M.

text: L.

text: T.

text: M.

text: O.

text: C.

text: M.

text: P.

text: Schammel

text: and

text: E.

text: M.

text: Department,

text: University

text: of

text: the

text: Western

text: Cape,

text: Bellville

text: South

text: Africa

text: Group,

text: Cavendish

text: Laboratory,

text: J.

text: J.

text: Thomson

text: Avenue,

text: Cambridge

text: International

text: Centre

text: for

text: Radio

text: Astronomy

text: Research

text: (ICRAR),

text: Curtin

text: University,

text: Bentley,

text: WA

text: Australia

text: Accepted

label: date

text: received

text: in

text: original

text: form

text: ABSTRACT

text: We

label: abstract

text: present

text: deep

text: observations

text: made

text: with

text: the

text: Arcminute

text: Microkelvin

text: Imager

text: Large

text: Array

text: in

text: two

text: ﬁelds

text: previously

text: observed

text: as

text: part

text: of

text: the

text: Tenth

text: Cambridge

text: survey.

text: These

text: observations

text: allow

text: the

text: source

text: counts

text: to

text: be

text: calculated

text: down

text: to

text: mJy,

text: a

text: factor

text: of

text: ﬁve

text: deeper

text: than

text: achieved

text: by

text: the

text: survey.

text: The

text: new

text: source

text: counts

text: are

text: consistent

text: with

text: the

text: extrapolated

text: ﬁt

text: to

text: the

text: source

text: count,

text: and

text: display

text: no

text: evidence

text: for

text: either

text: steepening

text: or

text: ﬂattening

text: of

text: the

text: counts.

text: There

text: is

text: thus

text: no

text: evidence

text: for

text: the

text: emergence

text: of

text: a

text: signiﬁcant

text: new

text: population

text: of

text: sources

text: (e.g.

text: starforming)

text: at

text: GHz

text: ﬂux

text: densities

text: above

text: mJy,

text: the

text: ﬂux

text: density

text: level

text: at

text: which

text: we

text: expect

text: starforming

text: galaxies

text: to

text: begin

text: to

text: contribute.

text: Comparisons

text: with

text: the

text: de

text: Zotti

text: et

text: al.

text: model

text: and

text: the

text: SKADS

text: Simulated

text: Sky

text: show

text: that

text: they

text: both

text: underestimate

text: the

text: observed

text: number

text: of

text: sources

text: by

text: a

text: factor

text: of

text: two

text: at

text: this

text: ﬂux

text: density

text: level.

text: We

text: suggest

text: that

text: this

text: is

text: due

text: to

text: the

text: ﬂat-

text: spectrum

text: cores

text: of

text: radio

text: galaxies

text: contributing

text: more

text: signiﬁcantly

text: to

text: the

text: counts

text: than

text: predicted

text: by

text: the

text: models.

text: Key

text: words:

text: galaxies:

text: active

text: radio

text: continuum:

text: galaxies

text: catalogues

text: surveys

text: INTRODUCTION

text: at

text: GHz.

text: The

text: observations,

text: mapping

text: and

text: source

text: extrac-

text: tion

text: are

text: described

text: in

text: AMI

text: Consortium:

text: Franzen

text: et

text: al.

text: (here-

text: The

text: high-frequency

text: radio

text: sky

text: (ν

text: GHz)

text: has

text: been

text: much

text: after

text: Paper

text: F)

text: and

text: the

text: source

text: counts

text: and

text: spectral

text: properties

text: are

text: less

text: widely

text: studied

text: than

text: the

text: population

text: at

text: lower

text: radio

text: frequencies

text: presented

text: in

text: AMI

text: Consortium:

text: Davies

text: et

text: al.

text: (hereafter

text: Pa-

text: (e.g.

text: GHz),

text: mainly

text: due

text: to

text: the

text: increased

text: time

text: required

text: to

text: sur-

text: per

text: D).

text: The

text: survey

text: is

text: complete

text: to

text: mJy

text: in

text: ten

text: diﬀerent

text: ﬁelds

text: vey

text: an

text: area

text: to

text: an

text: equivalent

text: depth

text: at

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