Demonstrating nonclassicality and non-Gaussianity of single-mode fields: Bell-type tests using generalized phase-space distributions

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by  Jiyong Park

Demonstrating nonclassicality and non-Gaussianity of single-mode fields: Bell-type
tests using generalized phase-space distributions
Jiyong Park1 and Hyunchul Nha1,2
1Department of Physics, Texas A&M University at Qatar, Education City, P.O.Box 23874, Doha, Qatar
2School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-722, Korea
(Dated: January 5, 2016)
WepresentBell-typetestsofnonclassicalityandnon-Gaussianityforsingle-modefieldsemploying
ageneralized quasiprobability function. Ournonclassicality testsarebasedon theobservation that
twoorthogonalquadraturesinphasespace(positionandmomentum)behaveasindependentrealistic
variablesforacoherentstate. Takingfour(three)pointsattheverticesofarectangle(righttriangle)
inphasespace,ourtestsdetecteverypurenonclassicalGaussianstateandarangeofmixedGaussian
6
states. ThesetestsalsosetanupperboundforallGaussianstatesandtheirmixtures,whichthereby
1
provide criteria for genuine quantum non-Gaussianity. We optimize the non-Gaussianity tests by
0
employinga squeezingtransformation in phase spacethat convertsa rectangle (right triangle) toa
2
parallelogram(triangle),whichenlargesthesetofnon-Gaussianstatesdetectableinourformulation.

n Weaddressfundamentalandpracticallimitsofourgeneralizedphase-spacetestsbylookingintotheir
a relation with decoherence under a lossy Gaussian channel and their robustness against finite data
J
andnon-optimalchoiceofphase-spacepoints. Furthermore,wedemonstratethatourparallelogram

3 test can identify useful resources for nonlocality testing in phasespace.

] PACSnumbers: 03.65.Ta,42.50.Dv,42.50.Ar
h
p
-
t I. INTRODUCTION quasiprobabilityfunctions. Wenotonlygivemoredetails
n
of the proposal in [16], but we also investigate other rel-
a
u Describingaquantumstateinphasespace[1]isavery evantaspects,e.g.,robustnessofourtests againstexper-
q usefulapproachtomakeacomparisonbetweenquantum imental imperfections including photon loss, finite data,
[ and a nonoptimal choice of phase-space points. Further-
mechanics and classical mechanics. It provides a valu-
more, we introduce an optimized test of genuine non-
1 ableinsightintothephenomenonofquantum-to-classical
Gaussianity employing three phase-space points, as an
v transition [2] and a powerful tool to manifest nonclassi-
9 cal effects in quantum optics [3] and continuous variable addition to the four-point test in [16]. We also make a
7 (CV) quantum informatics [4, 5]. One of the remark- direct connection between our single-mode test and the
2 BWnonlocalitytest,particularlyshowingthatthesingle-
able distinctions between quantum and classical phase-
0 mode nonclassical states detected under our parallelo-
space distributions is that a negative value is allowedfor
0
gram test can be a useful resource to make a two-mode
. aquantumstate. Althoughthenegativityinphasespace
1 state manifesting nonlocality under the BW test.
thusdemonstratesnonclassicalityimmediately,itenables
0
6 us to detect only a limited subset of nonclassical states. Ourstartingpointistheobservationthateverypairof
1 There exist nonclassicalstates with positive-definite dis- orthogonal quadratures in phase space behaves as inde-
: tributions, e.g., Gaussian states with squeezing, which pendent realistic variables for a coherent state. Exploit-
v
arereadilyaccessible withincurrenttechnologyand pro- ingit,weproposetwononclassicalityteststhattakefour
i
X videimportantpracticalresourcesforCVquantuminfor- andthreepointsattheverticesofarectangleandaright
r matics[5]. Itisfundamentallyandpracticallyimportant triangle, respectively. Our tests detect a broad range of
a to have a simple test manifesting nonclassicality [6, 7] nonclassical Gaussian states, including all pure states.
beyond the negativity in phase space. Identifying the upper bounds for all Gaussianstates and
In this respect, there was a seminal work by Banaszek their mixtures, we also propose tests for genuine quan-
and W´odkiewicz [8] (BW), who proposed a method to tumnon-Gaussianity. Non-Gaussianresourcesareknown
test the Bell nonlocality directly in phase space. Unlike to be essential for many quantum informatic tasks, in-
the Bell test using homodyne detection, which requires cluding universal CV quantum computation [17], entan-
thetransformationofaGaussianstatetoanon-Gaussian glement distillation [18], quantum error correction [19],
state having a nonpositive Wigner function [9, 10], BW and CV nonlocality testing [10, 20]. A simple method
formalism enables us to detect nonclassical correlation to obtain a non-Gaussian state would be to prepare a
even with a positive-definite Wigner function. It has finite mixture of Gaussian states. However, we cannot
been extended to generalized quasiprobability functions claim such a state as a genuine non-Gaussian resource.
[11] and multipartite systems [12–15]. Recently, we have Indealingwithnon-Gaussianityinquantumphasespace,
theoretically proposedand experimentally demonstrated it is important to distinguish a genuinely quantum non-
asingle-modenonclassicalitytestusingtheWignerfunc- GaussianstatefromamixtureofGaussianstates[21,22].
tion [16] in analogy with BW formalism. We further optimize our non-Gaussianitytests by taking
We here extend this recent work by using generalized pointsfromaparallelogram(triangle)insteadofarectan-
2
gle(righttriangle),whichessentiallyrealizesasqueezing
operationonagivenstatewithoutactuallyimplementing
it.
Wealsodiscussthefundamentalandthepracticallim-
its of our tests. Note that there exists a one-to-one cor-
respondence between a s-parametrized phase-space dis-
tribution and a loss mechanism, i.e., interaction with a
vacuum reservoir [11, 23]. This correspondence might
suggest that the limits of the s-parametrized distribu-
tionsarethesameasthoseoftheWignerfunction(s=0)
underalossyGaussianchannel. However,they canyield
different results for a nontrivial test. We show that the
FIG.1: (Coloronline)(a)NonclassicalitytestinEq.(6)takes
former can detect more states than the latter when a
four points at the vertices of a rectangle. Neglecting a point
test sets a bound varying with the parameter s, e.g., the
(x ,y ), the remaining three points form a right triangle in
0 0
case of our non-Gaussianity tests. We demonstrate that Eq.(9). (b)Optimalchoiceofrectanglefortestingasqueezed
our tests are able to detect a nonclassical state reliably state occurs at θ−φ = π (φ: squeezing axis), that is, when
4
evenwhen the number of data is finite andthe measure- the axes of the rectangle are oriented midway between the
ment setting deviates from the optimal setting. We also squeezed and the antisqueezed axes.
showthatourparallelogramtestcandetectgenuinenon-
Gaussianity for a range of superposition states with loss
the parameter s can have a positive value up to 1 (P
above 50%, at which the Wigner function becomes pos-
function), we only deal with a nonpositive s throughout
itive definite. Finally, we show that our parallelogram
testcanidentify usefulresourcesfor a nonlocalitytestin the paper as the eigenvalues of Tˆ(s) become unbounded
phase space. It may open a direction for future works, for s>0.
e.g., on a deeper understanding of the relation between Interestingly, the s-parametrized quasiprobability
nonclassicality and nonlocality. functionofacoherentstate α isfactorizedasaproduct
| i
oftwoGaussiandistributionsforeverypairoforthogonal
quadratures,
II. NONCLASSICALITY TESTS
π(1 s)
− W (x,y;s)
2 |αihα|
The s-parametrized quasiprobability function of a
2(x α )2 2(y α )2
quantum state ρ is defined as [3] =exp − x exp − y , (4)
(cid:18)− 1 s (cid:19) (cid:18)− 1 s (cid:19)
− −
2
Wρ(q,p;s)= tr[ρDˆ†(α)Tˆ(s)Dˆ(α)], (1) where (x,y)T = (θ)(q,p)T is a coordinate system ro-
π(1 s) R
− tated by an angle θ from (q,p) [Fig. 1 (a)], with a rota-
where Dˆ(α)=exp(αaˆ α aˆ) is the displacement oper- tion matrix
† ∗
−
ator with complex amplitude α = q+ip. The operator cosθ sinθ
Tˆ(s) is given by R(θ)=(cid:18) sinθ cosθ(cid:19), (5)
−
Tˆ(s) s+1 nˆ = ∞ s+1 n n n, (2) andαx =Re[αe−iθ]andαy =Im[αe−iθ]. Itisthuspossi-
≡(cid:18)s 1(cid:19) (cid:18)s 1(cid:19) | ih | ble to consider the generalized quasiprobability function
− nX=0 − of a coherent state as a product of two independent ran-
wuuhmichstyaiteeld0s, e0.gf.o,rasp=ar0ityanodper1a,tocror(r−es1p)onˆndanindgatovtahce- wdoitmhv0a<riaab,lbesa1a.ndb,thatis, π(12−s)W|αihα|(x,y;s)=ab
| ih | − ≤
well-knownWignerandQfunctions,respectively. Asthe
eigenvaluesofTˆ(s)are(s+1)n(n: non-negativeintegers),
s 1
the s-parametrizedquasi−probability function is bounded
as A. Rectangle test
s+1 π(1 s)
− W (q,p;s) 1 for 1 s 0, We then construct a linear sum of s parametrized
s 1 ≤ 2 ρ ≤ − ≤ ≤ functions at four phase-space points as −
−
π(1 s)
0< − Wρ(q,p;s) 1 for s< 1. (3) π(1 s)
2 ≤ − [ρ] − W (x ,y ;s)+W (x ,y ;s)
s ρ 0 0 ρ 1 0
J ≡ 2 {
This shows that the lower bound becomes minimum for
+W (x ,y ;s) W (x ,y ;s) , (6)
ρ 0 1 ρ 1 1
the Wigner function (s = 0) and approaches zero with − }
s decreasing. The above equation also clearly tells us wherethepointsconstitutesarectangleorientedatangle
that the Q function (s = 1) is non-negative. While θ in phase space as depicted in Fig. 1. We then obtain
−
3
=a b +a b +a b a b foracoherentstate,which asfollows: (i)Fora <a ,wehave 1< (a a )b <
s 0 0 1 0 0 1 1 1 0 1 1 0 1
J − − − −
has the same form as the Clauser-Horne-Shimony-Holt < a b 1. The minimum is achieved by a =
Js′ 1 0 ≤ 0
(CHSH) inequality [24]. Using 0 < a,b 1, we obtain b = 0 and a = b = 1. (ii) For a a , we have
0 1 1 0 1
≤ ≥
1 < 2 for a coherent state as follows. (i) For 0< <a +(a a ) 1. The maximum is achieved
− Js ≤ Js′ 1 0− 1 ≤
a <a ,wehave 1< (a a )b < <(a +a )b by a =b =a =b =1.
0 1 1 0 1 s 0 1 0 0 0 1 1
− − − J ≤
2. The minimum is given by, e.g., a = b = 0 and Therefore, similar to Eq. (7), we obtain another clas-
0 0
a = b = 1. (ii) For a a , we have 0 < sicality condition as
1 1 0 1 s
≥ J ≤
(a +a )+(a a ) 2. Themaximumisgivenby,e.g.,
0 1 0 1
− ≤
a0 =b0 =a1 =b1 =1. −1<Js′[ρcl]≤1. (10)
We extend the above result to an arbitrary classi-
cal state, i.e. a mixture of coherent states, ρ = Wenotethat,contrarytotherectangletestinEq.(6),it
cl
dλp(λ)λ λ, where λ is a coherent state. As the has no analogy with a nonlocality test, as a1b0+a0b1
| ih | | i −
Rcoherent amplitude λ behaves like a hidden variable, a1b1 is saturated by a hidden variable theory: a0 =b0 =
a = b = 1 yields 3, which are also the quantum
1 1
π(12−s)Wρcl(x,y;s)=Z dλp(λ)a(x|λ)b(y|λ), (7) −bounds due t±o (cid:12)π(12−s)W±s(x,y)(cid:12)≤1.
(cid:12) (cid:12)
(cid:12) (cid:12)
we obtain a classicality condition as
1. Invariance under displacement and phase-rotation
1< s[ρcl] 2. (8) operations
− J ≤
Inotherwords,theviolationofEq.(8)demonstratesthe
Note that the optimal values of [ρ] and [ρ] for a
nonclassicality of a single-mode state. Js Js′
given state ρ are invariant under displacement and ro-
At this point, it may be intriguing to ask how many
tation. Let us assume that a state ρ has an optimal
phase-space points should be considered to come up
valueatpoints x ,y ,x ,y ,andthenadisplacedstate
with a meaningful nonclassicality test, particularly to { 0 0 1 1}
Dˆ(α)ρDˆ (α) has the same optimum at shifted points
test a positive quasiprobability distribution. Can we †
x + α ,y + α ,x + α ,y + α . This is because
obtain a useful nonclassicality criterion employing less { 0 x 0 y 1 x 1 y}
the displacement operator only translates the center of
numbers of points than four in Eq. (6)? Of course,
the quasiprobability function while preserving its entire
verifying nonclassicality from an arbitrary set of points
profile. Similarly, if a state has the optimal value at
is impossible without specifications on the chosen
points x ,y ,x ,y wherethecoordinatesystemisori-
pwoitinhtsen{eqrjg,ypjc}o,nlsitkreaiantpirned[e2t2e]rmorineaddpeosisgitniaotned(osrhigaipne) ented a{t a0ng0le θ1, a1r}otated state eiϕaˆ†aˆρe−iϕaˆ†aˆ has the
(rectangle)in ourcase[16]. Forexample, ifwe construct same optimum in the coordinate system now orientedat
a test exploiting the values at fully arbitrary N points angle θ+ϕ, since the phase-rotation also preserves the
without specifying locations, that is, [v ,...,v ] where profileofthequasiprobabilityfunction. Theseinvariance
1 N
vj = π(12−s)Wρ(qj,pj;s) with j ∈{1,.F..,N}, every result pclraospsiecratliietsyctaenstbsefoursaefugilvteonssitmatpel.ify the analysis of non-
from a set of positive values (v ,...,v ) is mimicked by
1 N
a single vacuum state because the same values can be
found at (q ,p ) = (0, s 1logv ,s) for j 1,...,N .
j′ ′j q −2 j ∈ { } C. Gaussian states
In this sense, if we intend to introduce a specified shape
as a constraint, the least number of points is possibly
We first demonstrate how our tests and can
three with the shape of triangle, whereas the test in Js Js′
detect a wide range of Gaussian states. A single-mode
Eq. (6) adopts a rectangle with four points.
Gaussian state σ is fully characterized by its first-order
moments (averages) qˆ and pˆ , and second-order mo-
h i h i
ments represented by a covariance matrix Γ with ele-
ments
B. Right-triangle test
1
Γ = Qˆ Qˆ +Qˆ Qˆ Qˆ Qˆ , (j,k =1,2) (11)
Thus we also introduce a three-points (right triangle) jk 2h j k k ji−h jih ki
test as
whereQˆ (qˆ,pˆ)T withqˆ= 1(aˆ+aˆ )andpˆ= 1(aˆ aˆ ).
π(1 s) ≡ 2 † 2i − †
Js′[ρ]≡ 2− {Ws(x1,y0)+Ws(x0,y1) fIutsncst-ipoanraams etrized distribution is given by a Gaussian
W (x ,y ) , (9)
s 1 1
− }
W (q,p;s)
σ
which excludes one point (x ,y ) from Eq. (6). We then
0 0
2f 1
hstaavtee.aUstsrinugct0ur<e Ja,s′b= a11ba0g+aina,0bw1e−obat1abi1nfor1a<coheren1t = π(1 ss)exp(cid:20)− 2(Q−hQi)TΓ−s1(Q−hQi)(cid:21), (12)
≤ − Js′ ≤ −
4
FIG. 2: (Color online) (a, b) Maximum Js=0 and Js′=0 (test with Wigner function s=0) for Gaussian states with respect to
squeezing strength κ0 =2tanh2r. (c, d) Maximum values of Js and Js′, respectively, among all Gaussian states with respect
to s (generalized quasidistributions). These maximum values become critical bounds to test genuine non-Gaussianity for each
s inEq. (35). (e,f) Contourplots foroptimal Js and Js′ with respect tos andκ0. Black dashedlines representthesqueezing
strengthκ thatyieldsthemaximalvaluesshown in(c)and(d). (g,h)Contourplotsofacriticalparameters foraGaussian
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
where Q=(q,p)T, f = 1 s and Γ =Γ sI. Equation(15)showsthattheoverallfactorofW (q,p;s)
Alternatively, a sinsgle-4m√od−edteΓsGaussiasn stat−e σ4 can be in Eq. (12) is bounded by 0 < fs 1. Its mσaximum
≤
represented as a displaced squeezed thermal state, fs = 1 is achieved by every pure Gaussian state (n¯ =0)
for s = 0, as f = 1 represents the purity of a
σ =Dˆ(α)Sˆ(r,φ)σ (n¯)Sˆ (r,φ)Dˆ (α), (13) s=0 1+2n¯
th † † Gaussian state µ trσ2 = 1 . On the other hand,
whereSˆ(r,φ)=exp[ r(e2iφaˆ 2 e 2iφaˆ2)]isthesqueez- only a vacuum sta≡te attains1t+h2en¯ maximum f = 1 for
−2 † − − s
ing operator (r: squeezing strength, φ: squeezing axis), s < 0. In general, with r and s fixed, f increases with
s
and σth(n¯) = ∞n=0 (n¯+n¯1n)n+1|nihn| is a thermal state purity (n¯ decreasing).
with mean photPon number n¯. For a Gaussian state with Rewriting Eq.(12) using the rotatedquadraturesQ=
parameters α,r,φ,n¯ , its first moments are given by (x,y)T = (θ)(q,p)T, we obtain
{ } R e
qˆ = Re[α] and pˆ = Im[α], and the covariance matrix
h i h i W (x,y;s)
elements by σ
2f 1
1 1 = s exp (Q Q )TΓ 1(Q Q ) , (16)
Γ11 = n¯+ (cosh2r sinh2rcos2φ), π(1 s) (cid:20)− 2 −h i −s −h i (cid:21)
2(cid:18) 2(cid:19) − −
e e e e e
1 1 where Γ = (θ)Γ ( θ) is the covariance matrix in a
Γ22 = 2(cid:18)n¯+ 2(cid:19)(cosh2r+sinh2rcos2φ), rotated sframRe andsRQ−= (Re[αe iθ],Im[αe iθ]). From
e h i − −
1 1 now on, we set α = 0, as the displacement operation
Γ12 =Γ21 = n¯+ sinh2rsin2φ, (14) has no effect on the oeptimal values (Sec. II B 1). Every
−2(cid:18) 2(cid:19)
Gaussian function in the form of Eq. (16) can be recast
which yields to
1 s
fs = (1+2n¯)2+s2 −2(1+2n¯)scosh2r. (15) π(12−s)Wσ(x,y;s)=fsexp[−x2−y2+ksxy], (17)
−
p
e e ee
5
by introducing rescaled variables, κ decrease with the parameter s decreasing. We thus
s
observethat maximumvalues amongallGaussianstates
2√2fs s s for rectangle and right triangle tests, respectively, be-
(x,y)= Γ x, Γ y , (18)
1 s (cid:18)r 22− 4 r 11− 4 (cid:19) comesmallerwiththeparametersdecreasinginFig.2(c)
−
e e e e and2(d), respectively. Inaddition,althoughtheoptimal
and the parameter κ in Eq. (20) increases with the thermal photon n¯ for
s
s<0, the overallfactor f in Eq. (15) decreases with n¯,
s
2Γ12 which eventually makes the case of a pure state (n¯ = 0)
k =
s
optimal for given r and s.
(Γ es)(Γ s)
q 11− 4 22− 4
We plot the optimal values of [Fig. 2 (e)] and
e 2esinh2rsin2(θ φ) Js Js′
= − . (19) [Fig. 2 (f)] with respect to parameter s and squeezing
( s cosh2r)2 sinh22rcos22(θ φ) strength κ =2tanh2r. In these contour plots, we show
q 1+2n − − − howtheop0timalsqueezingκ foreachmaximumchanges
0
From the rescaled distributions in Eq. (17), we see that with s (black dashed lines). Interestingly, we note that
twoparameters,i.e. the overallfactorf andthe param- theoptimalviolationforanonzeros<0occursatafinite
s
eter k , determine optimal values and for a given squeezing, similar to the case of a two-mode nonlocality
s Js Js′
Gaussian state. In fact, we can show that these optimal test[11]. Withtheparametersdecreasing,themaximum
values monotonically increase with k as well as f [16]. values andthe correspondingsqueezingstrengthbecome
s s
We point out that the parameter k for every Gaussian smaller. At s = 1 and s = 2, there is no Gaussian
s − −
state is bounded by 2 k 2. In particular, with s state violating the rectangle test and the triangle test,
s
− ≤ ≤
and n fixed, the parameter k is bounded by respectively, which suggests that the right-triangle test
s
is more useful practically for detecting Gaussian states.
2sinh2r
|ks|≤ s +cosh2r ≡κs, (20) In Figs. 2(g) and 2(h), we also identify the range
−1+2n of mixed Gaussian states (colored region) that can be
detected under our nonclassicality tests. Specifically, we
where the upper bound κ is obtained at the choice of
s
angle θ φ = π (see Fig. 1). This optimal choice of plot the critical parameter sc for each Gaussian state
− 4 withpurityµ andsqueezingκ ,abovewhichitsnonclas-
angleintuitively makessense,asournonclassicalitytests 0
sicality can successfully be detected, i.e., in the range
rely on the nonfactorizability of the quasi-probability
s [s ,0]. As the purity µ decreases, we see that the
distributions in phase space. In contrast, if we take ∈ c
squeezing level κ required for a successful test becomes
θ φ=0 or π, the quasiprobabilityfunction in Eq. (16) 0
− 2 higher. In addition, the two contour plots (g) and (h)
is factorized to a form W (x,y;s) = W(x)W(y), where
σ
in comparisonshow that right triangle test detects more
the two quadratures x and y behave as independent
Gaussian states than the rectangle test. The range of
variables, yielding no violation of our nonclassicality
successful detection may be attributed to the ratio of
tests.
maximum Gaussian bound to classical bound in each
test. The rectangle test gives the ratio 8 /2 1.16,
39/8 ≈
whereas the right-triangle test gives 2/1(cid:0)=2,(cid:1)which may
account for the resilience of the latter test compared to
1. Maximum values of Js and Js′ for Gaussian states
the former. For instance, violating 2 and 1
J0 ≥ J0′ ≥
becomes impossible if the purity falls below the inverse
For s = 0 (Wigner function), the overall factor f =
1+12n¯ depends only on purity; thus large J0 and J0′0oc- obfeetnhenuramioesr,ica39l4/ly8 c≈on0fi.8rm6eadn.d 12, respectively, which has
cur for a pure state. On the other hand, the optimal
κ = 2tanh2r in Eq. (20) depend only on the de-
s=0
gree of squeezing. In Figs. 2(a) and 2(b), we show that
the optimal and monotonically increase with κ
J0 J0′ 0
(degree of squeezing). They rise up to the maximum
values = 8 2.32 and = 2 for the rectangle
J0 39/8 ≈ J0′
and the right triangle tests, respectively, both achieved
2. s-parametrized functions and loss mechanism
at an infinite squeezing κ = 2 (r ). (See Supple-
0
→ ∞
mental Matrerial of Ref. [16] for rigorous proofs of the
maximalvalues.) Note that the Gaussianbound 8 for Inaddition,Fig.2showsthatWignerfunction(s=0)
39/8
the four-points test coincides with the maximal value of is optimal among all s-parametrized distributions for
a Gaussian state for two-mode [25] and three-mode [15] both tests, which makes sense as the s-parametrized
nonlocality tests in phase space. quasiprobability function is closely related to loss dy-
Onthe otherhand, for s<0,the overallfactorf and namics under a Gaussian reservoir. The s-parametrized
s
the ratio κ involve both the purity µ and the squeezing quasiprobabilityfunctionsofaquantumstateρwithtwo
s
strengthr. Withpurityandsqueezingfixed,bothf and different parameters s and s (s <s ) are related by a
s 1 2 2 1
6
Gaussian convolution [3], practicalinteresttoidentifytheangletolerance∆,i.e.,a
successfuldetectionintherangeofangles θ φ π ∆.
Wρ(α;s2) This is a particularly important issue wh|e−n th−er4e|≤is n2o
= π(s 2 s )Z d2βWρ(β,s1)e−s1−2s2|α−β|2. (21) iInnfotrhmisactaiosne,oonutrhcehpohicaeseof(saqnugeleeziθngfoarnaglreeφct)aonfgtlhee(tsrtiaatne-.
1 2
− gle) in Fig. 1 becomes completely random. A worstcase
Equation(21) is similar to the actionofa loss channel would be the choice of θ φ=0 or π at which no viola-
L − 2
on a state ρ, tion occurs due to the factorizability of the phase-space
function, as explained below Eq. (20).
W (α;0)
[ρ]
L For given purity µ and squeezing κ , we may take
= 2 d2βWρ(β;0)e−1−2η|α−√ηβ|2, (22) a fixed dimension of rectangle (triangl0e), like the one
π(1 η)Z
− used for an optimal test with known phase, but con-
sider the angle θ randomly distributed over the whole
where the loss channel is modeled by mixing the in-
L range of π. We can then measure a success probabil-
put state ρ and a vacuum at a beam splitter with trans- 2
ity as P = ∆/(π). In Fig. 3, we show how faithfully
mittance η. It provides a direct connection between the s 2
our Wigner-function tests in (a,c,e,g) and in
generalized quasiprobability function and loss dynamics Js=0 Js′=0
(b,d,f,h) detect Gaussian states with unknown phase by
as [11, 23]
evaluatingP foradatanumberN =103 (a,b), N =104
s
1 α 1 s (c,d), N =105 (e,f), andN =106 (g,h). Weseethatour
W (α;s)= W ;1 − . (23)
L[ρ] η ρ(cid:18)√η − η (cid:19) tests can confidently detect a range of mixed squeezed
states with a practical number N = 103 106. In gen-
∼
Inverting Eq. (23) by setting s = 0 and s = 1 1, we eral, the angle tolerance ∆, and thus the success proba-
′ − η bility P , becomes large by increasing the data number
obtain s
N as well as the purity µ and squeezing strength κ. As
π(12−s′)Wρ(√1−s′α;s′)= π2WL[ρ](α;0), (24) acalrlelaimdyitsidoefntpiufireidtyinforFaig.suc2c(egs)sfaunldtes2t(ho)f, thaentdheo′reatrie-
J J
µ 0.83 and µ= 0.5, respectively, which are achievable
which reveals that and in Eqs. (6) and (9) ≈
Js Js′ with N growing. In Fig. 3, we already have similar lev-
correspond to loss dynamics of and , respectively,
with η = 1 . The results inJF0ig. 2 tJhu0′s identify the els of critical purity µ = 0.867 (e) and 0.516 (f) with a
1 s′ finite data N = 105. For the rectangle test, a high level
ultimate lim−it of our tests for Gaussian states under a
P 0.8 means that our test can be successful unless
lossy channel. That is, as the rectangle and the triangle s ∼
the randomlychosenangle is too close to the one for the
tests for Gaussian states have critical values s = 1
ands = 2,respectively,wehave3dB(η = 1c =−1) factorized Wigner function, θ−φ =0 or π2. For the tri-
and4c.77d−B (ηc = 31)loss limits,below whcicht1h−esrce exi2st athneglreatnegset,otfhemsiuxcecdessstaptreosbdaebtielicttyabPlseiissslmaraglleerr,thhaonwetvheart,
some Gaussian states detectable using our tests. The of the rectangle test.
different limits 3 dB and 4.77 dB manifest a practical
superiority of the triangle test to the rectangle test. In Fig. 4, we also show the results for a triangle test
basedontheQfunction(s= 1). Notethatwehave
eJx−′c1ludedtherectangletest ,si−ncenoviolationoccurs
1
as shown in Fig. 2. QuiteJn−aturally (see Sec. II C 2),
3. finite data and nonoptimal phase-space points both the detectable range of mixed Gaussian states and
the success probability significantly shrink compared to
Nowletus furtherinvestigatetowhatextentourtests the case of Wigner-function tests.
canbe usefulunderpracticalconditions. First,the num-
ber N of data to construct an averagevalue is always fi-
nite,incurringanerroroforderO( 1 ). Wethusrequire
√N
that the degree of violation is large enough to overcome
the statistical error as
D. Quantum bound beyond Gaussian states
∆
>B + J , (25)
c
hJi √N
We now go beyond the Gaussian regime and investi-
whˆ2ere hJˆki2 w=ithN1tPheNi=cl1aJssikica(lkity=bou1,n2d)Band=∆22aJnd=1 gstaatteesabmeyaoxnimdaGlapuossssiiabnlestvaatleuse. TofoJfinadmoountgthaellmqauxainmtuumm
c
hJ i−hJi
forthe rectangleandthe right-triangletest, respectively. and the minimum values, we solve eigenvalue equations
Moreover,although we have previously obtained the op- ψ = λψ for Hermitian operators and that
timalchoiceofangleθ φ= π asshowninFig. 1,itisof Hco|rreispond| tio rectangle and right triangHles tests,Hrs′espec-
− 4
7
FIG. 4: (Color online) Success probability P =∆/(π) for a
s 2
triangletestJs′= 1todetectthenonclassicalityofaGaussian
statewith purity−µandsqueezingstrength κwhen thephase
isunknown. WehaveusedthenumberofdataN inEq.(25)
as N =104 (a), N =105 (b),N =106 (c), and N =107 (d).
tively. These Hermitian operators are given by
s Dˆ(q0+ip0)Tˆ(s)Dˆ†(q0+ip0)
H ≡
+Dˆ(q +ip )Tˆ(s)Dˆ (q +ip )
0 1 † 0 1
+Dˆ(q1+ip0)Tˆ(s)Dˆ†(q1+ip0)
Dˆ(q +ip )Tˆ(s)Dˆ (q +ip ), (26)
1 1 † 1 1
−
and
Dˆ(q +ip )Tˆ(s)Dˆ (q +ip )
Hs′ ≡ 0 1 † 0 1
+Dˆ(q1+ip0)Tˆ(s)Dˆ†(q1+ip0)
Dˆ(q +ip )Tˆ(s)Dˆ (q +ip ). (27)
1 1 † 1 1
−
FIG. 3: (Color online) Success probability Ps =∆/(π2) for a From now on, without loss of generality, we set
rectangle test Js=0 [(a), (c), (e), and (g)] and a triangle test q0,p0,q1,p1 = 0,0,dq,dp since the optimal values in
Js′=0 [(b), (d), (f), and (h)] to detect the nonclassicality of a {our tests are}inva{riant under}displacement as mentioned
Gaussian state with purity µ and squeezing strength κ when
in Sec. II B 1.
the phase is unknown. We have used the number of data N
For the case of the Wigner function (s = 0), we can
inEq.(25)asN =103 (a,b),N =104 (c,d),N =105 (e,f),
and N =106 (g, h). solve the eigenvalue problem on the basis of coherent
states. We first construct a trial solution as
∞ ∞
ψ = C 2d n+2id m , (28)
n,m q p
| i | i
n=X mX=
−∞ −∞
where the coherent states 2d n+2id m with integers
q p
| i
n and m form a two-dimensional lattice with points
spaced by 2d and 2d along the q and p axes, re-
q p
spectively. Exploiting an identity Dˆ(α)Tˆ(0)Dˆ (α)γ =
†
| i
8
FIG.6: (Coloronline)(a,b)MaximalJs(Js′)and(c,d)min-
imal Js (Js′) among all quantum states against the param-
eter s (orange solid line), compared to the algebraic (black
dotted lines) and Gaussian bounds (brown dashed lines), re-
spectively. The quantum bounds reach the algebraic upper
bounds at s = 0 for each case. In (a) and (b), the gap be-
tween quantum and classical bounds disappears at s = −1
and s=−2 for Js and Js′, respectively.
FIG.5: (Coloronline)(a)Maximumpositive(redfilledcircle)
and minimum negative (black open circle) eigenvalues λ of
N
H in Eq. (26) that are obtained by truncating n and m up algebraic bounds are actually obtained by the states in
0
to N in the recurrence relation of Eq. (29). (b) Expectation Eq. (30). In addition, those states also achieve the alge-
valuesµ =hH iofatruncatedsuperposition of(2N+1)× braic bounds 3 for a right triangle test (Fig. 6).
N 0 ±
(2N+1)coherentstatesinEq.(30)comparedwithλN in(a). Ontheotherhand,forageneralizeddistributions<0,
As an example, we plot (c) the Wigner function and (d) its we solve the eigenvalue problems in the basis of Fock
contourplot forasuperposition of5×5coherentstates with states. To this aim, we first express Dˆ(α)Tˆ(s)Dˆ (α) as
†
d =d = √π, which achieves a value of J [ρ]≈3.70.
q p 2 0
Dˆ(α)Tˆ(s)Dˆ (α)= ∞ ∞ T n m, (31)
† n,m
| ih |
e αγ∗+α∗γ 2α γ , we derive a recurrence relation nX=0mX=0
−
| − i
λCn,m =C n, m+e−4id2mC n+1, m+e4id2nC n, m+1 whereTn,m(α)≡Tr[|mihn|Dˆ(α)Tˆ(s)Dˆ†(α)] forn≥mis
− − − − − − given by
e 4id2(m n)C , (29)
− − n+1, m+1
− − − m! s+1 m 2α2
where d2 dqdp refers to the area of unit cell in the Tn,m(α)=rn!(cid:18)s 1(cid:19) exp(cid:18)− 1| |s(cid:19)
lattice. In≡Fig. 5, we show the maximum and minimum − −
ienigethnvearlueceusrartetnacienarbelleatbioyntsakwinitghadt2ru=ncaRtπio+n nπum(Rb:eriNn- ×(cid:18)12αs(cid:19)n−mLm(n−m)(cid:18)14|α|s22(cid:19), (32)
2 4 − −
teger). It shows that the simple algebraic bounds 4
± andT forn<misobtainedbyT . UsingEq.(32),
for a rectangle test are asymptotically obtained by in- n,m m∗,n
we construct a density matrix for and , and have
creasing N. As a double check, we compare λN and Hs Hs′
µ ψN ψN / ψN ψN with d2 =d2 = π, where obtained the maximum and minimum eigenvalues for
N ≡h |H0| i h | i q p 4 each generalized quasiprobability function by extensive
numerical optimizations. In Fig. 6, the maximal and
N N s
J
|ψNi=n=XNmX= NCn,m|2dqn+2idpmi (30) Jdes′crfeoarseqsu,awnhtuicmh ismtaptleisesatphpartoathche wclhaossleicaseltboofuqnudasnatusms
− − states detectable under our tests shrinks with s decreas-
is given by plugging the coefficients C obtained from ing.
n,m
the recurrence relations. That is, we construct a state For comparison,we alsoplot the Gaussianbounds ob-
with certain numeric coefficients and caluclate its actual tained in the previous sections and the algebraic bounds
average value µ of . Figure 5 confirms that the obtainable from Eq. (3). The algebraically possible
N s=0
H
9
state is a mixture of Gaussian states, it must satisfy
1< [ρ = p σ ] max ,
s MG i i s
− J ≤ σ J
Xi
1< [ρ = p σ ] max , (35)
− Js′ MG i i ≤ σ Js′
Xi
whereσ isanarbitraryGaussianstateandthemaximum
i
values, max and max , are shown in Figs. 2(c)
σJs σJs′
and 2(d), respectively. In particular, the case of s = 0
FIG. 7: (Color online) (a, b) Optimal Js=1−η1 and Js′=1−η1 gives
for an even cat state |ψγi = (2+2e−2|γ|2)−1/2(|γi+|−γi) 8
withγ =2(bluesolidline),whichareequivalenttoJ0andJ0′ −1<J0[ρMG]≤ 39/8,
for a decohered state L[|ψ ihψ |] with loss parameter 1−η
γ γ
[Eq. (24)], respectively. On the other hand, black dashed −1<J0′[ρMG]≤2. (36)
lines denote the Gaussian bounds in Eq. (35) [the same as
Fig. 2(c) and 2(d)] for each s = 1− 1, above which gen- In Fig. 7, as an example, we plot the violation of in-
η
uinenon-Gaussianityisdetected. Foranon-Gaussianitytest, equalities (35) and (36) for an even cat state ψγ =
the detectable range of η = 11s based on an s-parametrzied (2 + 2e−2|γ|2)−1/2(γ + γ ) under a lossy c|haninel,
function for theoriginal pure−cat state (redshaded region) is which is represented| biy a|W−ignier function,
larger than the range of η based on the Wigner function for
the decohered state under loss (brown shaded region). See W (q,p;s=0)
main text. L[|ψγihψγ|]
= π21+e1 2γ2e−2q2−2p2{e−2ηγ2cosh(4√ηγq)
−
ranges of Js and Js′ are given by +e−2(1−η)γ2cos(4√ηγp)}, (37)
2s+4 2s 4 with γ real. We particularly compare (i) the Wigner-
− for 1 s 0,
s
s 1 ≤J ≤ s 1 − ≤ ≤ function-based tests and of the decohered
− − Js=0 Js′=0
−1<Js <3 for s<−1, (33) state L[|ψγihψγ|] and (ii) the s-parametrized-function-
based and of the original pure state ψ with
and s = 1Js 1, reJssp′ectively. In Sec. II C 2,| wγei have
− η
shown the equivalence of two nonclassicality tests—the
s+3 s 3
s 1 ≤Js′ ≤ s−1 for −1≤s≤0, Ws-piganraemr-feutnriczteidonfutnecsttiofnortetshtefodrectohheeroerdigisntaatlestaanted. thIne
− −
−1<Js′ <2 for s<−1. (34) contrast, there exists inequivalence for non-Gaussianity
tests—the latter detects non-Gaussianity in a broader
While the maximumandthe minimum algebraicbounds parameter regime than the former. It is due to the
aresaturatedbyquantumstatesfors=0,therearegaps factthatquantumnon-Gaussianitybounds,max and
σ s
between algebraic and quantum bounds for a nonzero max , vary with the parameter s, whereas thJe non-
s < 0. As for the lower bounds of Js and Js′, the van- classσicJasl′ity bounds are the same regardless of s. On a
ishinggapbetweenalgebraicandclassicalboundsclearly practicalside,itimpliesthatthenon-Gaussianitytestfor
indicatethatthereisnoquantumviolationbelows= 1. a decohered state under a lossy channel can be harder
−
On the other hand, as for the upper bounds, we observe than expected from the analysis only based on the s-
that the loss limits of Js and Js′ for the whole set of parametrized function of the original state.
quantum states are identical to the loss limits for Gaus-
sian states, 3 dB (s = 1) for and 4.77 dB (s = 2)
s
− J −
for Js′, respectively. That is, below those limits, there A. Optimizing non-Gaussianity tests
exist some quantum states, both Gaussian (Sec. II C 2)
and non-Gaussian, violating the classical bounds 2 and
The test of genuine non-Gaussianity in Eq. (35) can
1, respectively. be further enhanced by applying squeezing Sˆ on a given
state. Note that a mixture of Gaussian state remains to
beaGaussianmixtureunderGaussianoperations. Thus,
III. TESTING GENUINE NON-GAUSSIANITY if the state under squeezing shows violation of Gaus-
sian bounds, the original state must be genuinely non-
Asourtestsarelinearwithrespecttoaconvexmixture Gaussian. For example, we show the case of the mixed
of quantum states, i.e., [ p ρ ] = p [ρ ], the state f 0 0 +(1 f)2 2 in Figs.8(a) and8(b), which
Js i i i i iJs i | ih | − | ih |
gaps between the quantum aPnd GaussianPbounds enable showsthatthesqueezingoperationonthestateenhances
ustodetectquantumnon-Gaussianity. Thatis,ifagiven the detected region.
10
FIG. 9: (Color online) A squeezing transformation converts
(a) a parallelogram to (b) a rectangle while it squeezes the
entireprofile of Wigner function.
pendix):
FIG. 8: (Color online) (a, b) J0 and J0′ optimized over the π(1 s) π α
points of a rectangle (right triangle) for Sˆ{f|0ih0| + (1 − 2− Wρ(S[α];s)= 2WL′[SˆρSˆ†](cid:18)√1 s;0(cid:19), (39)
f)|2ih2|}Sˆ† asfunctionsofthevacuumfractionf. Bluesolid, −
purpledotted,andreddashedlinesdenotethecaseofsqueez- where [ρ] represents a beam-splitter interaction with
′
ingappliedtoagivenstate,withsqueezingr={0,0.5,1}(a) transmLittance η = 1 between a quantum state ρ
and r = {0,1,2} (b), respectively. We find no violation of 1 s
Eq.(36)foraninitialstate(r=0),whileastatewithasuffi- and a squeezed vacuu−m Sˆ0 0Sˆ†. Therefore, all s-
| ih |
cientlylargesqueezingviolatesEq.(36)(bluecoloredregion) parametrized functions taken at the vertices of a par-
even when its Wigner function is positive (f > 1). (c, d) alleogram can be understood as the Wigner function
2
Ns and Ns′ optimized overthepoints of aparallelogram (tri- taken at the vertices of a rectangle for the decohered
angle) in Eqs. (41) and (42) for f|0ih0|+(1−f)|2ih2| with state [SˆρSˆ ]. As the whole process [SˆρSˆ ] is Gaus-
′ † ′ †
respect to f. Black solid, orange dotted, brown dashed lines sian, iLt does not create non-GaussianLity. We then use
represents={0,−0.01,−0.02}(c)ands={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 quantumnon-Gaussianity.
functions.
We thus propose enhanced non-Gaussianity tests for
an arbitrary s as
Forthe caseofa Wignerfunction, wemayaddressthe
8
problem by using its interesting property [3]
1< [ρ ] ,
− Ns MG ≤ 39/8
W (α;0)=W (S[α];0), (38) −1<Ns′[ρMG]≤2, (40)
SˆρSˆ† ρ
with
where S[α] = αcoshr+α∗e2iφsinhr is the transforma- π(1 s)
tionofphase-spacepointsduetothesqueezingoperation s[ρ]= − Wρ(S[q0+ip0];s)+Wρ(S[q1+ip0];s)
N 2 {
Sˆ. As we illustrate in Fig. 9, under the squeezing trans-
+W (S[q +ip ];s) W (S[q +ip ];s) , (41)
formationS[α], a parallelogramin an unsqueezed profile ρ 0 1 − ρ 1 1 }
corresponds to a rectangle in a squeezed profile. Impor-
and
tantly, it means that we do not need to implement a
squeezing operation on a given state in order to have an π(1 s)
enhanced test. Instead, we may simply choose the four Ns′[ρ]= 2− {Wρ(S[q1+ip0];s)+Wρ(S[q0+ip1];s)
points atthe verticesofthe parallelogramcorresponding
W (S[q +ip ];s) , (42)
ρ 1 1
to the squeezingoperationanddo our tests for the given − }
initial state.
whichtakeintoaccounttheverticesofparallelogramand
On the other hand, for a nonzero s < 0, the above triangle,respectively,withS[α]=αcoshr+α e2iφsinhr.
∗
argument is not directly applicable. This is because Note that any values of r and φ for the squeezing trans-
the s-parametrized function of a state after squeezing formation S[α] can be used in Eqs. (41) and (42), as
is not simply obtained by reshaping of the original s- squeezing does not create non-Gaussianity. As an illus-
parametrizedfunction(squeezingoftheprofile). Instead, tration,weshowthecaseofthestatef 0 0+(1 f)2 2
| ih | − | ih |
we find the following identity (with its proof in the Ap- in Figs. 8(c) and 8(d) using the above tests.