Quantum non-Gaussianity and quantification of nonclassicality

B. Kuhn∗ and W. VogelArbeitsgruppe Quantenoptik, Institut fur Physik, Universitat Rostock, D-18051 Rostock, Germany

(Dated: May 21, 2018)

The algebraic quantification of nonclassicality, which naturally arises from the quantum superpo-sition principle, is related to properties of regular nonclassicality quasiprobabilities. The latter areobtained by non-Gaussian filtering of the Glauber-Sudarshan P function. They yield lower boundsfor the degree of nonclassicality. We also derive bounds for convex combinations of Gaussian statesfor certifying quantum non-Gaussianity directly from the experimentally accessible nonclassicalityquasiprobabilities. Other quantum-state representations, such as s-parametrized quasiprobabili-ties, insufficiently indicate or even fail to directly uncover detailed information on the properties ofquantum states. As an example, our approach is applied to multi-photon-added squeezed vacuumstates.

I. INTRODUCTION

Uncovering structural information on quantum statesis of fundamental importance for present quantum tech-nologies. A prominent example for such characteristicsis the classification of a state as classical or nonclassical[1]. Beyond this bivalent categorization, the amount andthe kind of nonclassicality provided by various quantumresources is of great interest to figure out optimal ex-perimental implementations of quantum technologies. Inparticular, a further state property—the quantum non-Gaussianity [2–5]—is important for various applicationsin quantum technology [6–8].

In the past decades distinct measures for the quantifi-cation of nonclassicality have been proposed for harmonicoscillator quantum systems, such as light, the quantizedmotion of trapped atoms, and others. The early attemptsrely on topological properties of quantum states; they arebased on the distance of a state to the set of classicalstates. Such an approach can be based on different no-tions of the distance, such as the trace-norm-induced dis-tance [9], the Monge distance [10], or the Hilbert-Schmidtdistance [11]. The crucial point of such quantificationsconsists in its ambiguity. On the other hand, a quan-tification based on the number of superpositions of co-herent states relies on the fundamental quantum super-position principle and yields an unambiguous quantifi-cation of nonclassicality [12]. Note that distance-basedmeasures have also been studied for the quantification ofnon-Gaussianity [13–15].

Alternatively, the quantification problem has been ad-dressed by using properties of quasiprobabilities [16, 17],which are full representations of the quantum states[18, 19]. They may certify nonclassicality of variousstates [20, 21]. A powerful tool is the nonclassicalityquasiprobability [22]—a regularized form of the Glauber-Sudarshan P function [23, 24]. It is designed such thatit identifies any nonclassical state through negativities.This could be experimentally shown even for the strongly

∗ benjamin.kuehn2@uni-rostock.de

singular squeezed states [25, 26]. In Ref. [27], the po-tential of a single-mode nonclassical state to generatequantum entanglement [28] by applying only linear op-tical elements was used to quantify nonclassicality. Analgebraic approach for the quantification of nonclassical-ity was introduced, defining the degree of nonclassical-ity by the number of quantum superpositions of coher-ent states [29]. Based on the Schmidt number, an al-gebraic quantification was also used for quantum entan-glement [28, 30, 31], which can be generalized to mul-tipartite scenarios [32]. A unified quantification of non-classicality and entanglement has been introduced [33],which directly relates the degree of nonclassicality to theSchmidt number of bipartite and even multipartite en-tangled states created by linear optical devices, such asan N splitter. This is an important relation as entan-glement is the basis for quantum technologies such asquantum information processing [34] and secure commu-nication protocols [35]. For practical applications, thedegree of nonclassicality can be verified both by a wit-ness [36] and through the structure of the characteris-tic function of the Glauber-Sudarshan P function [37].However, these methods have not been developed for thepurpose of uncovering quantum non-Gaussianity.

In the present paper we prove the usefulness of non-classicality quasiprobabilities for certifying both quan-tum non-Gaussianity and the degree of nonclassicality.The single class of phase-space distributions under studyuncovers both topological and algebraic quantificationsof nonclassicality. We may uncover dissimilar features ofarbitrary quantum states, such as nonclassicality, non-Gaussianity, and the minimal number of quantum super-positions forming a given state. As the required phase-space functions are experimentally accessible, the presentmethod also applies to state-of-the-art experiments.

II. NONCLASSICALITY VERSUS QUANTUMNON-GAUSSIANITY

The coherent states |α〉 are well known to resemble theclassical behavior of the harmonic oscillator and, hence,the classical character of light. Since mixing is a classical

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operation, a general state ρ is referred to as classical if itcan be written as a mixture of coherent states,

ρ =

∫d2αPcl(α) |α〉〈α|, (1)

with a classical probability distribution Pcl. In otherwords, classical states are elements of the convex hullof the set of coherent states. In fact, any quantum statecan be written in the coherent state basis [23, 24],

ρ =

∫d2αP (α) |α〉〈α|. (2)

However, the Glauber-Sudarshan function P (α) may failto show the properties of a classical probability. Suchstates are referred to as nonclassical ones [1]. On this ba-sis, any nonclassicality requires quantum superpositionsof coherent states,

|ψr〉 =

r∑j=1

µj |γj〉, (3)

where µj are nonzero complex numbers, |γj〉 are variouscoherent states, and r is the number of superpositions.Now we may define the degree of nonclassicality as

κ = r − 1 (4)

or as a monotonous function of this quantity [29]. Thisis an ambiguous quantification of nonclassicality, as anincrease of nonclassicality is caused by a larger number ofquantum superpositions. Pure states with nonclassicalitydegree r − 1 can be written in the form (3) with theminimal number of r superpositions of coherent states.

Extending these considerations, a general quantumstate ρ has a nonclassicality degree r − 1 if it is a classi-cal mixture of pure states of a nonclassicality degree ofat most r − 1, i.e.,

ρ =

∫dPcl(|ψr〉)|ψr〉〈ψr|. (5)

In the case r > 1, different such decompositions exist;thus, the nonclassicality degree refers to the minimal pos-sible r. The states with a degree of nonclassicality of atmost r− 1 form a closed, convex setMr. This definitionis straightforwardly generalized to multimode harmonicoscillator systems; see also Ref. [36].

A further important characteristics of quantum statesis the non-Gaussianity. The Gaussian state |Gu,Σ〉 isfully determined through the first and second moment,namely, the mean value u and the covariance matrix Σ.In principle, there are two types of non-Gaussianity. Onthe one hand, a non-Gaussian state can be obtained byproperly mixing Gaussian states according to a classicalprobability distribution Pcl, i.e.,

ρ =

∫dPcl(u,Σ) |Gu,Σ〉〈Gu,Σ|. (6)

These states form also a closed, convex set G (seeRef. [5]). However, for quantum technologies this kind ofnon-Gaussianity is rather useless, since it originates fromclassical noise, such as phase randomization of Gaussianquantum states [38]. Of greater interest are quantumnon-Gaussian states, ρ 6∈ G, whose non-Gaussianity isintrinsically quantum.

III. STRUCTURAL INFORMATION INPHASE-SPACE FUNCTIONS

It is possible to use a witness approach to formulatecriteria for the quantification of nonclassicality. A cor-responding method was proposed [36], where the lowerand upper bounds, gr and g

r, of the expectation value

of a given Hermitian operator L are determined with re-spect to Mr. If 〈L〉 > gr or 〈L〉 < g

rfor an unknown

state, the nonclassicality degree of this state is shownto be necessarily greater than or equal to r. Equiva-

lently, one introduces witness operators W r = gr1 − Land W r = L − g

r1, such that 〈W r〉 ≥ 0 and 〈W r〉 ≥ 0

for all states in Mr.For the case r = 1, a complete family of witness oper-

ators, W, is already known [39], which is able to showthat any state with nonclassicality degree r−1 > 0 (non-classical state) cannot be written as a convex combina-tion of pure nonclassicality degree-zero states (coherentstates). Thus, these witnesses uncover all nonclassicaleffects of single- and multimode harmonic oscillator sys-tems. In the single-mode case, they are of the form

Ww,α = D†αWwDα (7)

with

Ww =1

π2

∫d2β Ωw(β) eβa

†e−β

∗a, (8)

and the coherent displacement operator Dα =exp

[αa† − α∗a

]. Here a and a† are the bosonic annihila-

tion and creation operators, respectively. In total, thereare only three free real parameters, a positive quantityw and a complex number α. The so-called nonclassical-ity filter Ωw is chosen in such a way that the expecta-tion value of Ww,α exists for all states. Furthermore, itsspecific structure guarantees that this expectation valueis non-negative for all classical states. A necessary andsufficient condition for a state being nonclassical is theexistence of parameters w and α such that 〈Ww,α〉 < 0.

The witnesses Ww,α are associated with phase-spacefunctions, the latter being an established method to vi-sualize quantum effects through their negativities. Inparticular, the expectation value Pw(α) = 〈Ww,α〉 canbe regarded as a nonclassicality quasiprobability in phasespace; it holds

∫d2αPw(α) = 1. For any positive value

w this function is nonsingular, as Ww,α is a bounded

3

TABLE I. For parameters w in the range [wmin, wmax] in eachcolumn, a lower bound for the degree of nonclassicality ofup to κ can be verified. In this range the overall supremum(infimum) of the nonclassicality quasiprobability Pw(α) is at-tained by a Fock state |n〉 (|n〉).

wmin 1.200 1.550 1.795 2.027 2.239 2.436 · · ·wmax 1.550 1.795 2.027 2.239 2.436 2.619 · · ·n 0 2 2 4 4 6 · · ·n 1 1 3 3 5 5 · · ·κ 1 2 3 4 5 6 · · ·

operator, and, thus, it is in principle accessible in ex-periments. In the limit w → ∞ the function Pw ap-proaches the Glauber-Sudarshan P function, which is ourreference for nonclassicality. Note that s-parametrizedquasiprobabilities [18], which correspond to a Gaussianfilter Ωw in Eq. (8), are only regular if the s parameteris sufficiently small for the state under study. There isno s-parametrized quasiprobability which visualizes thenonclassicality of a squeezed vacuum state. Introducingnon-Gaussian filters that decay more strongly than anyGaussian and that have a non-negative Fourier transformresolve this problem.

The witnesses Ww,α in Eq. (7) together with Eq. (8)and, therefore, the associated nonclassicality quasiprob-abilities provide a full test for nonclassicality. However,it is unknown yet to what extent structural state charac-teristics, such as the nonclassicality degree and quantumnon-Gaussianity, can also be uncovered solely on the ba-sis of these specific quantities. In order to approach thisquestion, we combine the witness approach for quanti-fying nonclassicality on the basis of the quantum super-position principle [36] and the universal nonclassicality

witness operators Ww,α, corresponding to regular phase-space functions. In the following we use the compactsupport filter defined in Refs. [39, 40], since the associ-ated witness (8) has the closed form expression

Ww = :1

π

[J1

(2w√a†a)]2

a†a: , (9)

where J1(·) is the Bessel function of the first kind and: · : denotes normal ordering. This operator is diagonalin the Fock basis,

Ww =

∞∑n=0

cw,n|n〉〈n|, (10)

with the analytical coefficients [41]

cw,n =w2

π

n∑m=0

(−w2)m

[(m+ 1)!]2n!

(n−m)!

(2m+ 2

m

). (11)

Due to unitarity, the coherently displaced operatorWw,α in Eq. (7) has the same eigenvalues as the opera-

tor Ww. Therefore, the overall supremum and infimum

of the nonclassicality quasiprobability Pw(α) for a fixedvalue of w is attained for Fock states |n(w)〉 and |n(w)〉,respectively (see Appendix A). Table I shows that thesupremum is attained for even numbers n, while the in-fimum is attained for odd numbers n. These numbersincrease sequently and change alternately whenever theparameter w exceeds a critical value. Fock states |n〉have the nonclassicality degree κ = n, as they are rep-resentable as a quantum superposition of n+ 1 coherentstates [42],

|n〉 = limε→0Cn(ε)

n∑k=0

e−2πikn/(n+1)|εe2πik/(n+1)〉, (12)

weighted equally and their amplitudes located on a cir-cle in the complex plane with radius tending to zero;Cn provides the correct normalization. This is a maxi-mally nonclassical state in the topological picture. Con-sequently, for given w a nonclassicality degree up tomax [n(w), n(w)] can be certified by means of the non-classicality quasiprobability Pw(α).

Let us compare this result with the case of s-parametrized quasiprobabilities. In the parameter ranges ≤ 0, where these functions are always regular, the over-all supremum is attained by the vacuum state and theoverall infimum by the single-photon state. In fact, thesequasiprobabilities uncover only the presence of some non-classicality for s > −1 without resolving a nonclassi-cality degree. This clearly demonstrates, that the s-parametrized quasiprobabilities not only provide an in-complete nonclassicality test; they also do not yield morespecific structural insight.

Now, we determine the upper and lower bounds, grand g

r, of the nonclassicality quasiprobability Pw(α) as-

sociated with the witness operator (9) over the set Mr.Mixing cannot shift the optimum, thus, the optimizationcan run over pure states |ψr〉 defined in Eq. (3). Fur-thermore, the coherent displacement in Eq. (7), can beincluded in the states. In total, the quantity to be opti-mized reads as

〈ψr|Ww|ψr〉 =1

π

r∑`=1

r∑j=1

µ∗`µj

[J1

(2w√γ∗` γj

)]2 〈γ`|γj〉γ∗` γj

.

(13)

The complex coherent amplitudes (γ1, . . . , γr)T and com-

plex coefficients (µ1, . . . , µr)T are the optimization pa-

rameters, which have to fulfill the normalization con-straint 〈ψr|ψr〉 = 1. We perform our analysis up to r = 6,requiring to optimize in total up to 22 free real parame-ters, which is done by a genetic search [43] together witha gradient descent [44]. In a similar way, the upper andlower bounds, gG and g

G, of Pw(α) with respect to the

Gaussian convex hull G are obtained by an optimizationover pure Gaussian states. The latter are squeezed coher-ent states, defined by the complex squeezing parameterand complex coherent displacement amplitude.

The result is shown in Fig. 1 and gives a rich amountof insight. For better visualization of the relative effects,

4

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

-1.0

-0.5

0.0

0.5

1.0

w

grHGL

ÈgÈg

rHGLg

FIG. 1. (Color online) The maximal (minimal) values gr (gr)

for r = 1, . . . , 6 as a function of the parameter w. Solid lineswith positive (negative) function values from bottom to top(from top to bottom): r = 1 (orange), r = 2 (blue), r = 3(dark yellow), r = 4 (green), r = 5 (purple), and r = 6(red). We normalized the bounds to the overall supremum g(infimum g). The maximal (minimal) values gG (g

G) attained

by any mixture of Gaussian states for a given w are illustratedby the dashed blue lines, normalized in the same way. Thevertical dotted lines mark the critical values of w, where anext higher degree of nonclassicality κ can be certified.

gr(w) is normalized by the overall supremum, and gr(w)

by the modulus of the overall infimum of the nonclassi-cality quasiprobability. One observes a splitting of thesupremum (infimum) into three separated levels (trifur-cation) whenever the increasing parameter w traversescritical values, which turn out to be identical to the val-ues where the overall supremum (infimum) is attained byanother Fock state (see Table I). This systematics mostlikely persists also for w > 2.6, which allows us to arbi-trarily increase the resolvable nonclassicality degree. Theparameter w should be chosen as large as necessary butas small as possible, as experimental data sampling noiseincreases with increasing w.

Interestingly, in the case of r = 2, the bounds gr andgr

are attained by even and odd coherent states, |γ±〉 =

N± (|γ〉 ± | − γ〉), respectively, which are identified to bemaximally nonclassical in the sense of (distance-based)topological nonclassicality measures. The upper andlower bounds, gG and g

G, for the set G of mixtures of

Gaussian states are also contained in Fig. 1 and theyare well separated from the levels gr (g

r); for further

details on the optimal states see Appendix B. Since Gcontains also the nonclassical squeezed states, it holdsgG< g

1= 0, and gG also exceeds the classical level

g1. Note that, even for arbitrarily weak squeezing, thesestates are quantum superpositions of an infinite numberof coherent states, referring to them as maximally non-classical from the perspective of algebraic nonclassicalitymeasures. Our results clearly show, that the nonclassical-ity quasiprobabilities uncover both the degree of nonclas-sicality and quantum non-Gaussianity without additional

means.

IV. PHOTON-ADDED SQUEEZED STATES

The structure of the operator Ww,α in Eq. (7) allows usto certify especially Fock-like states—states having a highoverlap with a Fock state—to exceed the nonclassicalitydegree bounds gr and g

r. As an example, we consider the

non-Gaussian states obtained by multiphoton additions

(a)

(b)

FIG. 2. (Color online) Nonclassicality quasiprobabilityPw(α) of an m-photon-added squeezed vacuum state for oddm and squeezing parameter ξ = 0.1: (a) m = 1, w = 1.6; (b)m = 3, w = 2.0. The lower bounds g

rfor states with a non-

classicality degree of at most r− 1 are given by the transpar-ent orange planes. The lower bound for convex combinationsof Gaussian states, g

G, is indicated by the transparent pink

plane. For better visualization, we cut out the front-rightquadrant.

5

to an initial Gaussian squeezed state, i.e.,

|m, ξ〉 = Nm,ξ a†mS(ξ)|0〉, (14)

with the unitary squeezing operator S(ξ) =exp

[12

(ξ∗a2 − ξa†2

)]. The squeezing parameter is

chosen to be ξ = 0.1 and m is the number of addedphotons. The prefactor Nm,ξ ensures the correctnormalization.

(a)

(b)

FIG. 3. (Color online) Nonclassicality quasiprobability ofan m-photon-added squeezed vacuum state for even m andsqueezing parameter ξ = 0.1: (a) m = 2, w = 1.8; (b) m = 4,w = 2.2. The upper bounds gr for states with a nonclassical-ity degree of at most r−1 are given by the transparent orangeplanes. The upper bound for convex combinations of Gaus-sian states, gG, is indicated by the transparent pink plane.For better visualization, we cut out the front-right quadrant.

Figures 2(a) and 2(b) show the nonclassicalityquasiprobability of a single- and a three-photon-addedsqueezed vacuum state, respectively. For the single-photon addition, the phase-space function falls below

g1

= 0 but not below g2. Thus, nonclassicality is certified

(κ > 0), but no information on the nonclassicality degreeis extracted. One clearly observes that through the ad-dition of three photons the quasiprobability falls belowthe lower bound g

3. Accordingly, this state has at least

the nonclassicality degree κ = 3; i.e., it is composed ofa quantum superposition of at least four coherent states.Both states are certified to be quantum non-Gaussianones, since Pw(α) penetrates g

G.

In addition, we present in Figs. 3(a) and 3(b) the re-sults for photon-added squeezed vacuum states with twoand four added photons and the same squeezing param-eter as in Fig. 2. Since Pw(α) in Fig. 3(a) exceedsthe upper bound g2, the two-photon-added squeezed vac-uum state is shown to have a nonclassicality degree ofat least 2. The nonclassicality quasiprobability in Fig.3(b) exceeds the upper bound g4 and, accordingly, thefour-photon-added squeezed vacuum state has a nonclas-sicality degree of at least 4. Both states strongly exceedthe range allowed for convex combinations of Gaussianstates; they are therefore shown to be quantum non-Gaussian.

V. CONCLUSIONS

In conclusion, the nonclassicality quasiprobabilitieswere so far known to fully verify the nonclassicalityof quantum states. Beyond this, we have shown thatthey additionally uncover important structural quantumcharacteristics of the state. Beneficially, both the degreeof nonclassicality and the quantum non-Gaussianity areaccessible via this single quantity. The nonclassicalitydegree, detected by our method, is useful for applicationsin quantum technologies, as the nonclassicality degreeof a single-mode beam forwarded to the input of an Nsplitter coincides with the amount of multipartite entan-glement produced in the outputs. Most importantly, thestructure of nonclassicality quasiprobabilities uncoversboth algebraic and topological aspects of nonclassicality.

ACKNOWLEDGMENTS

The authors are grateful to S. Ryl and J. Sperlingfor enlightening discussions. This work has been sup-ported by the European Commission through the projectQCUMbER (Quantum Controlled Ultrafast MultimodeEntanglement and Measurement), Grant No. 665148.

6

APPENDIX

Appendix A: Overall boundaries of nonclassicalityquasiprobabilities

In this section the overall maximization and minimiza-tion of the nonclassicality quasiprobability Pw(α) withrespect to all quantum states is studied. For reasonsdescribed in the main text, the overall supremum g(w)and infimum g(w) of the nonclassicality quasiprobabilityPw(α) for a fixed value of w is attained for Fock states|n(w)〉 and |n(w)〉, respectively, with

n(w) = arg maxn

[cw,n] ,

n(w) = arg minn

[cw,n] ,

g(w) = maxn

[cw,n] ,

g(w) = minn

[cw,n] ,

where the coefficients cw,n are defined in Eq. (11) of themain text.

Èn\=È0\ Èn\=È2\Èn\=È4\

Èn\=È1\ Èn\=È3\ Èn\=È5\1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

0.1

1

10

100

1000

104

w

10

´g

,ÈgÈ

FIG. 4. (Color online) The supremum g (dashed orange line)and the modulus of the infimum g (solid blue line) of Pw,which are attained by Fock states |n〉 and |n〉 as a function ofthe parameter w. Note that the supremum is depicted witha factor of 10 for better visual separation of both curves.

These quantities are illustrated in Fig. 4 on a loga-rithmic scale. Interestingly, one observes a well-orderedstructure. The supremum is attained for Fock states witheven number, while the infimum is attained for odd ones.The change of n (n) at the critical values accompanies adiscontinuity of the derivative of g(w) (g(w)) with respectto w.

Appendix B: Optimal states

It turns out that even (odd) coherent states,

|γ±〉 =1√

2(1± e−2|γ|2

) (|γ〉 ± | − γ〉) , (B1)

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.60.0

0.5

1.0

1.5

2.0

w

ÈΓÈ

FIG. 5. (Color online) Dashed orange line (solid blue line):Amplitude |γ| of the even (odd) coherent state, which maxi-mizes (minimizes) the nonclassicality quasiprobability Pw(α)with respect to the set M2 of states with a nonclassicalitydegree of κ = 1 as a function of the parameter w.

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.60.0

0.5

1.0

1.5

2.0

w

ÈΞÈ

FIG. 6. (Color online) Dashed orange line (solid blue line):Absolute value of the squeezing parameter ξ of the squeezedvacuum state, which maximizes (minimizes) the nonclassical-ity quasiprobability Pw(α) with respect to the set G of mix-tures of Gaussian states as a function of the parameter w.

where the “+” corresponds to even ones and the “−”to odd ones, maximize (minimize) the nonclassicalityquasiprobability Pw(α) with respect to the set M2 ofstates with nonclassicality degree κ = 1. The absolutevalue of the amplitude γ, for which the maximum andminimum are obtained, is shown in Fig. 5 as a functionof w. The two curves approach each other for increas-ing parameter w. Accordingly, for large w the maximumand minimum are attained for even and odd coherentstates with nearly the same amplitude γ. This optimalamplitude increases with increasing w. For w < 1.55the optimal amplitude of the even coherent state, cor-responding to the maximum of Pw(α), is γ = 0 in Eq.(B1), which is the vacuum state. For w < 1.795 the am-plitude of the odd coherent state providing the minimumis γ = 0, which according to Eq. (B1) coincides with thesingle-photon state.

The nonclassicality quasiprobability is optimized withrespect to the set G of mixtures of Gaussian states for

7

squeezed vacuum states [Eq. (14) with m = 0 in themain text]. The absolute value of the squeezing parame-ter ξ, which yields the maximum and minimum of Pw(α),is illustrated in Fig. 6 as a function of w. In the an-alyzed range of w, the minimum of the nonclassicalityquasiprobability is attained for larger |ξ| than the maxi-

mum. Furthermore, for w < 1.55 one obtains the maxi-mum for the vacuum state, which corresponds to ξ = 0.We also observe that the maximum and minimum areobtained for the same increasing squeezing parameter inthe limit of a large parameter w.

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