Measure of the non-gaussian character of a quantum state

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1 PHYSICAL REVIEW A 76, Measure of the o-gaussia character of a quatum state Marco G. Geoi, 1 Matteo G. A. Paris, 1,, * ad Korad Baaszek 3 1 Dipartimeto di Fisica dell Uiversità di Milao, I-0133, Milao, Italy Istitute for Scietific Iterchage, I Torio, Italy 3 Istitute of Physics, Nicolaus Copericus Uiversity, PL Toruń, Polad Received 19 April 007; revised mauscript received 1 July 007; published October 007 We address the issue of quatifyig the o-gaussia character of a bosoic quatum state ad itroduce a o-gaussiaity measure based o the Hilbert-Schmidt distace betwee the state uder examiatio ad a referece Gaussia state. We aalyze i detail the properties of the proposed measure ad exploit it to evaluate the o-gaussiaity of some relevat sigle-mode ad multimode quatum states. The evolutio of o- Gaussiaity is also aalyzed for quatum states udergoig the processes of Gaussificatio by loss ad de- Gaussificatio by photo-subtractio. The suggested measure is easily computable for ay state of a bosoic system ad allows oe to defie a correspodig measure for the o-gaussia character of a quatum operatio. DOI: 1103/PhysRevA I. INTRODUCTION Gaussia states play a crucial role i quatum iformatio processig with cotiuous variables. This is especially true for quatum optical implemetatios sice radiatio at thermal equilibrium, icludig the vacuum state, is itself a Gaussia state ad most of the Hamiltoias achievable withi the curret techology are at most biliear i the field operators i.e., preserve the Gaussia character 1 3. Asa matter of fact, usig sigle-mode ad etagled Gaussia states, liear optical circuits, ad Gaussia operatios, like homodye detectio, several quatum iformatio protocols have bee implemeted, icludig teleportatio, dese codig, ad quatum cloig. O the other had, quatum iformatio protocols required for log-distace commuicatio, such as, for example, etaglemet distillatio ad etaglemet swappig, rely o o-gaussia operatios. I additio, it has bee demostrated that teleportatio 5 7 ad cloig 8 of quatum states may be improved by usig o-gaussia states ad o-gaussia operatios. Ideed, de- Gaussificatio protocols for sigle-mode ad two-mode states have bee proposed 5 7 ad realized 9. It should be also oticed that ay strogly superadditive fuctio is miimized, at fixed covariace matrix, by Gaussia states. This is crucial to prove the extremality of Gaussia states ad Gaussia operatios 10,11 for what cocers various quatities such as chael capacities 1, multipartite etaglemet measures 13, ad distillable secret keys i quatum key distributio protocols. Sice i most cases these quatities ca be computed oly for Gaussia states, a o- Gaussiaity measure may serve as a guidelie to quatify them for the class of o-gaussia states. Overall, o- Gaussiaity is revealig itself as a resource for cotiuous variable quatum iformatio, ad thus we urge a measure able to quatify the o-gaussia character of a quatum state. I this paper we itroduce a quatity, the o-gaussiaity ϱ of a quatum state, which quatifies how much a state *matteo.paris@fisica.uimi.it PACS umber s : a, Ta,.50.Dv fails to be Gaussia. Our measure, which is based o the Hilbert-Schmidt distace betwee the state itself ad a referece Gaussia state, ca be easily computed for ay state, either sigle-mode or multimode. The paper is structured as follows. I the ext sectio we itroduce otatio ad review the basic properties of Gaussia states. The, i Sec. III we itroduce the formal defiitio of ϱ ad study its properties i details. I Sec. IV we evaluate the o-gaussiaity of relevat quatum states, whereas i Sec. V we aalyze the evolutio of o- Gaussiaity for kow Gaussificatio ad de-gaussificatio maps. Sectio VI closes the paper with some cocludig remarks. II. GAUSSIAN STATES For cocreteess, we will use here the quatum optical termiology of modes carryig photos, but our theory applies to geeral bosoic systems. Let us cosider a system of modes described by mode operators a k, k=1,...,, satisfyig the commutatio relatios a k,a j = kj. A quatum state ϱ of modes is fully described by its characteristic fuctio 1 ϱ =Tr ϱd, where D = k=1 D k k is the -mode displacemet operator, with = 1,..., T, k C, ad where D k k = exp k a k k a k is the sigle-mode displacemet operator. The caoical operators are give by q k = 1 a k + a k, p k = 1 a i k a k, with commutatio relatios give by q j, p k =i jk. Upo itroducig the real vector R= q 1, p 1,...,q, p T, the commutatio relatios ca be rewritte as /007/76 / The America Physical Society

2 GENONI, PARIS, AND BANASZEK R k,r j = i kj, where kj are the elemets of the symplectic matrix =i k=1, beig the y Pauli matrix. The covariace matrix ϱ ad the vector of mea values X X ϱ of a quatum state ϱ are defied as X j = R j, kj = 1 R k,r j R j R k, where A,B =AB+BA deotes the aticommutator ad O =Tr ϱo is the expectatio value of the operator O. A quatum state ϱ G is referred to as a Gaussia state if its characteristic fuctio has the Gaussia form ϱ G = exp 1 T + X T, where is the real vector = Re 1,Im 1,...,Re,Im T. Of course, oce the covariace matrix ad the vector of mea values are give, a Gaussia state is fully determied. For a sigle-mode system the most geeral Gaussia state ca be writte as ϱ G = D S t S D, D beig the displacemet operator, S =exp 1 a 1 a the squeezig operator,, C, ad t = 1+ t 1 t / 1+ t a a a thermal state with a t average umber of photos. III. MEASURE OF THE NON-GAUSSIAN CHARACTER OF A QUANTUM STATE I order to quatify the o-gaussia character of a quatum state ϱ we use a quatity based o the distace betwee ϱ ad a referece Gaussia state, which itself depeds o ϱ. Specifically, we defie the o-gaussiaity ϱ of the state ϱ as ϱ = D HS ϱ,, ϱ where D HS ϱ, deotes the Hilbert-Schmidt distace betwee ϱ ad, D HS ϱ, = 1 Tr ϱ = 1 ϱ + ϱ,, 3 with ϱ =Tr ϱ ad ϱ, =Tr ϱ deotig the purity of ϱ ad the overlap betwee ϱ ad, respectively. The Gaussia referece is the Gaussia state such that X ϱ = X, ϱ = ; i.e., is the Gaussia state with the same covariace matrix ad the same vector X of the state ϱ. The relevat properties of ϱ, which cofirm that it represets a good measure of the o-gaussia character of ϱ, are summarized by the followig lemmas. Lemma 1. ϱ =0 if ad oly if ϱ is a Gaussia state. Proof. If ϱ =0, the ϱ= ad thus it is a Gaussia state. If ϱ is a Gaussia state, the it is uiquely idetified by its first ad secod momets ad thus the referece Gaussia state is give by =ϱ, which, i tur, leads to D HS ϱ, =0 ad thus to ϱ =0. Lemma. If U is a uitary map correspodig to a symplectic trasformatio i phase space i.e., if U=exp ih with Hermitia H that is at most biliear i the field operators the UϱU = ϱ. This property esures that displacemet ad squeezig operatios do ot chage the Gaussia character of a quatum state. Proof. Let us cosider ϱ =UϱU. The the covariace matrix trasforms as ϱ = ϱ T, beig the symplectic trasformatio associated to U. At the same time the vector of mea values simply traslates to X =X+X 0, where X 0 is the displacemet geerated by U. Sice ay Gaussia state is fully characterized by its first ad secod momets, the referece state must ecessarily trasform as =U U i.e., with the same uitary trasformatio U. Sice the Hilbert-Schmidt distace ad the purity of a quatum state are ivariat uder uitary trasformatios, the lemma is proved. Lemma 3. ϱ is proportioal to the squared L C distace betwee the characteristic fuctios of ϱ ad of the referece Gaussia state. I the formula, ϱ d ϱ. Sice the otio of Gaussiaity of a quatum state is defied through the shape of its characteristic fuctio ad sice the characteristic fuctio of a quatum state belogs to the L C space 1, we address L C distace to as a good idicator of the o Gaussia character of ϱ. Proof. Sice characteristic fuctios of self-adjoit operators are eve fuctios of ad by meas of the idetity we obtai PHYSICAL REVIEW A 76, Tr O 1 O = d O 1 O, D HS ϱ, = 1 d ϱ. Lemma. Cosider a bipartite state ϱ=ϱ A ϱ G.Ifϱ G is a Gaussia state, the ϱ = ϱ A. Proof. We have ϱ = ϱ A ϱ G, = A G, ϱ, = ϱ A, A ϱ G,ϱ G. Therefore, sice ϱ G,ϱ G = ϱ G, we arrive at 037-

3 MEASURE OF THE NON-GAUSSIAN CHARACTER OF A PHYSICAL REVIEW A 76, ϱ = ϱ A ϱ G + A ϱ G ϱ A, A ϱ G,ϱ G ϱ A ϱ G = ϱ A. 5 The four properties illustrated by the above lemmas are the atural properties required for a good measure of the o- Gaussia character of a quatum state. Notice that by usig the trace distace D T ϱ, = 1 Tr ϱ istead of the Hilbert- Schmidt distace, we would lose lemmas 3 ad ad that the ivariace expressed by lemma holds thaks to the reormalizatio of the Hilbert-Schmidt distace through the purity ϱ. We stress the fact that our measure of o- Gaussiaity is a computable oe: It may be evaluated for ay quatum state of modes by the calculatio of the first two momets of the state, followed by the evaluatio of the overlap with the correspodig Gaussia state. Notice that ϱ is ot additive or multiplicative with respect to the tesor product. If we cosider a separable multipartite quatum state i the product form ϱ= k=1 ϱ k, the o-gaussiaity is give by ϱ = k=1 ϱ k + k=1 k k=1 k=1 ϱ k ϱ k, k, 6 where k is the Gaussia state with the same momets of ϱ k. I fact, sice the state ϱ is factorizable, we have that the correspodig Gaussia is a factorizable state too. IV. NON-GAUSSIANITY OF RELEVANT QUANTUM STATES Let us ow exploit the defiitio to evaluate the o- Gaussiaity of some relevat quatum states. At first we cosider Fock umber states p of a sigle mode as well as multimode factorizable states p made of copies of a umber state. The referece Gaussia states are a thermal state p = p with average photo umber p ad a factorizable thermal state N = p with average photo umber p i each mode 15. No-Gaussiaity may be aalytically evaluated, leadig to p p = 1 1+ p p = p +1 1 p +1 1 p +1 p 1 p, p +1 p +1 p p p +1 I the multimode case of p, we seek the umber of copies that maximizes the o-gaussiaity. I Fig. 1 we show both p p p ad p=max p p as a fuctio of p. As is apparet from the plot, the o-gaussiaity of the Fock states p icreases mootoically with the umber of photos, p, with the limitig value p =1/ obtaied for p. Upo cosiderig multimode copies of Fock states we obtai a larger value of o-gaussiaity: p is a decreasig fuctio of p, approachig =1/ from above. The value of p correspods to =3 for p 6 ad to = for 7 p Aother example is the superpositio of coheret states: S = N 1/ cos + si, with ormalizatio N=1+si exp, which for =± / reduces to the so-called Schrödiger cat states ad whose referece Gaussia state is a displaced squeezed thermal state S =D C S r N S r D C, where the real parameters C, r, ad N are aalytical fuctios of ad. Fially, we evaluate the o-gaussiaity of the two-mode Bell-like superpositios of Fock states: = cos 0,0 + si 1,1, = cos 0,1 + si 1,0, which for =± / reduces to the Bell states ± ad ±. The correspodig referece Gaussia states are, respectively, a two-mode squeezed thermal state =S N N S, where S =exp a 1 a ab deotes the two-mode squeezig operator ad =R N 1 N R amely, the correlated twomode state obtaied by mixig two thermal states at a beam splitter of trasmissivity cos, i.e., R =exp i a 1 a +a a 1. All the parameters ivolved i these referece Gaussia states are aalytical fuctios of the superpositio parameter. No-Gaussiaities are thus evaluated by meas of ad are reported i Fig. 1 as a fuctio of the parameter. As is apparet from the plot, the o-gaussiaity of Φ FIG. 1. Top No-Gaussiaity of sigle mode Fock states gray lie p ad of multimode Fock states p black lie as a fuctio of p. No-Gaussiaity for multimode states has bee maximized over the umber of copies,. Bottom No-Gaussiaity, as a fuctio of the parameter, for the two-mode superpositios dashed gray lie, solid gray lie, ad for the sigle-mode superpositio of coheret states, S, for =0.5 solid black lie ad =5 dashed black lie

4 GENONI, PARIS, AND BANASZEK PHYSICAL REVIEW A 76, sigle-mode states does ot surpass the value =1/, ad this fact is cofirmed by other examples ot reported here. As cocers the Schrödiger-cat-like states, we otice that for small values of the o-gaussiaity of the superpositio S shows a differet behavior for positive ad egative values of the parameter : for 0 ad =0.5 we have almost zero, while higher values are achieved for 0. For higher values of =5 i Fig. 1, o- Gaussiaity becomes a eve fuctio of. This differet behavior ca be uderstood by lookig at the Wiger fuctios of eve ad odd Schrödiger cat states for differet values of : for small values of the eve cat s Wiger fuctio is similar to a Gaussia fuctio, while the odd cat s Wiger fuctio shows a o-gaussia hole i the origi of phase space; icreasig the value of, the Wiger fuctios of the two kid of states become similar ad deviate from a Gaussia fuctio. We have also doe a umerical aalysis of the o- Gaussiaity of sigle-mode quatum states represeted by a fiite superpositio of Fock states: d ϱ d =,k=0 ϱ k k. To this aim we geerate radom quatum states i a fiite-dimesioal subspace, dim H d+1 1, followig the algorithm proposed by Życzkowski et al. 16,17 i.e., by geeratig a radom diagoal state i.e., a poit o the simplex ad a radom uitary matrix accordig to the Haar measure. I Fig. we report the distributio of o- Gaussiaity ϱ d, as evaluated for 10 5 radom quatum states, for three differet values of the maximum umber of photos, d. As is apparet from the plots, the distributio of ϱ d becomes Gaussia-like for icreasig d. I the fourth pael of Fig. we thus report the mea values ad variaces of the distributios as a fuctio of the maximum umber of photos, d. The mea value icreases with the dimesio, whereas the variace is a mootoically decreasig fuctio of d. Also for fiite superpositios simulatios we did ot observe o-gaussiaity higher tha 1/. Therefore, although we have o proof, we cojecture that =1/ is a limitig value for the o-gaussiaity of a sigle-mode state. Higher values are achievable for two-mode or multimode quatum states e.g., =/3 for the Bell states ±. V. GAUSSIFICATION AND DE-GAUSSIFICATION PROCESSES 8 FIG.. Distributio of o-gaussiaity ϱ d as evaluated for 10 5 radom quatum states, for three differet value of the maximum umber of photos, d. Top: d= left, d=10 right. Bottom: d=0 left. Bottom right Mea values ad variaces of the o- Gaussiaities evaluated for 10 5 radom quatum states, as a fuctio of the maximum umber of photos, d. We have also studied the evolutio of o-gaussiaity of quatum states udergoig either Gaussificatio or de- Gaussificatio processes. First, we have cosidered the Gaussificatio of Fock states due do the iteractio of the system with a bath of oscillators at zero temperature. This is perhaps the simplest example of a Gaussificatio protocol. I fact, the iteractio drives asymptotically ay quatum state to the vacuum state of the harmoic system, which, i tur, is a Gaussia state. The evolutio of the system is govered by the Lidblad master equatio ϱ = L a ϱ, where ϱ deotes time derivative, is the dampig factor, ad the Lidblad superoperator acts as follows: L a ϱ=a ϱa a aϱ ϱa a. Upo writig =e t the solutio of the master equatio ca be writte as ϱ = V m ϱv m, m V m = 1 m /m! 1/ a m 1/ a a m, 9 where ϱ is the iitial state. I particular, for the system iitially prepared i a Fock state, ϱ p = p p, we obtai, after evolutio, the mixed state ϱ p = V m ϱ p V m = l,p l l, 10 m l=0 with l,p = p l 1 p l l. The referece Gaussia state correspodig to ϱ p is a thermal state p = p with average photo umber p. No-Gaussiaity of ϱ p ca be evaluated aalytically; we have 1 p ϱ p = 1 m F 1 m, m,1; 1 1 m F 1 m, m,1; + 1+m 1 1 p 1+ m 1 m 1+m m+1,

5 MEASURE OF THE NON-GAUSSIAN CHARACTER OF A F 1 a,b,c;x beig a hypergeometric fuctio. We show the behavior of p i Fig. 3 as a fuctio of 1 for differet values of p. As is apparet from the plot, p is a mootoically decreasig fuctio of 1 as well as a mootoically icreasig fuctio of p. That is, at fixed time t, the higher the iitial photo umber p is, the larger the resultig o- Gaussiaity. Let us ow cosider the de-gaussificatio protocol obtaied by the process of photo subtractio. Icoclusive photo subtractio IPS has bee itroduced for siglemode ad two-mode states i 6,7,18 ad experimetally realized i 9. I the IPS protocol a iput state ϱ i is mixed with the vacuum at a beam splitter BS with trasmissivity T ad the, o ad off photodetectio with quatum efficiecy is performed o the reflected beam. The process ca be thus characterized by two parameters: the trasmissivity T ad the detector efficiecy. Sice the detector ca oly discrimiate the presece from the absece of light, this measuremet is icoclusive; amely it does ot resolve the umber of detected photos. Whe the detector clicks, a ukow umber of photos is subtracted from the iitial state ad we obtai the coditioal IPS state ϱ IPS. The coditioal map iduced by the measuremet is o- Gaussia 7, ad the output state is de-gaussified. Upo applyig the IPS protocol to the Gaussia sigle-mode squeezed vacuum S r 0 r R, where S r is the real squeezig operatio, we obtai 18 the coditioal state ϱ IPS, whose characteristic fuctio ϱ IPS is a sum of two Gaussia fuctios ad therefore is o loger Gaussia. The correspodig Gaussia referece state is a squeezed thermal state IPS =S IPS N IPS S IPS where the parameters IPS ad N IPS are aalytic fuctios of r, T, ad. No-Gaussiaity IPS = IPS T,,r has bee evaluated, ad i Fig. 3 bottom we report IPS for r=0.5 as a fuctio of the trasmittivity T for differet values of the quatum efficiecy. As is apparet from the plot, the IPS protocol ideed de-gaussifies the iput state; i.e., ozero values of the o-gaussiaity are obtaied. We foud that IPS is a icreasig fuctio of the trasmissivity T which is the relevat parameter, while the quatum efficiecy oly slightly affects the o-gaussia character of the output state. The highest value of o-gaussiaity is achieved i the limit of uit trasmissivity ad uit quatum efficiecy: lim IPS = 1 1 = S r 1 1 S r, T, 1 where the last equality is derived from lemma. This result is i agreemet with the fact that a squeezed vacuum state udergoig the IPS protocol is drive toward the target state S r 1 i the limit of T, Fially, we otice that for T, 1 ad for r the o-gaussiaity vaishes. I tur, this correspods to the fact that oe of the coefficiets of the two Gaussias of ϱ IPS vaishes; i.e., the output state is agai a Gaussia oe. VI. CONCLUSION AND OUTLOOK PHYSICAL REVIEW A 76, Η T FIG. 3. Color olie Top No-Gaussiaity of Fock states p udergoig Gaussificatio by the loss mechaism due to the iteractio with a bath of oscillators at zero temperature. We show p as a fuctio of 1 for differet values of p: from bottom to top, p=1,10,100,1000. Bottom No-Gaussiaity of ϱ IPS as a fuctio of T for r=0.5 ad for differet values of =0.,,0.6,0.8 from bottom to top. IPS results to be a mootoous icreasig fuctio of T, while oly slightly chages the o-gaussia character of the state. Havig at our disposal a good measure of o- Gaussiaity for the quatum state allows us to defie a measure of the o-gaussia character of a quatum operatio. Let us deote by G the whole set of Gaussia states. A coveiet defiitio for the o-gaussiaity of a map E reads as follows: E =max ϱ G E ϱ, where E ϱ deotes the quatum state obtaied after the evolutio imposed by the map. Ideed, for a Gaussia map E g, which trasforms ay iput Gaussia state ito a Gaussia state, we have E g =0. Work alog this lie is i progress, ad results will be reported elsewhere. I coclusio, we have proposed a measure of the o- Gaussia character of a CV quatum state. We have show that our measure satisfies the atural properties expected from a good measure of o-gaussiaity ad have evaluated the o-gaussiaity of some relevat states, i particular of states udergoig Gaussificatio ad de-gaussificatio protocols. Usig our measure, a aalog o-gaussiaity measure for quatum operatios may be itroduced. ACKNOWLEDGMENTS This work has bee supported by MIUR Project No. PRIN , the EC Itegrated Project QAP Cotract No , ad Polish MNiSW Grat No. 1 P03B APPENDIX: GAUSSIAN REFERENCE WITH UNCONSTRAINED MEAN VALUE As we have see from the above examples, ϱ of Eq. represets a good measure of the o-gaussia character 037-5

6 GENONI, PARIS, AND BANASZEK of a quatum state. A questio arises at to whether differet choices for the referece Gaussia state may lead to alterative, valid, defiitios. For example for sigle-mode states, we may defie ϱ = mi Ρ 0. Ρ 0. D HS ϱ, / ϱ, A1 where =D C S N S D C is a Gaussia state with the same covariace matrix of ϱ ad ucostraied vector of mea values X= Re C,Im C used to miimize the Hilbert- Schmidt distace. Here we report a few examples of the compariso betwee the results already obtaied usig with that comig from A1. As we will see, either the two Φ Φ FIG.. No-Gaussiaity of a Schrödiger-cat-like state as a fuctio of the superpositio parameter, with either C obtaied by umerical miimizatio solid lie or with C=Tr aϱ dotted lie. Left =0.5. Right =5. PHYSICAL REVIEW A 76, defiitios coicide or ad are mootoe fuctios of each other. Sice the defiitio correspods to a easily computable measure, we coclude that it represets the most coveiet choice. Let us first cosider the Fock state ϱ= p p. Accordig to A1, the referece Gaussia state is give by a displaced thermal state =D C p D C. The overlap betwee ϱ ad is give by p p, = 1 1+p exp 1+p 1+p C p p C L p p 1+p. A The maximum of A is achieved for C=0, which coicides with the assumptios C=Tr a p p. Let us cosider the quatum state 10 obtaied as the solutio of the loss master equatio for a iitial Fock state p p. The ucostraied Gaussia referece is agai a displaced thermal state =D C p D C, ad the overlap is give by ϱ p, =Tr ϱ p 1+ p 1 p = 1+p p+1 L p C 1+p 1 p 1 e C /1+p. Agai, sice the overlap is maximum for C=Tr aϱ p =0, both defiitios give the same results for the o- Gaussiaity. Let us ow cosider the Schrödiger-cat-like states of 7. The referece Gaussia state is a displaced squeezed thermal state, with squeezig ad thermal photos as calculated before. The optimizatio over the free parameter C may be doe umerically. I Fig. we show the o-gaussiaitiy, both as resultig from A1 ad by choosig C=Tr aϱ S as i, as a fuctio of. The two curves are almost the same, with o qualitative differeces. 1 A. Ferraro, S. Olivares, ad M. G. A. Paris, Gaussia States i Quatum Iformatio Bibliopolis, Napoli, 005. J. Eisert ad M. B. Pleio, It. J. Quatum If. 1, F. Dell Ao et al., Phys. Rep. 8, S. L. Braustei ad P. va Loock, Rev. Mod. Phys. 77, T. Opatry, G. Kurizki, ad D. G. Welsch, Phys. Rev. A 61, P. T. Cochrae, T. C. Ralph, ad G. J. Milbur, Phys. Rev. A 65, S. Olivares, M. G. A. Paris, ad R. Boifacio, Phys. Rev. A 67, ; S. Olivares ad M. G. A. Paris, Laser Phys. 16, N. J. Cerf, O. Kruger, P. Navez, R. F. Werer, ad M. M. Wolf, Phys. Rev. Lett. 95, J. Weger, R. Tualle-Brouri, ad P. Gragier, Phys. Rev. Lett. 9, ; A. Ourjoumtsev et al., Sciece 31, M. M. Wolf, G. Giedke, ad J. I. Cirac, Phys. Rev. Lett. 96, M. M. Wolf, D. Perez-Garcia, ad G. Giedke, Phys. Rev. Lett. 98, A. S. Holevo ad R. F. Werer, Phys. Rev. A 63, L. M. Dua, G. Giedke, J. I. Cirac, ad P. Zoller, Phys. Rev. Lett. 8, ; R. F. Werer ad M. M. Wolf, ibid. 86, K. E. Cahill ad R. J. Glauber, Phys. Rev. 177, P. Maria ad T. A. Maria, Phys. Rev. A 7, K. Życzkowski ad M. Kús, J. Phys. A 7, K. Życzkowski, P. Horodecki, A. Sapera, ad M. Lewestei, Phys. Rev. A 58, S. Olivares ad M. G. A. Paris, J. Opt. B: Quatum Semiclassical Opt. 7, S