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PHYSICAL REVIEW A 88, 063837 (2013) Distillation of quantum squeezing Radim Filip Department of Optics, Palack´ y University, 17 Listopadu 1192/12, 771 46 Olomouc, Czech Republic (Received 24 September 2013; published 20 December 2013) Squeezed states of various types of quantum oscillators are important nonclassical resources for quantum physics. We derive how much squeezing is universally distillable from multiple copies of non-Gaussian nonclassical states. We extend the approach to more efficient nonuniversal squeezing distillation, transforming hidden and scattered quantum nonclassicality to applicable squeezing. We demonstrate the distillation of squeezing from the mixtures of Gaussian states and also from quantum non-Gaussian states, incompatible with such the mixtures of Gaussian states. DOI: 10.1103/PhysRevA.88.063837 PACS number(s): 42.50.Dv, 42.50.Lc, 42.50.Xa I. INTRODUCTION Squeezed states of quantum harmonic oscillators are fundamentally nonclassical states of quantum physics. The nonclassical means incompatible with any mixture of coherent states, which are quantum analogs of classical states with maximum coherence [1]. Squeezed means that the variance of a linear combination X(θ ) = cos θX + sin θP of the co- ordinate X and momentum P is suppressed, for an optimal phase θ , below the level corresponding to the ground state of the quantum harmonic oscillator [2]. The squeezed states are necessary quantum resources for quantum processing with continuous variables [3], as well for a hybrid approach to quantum information processing [4]. More squeezing is specifically required to implement Gaussian operations [5], quantum metrology [6], and also for robust continuous- variable quantum key distribution [7]. The squeezing does not vanish for any loss during propagation or storage; however, it is very sensitive to noise. The squeezing affected by the noise can be completely lost and in the worst case, the state becomes classical [1]. In other cases, the noise can still completely erase the squeezing, but more positively, the state can remain nonclassical. A procedure which can conditionally increase the squeezing of quantum states without the help of any quantum nonlinearity is squeezing distillation. Although the squeezing distillation of ideal Gaussian states is physically prohibited [8], the squeezing distillation of various realistic non-Gaussian states is possible. During the last decade, both nonuniversal single-copy squeezing distillation [9] and universal multicopy squeezing distillation [1013] were proposed and experimentally tested for complementary types of non-Gaussian amplitude and phase fluctuations disturbing the squeezed state. Note, univer- sal distillation is a fixed procedure that does not require any knowledge about the input states. A similar approach was used to improve stability of the squeezing using universal quantum averaging, which produces a harmonic mean of variances of input states [14]. All these procedures belong to the general class of Gaussifiers [15], since they asymptotically converge to the Gaussian states as a consequence of the quantum central limit theorem [16]. These probabilistic methods are applicable for even weakly non-Gaussian states produced by the sources of squeezed states, in which a high squeezing can be masked by the non-Gaussian noise. In this paper, we derive the maximal squeezing achievable by universal distillation in an asymptotic limit of many copies of the marginal probability density of a quantum state. In this limit, the universal distiller extracts squeezing from the local uncertainty of the marginal probability density described by relative concavity around a global maximum. We further introduce nonuniversal squeezing distillation, combining the advantages of both approaches [9] and [1013], which can produce larger squeezing than the universal method. It is a witness of an interesting type of quantum nonclassicality nontrivially hidden and scattered in the marginal probability density. In particular, we apply it to conditionally distill squeezing from a mixture of squeezed states and from the quantum non-Gaussian states. The manuscript is organized as follows. In Sec. II, we describe basic features of the squeezing distillation as the prob- ability density filter, which are subsequently used in Sec. III to derive the minimal variance achievable by a universal version of the squeezing distillery. In the same section, the nonuniversal squeezing distillery is introduced. In Secs. IV and V, both the universal and nonuniversal squeezing distilla- tions are applied to the mixture of the Gaussian states and to the quantum non-Gaussian states. The conclusion summarizes the achieved results and discusses possible outlooks. II. SQUEEZING DISTILLATION AND LOCAL UNCERTAINTY The elementary squeezing distillation setup uses two copies of the state with the identical uncorrelated probability distributions P in (x 1 ) and P in (x 2 ) of the generalized coordinate variables x 1 and x 2 [10]. First, the beam splitter coupling applies the symmetrical linear transformation [17], x 1 = (x 1 x 2 )/ 2 and x 2 = (x 1 + x 2 )/ 2, to the arguments in P in (x 1 ) and P in (x 2 ). It produces a joint probability distribution P in ( x 1 ¯ x 2 2 )P in ( x 1 + ¯ x 2 2 ), where ¯ x 2 is a coordinate which is subsequently measured on one output of the coupling. By selecting the measured values ¯ x 2 in a narrow interval around ¯ x , the conditional probability density of the output approaches the distribution P out (x | ¯ x ) = 1 S(x) P in ( x+ ¯ x 2 )P in ( x¯ x 2 ). By the proper displacements of input states, it can be transformed into P out (x ) = 1 S (x ) P in x 2 P in x 2 + x , (1) S (x ) = 2R(x ) = 2 −∞ P in (x )P in (x + x )dx, 063837-1 1050-2947/2013/88(6)/063837(5) ©2013 American Physical Society

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Page 1: Distillation of quantum squeezing

PHYSICAL REVIEW A 88, 063837 (2013)

Distillation of quantum squeezing

Radim FilipDepartment of Optics, Palacky University, 17 Listopadu 1192/12, 771 46 Olomouc, Czech Republic

(Received 24 September 2013; published 20 December 2013)

Squeezed states of various types of quantum oscillators are important nonclassical resources for quantumphysics. We derive how much squeezing is universally distillable from multiple copies of non-Gaussiannonclassical states. We extend the approach to more efficient nonuniversal squeezing distillation, transforminghidden and scattered quantum nonclassicality to applicable squeezing. We demonstrate the distillation ofsqueezing from the mixtures of Gaussian states and also from quantum non-Gaussian states, incompatiblewith such the mixtures of Gaussian states.

DOI: 10.1103/PhysRevA.88.063837 PACS number(s): 42.50.Dv, 42.50.Lc, 42.50.Xa

I. INTRODUCTION

Squeezed states of quantum harmonic oscillators arefundamentally nonclassical states of quantum physics. Thenonclassical means incompatible with any mixture of coherentstates, which are quantum analogs of classical states withmaximum coherence [1]. Squeezed means that the varianceof a linear combination X(θ ) = cos θX + sin θP of the co-ordinate X and momentum P is suppressed, for an optimalphase θ , below the level corresponding to the ground stateof the quantum harmonic oscillator [2]. The squeezed statesare necessary quantum resources for quantum processingwith continuous variables [3], as well for a hybrid approachto quantum information processing [4]. More squeezing isspecifically required to implement Gaussian operations [5],quantum metrology [6], and also for robust continuous-variable quantum key distribution [7]. The squeezing does notvanish for any loss during propagation or storage; however, itis very sensitive to noise. The squeezing affected by the noisecan be completely lost and in the worst case, the state becomesclassical [1]. In other cases, the noise can still completelyerase the squeezing, but more positively, the state can remainnonclassical. A procedure which can conditionally increase thesqueezing of quantum states without the help of any quantumnonlinearity is squeezing distillation. Although the squeezingdistillation of ideal Gaussian states is physically prohibited [8],the squeezing distillation of various realistic non-Gaussianstates is possible.

During the last decade, both nonuniversal single-copysqueezing distillation [9] and universal multicopy squeezingdistillation [10–13] were proposed and experimentally testedfor complementary types of non-Gaussian amplitude andphase fluctuations disturbing the squeezed state. Note, univer-sal distillation is a fixed procedure that does not require anyknowledge about the input states. A similar approach was usedto improve stability of the squeezing using universal quantumaveraging, which produces a harmonic mean of variances ofinput states [14]. All these procedures belong to the generalclass of Gaussifiers [15], since they asymptotically convergeto the Gaussian states as a consequence of the quantum centrallimit theorem [16]. These probabilistic methods are applicablefor even weakly non-Gaussian states produced by the sourcesof squeezed states, in which a high squeezing can be maskedby the non-Gaussian noise.

In this paper, we derive the maximal squeezing achievableby universal distillation in an asymptotic limit of many copies

of the marginal probability density of a quantum state. Inthis limit, the universal distiller extracts squeezing from thelocal uncertainty of the marginal probability density describedby relative concavity around a global maximum. We furtherintroduce nonuniversal squeezing distillation, combining theadvantages of both approaches [9] and [10–13], which canproduce larger squeezing than the universal method. It isa witness of an interesting type of quantum nonclassicalitynontrivially hidden and scattered in the marginal probabilitydensity. In particular, we apply it to conditionally distillsqueezing from a mixture of squeezed states and from thequantum non-Gaussian states.

The manuscript is organized as follows. In Sec. II, wedescribe basic features of the squeezing distillation as the prob-ability density filter, which are subsequently used in Sec. IIIto derive the minimal variance achievable by a universalversion of the squeezing distillery. In the same section, thenonuniversal squeezing distillery is introduced. In Secs. IVand V, both the universal and nonuniversal squeezing distilla-tions are applied to the mixture of the Gaussian states and tothe quantum non-Gaussian states. The conclusion summarizesthe achieved results and discusses possible outlooks.

II. SQUEEZING DISTILLATION ANDLOCAL UNCERTAINTY

The elementary squeezing distillation setup usestwo copies of the state with the identical uncorrelatedprobability distributions Pin(x1) and Pin(x2) of the generalizedcoordinate variables x1 and x2 [10]. First, the beam splittercoupling applies the symmetrical linear transformation [17],x1 = (x ′

1 − x ′2)/

√2 and x2 = (x ′

1 + x ′2)/

√2, to the arguments

in Pin(x1) and Pin(x2). It produces a joint probabilitydistribution Pin( x ′

1−x ′2√

2)Pin( x ′

1+x ′2√

2), where x2 is a coordinate

which is subsequently measured on one output of the coupling.By selecting the measured values x2 in a narrow interval aroundx, the conditional probability density of the output approachesthe distribution Pout(x|x) = 1

S(�x)Pin( x+x√2

)Pin( x−x√2

). By theproper displacements of input states, it can be transformed into

Pout(x) = 1

S(�x)Pin

(x√2

)Pin

(x√2

+ �x

),

(1)

S(�x) =√

2R(�x) =√

2∫ ∞

−∞Pin(x)Pin(x + �x)dx,

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RADIM FILIP PHYSICAL REVIEW A 88, 063837 (2013)

where �x = √2x and S(�x) is the success probability

density. The multiplication of the input probabilitydistributions arising from their interference is a keystep to distillation of the squeezing.

To understand the elementary distillation, we redefinethe output distribution Pout(x) by deamplification of outputpositions using the transformation x → √

2x. After this step,the distillation simply works as a filter,

Pout(x) = HF (x)Pin(x), HF (x) = Pin(x + �x)

R(�x), (2)

on the probability density Pin(x), where the filtering functionHF (x) proportional to the distribution Pin(x + �x) of anothercopy can be optimized by a proper choice of postselectionvalue �x. For �x = 0, we obtain universal squeezing distil-lation, where the distillery is fixed. For the universal distil-lation, the positions xmax

out of all local maxima in deamplifieddistribution Pout(x) are equal to the positions xmax

in of maximain the input distributions Pin(x). It directly follows from thecondition d

dxP 2(x) = 2P (x) d

dxP (x) = 0, which is equivalent

to ddx

P (x) = 0, both required for maxima. The filtration (2)then suppresses the probability density of less probable valuesof x and the distribution Pout(x) is therefore peaked more aboutits maxima. The elementary step of the distillation increasesa difference between probability density in the local maxima;the probability density in the global maximum becomes higher.If two global maxima in different points are exactly equal (ifit can really happen), they theoretically still remain equal.Due to the simultaneous amplification effect, global maximaare additionally more separated and therefore, they are moredistinguishable. All these effects are translational invariant. Inthis way, a local uncertainty in the distribution Pout(x) aroundthe global maxima is reduced by the simplest distillation step,although global variance of Pout(x) may even increase.

A multicopy version of the universal distillation [10,12,13]

uses a series of linear transformations x ′j =

√j−1j

x ′j−1 +

1√jxj , x ′

1 = x1 of the coordinates provided by the beam splitterinteractions, where j = 1, . . . ,N (N � 2). The measuredvalues x ′

2, . . . ,x′N of the generalized coordinates x ′

2, . . . ,x′N

receive from all the outputs except the one described by x ′1,

or postselected in a tiny volume around x ′2 = . . . = x ′

N = 0.In this limit, the universal distillation produces the outputprobability density

Pout(x) =[Pin

(x√N

)]N

S(N),

(3)

S(N) =√

NR(N) =√

N

∫ ∞

−∞[Pin(x)]Ndx,

where S(N) denotes the success probability density integratedover variables x = (x1, . . . ,xn). Continuing the previous anal-ysis, after N − 1 steps of the multicopy distillation we obtainthe rescaled output probability density

P(N)out (x) = H

(N−1)F Pin(x), H

(N−1)F = [P (x)](N−1)

R(N), (4)

where for N − 1 steps, the distributions P(N)out (x) are scaled by

x → √Nx and renormalized. As N increases, the filter H

(N−1)F

can concentrate the distribution Pout(x) more around theglobal maxima, which become more separated. The filtrationfurther flattens other parts of the distribution. If two maximaare even weakly different, a single global maximum willbe asymptotically always more preferred in the distillationprocess. Only if more global maxima in different points arereally identical will the distillation theoretically keep all ofthem equal, but they are arbitrarily well separated due to theamplification effect.

Remarkably, a relative concavity of the probability dis-tribution around the maxima [absolute value of the secondderivative of P (x) near maxima normalized by value P (xmax)]satisfies

P′′(N)out

(xmax

out

)P

(N)out

(xmax

out

) = P′′(

xmaxin

)P

(xmax

in

) , (5)

as can be proven from (3). The relative concavity becomes aninvariant of the universal multicopy distillation process, whichis important later.

III. DISTILLABLE SQUEEZING

In the asymptotic limit of a large number of copies N ,because of the central limit theorem [16] we approach eithersingle Gaussian distribution or theoretically, a combination ofmultiple arbitrarily well-separated Gaussian distributions withvanishing overlap. In the latter case, we can use a single-copyprocedure [9] to conditionally extract just a single Gaussiandistribution. To find an asymptotic limit of the variance ofthis single Gaussian distribution for large N , we can expand[P ( x√

N)]N from (3) to the following Taylor series:

P (a)N(

1 + P ′(a)

P (a)

x − a√N

+ P ′′(a)

2P (a)

(x − a)2

N+ · · ·

)N

, (6)

around the global maximum x = a, where the distribution hasa tendency to concentrate. Since P ′′(a) = −|P ′′(a)| [concaveP (x) around the global maximum], in the limit of a large N , theTaylor series approaches the Gaussian probability distributionwith the distillable variance

Vd = P (a)

|P ′′(a)| . (7)

The distillation process therefore approaches (7), being aninverse of relative local concavity around a global maximumlocated in position x = a. The mean value of distributionproduced by the distillation can increase asymptotically forlarge N . However, for any finite N , it can be alwayscompensated by coherent displacements of output distributionor input distributions, shifting the global maximum to x = 0.The variance Vd without a residual output displacement cantherefore always be approached.

If the distribution P (x) has a more global maxima, wecan shift one from the rest of the maxima to the originx = 0, before the universal distillation on all shifted copiesis implemented. The relative concavity around that globalmaximum in the origin is then preserved, whereas other globalmaxima are shifted far from the origin, as the number of thecopies used in the distillation increases. To isolate this singleglobal maximum in the origin, a single-copy distillation [9]

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DISTILLATION OF QUANTUM SQUEEZING PHYSICAL REVIEW A 88, 063837 (2013)

adjusted to symmetrically filter out the distribution aroundthe origin can be simply used. This symmetrical single-copy filter requires only a variable beam-splitter interactionwith the transmittancy T with the reflected part detected inposition and selected in a tiny interval around the origin. Itconditionally transforms the output distribution Pout(x) fromthe universal distillation to the distribution proportional toP ′

out(x) ∝ exp(− x2

21−T

)Pout(√

T x). The variance of the filter can

be controlled by the transmittancy T . When the global maximaare well separated, the filter with T arbitrarily close to unitycan separate the single global maximum in the origin almostwithout any damping of the distribution Pout(x). Then, if Vd isbelow the variance of the ground state, the squeezing is reducedby the single-copy filter. The local uncertainty described byrelative concavity around the global maximum then turns todistillable squeezing. Since the relative local curvature |P ′′(x)|

P (x)in the global maximum is an invariant of the Gaussifyingdistillation procedure and it finally determines variance of theGaussian distribution, why it determines the variance Vd (7) issimply explained. It was recently proven that such processesgenerally converge to the Gaussian state [16]. Here we presenta lower bound on universally distillable squeezing.

The described method can be considered as a universalsqueezing distillation, since we can turn it to a universalprecept: after a displacement of a global maximum of P (x)to the origin, the distillery is fixed and independent ona detailed structure of the distribution P (x). However, formany copies P (x1) . . . P (xN ) of known and stable positiondistribution, we could asymptotically obtain larger quantumsqueezing or enlarge the class of distillable nonclassicalstates, if the universality of the distillation is abandoned.To reach a more general, nonuniversal squeezing distilla-tion, we can coherently displace all the input distributionsP (x1) . . . P (xN ) to P (x1 + a1) . . . P (xN + aN ) with optimizeddifferent a1, . . . ,aN and obtain, after the distillation, the outputprobability density P

(N)out (x) = 1

S(N)(a1,...,aN )

∏Nn=1 P ( x√

N+ aN ),

where S(N)(a1, . . . ,aN ) = ∫ ∞−∞

∏Ni=1 P ( x√

N+ ai)dx. Alterna-

tively, we can optimize the values of x ′2, . . . ,x

′N of coordinate

measurements, depicted in Fig. 1, which postselect in thedistillation described above, and abandon the predisplacementof the distributions. For this nonuniversal version of thedistillation we need to numerically search for an optimalsetting of a1, . . . ,aN (or x ′

2, . . . ,x′N ) to find the smallest

FIG. 1. N -copy squeezing distillery consisting of the beam-splitter couplings with a transmittancy Ti (Ti = 1

2 , 23 , . . . , N−1

N) and

the measurements of coordinate variables x ′2,x

′3, . . . ,x

′N .

variance Vd . The optimal setting then compromises betweenmany possible strategies how the squeezing can be extractedfrom nonclassicality. It is therefore the only numerical wayto completely prove that nonclassicality can be distilled toquantum squeezing.

IV. SQUEEZING DISTILLATION FROM A MIXTUREOF GAUSSIAN STATES

To simply demonstrate the squeezing distillation andapplicability of new nonclassicality measures, we can considerthe squeezed state with zero-mean Gaussian distributionsqueezed in the position with the variance Vs < 1/4, whichis randomly degraded to a noisy zero-mean Gaussian distri-bution with the variance Vn > 1/4. The resulting probabilitydensity is a mixture pPs(x) + (1 − p)Pn(x), where p isthe probability that Ps(x) passes the channel. The mixturewith p < Vn−1

4(Vn−Vs ) does not have any observable squeezing.Numerically checked, for Vs < 1/4 and for any finite Vn, atleast some u exists that |C(u)| > exp(− 1

2u2), even though thevariance Vx = pVs + (1 − p)Vn does not show any squeezing,proving nonclassicality [18]. However, we do not know whensqueezing can be obtained from nonclassicality. After theuniversal distillation, the variance

Vd = VsVn

p√

Vn + (1 − p)√

Vs

p√

V 3n + (1 − p)

√V 3

s

(8)

determines it directly. It can be approximated by Vd ≈ Vs +(1−p)V 3/2

s√Vn

for small Vs , and the squeezing is therefore distillableeven for small p and large Vn, which do not allow observationof the squeezing directly.

As an explicit example of the nonuniversal distillation,the previous mixture of two Gaussian distributions P (x) =pPs(x) + (1 − p)Pn(x) can be modified by considering thatthe mean value 〈xs〉 of PS(x) is now shifted to 〈xs〉 = 1(see Fig. 2). The variances are Vs = 0.05, Vn = 0.3, andp = 0.1. For these parameters, the distribution P (x) has asingle maximum at x = 0, where the universal distillationasymptotically produces Vd = 0.3 > 1/4. However, alreadytwo-copy nonuniversal distillation extracts squeezing Vd =0.19. It can be further pronouncedly increased using morecopies, as is visible in Fig. 1. It is a clear witness of quantum

0 5 10 15N

0.1

0.2

0.3

0.4

0.5Vd

FIG. 2. Squeezing distillation. Variance Vd from the universal dis-tillation (black squares) and after optimized nonuniversal distillation(black diamonds) for 〈xs〉 = 1, Vs = 0.05, Vn = 0.3, and p = 0.1.Gray circles indicate variance of the ground state.

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RADIM FILIP PHYSICAL REVIEW A 88, 063837 (2013)

nonclassicality nontrivially spread over the distribution P (x)and not accessible by the universal distiller. Simultaneously,this hidden and scattered nonclassicality remains a very goodand accessible resource which can be directly distillable to thesqueezing.

V. DISTILLATION FROM QUANTUMNON-GAUSSIAN STATES

Both a different and interesting example of nonuniversaldistillation appears for non-Gaussian quantum states ρ =p|1〉〈1| + (1 − p)|0〉〈0|, where |n〉 is an energy eigenstate ofthe quantum harmonic harmonic oscillator. In particular, it isvery interesting for quantum optics, where such a state is anoutput of realistic single-photon sources [19]. They exhibita negativity of Winger function only for p > 0.5, but theyare still nonclassical for any p > 0. Despite the positivity ofthe Wigner function for p � 0.5, they are still not a mixtureof Gaussian states for any p > 0 [20–23], in contrast to theprevious two examples. The coordinate distribution P (x) =〈x|ρ|x〉 =

√2π

exp(−2x2)[1 + p(4x2 − 1)] exhibits for p >

1/3 two equal global maxima placed at x = ±√

3p−12√

paround

the origin; otherwise the global maximum is in the center. Thevariance V = 1

4 (1 + 2p) of P (x) never exhibits any squeezing.For p � 1

3 , the universal distillation uses a = 0 and it

converges to a state with the variance Vd = 14

1−p

|3p−1| > V �1/4 without any squeezing. For p > 1

3 , when two equalmaxima appear, the universal distillation produces the varianceVd = 1

4p

|3p−1| after one Gaussian distribution is selected bya single-copy distillation [9], as we have discussed above.For all p > 1

2 , when the Winger function is negative theuniversal distillation produces the squeezing, which for p = 1approaches variance Vd = 0.125, corresponding to a half ofthe variance of the ground state. However, for the nonclassicalstates with a positive Wigner function, no squeezing canbe achieved by the universal distillation method. On theother hand, two-copy nonuniversal squeezing distillation

approaches the variance Vd = 5p−8+√2√

8−p(4+p)4(3p−4) . The variance

Vd < 14 is always squeezed for any p > 0. Remarkably,

squeezing is even distilled from all the nonclassical states witha positive Wigner function. Imperfect single-photon states cantherefore be conditional to the squeezed states and used forother applications. The resulting squeezing is not generallypure for p < 1; however, it is not limiting for many of theapplications of squeezed light. An extension to the multicopy,nonuniversal distillation of squeezing from the single photonsrequires numerical optimization of the values x ′

2, . . . ,x′N−1,

which leads to large complexity. This shows how complex it isto turn the nonclassicality in the quantum non-Gaussian stateto the observable squeezing.

VI. CONCLUSION

On one hand, we have derived a distillable squeezing(7) accessible from universal distillation protocols [10,11].On the other hand, we have clearly demonstrated that anonuniversal approach, which can pick up nonclassicalityscattered in the probability density, can be substantiallybetter. It reflects both the nontrivial and complex characterof quantum non-classicality, which is very fascinating subjectof research. In general, our approach is to address quantumnonclassicality distillable to a resource. However, the questionof whether nondistillable, quantum nonclassicality could existstill remains. To test experimentally the existence of the specialforms of hidden nonclassicality of quantum states discussedhere is feasible in an extension of the schemes [11–13]. Thiswill be a very interesting topic for further study of the physicsof quantum resources.

ACKNOWLEDGMENTS

I acknowledge financial support from Grant No.P205/12/0577 of the Czech Science Foundation. The researchleading to these results has also received funding from theEuropean Union Seventh Framework Programme under GrantAgreement No. 308803 (Project No. BRISQ2) and support ofthe Czech-Japan bilateral grant of MSMT CR No. LH13248(KONTAKT).

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[17] The beam splitter type of coupling is described by the interactionHamiltonian HBS = h κ

2 (X1P2 − P1X2) of the generalized posi-tions X1, X2 and momenta P1, P2, where κ is an interactionstrength. It can easily be obtained for optical, mechanical,and atomic harmonic oscillators. Since for many experimentalplatforms the beam-splitter type of linear coupling [17] and thelinear generalized position and momentum measurement are the

most available tools (which never produce squeezing from anyclassical Gaussian states), we reasonably limit our discussionof the squeezing distillation only to a combination of them.Otherwise, the squeezing can be generated by operation itselfand it will complicate a way of recognizing the distillation effect.The distiller does not use any active quantum nonlinearity in thegeneralized position or momentum.

[18] W. Vogel, Phys. Rev. Lett. 84, 1849 (2000); Th. Richter andW. Vogel, ibid. 89, 283601 (2002).

[19] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek,and S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).

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R. Filip, Phys. Rev. Lett. 107, 213602 (2011).[22] M. Jezek, A. Tipsmark, R. Dong, J. Fiurasek, L. Mista, Jr.,

R. Filip, and U. L. Andersen, Phys. Rev. A 86, 043813 (2012).[23] H. Song, K. B. Kuntz, and E. H. Huntington, New J. Phys. 15,

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