arX

iv:1

009.

6155

v2 [

quan

t-ph

] 24

Nov

201

0

Teleportation of squeezing: Optimization using non-Gaussian resources

Fabio Dell’Anno,1 Silvio De Siena,1 Gerardo Adesso,2 and Fabrizio Illuminati1,∗

1Dipartimento di Matematica e Informatica, Universita degli Studi di Salerno, CNISM, Unita di Salerno,and INFN, Sezione di Napoli - Gruppo Collegato di Salerno, Via Ponte don Melillo, I-84084 Fisciano (SA), Italy

2School of Mathematical Sciences, University of Nottingham,University Park, Nottingham NG7 2RD, United Kingdom

(Dated: November 24, 2010)

We study the continuous-variable quantum teleportation ofstates, statistical moments of observables, andscale parameters such as squeezing. We investigate the problem both in ideal and imperfect Vaidman-Braunstein-Kimble protocol setups. We show how the teleportation fidelity is maximized and the differencebetween output and input variances is minimized by using suitably optimized entangled resources. Specifically,we consider the teleportation of coherent squeezed states,exploiting squeezed Bell states as entangled resources.This class of non-Gaussian states, introduced in References [1, 2], includes photon-added and photon-subtractedsqueezed states as special cases. At variance with the case of entangled Gaussian resources, the use of entan-gled non-Gaussian squeezed Bell resources allows one to choose different optimization procedures that lead toinequivalent results. Performing two independent optimization procedures one can either maximize the stateteleportation fidelity, or minimize the difference betweeninput and output quadrature variances. The two dif-ferent procedures are compared depending on the degrees of displacement and squeezing of the input states andon the working conditions in ideal and non-ideal setups.

PACS numbers: 03.67.Hk, 03.67.Mn, 42.50.Pq

I. INTRODUCTION

Non-Gaussian quantum states, endowed with properly en-hanced nonclassical properties, may constitute powerful re-sources for the efficient implementation of quantum informa-tion, communication, computation and metrology tasks [1–13]. Indeed, it has been shown that, at fixed first and secondmoments, Gaussian statesminimizevarious nonclassical prop-erties [14, 15]. Therefore, many theoretical and experimentalefforts have been made towards engineering and controllinghighly nonclassical, non-Gaussian states of the radiationfield(for a review on quantum state engineering, see e.g. [16]).In particular, several proposals for the generation of non-Gaussian states have been presented [17–23], and some suc-cessful ground-breaking experimental realizations have beenalready performed [24–29]. Concerning continuous-variable(CV) quantum teleportation, to date the experimental demon-stration of the Vaidman-Braunstein-Kimble (VBK) teleporta-tion protocol [30, 31] has been reported both for input coher-ent states [32–36], and for squeezed vacuum states [37, 38].Inparticular, Ref. [38] has reported the teleportation of squeez-ing, and consequently of entanglement, between upper andlower sidebands of the same spatial mode. It is worth to re-mark that the efficient teleportation of squeezing, as well asof entanglement, is a necessary requirement for the realiza-tion of a quantum information network based on multi-stepinformation processing [39].

In this paper, adopting the VBK protocol, we study infull generality, e.g. including loss mechanisms and non-unitygain regimes, the teleportation of input single-mode coherentsqueezed states using as non-Gaussian entangled resourcesa

∗Corresponding author. Electronic address: illuminati@sa.infn.it

class of non-Gaussian entangled quantum states, the class ofsqueezed Bell states [1, 2]. This class includes, for specificchoices of the parameters, non-Gaussian photon-added andphoton-subtracted squeezed states. In tackling our goal, weuse the formalism of the characteristic function introduced inRef. [40] for an ideal protocol, and extended to the non-idealinstance in Ref. [2]. Here, in analogy with the teleportationof coherent states, we first optimize the teleportation fidelity,that is, we look for the maximization of the overlap betweenthe input and the output states. But the presence of squeez-ing in the unknown input state to be teleported prompts alsoan alternative procedure, depending on the physical quanti-ties of interest. In fact, if one cares about reproducing in themost faithful way the initial state in phase-space, then thefi-delity is the natural quantity that needs to be optimized. Onthe other hand, one can be interested in preserving as muchas possible the squeezing degree at the output of the teleporta-tion process, even at the expense of the condition of maximumsimilarity between input and output states. In this case, oneaims at minimizing the difference between the output and in-put quadrature averages and the quadrature variances. It isim-portant to observe that this distinction makes sense only ifoneexploits non-Gaussian entangled resources endowed with tun-able free parameters, so that enough flexibility is allowed torealize different optimization schemes. Indeed, it is straight-forward to verify that this is impossible using Gaussian en-tangled resources. We will thus show that exploiting non-Gaussian resources one can identify the best strategies fortheoptimization of different tasks in quantum teleportation,suchas state teleportation vs teleportation of squeezing. Compari-son with the same protocols realized using Gaussian resourceswill confirm the greater effectiveness of non-Gaussian statesvs Gaussian ones as entangled resources in the teleportationof quantum states of continuous variable systems.

The paper is organized as follows. In Section II, we intro-

2

duce the single-mode input states and the two-mode entangledresources, and we recall the basics of both the ideal and theimperfect VKB quantum teleportation protocols. With respectto the instance of Gaussian resources (twin beam), the furtherfree parameters of the non-Gaussian resource (squeezed Bellstate) allow one to undertake an optimization procedure to im-prove the efficiency of the protocols. In Section III we inves-tigate the optimization procedure based on the maximizationof the teleportation fidelity. We then analyze an alternativeoptimization procedure leading to the minimization of the dif-ference between the quadrature variances of the output andinput fields. This analysis is carried out in Section IV. Weshow that, unlike Gaussian resources, in the instance of non-Gaussian resources the two procedures lead to different resultsand, moreover, always allow one to improve on the optimiza-tion procedures that can be implemented with Gaussian re-sources. Finally, in Section V we draw our conclusions anddiscuss future outlooks.

II. TELEPORTATION OF SQUEEZED COHERENTSTATES USING SQUEEZED BELL RESOURCES

In this Section, we briefly recall the basics of the ideal andimperfect VBK CV teleportation protocols (for details seeRef. [2]). The scheme of the (CV) teleportation protocol isthe following. Alice wishes to send to Bob, who is at a re-mote location, a quantum state, drawn from a particular setaccording to a prior probability distribution. The set of in-put states and the prior distribution are known to Alice andBob, however the specific state to be teleported that is pre-pared by Alice remains unknown. Alice and Bob share a re-source, e.g. a two-mode entangled state. The input state andone of the modes of the resource are available for Alice, whilethe other mode of the resource is sent to Bob. Alice performsa suitable (homodyne) Bell measurement, and communicatesthe result to Bob exploiting a classical communication chan-nel. Then Bob, depending on the result communicated by Al-ice, performs a local unitary (displacement) transformation,and retrieves the output teleported state. The non-ideal (real-istic) teleportation protocol includes mechanisms of lossandinefficiency: the photon losses occurring in the realistic Bellmeasurements, and the noise arising in the propagation of op-tical fields in noisy channels (fibers) when the second modeof the resource is sent to Bob. The photon losses occurringin the realistic Bell measurements are modeled by placing infront of an ideal detector a fictitious beam splitter with non-unity transmissivityT 2 (and corresponding non-zero reflec-tivity R2 = 1 − T 2) [41]. The propagation in fiber is mod-eled by the interaction with a Gaussian bath with an effectivephoton numbernth, yielding a damping process with inverse-time rateγ [42, 43]. Denoting byin the input field mode,and by1 and2, respectively, the first and the second modeof the entangled resource, the decoherence due to imperfectphoto-detection in the homodyne measurement performed byAlice involves the input field modein, and one mode of theresource, e.g. mode1. Throughout, we assume a pure entan-gled resource. Indeed, it is simple to verify that considering

mixed (impure) resources is equivalent to a consider a suit-able nonvanishing detection inefficiencyR [2]. The degra-dation due to propagation in fiber affects the other mode ofthe resource, e.g. mode2, which has to reach Bob’s remoteplace at the output stage. Denoting now byρin = |φ〉in in〈φ|andρres = |ψ〉12 12〈ψ| the projectors corresponding, respec-tively, to a generic pure input single-mode state and a genericpure two-mode entangled resource, the characteristic functionχout of the single-mode output fieldρout can be written as [2]:

χout(α) = Tr[Dout(α)ρout]

= e−Γτ,R|α|2χin (gT α)χres

(

gT α∗; e−τ2 α

)

,(1)

whereDout(α) = eαa†out+α∗aout is the Glauber displace-

ment operator,χin(α) = Tr[Din(α)ρin] is the char-acteristic function of the input state,χres(α1, α2) =Tr[D1(α1)D2(α2)ρres] is the characteristic function of theresource,g is the gain factor of the protocol [44],τ ≡ γt isthe scaled dimensionless time proportional to the fiber propa-gation length, and the functionΓτ,R is defined as:

Γτ,R = (1− e−τ )

(

1

2+ nth

)

+ g2R2 . (2)

We assume in principle to have some knowledge about thecharacteristics of the experimental apparatus: the inefficiencyR (orT ) of the photo-detectors, and the loss parametersτ andnth of the noisy communication channel.

We consider as input state a single-mode coherent andsqueezed (CS) state|ψCS〉in with unknown squeezing param-eter(ε = s eiϕ) and unknown coherent amplitudeβ. We thenconsider as non-Gaussian entangled resource the two-modesqueezed Bell (SB) state|ψSB〉12, defined as [1, 2]:

|ψCS〉in = Din(β)Sin(ε)|0〉in , (3)

|ψSB〉12 = S12(ζ){cos δ|0, 0〉12 + eiθ sin δ|1, 1〉12} . (4)

Here Din(β) = eβa†in

+β∗ain is, as before, the displace-

ment operator,Sin(ε) = e−1

2εa†2

in+ 1

2ε∗a2

in is the single-mode

squeezing operator,S12(ζ) = e−ζa†1a†2+ζ∗a1a2 is the two-

mode squeezing operator(ζ = reiφ), with aj denoting theannihilation operator for modej (j = in, 1, 2), |m,n〉12 ≡|m〉1 ⊗ |n〉2 is the two-mode Fock state (of modes 1 and 2)with m photons in the first mode andn photons in the secondmode, andθ andδ are two intrinsic free parameters of the re-source entangled state, in addition tor andφ, which can beexploited for optimization. Note that particular choices of theangleδ in the class of squeezed Bell states Eq. (4) allow oneto recover different instances of two-mode Gaussian and non-Gaussian entangled states: forδ = 0 the Gaussian twin beam(TwB); for δ = arccos

[

(cosh 2r)−1/2 sinh r]

andθ = φ− πthe two-mode photon-added squeezed (PAS) state|ψPAS〉12;for δ = arccos

[

(cosh 2r)−1/2 cosh r]

andθ = φ−π the two-mode photon-subtracted squeezed (PSS) state|ψPSS〉12. The

3

object parameter description

input stateβ displacement

s squeezing degree

ϕ squeezing phase

two-mode resource state

r squeezing degree

φ squeezing phase

δ mixing angle

θ mixing phase

teleportation apparatus

R=√1− T 2 detection inefficiency

τ fiber loss factor

nth fiber bath temperature

g gain of the protocol

TABLE I: Summary of the notation employed throughout this workto describe the different parameters that characterize theinput coher-ent squeezed (CS) states [Eq. (3)], the shared entangled two-modesqueezed Bell (SB) resources [Eq. (4)], and the characteristics ofnon-ideal teleportation setups [2]. See text for further details on therole of each parameter.

last two non-Gaussian states are defined as:

|ψPAS〉12 = (cosh 2r)−1/2a†1a†2S12(ζ)|0, 0〉12 , (5)

|ψPSS〉12 = (cosh 2r)−1/2a1a2S12(ζ)|0, 0〉12 , (6)

and are already experimentally realizable with current tech-nology [24–28].

In the following Section we study, in comparison with theinstance of two-mode Gaussian entangled resources, the per-formance of the optimized two-mode squeezed Bell stateswhen used as entangled resources for the teleportation of in-put single-mode coherent squeezed states. For completeness,

in the same context we make also a comparison with the per-formance, as entangled resources, of the more specific real-izations (5), (6). The characteristic functions of states (3), (4),(5), and (6) are computed and their explicit expressions aregiven in Appendix A.

For ease of reference, table I provides a summary of the pa-rameters associated with the input states, the shared resources,and the sources of noise in the teleportation protocol.

III. OPTIMAL TELEPORTATION FIDELITY

The commonly used measure to quantify the performanceof a quantum teleportation protocol is the fidelity of telepor-tation [45], F = Tr[ρinρout], which amounts to the over-lap between a pure input stateρin and the (generally mixed)teleported stateρout. In the formalism of the characteristicfunction the fidelity reads

F =1

π

∫

d2α χin(α)χout(−α) , (7)

whereχin(α) is the characteristic function of the single-modeinput stateρin = |ψCS〉in in〈ψCS |, Eq. (3), andχout(α) is thecharacteristic function for the output teleported state, Eq. (1).In this Section, we will make use of Eq. (7) to analyze theefficiency of the CV teleportation protocol.

In the instance of non-Gaussian squeezed Bell resources(4), at fixed squeezing parameter, the optimization procedureamounts to the maximization of the teleportation fidelity (7)over the free parameters of the entangled resource. It can beshown that the optimal choice for the phasesφ andθ is φ = πandθ = 0. The analytic expression for the fidelityFCS of thenon-ideal quantum teleportation of coherent squeezed statesusing squeezed Bell resources reads

FCS =4√Λ1Λ2

eω21

Λ1−ω2

2

Λ2

{

1 + e−(2r+τ) sin δ(∆2 cos δ −∆1 sin δ)

[

1

Λ1

(

1 +2ω2

1

Λ1

)

+1

Λ2

(

1− 2ω22

Λ2

)]

+1

4e−2(2r+τ)∆2

2 sin2 δ

[

1

Λ21

(

3 +12ω2

1

Λ1+

4ω41

Λ21

)

+1

Λ22

(

3− 12ω22

Λ2+

4ω42

Λ22

)

(8)

+2

Λ1Λ2

(

1 +2ω2

1

Λ1− 2ω2

2

Λ2− 4ω2

1ω22

Λ1Λ2

)]}

,

where, introducingg = gT , the quantitiesΛ1, Λ2, ∆1, ∆2, ω1, andω2 are defined by the following relations:

∆1 = 1+ e4r + 2eτ2 (1− e4r)g + eτ (1 + e4r)g2 ,

∆2 = 1− e4r + 2eτ2 (1 + e4r)g + eτ (1− e4r)g2 ,

Λ1 = e−2r−τ∆1 + 2e2s(1 + g2) + 4Γτ,R ,

Λ2 = e−2r−τ∆1 + 2e−2s(1 + g2) + 4Γτ,R ,

ω21 = (1− g)2(β − β∗)2 , ω2

2 = (1− g)2(β + β∗)2 ,

g = gT . (9)

4

For different choices ofδ in Eq. (9), see Section II, oneobtains the teleportation fidelities associated to photon-addedand photon-subtracted squeezed resource states. Let us ob-serve that the fidelity in Eq. (9) depends both on the inputcoherent amplitudeβ, and on the input single-mode squeez-ing parameters, while it is independent of the input squeezingphaseϕ. Once again, it is worth stressing that, in the telepor-tation paradigm, the input state is unknown and only partial(probabilistic) knowledge on the alphabet of input states isadmitted. It is thus required, in principle, to assume teleporta-tion protocols independent of the input parameters, as it turnsout to be the case for the VBK protocol with Gaussian entan-gled resources and input coherent states. However, in moregeneral cases, one can study the behavior of the so-called one-shot fidelity, that is the teleportation fidelity at specific valuesof the input parameters. Suitable averages of the one-shot fi-delity over the set of input states and parameters, according toan assigned prior distribution, will then result in the averagequantum teleportation fidelity. The latter quantity can then beconfronted with so-called classical fidelity thresholds (bench-marks) that correspond to the maximum achievable averagefidelity between the input state (measured by Alice in order toachieve an optimal estimation of it) and the output state (pre-pared by Bob according to Alice’s measurement outcomes),without the use of any shared entanglement [45]. While tele-portation benchmarks are available for the cases of coher-ent input states (with completely unknownβ) [46], purelysqueezed input states (withβ = ϕ = 0 and completely un-knowns) [47], as well as for states with known squeezing de-gree and unknown displacement and phase [48], a benchmarkfor the case of input states with totally unknown displacementand squeezing has not yet been derived, and stands as a chal-lenging problem in quantum estimation theory.

Henceforth, assuminga priori that the input parameters(displacement and squeezing degree) are completely random,we adopt then the following approach to optimize the quan-tum teleportation fidelity. We exploit a non-unity gain strategyto remove at least theβ-dependence in the one-shot fidelity;then, we study the behavior of theβ-independent one-shot fi-delity for specific values of the input squeezing parameters,in order to identify an effective,s-independent approximation.Indeed, fixing the gaing at the valueg = 1/T (g = 1) inEq. (9) yields theβ-independent fidelityFS :

FS =4√Λ1Λ2

{

e−2(2r+τ)

4∆2

2 sin2 δ

(

3

Λ21

+3

Λ22

+2

Λ1Λ2

)

+e−(2r+τ) sin δ(∆2 cos δ −∆1 sin δ)

(

1

Λ1+

1

Λ2

)

+1

}

, (10)

where the quantitiesΛ1, Λ2, ∆1, ∆2, ω1, andω2 are definedin Eq. (9). For different choices ofδ (see Section II), one ob-tains the teleportation fidelities associated to the use of differ-ent Gaussian and non-Gaussian entangled resources: the twinbeam, the photon-added, and the photon-subtracted squeezedstates. For such resources no optimization procedure is possi-ble asδ is a specific function ofr. Instead, the optimization of

FIG. 1: (Color online) One-shot fidelityFS at fixed s, and asa function of the angleδ parameterized bys, expressed in dB,i.e. δ(s) = δsubopt, see Eq. (13), both in the instance of the idealprotocolτ = nth = R = 0 (full lines), and of a non-ideal protocol,with τ = 0.1, nth = 0, andR2 = 0.05 (dashed lines). The one-shotfidelities are drawn for three different values of the input squeezing:s = 0, 5, 10 dB. The curves are ordered from top to bottom forincreasings.

the fidelity (10) with respect to the free non-Gaussian param-eterδ identifies the optimal squeezed Bell resource associatedto the optimal value:

δopt=1

2arctan

[

4∆2Λ1Λ2(Λ1+Λ2)4∆1Λ1Λ2(Λ1+Λ2)−e−2r−τ∆2

2(3Λ2

1+2Λ1Λ2+3Λ2

2)

]

.

(11)Let us notice that, forτ = nth = R = 0 (ideal protocol) ands = 0 (input coherent states), Eq. (11) reduces to [1]:

δopt =1

2arctan

[

1 + e−2r]

. (12)

The displacement-independent one-shot fidelityFS and theoptimal angleδopt are still dependent ons, the input squeez-ing. Unfortunately, the optimization of the non-Gaussianresource based on the choice (11) as optimal angle wouldbe practically unfeasible because the input squeezing is notknown. In order to circumvent this problem, we introduce asub-optimal angleδsubopt such that

δsubopt ≡ δopt∣

∣

s=s, (13)

wheres is a fixed effective value of the input squeezing cho-sen, according to a suitable criterion that will be clarifiedbe-low, in the range of possible values of the squeezing parameters.

In the following we will express the squeezing parametersr ands in decibels, according to the relation [49]:

κ (dB) = 10 log10 e2κ , κ = s, r . (14)

The practical rationale for introducing a sub-optimal charac-terization in the maximization of the output fidelity is basedon the observation that the assumption of a completely ran-dom degree of input squeezings is clearly unrealistic. It is

5

instead very sensible to consider that the range of possiblevalues ofs falls in a window[0, smax] dB. Indeed, to date, theexperimentally reachable values of squeezing fall roughlyinsuch a range withsmax ≃ 10 dB [50].

We can then study the behavior ofFS corresponding to theangleδsubopt as a function of the effective input squeezing pa-rameters, at fixed squeezing parameters of the resource andof the input state, respectivelyr ands, and at fixed loss param-etersτ , nth, andR. Fig. 1 shows thatFS is quite insensitiveto the value ofs. Assuming the realistic ranges ∈ [0, 10] dB,the choice of a sub-optimal angle such thats = 5 dB (aver-age value of the interval), leads to a decrease of the optimizedfidelity, compared to the choice ofδopt, of at most0.3% inideal conditions, and even smaller in realistic conditions. Inother words, the teleportation fidelity is essentially constantin the considered interval of variability for the angleδ. There-fore, throughout in the following, we fixs = s = 5 dB in theexpression Eq. (11) to make its-independent.

In Fig. 2, we plot the teleportation fidelity associated tothe various considered resources (Gaussian twin beam, opti-mized two-mode squeezed Bell-like state, two-mode squeezedphoton-subtracted state) both for the ideal protocol (panel I)and for the non-ideal protocol (panel II). We see that, at fixed(finite) squeezingr of the resource, the Gaussian twin beam isalways outperformed by the optimal non-Gaussian squeezedBell resource in the ideal protocol. It is worth to remark thatfor very high values of the squeezingr, the advantage of thenon-Gaussian resources fades and Gaussian twin beams per-form in practice equally well for the teleportation of the con-sidered input states. This reflects the well known fact that,us-ing the ideal VBK protocol and an ideal Einstein–Podolsky–Rosen resource (corresponding, e.g., to a twin beam in thelimit r → ∞), anyquantum state can be unconditionally tele-ported with unit fidelity [30]. All the one-shot fidelities de-crease for increasing squeezings of the input and, interest-ingly, in the non-ideal protocol they achieve a maximum at afinite valuerm of the squeezingr of the resource. The opti-mal squeezed Bell resource and the twin beam share the samerm ≃ 16 dB and coincide at that point. In Fig. 2 we also plotthe one-shot fidelities associated with the two-mode photon-subtracted squeezed states, Eq. (6). The two-mode photon-subtracted squeezed state always outperforms the twin beamin the ideal protocol, and at low and intermediate values of theresource squeezingr in the non-ideal case. It is always out-performed by the optimized squeezed Bell resource. We notethat, on the other hand, the two-mode photon-added squeezedstates always exhibit a performance worse than the two-modephoton-subtracted squeezed states and the squeezed Bell re-sources (the corresponding fidelities are omitted in the plotsfor clarity). In a given range of the squeezingr, |ψPSS〉12and|ψSB〉12 exhibit comparable levels in the fidelity of tele-portation. In conclusion, properly optimized non-Gaussian re-sources maximize the fidelity of teleportation of squeezed co-herent states both in the ideal and imperfect VBK protocols,outperforming the corresponding Gaussian resources. In thenext Section we carry out a similar analysis with the aim ofidentifying the optimal strategy that maximizes the reproduc-tion at the output of the input squeezing.

IV. TELEPORTATION OF QUADRATURE MOMENTS

In this Section, we introduce a different approach to the op-timization of the teleportation protocol, aimed at retaining andfaithfully reproducing at the output the variances and thusthesqueezing of the input state. The strategy is to constrain thefirst and second order moments of the output field to repro-duce the ones of the input field, by exploiting the free param-eters of the non-Gaussian resources. We introduce the meanvalues〈Zj〉 = Tr[Zjρj ], with Zj = Xj , Pj (j = in, out),and the variances〈∆Z2

j 〉 = Tr[Z2j ρj ] − Tr[Zjρj ]

2, and〈∆(XjPj)S〉 = Tr[(XjPj +PjXj)ρj ]− 2Tr[Xjρj ]Tr[Pjρj ](the cross-quadrature variance, withS denoting the sym-metrization) of the quadrature operatorsXj = 1√

2(aj + a†j),

Pj = i√2(a†j − aj), associated with the single-mode input

stateρin and the output stateρout of the teleportation proto-col. The explicit expressions for the quantities〈Zj〉, 〈∆Z2

j 〉,and〈∆(XjPj)S〉 are reported in the Appendix B.

The quantities measuring the deviation of the output fromthe input are the differences between the output and input firstand second quadrature moments:

D(X) ≡ 〈Xout〉 − 〈Xin〉 = (g − 1)〈Xin〉 ,D(P ) ≡ 〈Pout〉 − 〈Pin〉 = (g − 1)〈Pin〉 ,D(∆X2) ≡ 〈∆X2

out〉 − 〈∆X2in〉 = (g2 − 1)〈∆X2

in〉+Σ ,

D(∆P 2) ≡ 〈∆P 2out〉 − 〈∆P 2

in〉 = (g2 − 1)〈∆P 2in〉+Σ ,

D(∆(XP )S) ≡ 〈∆(XoutPout)S〉 − 〈∆(XinPin)S〉= (g2 − 1)〈∆(XinPin)S〉 ,

with Σ given by Eq. (B11). From the above equations, we seethat the assumptiong = 1 (i.e. g = 1/T ) yieldsD(X) =D(P ) = D(∆(XP )S) = 0 andD(∆X2) = D(∆P 2) =Σ∣

∣

g=1. Therefore, forg = 1, the input and output fields pos-

sess equal average position and momentum (equal first mo-ments),and equal cross-quadrature variance; then, the opti-mization procedure reduces to the minimization of the quan-tity Σ

∣

∣

g=1with respect to the free parameters of the non-

Gaussian squeezed Bell resource, i.e.minδ,θ,φΣ∣

∣

g=1. More-

over, as for the optimization procedure of Section III, it canbe shown that the optimal choice forφ andθ is, once again,φ = π andθ = 0. The optimization on the remaining freeparameterδ yields the optimal valueδoptvar:

δoptvar =1

2arctan

[

(1 + eτ2 )2 − e4r(1 − e

τ2 )2

(1 + eτ2 )2 + e4r(1 − e

τ2 )2

]

. (15)

The optimal angleδoptvar , corresponding to the minimizationof the differencesD(∆X2) andD(∆P 2) between the out-put and input quadrature variances, is independent ofR, atvariance with the optimal valueδopt, Eq. (11), correspond-ing to the maximization of the teleportation fidelity. It is alsoimportant to note that in this case there are no questions re-lated to a dependence on the input squeezings. For τ = 0Eq. (15) reduces toδoptvar = π/8. Such a value is equalto the asymptotic value given by Eq. (12) forr → ∞, sothat, in this extreme limit the two optimization proceduresbe-come equivalent. In the particular cases of photon-added and

6

FIG. 2: (Color online) One-shot fidelityFS as a function of the squeezing parameterr of the entangled resource, expressed in dB, for thesub-optimal squeezed Bell resource (full black line), for the photon-subtracted squeezed resource (dotted red line),and for the twin beamresource (dashed blue line), in the instance of the ideal protocol (panel I),τ = nth = R = 0, and the non-ideal protocol (panel II), withτ = 0.1, nth = 0, andR2 = 0.05. The one-shot fidelities are drawn for three different values of the input squeezing:s = 0, 5, 10 dB. In theplots, the fidelity corresponding to the photon-added squeezed resource has been omitted as it is always lower than the ones corresponding tothe photon-subtracted squeezed resource and the squeezed Bell resource. The curves are ordered from top to bottom for increasings.

photon-subtracted resources, no optimization procedure canbe carried out, and the parameterδ is simply a given specificfunction of r (see Section II). We remark that, having au-tomatically zero difference in the cross-quadrature varianceat g = 1, finding the angles that minimizeD(∆X2) andD(∆P 2) precisely solves the problem of achieving the op-timal teleportation of both the first moments and the full co-variance matrix of the input state at once.

In order to compare the performances of the Gaussianand non-Gaussian resources, and to emphasize the improve-ment of the efficiency of teleportation with squeezed Bell-likestates, we consider first the instance of ideal protocol (τ = 0,nth = 0, R = 0), and compute, and explicitly report below,the output variances〈∆Z2

out〉f of the teleported state asso-ciated with non-Gaussian resources (i.e. optimized squeezedBell-like states(f = SB), photon-added squeezed states(f = PAS), photon-subtracted squeezed states(f = PSS)),and with Gaussian resources, i.e. twin beams(f = TwB).From Eqs. (B8)–(B13), we get:

〈∆Z2out〉TwB = 〈∆Z2

in〉+ e−2r . (16)

〈∆Z2out〉PAS = 〈∆Z2

in〉+ e−2r

{

1 +2e−2r(1 + e−2r)

1 + e−4r

}

,

(17)

〈∆Z2out〉PSS = 〈∆Z2

in〉+ e−2r

{

1− 2e−2r(1− e−2r)

1 + e−4r

}

,

(18)

〈∆Z2out〉SB = 〈∆Z2

in〉+ e−2r(2−√2) , (19)

Eq. (19) is derived exploiting the optimal angle (15), whichreduces to Eq. (12) in the ideal case. Independently of the

resource, the teleportation process will in general resultinan amplification of the input variance. However, the use ofnon-Gaussian optimized resources, compared to the Gaussianones, reduces sensibly the amplification of the variances atthe output. Looking at Eq. (16), we see that the teleporta-tion with the twin beam resource produces an excess, quanti-fied by the exponential terme−2r, of the output variance withrespect to the input one. On the other hand, the use of thenon-Gaussian squeezed Bell resource Eq. (19) yields a reduc-tion in the excess of the output variance with respect to theinput one by a factor(2 −

√2). Let us now analyze the be-

haviors of the photon-added squeezed resources and of thephoton-subtracted squeezed resources, Eqs. (17) and (18),re-spectively. We observe that, in analogy with the findings of theprevious Section, the photon-subtracted squeezed resourcesexhibit an intermediate behavior in the ideal protocol; indeedfor low values ofr they perform better than the Gaussian twinbeam, but worse than the optimized squeezed Bell states. Thephoton-addedsqueezed resources perform worse than both thetwin beam and the other non-Gaussian resources. These con-siderations follow straightforwardly from a quantitativeanal-ysis of the terms associated with the excess of the output vari-ance in Eqs. (17) and (18). Moreover, again in analogy withthe analysis of the optimal fidelity, for low values ofr, thereexists a region in which the performance of photon-subtractedsqueezed states and optimized squeezed Bell states are com-parable. Finally, again in analogy with the case of the fidelityoptimization, the output variance associated with the Gaussiantwin beam and with the optimized squeezed Bell states coin-cide at a specific, large value ofr, at which the two resourcesbecome identical.

The input variances〈∆Z2in〉 (B3) and (B4), and the output

variances〈∆Z2out〉, are plotted in panels I and II of Fig. 3 for

the ideal VKB protocol and in panels III and IV of Fig. 3 forthe non-ideal protocol.

7

FIG. 3: (Color online) Output variances〈∆X2

out〉 and〈∆P 2

out〉, as a function of the squeezing parameterr of the resource, expressed in dB,for the optimized squeezed Bell resource (full black line),for the photon-subtracted squeezed resource (dotted red line), and for the twin beamresource (dashed blue line). Panels I and II: ideal protocol. Panels III and IV: non-ideal protocol. The various curves are to be compared withthe given input variances of the input single-mode squeezedcoherent state (horizontal solid lines). The squeezing of the input state is fixed ats = 5 dB andϕ = 0. In the non-ideal protocol, the experimental parameters are fixed atτ = 0.1, nth = 0, andR2 = 0.05. In the plots, thevariances associated with the photon-added squeezed resource have been omitted as they are always larger than the ones corresponding to thephoton-subtracted squeezed resource and the squeezed Bellresource.

In the instance of realistic protocol, for small resourcesqueezing degreer, similar conclusions can be drawn, leadingto the same hierarchy among the entangled resources. How-ever, analogously to the behavior of the teleportation fidelity,for high values ofr the photon-subtracted squeezed resourcesare very sensitive to decoherence. In fact, such resources per-form worse and worse than the Gaussian twin beam forrgreater than a specific finite threshold value.

Rather than minimizing the differences between output andinput quadrature variances, one might be naively tempted toconsider minimizing the difference between the ratio of theoutput variances〈∆X2

out〉/〈∆P 2out〉 and the ratio of the in-

put variances〈∆X2in〉/〈∆P 2

in〉. This quantity might appearto be of some interest because it is a good measure of howwell squeezing is teleported in all those cases in which the in-put and output quadrature variances are very different, that isthose situations in which the statistical moments are teleportedwith very low efficiency. However, it is of little use to preserveformally a scale parameter if the noise on the quadrature av-erages grows out of control. The procedure of minimizing thedifference between output and input quadrature statistical mo-ments is the only one that guarantees the simultaneous preser-

vation of the squeezing degree and the reduction of the excessnoise on the output averages and statistical moments of thefield observables.

V. CONCLUSIONS

We have studied the efficiency of the VBK CV quantumteleportation protocol for the transmission of quantum statesand averages of observables using optimized non-Gaussianentangled resources. We have considered the problem ofteleporting Gaussian squeezed and coherent states, i.e. inputstates with two unknown parameters, the coherent amplitudeand the squeezing. The non-Gaussian resources (squeezedBell states) are endowed with free parameters that can betuned to maximize the teleportation efficiency either of thestate or of physical quantities such as squeezing, quadratureaverages, and statistical moments. We have discussed two dif-ferent optimization procedures: the maximization of the tele-portation fidelity of the state, and the optimization of the tele-portation of average values and variances of the field quadra-tures. The first procedure maximizes the similarity in phase

8

space between the teleported and the input state, while thesecond one maximizes the preservation at the output of thedisplacement and squeezing contents of the input.

We have shown that optimized non-Gaussian entangled re-sources such as the squeezed Bell states, as well as other moreconventional non-Gaussian entangled resources, such as thetwo-mode squeezed photon-subtracted states, outperform,inthe realistic intervals of the squeezing parameterr of the en-tangled resource achievable with the current technology, en-tangled Gaussian resources both for the maximization of theteleportation fidelity and for the maximal preservation of theinput squeezing and statistical moments. These findings areconsistent and go in line with previous results on the im-provement of various quantum information protocols replac-ing Gaussian with suitably identified non-Gaussian resources[1, 2, 6–9]. In the process, we have found that the two optimalvalues of the resource angleδ associated with the two opti-mization procedures are different and identified, respectively,by Eqs. (11) and (15). This inequivalence is connected to thefact that, when using entangled non-Gaussian resources withfree parameters that are amenable to optimization, the fidelityis closely related to the form of the different input propertiesthat one wishes to teleport, e.g. quasi-probability distributionin the phase space, squeezing, statistical moments of higherorder, and so on. Different quantities correspond to differentoptimal teleportation strategies.

Finally, regarding the VBK protocol, it is worth remarkingthat the maximization of the teleportation fidelity correspondsto the maximization of the squared modulus of the overlap be-tween the input and the output (teleported) state, without tak-ing into account the characteristics of the output with respectto the input state. Therefore, part of the non-Gaussian charac-ter of the entangled resource is unavoidably transferred totheoutput state. The latter then acquires unavoidably a certain de-gree of non-Gaussianity, even if the presence of pure Gaussianinputs. Moreover, as verified in the case of non-ideal proto-cols, the output state is also strongly affected by decoherence.Thus, in order to recover the purity and the Gaussianity ofthe teleported state,purification and Gaussification protocolsshould be implemented serially after transmission throughtheteleportation channel is completed [51]. If the second (squeez-ing preserving) procedure is instead considered, the possibledeformation of the Gaussian character is not so relevant, be-cause the shape reproduction is not the main goal, while pu-rification procedures are again needed to correct for the extranoise added during teleportation when finite entanglement andrealistic conditions are considered.

An important open problem is determining a proper telepor-tation benchmark for the class of Gaussian input states withunknown displacement and squeezing. Such a benchmark isexpected to be certainly smaller than50% in terms of tele-portation fidelity, the latter being the benchmark for purelycoherent input states with completely random displacementinphase space [45, 46]. Our results indicate that optimized non-Gaussian entangled resources will allow one to beat the clas-sical benchmark, thus achieving unambiguous quantum statetransmission via a truly quantum teleportation, with a smalleramount of nonclassical resources, such as squeezing and en-

tanglement, compared to the case of shared Gaussian twinbeam resources. In this context, Fig. 2 provides strong andencouraging evidence that suitable uses of non-Gaussianity intailored resources, feasible with current technology [24–28],may lead to a genuine demonstration of CV quantum telepor-tation of displaced squeezed states in realistic conditions ofthe experimental apparatus. This would constitute a crucialstep forward after the successful recent experimental achieve-ment of the quantum storage of a displaced squeezed thermalstate of light into an atomic ensemble memory [52].

Acknowledgments

We acknowledge financial support from the EuropeanUnion under the FP7 STREP Project HIP (Hybrid Informa-tion Processing), Grant Agreement No. 221889.

Appendix A: Input states, entangled resources, and output states

Here we report the characteristic functions for the single-mode input states and for the two-mode entangled resources.The characteristic function for the coherent squeezed states(3), i.e.χin(α) = in〈ψCS |Din(α)|ψCS〉in , reads:

χin(α) = e1

2(αβ∗−α∗β)− 1

2 |(α+β) cosh s+(α∗+β∗)eiϕ sinh s|2 .(A1)

The characteristic function for the squeezed Bell-like resource(4), i.e. χSB(α1, α2) = 12〈ψSB|D1(α1)D2(α2)|ψSB〉12 ,reads:

χSB(α1, α2) = e−1

2(|ξ1|2+|ξ2|2)

×[

1 + sin δ cos δ(eiθξ1ξ2 + e−iθξ∗1ξ∗2 )

+ sin2 δ(|ξ1|2|ξ2|2 − |ξ1|2 − |ξ2|2)]

,

(A2)

where the complex variablesξk are defined as:

ξk = αk cosh r + α∗l e

iφ sinh r (k, l = 1, 2; k 6= l). (A3)

It is worth noticing that, forδ = 0, Eq. (A2) reduces tothe well-known Gaussian characteristic function of the twinbeam. Given the characteristic functions for the single-modethe input state and for the two-mode entangled resource,Eqs. (A1) and (A2), respectively, it is straightforward to obtainthe characteristic function for the single-mode output state ofthe teleportation protocol by using Eq. (1) and replacingχres

with χSB.

Appendix B: Mean values and variances of the quadratures

In this Appendix, we report the analytical expressions forthe mean values〈Zj〉 = Tr[Zjρj ], with Zj = Xj , Pj (j =

9

in, out), and the variances〈∆Z2j 〉 = Tr[Z2

j ρj ] − Tr[Zjρj ]2

of the quadrature operatorsXj = 1√2(aj + a†j), Pj =

i√2(a†j − aj), associated with the single-mode input stateρin

and the output stateρout of the teleportation protocol. Wealso compute the cross-quadrature variance〈∆(XjPj)S〉 =Tr[(XjPj +PjXj)ρj ]− 2Tr[Xjρj ]Tr[Pjρj ], associated withthe non-diagonal term of the covariance matrix of the densityoperator, where the subscriptS denotes the symmetrization.The mean values and the variances associated with the inputsingle-mode coherent squeezed state (3) can be easily com-puted:

〈Xin〉 =1√2(β + β∗), (B1)

〈Pin〉 =i√2(β∗ − β), (B2)

and

〈∆X2in〉 =

1

2(cosh 2s− cosϕ sinh 2s), (B3)

〈∆P 2in〉 =

1

2(cosh 2s+ cosϕ sinh 2s), (B4)

〈∆(XinPin)S〉 = − sinϕ sinh 2s. (B5)

The mean values and the variances associated with the outputsingle-mode teleported state, described by the characteristicfunction (1) read:

〈Xout〉 =g√2(β + β∗), (B6)

〈Pout〉 =ig√2(β∗ − β), (B7)

and

〈∆X2out〉 = g2〈∆X2

in〉+Σ , (B8)

〈∆P 2out〉 = g2〈∆P 2

in〉+Σ , (B9)

〈∆(XoutPout)S〉 = −g2 sinϕ sinh 2s , (B10)

with

Σ = Γτ,R + e−τ2 g sin(θ − φ) sin φ sin 2δ − 1

4e−2r−τ (1 + eτ g2 − 2e

τ2 g cosφ)[cos 2δ − cos(θ − φ) sin 2δ − 2]

− 1

4e2r−τ (1 + eτ g2 + 2e

τ2 g cosφ)[cos 2δ + cos(θ − φ) sin 2δ − 2] . (B11)

For the particular choicesg = 1, φ = π, andθ = 0, Eq. (B11) reduces to:

Σ∣

∣

g=1= Γτ,R

∣

∣

g=1/T− 1

4e−2r−τ(1 + e

τ2 )2[cos 2δ + sin 2δ − 2]− 1

4e2r−τ (1− e

τ2 )2[cos 2δ − sin 2δ − 2] . (B12)

In the instance of Gaussian resource(δ = 0), such quantitysimplifies to:

ΣG = Γτ,R

∣

∣

g=1/T+

1

4e−2r−τ(1+e

τ2 )2+

1

4e2r−τ (1−e τ

2 )2 .

(B13)

For suitable choices ofδ in Eq. (B12), see Section II, onecan easily obtain the output variances associated with photon-added and photon-subtracted squeezed states.

[1] F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati, Phys.Rev. A76, 022301 (2007).

[2] F. Dell’Anno, S. De Siena, and F. Illuminati, Phys. Rev. A81,012333 (2010).

[3] M. S. Kim, W. Son, V. Buzek, and P. L. Knight, Phys. Rev. A65, 032323 (2002).

[4] V. V. Dodonov and L. A. de Souza, J. Opt. B: Quantum Semi-class. Opt.7, S490 (2005).

[5] N. J. Cerf, O. Kruger, P. Navez, R. F. Werner, and M. M. Wolf,Phys. Rev. Lett.95, 070501 (2005).

[6] T. Opatrny, G. Kurizki, and D.-G. Welsch, Phys. Rev. A61,032302 (2000).

[7] P. T. Cochrane, T. C. Ralph, and G. J. Milburn, Phys. Rev. A65, 062306 (2002).

[8] S. Olivares, M. G. A. Paris, and R. Bonifacio, Phys. Rev. A67,032314 (2003).

10

[9] A. Kitagawa, M. Takeoka, M. Sasaki, and A. Chefles, Phys.Rev. A73, 042310 (2006).

[10] F. Dell’Anno, S. De Siena, L. Albano, and F. Illuminati,Eur.Phys. J. Special Topics160, 115 (2008).

[11] Y. Yang and F.-L. Li, Phys. Rev. A80, 022315 (2009).[12] G. Adesso, F. Dell’Anno, S. De Siena, F. Illuminati, andL. A.

M. Souza, Phys. Rev. A79, 040305(R) (2009).[13] N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C.

Ralph, and M. A. Nielsen, Phys. Rev. Lett.97, 110501 (2005).[14] M. M. Wolf, G. Giedke, and J. I. Cirac, Phys. Rev. Lett.96,

080502 (2006).[15] M. G. Genoni and M. G. A. Paris, e–print arXiv:1008.4243

(2010).[16] F. Dell’Anno, S. De Siena, and F. Illuminati, Phys. Rep.428,

53 (2006).[17] T. Tyc and N. Korolkova, New J. Phys.10, 023041 (2008).[18] G. S. Agarwal and K. Tara, Phys. Rev. A43, 492 (1991).[19] G. Bjork and Y. Yamamoto, Phys. Rev. A37, 4229 (1988).[20] Z. Zhang and H. Fan, Phys. Lett. A165, 14 (1992).[21] M. Dakna, T. Opatrny, L. Knoll, and D. G. Welsch, Phys.Rev.

A 55, 3184 (1997).[22] M. S. Kim, E. Park, P. L. Knight, and H. Jeong, Phys. Rev. A

71, 043805 (2005).[23] D. Menzies and R. Filip, Phys. Rev. A79, 012313 (2009).[24] A. Zavatta, S. Viciani, and M. Bellini, Science306, 660 (2004).[25] A. I. Lvovsky and S. A. Babichev, Phys. Rev. A66, 011801

(2002).[26] J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev. Lett.

92, 153601 (2004).[27] A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and P. Grangier,

Phys. Rev. Lett.98, 030502 (2007).[28] V. Parigi, A. Zavatta, M. Kim, and M. Bellini, Science317,

1890 (2007).[29] A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier,

Nature448, 784 (2007).[30] L. Vaidman, Phys. Rev. A49, 1473 (1994).[31] S. L. Braunstein and H. J. Kimble, Phys. Rev. Lett.80, 869

(1998).[32] A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H.

J. Kimble, and E. J. Polzik, Science282, 706 (1998).[33] W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph,

H. A. Bachor, T. Symul, and P. K. Lam, Phys. Rev. A67,032302 (2003).

[34] T. C. Zhang, K. W. Chou, P. Lodahl, and H. J. Kimble, Phys.Rev. A67, 033802 (2003).

[35] N. Takei, H. Yonezawa, T. Aoki, and A. Furusawa, Phys. Rev.Lett. 94, 220502 (2005).

[36] J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K.Ham-

merer, J. I. Cirac, and E. S. Polzik, Nature443, 557 (2006).[37] N. Takei, T. Aoki, S. Koike, K.-I. Yoshino, K. Wakui, H.

Yonezawa, T. Hiraoka, J. Mizuno, M. Takeoka, M. Ban, andA. Furusawa, Phys. Rev. A72, 042304 (2005).

[38] H. Yonezawa, S. L. Braunstein, and A. Furusawa, Phys. Rev.Lett. 99, 110503 (2007).

[39] S. L. Braunstein and P. van Loock, Rev. Mod. Phys.77, 513(2005).

[40] P. Marian and T. A. Marian, Phys. Rev. A74, 042306 (2006).[41] U. Leonhardt and H. Paul, Phys. Rev. A48, 4598 (1993).[42] D. Walls and G. Milburn,Quantum Optics(Berlin, Springer,

1994).[43] A. Serafini, M. G. A. Paris, F. Illuminati, and S. De Siena, J.

Opt. B: Quantum Semiclass. Opt.7, R19 (2005).[44] W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel, T. C. Ralph,

T. Symul, and P. K. Lam, IEEE J. Sel. Top. Quant.9, 1519(2003).

[45] S. L. Braunstein, C. A. Fuchs, and H. J. Kimble, J. Mod. Opt.47, 267 (2000).

[46] K. Hammerer, M. M. Wolf, E. S. Polzik, and J. I. Cirac, Phys.Rev. Lett.94, 150503 (2005).

[47] G. Adesso and G. Chiribella, Phys. Rev. Lett.100, 170503(2008).

[48] M. Owari, M. B. Plenio, E. S. Polzik, A. Serafini, and M. M.Wolf, New J. Phys.10, 113014 (2008).

[49] P. Kok and B.W. Lovett,Introduction to Optical Quantum In-formation Processing(Cambridge: Cambridge University Press2010)

[50] S. Suzuki, H. Yonezawa, F. Kannari, M. Sasaki, and A. Furu-sawa, Appl. Phys. Lett.89, 061116 (2006); H. Vahlbruch, M.Mehmet, N. Lastzka, B. Hage, S. Chelkowski, A. Franzen, S.Gossler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett.100,033602 (2008).

[51] O. Glockl, U. L. Andersen, R. Filip, W. P. Bowen, and G.Leuchs, Phys. Rev. Lett.97, 053601 (2006); J. Heersink, Ch.Marquardt, R. Dong, R. Filip, S. Lorenz, G. Leuchs, and U.L. Andersen, Phys. Rev. Lett.96, 253601 (2006); A. Franzen,B. Hage, J. DiGuglielmo, J. Fiurasek, and R. Schnabel, Phys.Rev. Lett. 97, 150505 (2006); B. Hage, A. Samblowski, J.DiGuglielmo, A. Franzen, J. Fiurasek, and R. Schnabel, Na-ture Phys.4, 915 (2008); R. Dong, M. Lassen, J. Heersink, Ch.Marquardt, R. Filip, G. Leuchs, and U. L. Andersen, NaturePhys.4, 919 (2008).

[52] K. Jensen, W. Wasilewski, H. Krauter, T. Fernholz, B. M.Nielsen, A. Serafini, M. Owari, M. B. Plenio, M. M. Wolf,and E. S. Polzik, e–print arXiv:1002.1920 (2010), Nature Phys.(Advance Online Publication, doi:10.1038/nphys1819).