Experimental quantification of the entanglement of noisy twin beams
aa r X i v : . [ qu a n t - ph ] J u l Experimental quantification of the entanglement of noisy twin beams
V´aclav Mich´alek, ∗ Jan Peˇrina, Jr., † and Ondˇrej Haderka Joint Laboratory of Optics of Palack´y University and Institute of Physics of CAS,Faculty of Science, Palack´y University, 17. listopadu 50a, 771 46 Olomouc, Czech Republic Institute of Physics of Academy of Sciences of the Czech Republic,Joint Laboratory of Optics of Palack´y University and Institute of Physics of AS CR,17. listopadu 12, 772 07 Olomouc, Czech Republic
Gradual loss of the entanglement of a twin beam containing around 25 photon pairs with theincreasing external noise is experimentally investigated. The entanglement is quantified by the non-classicality depths and the non-classicality counting parameters related to several non-classicalitycriteria. The reduction of intensity moments of the analyzed multi-mode twin beams to single-modeones allows to determine the negativity as another quantifier of the entanglement. Both the rawphotocount histograms and the reconstructed photon-number distributions are analyzed in parallel.
PACS numbers: 42.65.Lm,42.50.Ar
I. INTRODUCTION
Twin beams (TWBs) ideally composed of photon pairshave very interesting quantum properties: They exhibitthe entanglement between the photons belonging to thesame photon pair that occurs in different degrees of free-dom including frequencies, polarizations or propagationdirections. At the same time, however, the TWBs con-taining on average typically more than one photon pairexhibit perfect correlations between the numbers of thesignal and idler photons, that represent another attributeof the TWB quantumness. The entanglement in theTWB, as the TWB prominent feature, finds its appli-cations in metrology (measurement of ultra-short timeintervals, absolute detector calibration [1, 2]), quantumcommunications (reduction of noise, quantum cryptog-raphy) and various quantum-information protocols [3].Quantum states with specific properties may be obtainedusing various types of post-selection realized on the TWB[4]. However, the noise superimposed on the TWB occursin a smaller or greater amount in all these applications.For example, in the quantum-communication applica-tions the noise increases linearly with the distance [5].As certain minimal amount of the entanglement is in-dispensable for all applications of TWBs, restriction tothe maximal tolerable amount of the noise occurs. Thisbrings the need to quantify the TWB entanglement andits relationship to the noise. The noise may originate ei-ther in the sources outside the TWB or in photon pairs ofthe TWB being partly absorbed during their propagation(typically in optical fibers). In this contribution, we sug-gest three theoretical concepts how to quantify the TWBentanglement. We verify these concepts experimentally:We generate a TWB with around 25 photon pairs onaverage and superimpose an additional noise with theincreasing intensity onto both signal and idler beams. ∗ [email protected] † [email protected] Quantification of the entanglement of TWBs is not aneasy task because the TWBs are typically (spectrally andspatially) multi-mode and as such they are properly char-acterized by quasi-distributions of the overall signal andidler (integrated) intensities, instead of amplitudes. Thiscomes from the fact that the multi-mode character ofthe fields makes the information about the phases of in-dividual spatio-spectral modes as well as their individualintensities unimportant. A larger number of modes pre-vents the application of the homodyne tomography [6, 7]in the experimental investigations of TWBs, as well asthe use of the entanglement witnesses based on the mo-ments of fields’ amplitudes [8–11]. Quantification of theentanglement of multi-mode optical fields represents a se-rious and demanding problem even in specific cases whenindividual modes and their inter-modal correlations aremeasured [12, 13]. In the case of multi-mode TWBs, wedo not have access to the properties of individual modes.However, we know that the reduced states of the signaland idler beams are multi-mode thermal [14], i.e. theyare purely classical, as a consequence of the spontaneousemission of photon pairs in the process of spontaneousparametric down-conversion [15]. This means that thequantification of TWB entanglement can be mapped ontothe quantification of the TWB non-classicality.In general, the non-classicality of a state is recog-nized by the negative values of quasi-distributions ofintensities (even being in the form of generalized func-tions) [16, 17]. In the case of multi-mode TWBs, theproblem of non-classicality identification can be consid-erably simplified when applying suitable non-classicalityidentifiers/witnesses (NI) [18–20] that are convenientlybased on the intensity moments. The fields’ intensitiesand their moments can be measured by photon-number-resolving detectors that provide the corresponding pho-tocount distributions [4, 13, 21–24]. We note that alsothe NIs based directly on the elements of photocount(or photon-number) distributions may also be used forthis purpose [18, 25–27]. The quantification of non-classicality/entanglement is then reached by applying theconcept of the Lee non-classicality depth [28] or the ap-proach leading to the non-classicality counting parameter[29].Here, we suggest and verify an alternative approachin which we first determine the intensity moments ap-propriate to one typical (paired) mode and then we usethese intensity moments in the formula for the negativityof a Gaussian two-mode field [30, 31] to directly quantifythe TWB entanglement. The negativity [32] exploits theproperties of the partially-transposed statistical operator[33, 34] to quantify the amount of the entanglement in acomposed quantum system.The paper is organized as follows. Non-classicalityand entanglement identifiers and quantifiers are theoret-ically introduced in Sec. II. The experimental setup, per-formed experiment and the reconstruction method forrevealing a TWB joint photon-number distribution fromthe experimental photocount histogram are described inSec. III. Degradation of the non-classicality and entan-glement caused by an additional noise with the increasingintensity is discussed in Sec. IV using the theoretical toolsof Sec. II. Sec. V brings conclusions.
II. NON-CLASSICALITY ANDENTANGLEMENT IDENTIFICATION ANDQUANTIFICATION
For TWBs, the noise-reduction-factor R is the com-monly determined quantity that may also indicate theirnon-classicality: R = 1 + h [∆( W s − W i )] ih W s i + h W i i , (1)where W s ( W i ) denotes the signal- (idler-) field (inte-grated) intensity and ∆ W = W − h W i . According to itsdefinition the noise-reduction-factor R quantifies pairingof the photons in a TWB. For an ideal TWB composedof only photon pairs, it equals to zero. If an additionalnoise on the top of the paired photons is present in theTWB, R >
0. The larger the amount of the noise, thegreater the value of R . It can be shown that the TWBswith R < R as well as othercharacteristics of the TWBs are commonly derived fromthe moments of the reconstructed photon-number dis-tribution p ( n s , n i ). This distribution is obtained by thereconstruction from the experimental photocount his-togram f ( c s , c i ). The intensity moments h W k s W l i i rep-resent the normally-ordered photon-number moments.They are derived from the usual photon-number mo-ments h n i s n j i i using the following linear relations validfor one effective bosonic mode with the operators fulfill-ing the canonical commutation relations ( k, l = 1 , , . . . )[14, 17, 35]: h W k s W l i i = k X m =0 S ( k, m ) l X j =0 S ( l, j ) h n m s n j i i . (2) In Eq. (2), symbol S stands for the Stirling numbers ofthe first kind [36].The reconstruction of a photon-number distribution re-moves the ’distortions’ in the experimental photocounthistogram caused by the detector. As such it improves ingeneral the characteristics of the analyzed field, especiallyits non-classicality. To assess the parameters/quality ofthe directly measured photocount histogram, we may as-sume that it was obtained by an ideal detector whoseoperation does not require any correction. In this case,we may consider in the r.h.s. of Eq. (2) the photoucountmoments h c i s c j i i instead of the photon-number moments h n i s n j i i and determine the corresponding intensity mo-ments. Such intensity moments derived from the photo-count moments can then be used in parallel to the usualintensity moments of Eq. (2) to determine the quanti-ties of interest and discuss the related properties. Wenote that we systematically use the quantities c s and c i to count the numbers of detected electrons (photocounts)whereas the numbers n s and n i quantify photon numbersin the reconstructed TWB.The real experimental quantification of the TWB non-classicality can be based upon suitable NIs for which thenon-classicality depths τ introduced in [28] or the non-classicality counting parameters ν defined in [29] are de-termined (for details, see below). Following the compre-hensive analysis of NIs based on the intensity momentsof TWBs [18], we consider the following three represen-tative NIs: M ≡ h W ih W i − h W s W i i < ,E ≡ h W i + h W i − h W s W i i < ,E ≡ h W i + h W i − h W W i i − h W s W i < . (3)The NI M has a privileged position among other NIsbased on the intensity moments as it only identifies thenon-classicality in an arbitrary single-mode TWB [19].Whereas the NI M contains the intensity moments inthe cumulative fourth order, the other considered NI E uses just the second-order intensity moments. For thisreason, the most commonly applied NI E is determinedwith better experimental precision than the NI M . Wenote that for a balanced TWB with h W s i = h W i i , E < R <
1. In general, the condition
R < E + ( h W s i − h W i i ) < E is stronger in identifying the non-classicality than the noise-reduction-factor R . On theother hand, the last considered NI E directly involvesthe third-order intensity moments and as such it moni-tors the higher (third) -order non-classicality.The performance of the above NIs can directly be com-pared for single-mode fields. In this case, a TWB is non-classical provided that Q ≡ h W s ih W i i−h W s W i i < h W a i = 2 h W a i , h W a i = 6 h W a i , a = s , i, h W W i i = 2 h W s W i ih W s i , and h W s W i =2 h W s W i ih W i i valid for the single-mode Gaussian fields,we rewrite Eqs. (3) in the form: M = Q (2 h W s ih W i i + h W s W i i ) < ,E = 2 Q + 2( h W s i − h W i i ) < ,E = 2 Q ( h W s i + h W i i ) + 2( h W s i + h W i i )+ 4( h W s i − h W i i ) ( h W s i + h W i i ) < . (5)According to Eqs. (5), the NI M identifies all nonclassi-cal single-mode TWBs, whereas the NIs E and E areweaker than the condition Q <
0. We note that non-classical balanced TWBs are also completely identifiedby the NI E .The concept of the non-classicality depth (ND) τ [28]is based upon the behavior of quasi-distributions in thephase space of an optical field in relation to differentfield-operator orderings. It uses the fact that the amountof non-classicality decreases as we move from the nor-mal ordering, that corresponds to the usual detection byquadratic intensity detectors, to the anti-normal order-ing, in which any optical field exhibits only the classicalproperties. The ND τ gives the distance on the ordering-parameter axis s between the point at which the non-classicality is lost s th and the point of the normal order-ing s = 1: τ = (1 − s th ) / . (6)The threshold ordering parameter s th is determinedso that the corresponding s -ordered intensity moments h W k s W l i i s nullify the corresponding NI. The s -orderedintensity moments are given as [17]: h W k s W l i i s = (cid:18) − s (cid:19) k + l (cid:28) L k (cid:18) W s s − (cid:19) L l (cid:18) W i s − (cid:19)(cid:29) (7)and L k denotes the k -th Laguerre polynomial [36].Whereas we have 0 ≤ τ ≤ τ of any nonclassical Gaussian beam cannotexceed 1/2.On the other hand, the non-classicality counting pa-rameter (NCP) ν ≥ p ν , p ν ( n ′ s , n ′ i ; ν ) = P n ′ s n s =0 P n ′ i n i =0 p ( n s , n i ) × p th ( n ′ s − n s ; ν, p th ( n ′ i − n i ; ν, , (8)that is applied in the above discussed NIs. The photon-number distribution p th for a K -mode thermal field with h n i mean photons is given by the Mandel-Rice formula: p th ( n ; h n i , K ) = Γ( n + K ) n !Γ( K ) h n i n (1 + h n i ) n + K ; (9) Γ stands for the gamma function.Provided that the numbers K s and K i of modes inthe signal and idler beams, respectively, are close andare determined by the formula for a multi-mode thermalfield [17], K a = h W a i h (∆ W a ) i , a = s , i , (10)we may derive single-mode moments h w k s w l i i s . Theycharacterize a typical paired mode and the whole TWB isthen considered as composed of a given number of identi-cal typical paired modes. As the analyzed TWBs containseveral tens of spatio-spectral modes, this approximateTWB decomposition is well justified. The mean single-mode intensities h w s i and h w i i are given as: h w a i = h W a i K , a = s , i , (11)where K = ( K s + K i ) / w s and ∆ w i : h (∆ w s ) k (∆ w i ) l i = h (∆ W s ) k (∆ W i ) l i K . (12)Using the relations in Eq. (12) the single-mode intensitymoments are determined step by step starting from thosefor the lowest orders, i.e., from h w a i for a = s , i and h w s w i i .The single-mode intensity moments then allow us todirectly determine the negativity E N [30, 31], that is agenuine entanglement quantifier, along the formula: E N = (cid:8) b p − ( b s + b i )(4 b p + 1) − b s b i + q ( b s − b i ) + 4 b p ( b p + 1) (cid:9) × (cid:8) b s + b i )(2 b p + 1) + 8 b s b i + 2 (cid:9) − (13)in which b p = − / p / − h ∆ w s ∆ w i ) i and b a = h w a i− b p for a = s , i. We note that nonzero negativity E N of an entangled two-mode beam implies the fulfillment ofthe commonly used NIs for such beams [8, 9, 37]. III. EXPERIMENTAL SETUP ANDTWIN-BEAM RECONSTRUCTION
In the experiment whose scheme in shown in Fig. 1(a),a noiseless TWB was generated in a 5-mm-long type-I β -barium-borate crystal (BaB O , BBO) cut for a slightlynon-collinear geometry. Parametric down-conversion waspumped by pulses originating in the third harmonic(280 nm) of a femtosecond cavity-dumped Ti:sapphirelaser (pulse duration 180 fs at the central wavelength of840 nm, repetition rate 50 kHz, pulse energy 20 nJ atthe output of the third harmonic generator). The exter-nal noise was produced by a bulb lamp with variable light (a) S c I f(c S ,c I ) 0 20 4002040 n S n I p(n S ,n I ) (b) (c) FIG. 1. (a) Scheme of the experimental setup: nonlinear crys-tal BBO producing a TWB; mirror HR reflecting the idlerbeam; light bulb LB emitting the noisy field with definedintensity uniform over the iCCD; bandpass interference filterIF; intensified CCD camera iCCD; detector D used for pump-beam stabilization. (b) Normalized experimental photocounthistogram f ( c s , c i ) giving the number of realizations with c s and c i registered electrons (photocounts) and (c) the corre-sponding reconstructed photon-number distribution p ( n s , n i )of the least-noisy TWB. intensity. The signal, idler and noise fields were detectedin three different equally-sized detection regions (in theform of strips) on the photocathode of an iCCD cam-era Andor DH 345-18U-63. The camera set for the 4 ns-long detection window was driven by the synchronizationelectronic pulses from the laser and it operated at 14 Hzframe rate. Whereas two detection regions that moni-tored the signal and idler beams contained both photonsfrom pairs and the noise photons, the third detection re-gion was illuminated only by the noise photons thus gavethe intensity of the superimposed noise field. The pho-tons of all three fields impinging on the camera were fil-tered by a 14-nm-wide bandpass interference filter withthe central wavelength at 560 nm. As the bandwidthof the spectral intensity cross-correlation function of theTWB equals around 2 nm under the used conditions, theedge effects of the filters causing losses of photons fromphoton pairs did not have to be explicitly considered.The pump intensity, and thus also the TWB intensity,was actively stabilized by means of a motorized half-waveplate followed by a polarizer and a detector that moni-tored the actual intensity.In the experiment, we first investigated the TWB with-out an additional noise. The experimental photocounthistogram f ( c s , c i ) obtained after 10 measurement repe-titions as well as the reconstructed photon-number distri-bution p ( n s , n i ) are plotted in Figs. 1(b) and 1(c). ThisTWB caused on average h c i = 5 . h n i = h W i = 24 . f is smeared from the diagonal given as c s = c i .The reconstruction tends to eliminate this smearing, butstill a typical droplet shape is observed for the photon-number distribution p . The maximum-likelihood ap-proach was applied to arrive at the photon-number dis-tribution p ( n s , n i ) in the form of a steady state of thefollowing iteration procedure [38, 39] ( l = 0 , , . . . ): p ( l +1) ( n s , n i ) = p ( l ) ( n s , n i ) × X c s ,c i f ( c s , c i ) T s ( c s , n s ) T i ( c i , n i ) P n ′ s ,n ′ i T s ( c s , n ′ s ) T i ( c i , n ′ i ) p ( l ) ( n ′ s , n ′ i ) . (14)The positive-operator-valued measures T a , a = s , i, char-acterize detection in the region with beam a . We havefor an iCCD camera with N a active pixels, detection effi-ciency η a and mean dark count number per pixel D a [39]: T a ( c a , n a ) = (cid:18) N a c a (cid:19) (1 − D a ) N a (1 − η a ) n a ( − c a × c a X l =0 (cid:18) c a l (cid:19) ( − l (1 − D a ) l (cid:18) lN a η a − η a (cid:19) n a . (15)Calibration of our iCCD camera [2] gave us the followingparameters η s = 0 . ± . η i = 0 . ± . N s = N i = 4096, D s N s = D i N i = 0 . ± .
001 for the signal(s) and idler (i) detection regions.
IV. NON-CLASSICALITY ANDENTANGLEMENT DEGRADATION CAUSED BYTHE INCREASING NOISE
To investigate degradation of the TWB entanglementas well as to analyze the performance of the above en-tanglement quantifiers when the noise in the TWB in-creases, the noise with multi-thermal photon statistics,originating in a bulb lamp, was superimposed equallyonto the signal and idler beams. An increasing voltageapplied to the bulb lamp leads to the increasing meanphoton numbers h n i n of the noise field. 36 TWBs withdifferent levels of the noise were analyzed: Their meanphotocount numbers h c s i and h c i i in the signal and idlerdetection regions, respectively, as well as the mean pho-tocount numbers h c i n of the noise field measured in theindependent detection are plotted in Fig. 2(a).We first roughly estimate the amount of non-classicality by applying the noise-reduction-factor R [24]in Eq. (1) that, in fact, quantifies the relative amountof paired photons in a TWB. The gradual decrease ofthe relative amount of paired photons in the measuredTWBs with the increasing noise is monitored in Fig. 2(b)by the increasing values of the noise-reduction-factors R c and R n determined from the photocount histograms andreconstructed photon-number distributions of the ana-lyzed TWBs, respectively. According to the graphs inFig. 2(b), the TWBs with the mean noise photocountnumbers h c i n smaller than 5 are nonclassical ( R c , R n < (a) (b) FIG. 2. (a) Mean experimental photocount numbers h c s i (blue △ ), h c i i (red ◦ ) and h c i n ( ∗ ) in, in turn, signal-beam,idler-beam and noise detection region versus the number i identifying a TWB. (b) Noise-reduction-factors R c ( ∗ ) and R n (blue △ ) determined for the experimental photocount his-tograms and reconstructed photon-number distributions ofTWBs, respectively, as they depend on mean noise photo-count number h c i n . Experimental data are plotted as isolatedsymbols with error bars derived from the number of measure-ment repetitions. Relative errors in (b) estimated from thedata scattering are better than 3 %. In (a), experimental er-rors are smaller than the plotted symbols. In (b), theoreticalsolid curves with appropriate symbols originate in the model,dashed line R = 1 indicates the non-classicality border. R c and R n mutually cross at R = 1where the transition to the classical region of R occurs.The experimental results for the noisy TWBs are com-pared with the predictions of the model that convolvesthe photocount (photon-number) distributions of the in-dependent noisy fields present in both the signal andidler beams with the photocount histogram f n − l (photon-number distribution p n − l ) of the original TWB withoutan additional noise using the formula analogous to thatin Eq. (8). The distributions of the noisy fields are givenby Eq. (9) in which we consider h c i n ( h n i n = h c i n /η )mean photocount numbers (photon numbers) distributedinto N c ( N n ) equally populated modes. Comparison withthe experimental results suggests K c = 110 independentmodes in the noise fields to explain the loss of non-classicality of the experimental photocount histograms f . The slightly smaller number K n = 90 of indepen-dent modes is appropriate in the case of the reconstructedphoton-number distributions p . This is related to the factthat the reconstruction with the positive-operator-valuedmeasures T a in Eq. (15) partially reduces the noise.The experimental as well as the theoretical values ofboth NDs τ and NCPs ν drawn for different values ofthe mean noise photocount number h c i n in Fig. 3 confirmthe best performance of the NI M in revealing the non-classicality of a whole multi-mode TWB. On the otherhand, the NI E involving the third-order intensity mo-ments gives the worst results, in agreement with the find-ings of Ref. [18]. Whereas the NI M identifies the non-classicality of the TWB up to h c i n ≈
6, the third-order intensity moments of NI E lose their ability to revealthe non-classicality around h c i n ≈
4. It is worth notingthat the commonly used noise-reduction factors R per-form up to h c i n ≈
5. The comparison of NCPs ν drawn inFigs. 3(c,d) with the NDs τ plotted in Figs. 3(a,b) showscomparable sensitivity of the NCPs in quantification ofthe non-classicality from the point of view of the exper-imental errors under our conditions. We note, however,that the NCPs cannot quantify the non-classicality ofhighly quantum states [29]. On the other hand, the inten-sity moments do not have to be involved et all in the de-termination of NCPs if the NIs based on the photocount(photon-number) probabilities are applied [18, 29]. Inthis case the commutation relations, that depend on thenumber of field’s modes, are not needed. Substantial im-provement of the amount of TWB non-classicality afterthe reconstruction is evident when we compare the NDs τ and NCPs ν drawn in Figs. 3(a,c) for the experimen-tal photocount histograms f with those in Figs. 3(b,d)appropriate for the reconstructed photon-number distri-butions p . The increase of non-classicality in the recon-struction is due to partial elimination of the noise and,mainly, correction for the finite detection efficiencies thatbrake the photon pairs from which the non-classicalityoriginates. The values of NDs τ and NCPs ν are around4-5 times larger after the reconstruction. This factor isroughly proportional to 1 /η which is a signature of thefact that the mean photocount and photon numbers perone mode are smaller or comparable to 1. For strongerfields, the mapping between the NDs τ (NCPs ν ) belong-ing to the photocount histograms and the reconstructedphoton-number distributions is nonlinear (compare thecondition τ ≤ / E for quantificationof the non-classicality. On the other hand, the negativ-ity E N determined from up-to the second-order intensitymoments can directly be used as an entanglement quanti-fier, as documented in Figs. 4(a,b). Alternatively, it canbe considered as another NI and then the correspondingNDs τ E N [see Figs. 4(c,d)] and NCPs ν E N can be calcu-lated. In both cases, it identifies the measured TWBsas entangled up to h c i n ≈
6. The comparison of NDs τ m and τ e [Figs. 4(c,d)] belonging to the NIs M and E applied to single-mode moments with those valid forthe whole TWBs [Figs. 3(a,b)] shows that the low-ordersingle-mode intensity moments successfully maintain theinformation about the resistance of TWB non-classicalityagainst the noise.At the end, we note that the error bars plotted inthe figures were determined solely from the number of (a) (b)(c) (d) FIG. 3. Non-classicality depths τ (a,b) and non-classicalitycounting parameters ν (c,d) for NIs M (black ∗ ), E (blue △ )and E (red ◦ ) for photocount histograms (a,c) and photon-number distributions (b,d) as they depend on mean noise pho-tocount number h c i n . Experimental data are plotted as iso-lated symbols with error bars derived from the number ofmeasurement repetitions, solid curves with appropriate sym-bols come from the model. Relative errors in (a,b) [(c,d)]estimated from the data scattering are better than 10 % [5 %]. measurement repetitions. As such they do not reflectinstabilities and imperfections in the setup occurringduring the measurements of TWBs with different lev-els of the noise (one hour was typically needed to char-acterize one TWB). Slow pump-beam intensity fluctu-ations, pump-beam misalignment (temperature-inducedposition shifts) in the setup, temperature stabilization ofthe iCCD camera and its synchronization with the lasersource were responsible for the main detrimental effects.The corresponding errors were estimated from the exper-imental points in the graphs of Figs. 2, 3 and 4: Aver-age relative errors were obtained by considering all pairsof neighbor experimental points on a given experimentalcurve and determining the mean value and the relativedeclination for each pair. V. CONCLUSIONS
We have experimentally investigated deterioration ofthe entanglement of a twin beam caused by an in- creasing external noise. We have suggested, verifiedand mutually compared three experimentally feasibleways for quantifying the twin-beam entanglement. Thefirst two are based upon the non-classicality depths andthe non-classicality counting parameters of suitable non- (a) (b)(c) (d)
FIG. 4. Negativity E N (a,b) and non-classicality depth τ (c,d)for NIs M (black ∗ ), E (blue △ ) and E N (red ◦ ) for photo-count histograms (a,c) and photon-number distributions (b,d)’reduced’ to a single-mode along Eq. (12) as they depend onmean noise photocount number h c i n . Experimental data areplotted as isolated symbols with error bars derived from thenumber of measurement repetitions, solid curves with appro-priate symbols come from the model. Relative errors esti-mated from the data scattering are better than 10 % for allplotted quantities. classicality identifiers. In the third way, the negativityis directly determined for one typical mode of the TWB.The three entanglement quantifiers perform comparably.They may be applied in any metrology, quantum-imagingor quantum-information scheme that uses the twin beamsand whose sensitivity to the noise has to be quantified. ACKNOWLEDGEMENTS
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