Quantum characterization of bipartite Gaussian states
D. Buono, G. Nocerino, V. D'Auria, A. Porzio, S. Olivares, M. G. A. Paris
aa r X i v : . [ qu a n t - ph ] J un Quantum characterization of bipartite Gaussian states
D. Buono, G. Nocerino, V. D’Auria, A. Porzio,
3, 4, ∗ S. Olivares,
5, 6 and M. G. A. Paris
6, 5 Dipartimento di Scienze Fisiche Universit`a “Federico II”,Monte Sant’Angelo, via Cintia, I-80126 Napoli, Italy. Laboratoire Kastler Brossel, Ecole Normale Sup´erieure,Universit´e Pierre et Marie Curie, CNRS,4 place Jussieu, 75252 Paris, France. CNISM UdR Napoli Universit`a, Napoli, Italy. CNR–SPIN, Monte Sant’Angelo, via Cintia, I-80126 Napoli, Italy. CNISM UdR Milano Universit`a, I-20133 Milano, Italy. Dipartimento di Fisica dell’Universit`a degli Studi di Milano, I-20133 Milano, Italy. (cid:13)
OCIS codes: ∗ Corresponding author: [email protected] . Introduction The quantum characterization of physical systems has a fundamental interest in its ownand represents a basic tool for the design of quantum protocols for information processing inrealistic conditions. In particular, the full experimental reconstruction, at the quantum level,of optical systems opens the way not only to high fidelity encoding/transmission/decodingof information, but also to the faithful description of real communication channels and toprecise tests of the foundations of quantum mechanics [1–4].Among the systems of interest for quantum information processing we focus on the classof bipartite optical states generated by parametric processes in nonlinear crystals. These areGaussian states and play a crucial role in quantum information processing with continuousvariables [5–8]. Indeed, using single- and two-mode Gaussian states, linear optical circuitsand Gaussian operations, like homodyne detection, several quantum information protocolshave been implemented, including teleportation, dense coding and quantum cloning [9]. Inparticular, Gaussian entangled states have been successfully generated in the laboratories bytype-II optical parametric oscillators (OPO) below threshold [10–14]. In these OPO systemsthe parametric process underlying the dynamics is well described, at least not too close tothe threshold, by bilinear Hamiltonian, thus the output states are Gaussian and they arecompletely characterized by the first and second moments of their quadratures, i.e. thecovariance matrix.In this paper we address characterization of bipartite Gaussian states and review indetails a scheme to fully reconstruct the Gaussian output from an OPO below threshold,which has been proposed in the recent years [15, 16] successfully implemented experimentally[17]. In the present contibution we give a more accurate description of the experiment anddata analisys and, in particular, we pay attention to advanced Gaussianity test beyond thesimple check of Kurtosis. Our method relies on a single homodyne detector: it provides thefull reconstruction of the covariance matrix (CM) by exploiting the possibility of opticallycombining the two frequency degenerate OPO signal and idler beams and then measuringsuitable quadratures on the obtained auxiliary modes. Once the CM is obtained one mayretrieve all the quantities of interest on the state under investigation, e.g. energy andsqueezing, including those not corresponding to any observable quantity like purity, entropy,entanglement, and mutual information. Quantum properties are discussed in view of thepossible use of these states in quantum communication protocols. In particular, we address3he dependence of mutual information as a function of the bi–partite system total energy.Of course, a bipartite state is fully characterized by its covariance matrix if and only if it isa truly Gaussian one. Usually one assumes that the state to be processed has a Gaussiancharacter because the interaction Hamiltonians are approximated by bilinear ones and thisis often an excellent approximation [18]. In turn, the resulting evolution corresponds aGaussian operations. On the other hand, it is known that nonGaussian dynamics mayoccur when the OPO approaches the threshold [19, 20] and when phase diffusion [21, 22]is present during the propagation and/or the detection stages. Therefore, in order to avoidany possible experimental issue [23–26], a preliminary check on the Gaussian character ofthe signal is crucial to ensure that the actual measured CM fully characterizes the quantumstate. For the first time, in this paper, CM data analisys includes advanced statistical teststo assess Gaussianity [19, 27] of the state. The complete characterization strategy representsa powerful tool to study bipartite Gaussians state from the generation stage to the detectionone.The paper is structured as follows. In section 2 we introduce the formalism used through-out the paper and, in particular, we review two-mode Gaussian states and their covariancematrix as well as the relations among the CM elements and some physical quantities ofinterest, such as the purity, the entropy and the entanglement. The method to reconstructthe CM is described in section 3, while section 4 is devoted to the details of our experimentalimplementation. The analysis of the data and the results are discussed in details in Sections5, 6 and 7. In particular, test of Gaussianity are illustrated in Section 5 and results fromfull quantum tomography in Section 6. Section 8 closes the paper with some concludingremarks.
2. Two-mode Gaussian states A n -mode state ̺ of a bosonic system is Gaussian if its characteristic function χ [ ̺ ]( λ ) =Tr[ ̺D ( λ )] has a Gaussian form, D ( λ ) = N nk =1 D k ( λ k ) being the n -mode displacement op-erator with λ = ( λ , . . . , λ n ), λ k ∈ C , and D k ( λ k ) = exp { λ k a † k − λ ∗ k a k } denoting single-modedisplacement operators [7]. Gaussian states are completely characterized by the first andsecond statistical moments of the quadrature field operators, i.e. by the vector of mean val-ues and by the covariance matrix (CM). Since in this paper we focus on two-mode Gaussianstates of the radiation field, in this section we review the suitable formalism to describe the4ystem. We also assume, since this is the case in our experimental implementation, thatthe mean values of quadratures are zero. Upon introducing the vector of canonical oper-ators R = ( x a , y a , x b , y b ), in terms of the mode operators ˆ a k , k = a, b , ˆ x k = √ (ˆ a † k + ˆ a k ),ˆ y k = i √ (ˆ a † k − ˆ a k ) the CM σ of a bipartite state is the real symmetric definite positive blockmatrix: σ = A CC T B (1)with σ hk = h{ R k , R h }i − h R k ih R h i being { f, g } = f g + gf . Matrices A , B and C are 2 × a and b andtheir mutual correlation matrix. It can be observed that each block A , B and C can bewritten as the sum of two matrices, one containing the product of mean values of h R k i andthe other contain the mean value of products of operators h{ R k , R h }i .Once the CM is known, all the properties of ̺ may be described and retrieved. As forexample, the positivity of ̺ , besides positivity of the CM itself, impose the constraint σ + i Ω ≥ , (2)where Ω = ω ⊕ ω is the two-mode symplectic matrix, given in terms of ω ≡ adiag[1 , − σ is a bona fide CM.A relevant result concerning the actual expression of a CM is that for any two-mode CM σ , there exists a (Gaussian) local symplectic operation S = S ⊕ S that brings σ in itsstandard form, namely [28, 29]: S T σ S = ˜ A ˜ C ˜ C T ˜ B , (3)where ˜ A = diag[ n, n ], ˜ B = diag[ m, m ], ˜ C = diag[ c , c ], with n , m , c and c determined bythe four local symplectic invariants I ≡ det( A ) = n , I ≡ det( B ) = m , I ≡ det( C ) = c c , I ≡ det( σ ) = ( nm − c ) ( nm − c ). If n = m , the matrix is called symmetric andrepresents a symmetric bipartite state where the energy is equally distributed between thetwo modes.By using the symplectic invariants the uncertainty relation (2) can be expressed as: I + I + 2 I ≤ I + 14 . (4)5t is useful to introduce the symplectic eigenvalues, denoted by d ± with d − ≤ d + , which interms of symplectic invariants read as follows [30] d ± = s ∆( σ ) ± p ∆( σ ) − I , (5)where ∆( σ ) ≡ I + I + 2 I . In this way, the inequality (2) re-writes as: d − ≥ /
2. (6)A real symmetric definite positive matrix satisfying d − ≥ / A. Purity and entropies
The purity of the two-mode Gaussian state ̺ , may be expressed as a function of the CM (1)as follows [31]: µ ≡ µ = Tr[ ̺ ] = (16 I ) − . (7)Another quantity, characterizing the degree of mixedness of ̺ , is the von Neumann entropy S ( ̺ ) = − Tr ( ̺ log ̺ ). If the state is pure the entropy is zero ( S = 0), otherwise it is positive( S >
0) and for two-mode Gaussian states it may be written as [7, 30]: S ( ̺ ) ≡ S ( σ ) = f ( d + ) + f ( d − ) where the symplectic eigenvalues d ± are given in (5) the function f ( x ) =( x + 1 /
2) log( x + 1 / − ( x − /
2) log( x − / single mode Gaussian state the von Neumann entropy is a function of the purity alone [32]: S ( ̺ ) = 1 − µ µ log (cid:18) µ − µ (cid:19) − log (cid:18) µ µ (cid:19) , (8)whereas for a two-mode state all the four symplectic invariants are involved.For a two-mode state ̺ it is of interest to assess how much information about ̺ one canobtain by addressing the single parties. This is of course related to the correlation betweenthe two modes and can be quantified by means of the quantum mutual information or theconditional entropies [33]. Given a two-mode state ̺ the quantum mutual information I ( ̺ )is defined starting from the von Neumann entropies as: I ( ̺ ) = S ( ̺ ) + S ( ̺ ) − S ( ̺ ) , where ̺ k = Tr h ( ̺ ), with k, h = 1 , h = k , are the partial traces, i.e. the densitymatrices of mode k , as obtained tracing over the other mode. I ( ̺ ) can be easily expressed6n terms of the blocks of σ and its symplectic eigenvalues. One has I ( σ ) = f (cid:16)p I (cid:17) + f (cid:16)p I (cid:17) − f ( d + ) − f ( d − ) , (9)where f ( x ) is reported above. The conditional entropies are defined accordingly as [33]: S (1 |
2) = S ( ̺ ) − S ( ̺ ) , (10) S (2 |
1) = S ( ̺ ) − S ( ̺ ) . (11)If S (1 | ≥ S (2 | ≥ ̺ . If S (1 | < S (2 | < − S (1 |
2) or − S (2 |
1) bits ofentanglement, respectively. This has been proved for the case of discrete variable quantumsystems [34] and conjectured [35] for infinite dimensional ones.
B. Entanglement
A two-mode quantum state ρ is separable if and only if it can be expressed in the followingform: ρ = P k p k (cid:16) ρ ( a ) k ⊗ ρ ( b ) k (cid:17) , with p k > P k p k = 1 and ρ ( a ) k and ρ ( b ) k are single-modedensity matrices of the two modes a and b , respectively. Viceversa if the state is not sep-arable, it is entangled. A general solution to the problem of separability for mixed statehas not been found yet. For two-mode Gaussian states there exist necessary and sufficientconditions to assess whether a given state is entangled or not. In particular, there are twoequivalent criteria, usually referred to as Duan criterion and Peres-Horodecki-Simon crite-rion, which found an explicit form in term of the CM elements. The criteria provide a testfor entanglement, whereas to assess quantitatively the entanglement content of a state onemay use the logarithmic negativity or the negativity of the conditional entropies, as we seebelow.
1. Duan criterion
This criterion [29] is based on the evaluation of the sum of the variances associated toa pair of EPR-like operators defined on the two different subsystems. For any separablecontinuous variable state, the total variance is bounded by twice the uncertainty product.For entangled states this bound can be exceeded and the violation provides a necessaryand sufficient condition for entanglement. The criterion leads to an inequality that can be7xpressed in terms of standard form CM elements: β D = na + ma − | c | − | c | < a + 1 a , (12)with a = q n − m − . A separable state will not satisfy the above inequality. The criterionraises from the fact that for an entangled state it is possible to gain information on one ofthe subsystems suitably measuring the other one.
2. Peres-Horodecki-Simon criterion (PHS)
Also PHS criterion establishes a necessary and sufficient condition for separability of bipar-tite Gaussian states [28]. Given the CM σ , the corresponding two-mode Gaussian state isnot separable iff: ˜ σ + i Ω < , (13)where ∆ = diag[1 , , , −
1] and ˜ σ = ∆ σ ∆ is the CM associated with the partially trans-posed density matrix. Thanks to the symplectic invariants { I , I , I , I } the inequality(13) can be written in a form that resembles the uncertainty relation (4) [7]: I + I + 2 | I | > I + 14 , (14)or, in terms of standard form CM elements, as: n + m + 2 | c c | − (cid:0) nm − c (cid:1) (cid:0) nm − c (cid:1) ≤ , (15)or simply as: ˜ d − < / , (16)where : ˜ d ± = vuut ˜∆( σ ) ± q ˜∆( σ ) − I , (17)are the symplectic eigenvalues of ˜ σ and ˜∆( σ ) = I + I − I . Therefore, iff ˜ d − < / d − < / d − > E ( σ ) = max n , − log 2 ˜ d − o , (18)8nd it is a simple increasing monotone function of the minimum symplectic eigenvalue ˜ d − (for˜ d − < / C. EPR correlations
This way of assessing quantum correlations between two modes is named after the analogywith the EPR correlation defined for a system undergone to a quantum nondemolitionmeasurement (QND)[13]. Let us consider two subsystems a and b , QND establishes, inprinciple, that measurement performed on subsystem b , does not affect system a . Thiscriterion is equivalent to state that the conditional variance V a | b of a quadrature of beam a , knowing beam b , takes a value smaller than the variance a would have on its own. Theconditional variance can be expressed in terms of the unconditional variance V a of subsystem a ( i.e. the variance that the same quantity has in the subsystem a space) and normalizedcorrelation C ab between the two [14, 42]: V a | b = V a (1 − C ab ) , an analogous relation holdsfor V b | a . A bipartite state is said to be EPR correlated if it verifies the following inequality: V a | b V b | a < / , (19)that can be rewritten in terms of standard form CM elements as follows: β E = nm (cid:16) − c nm (cid:17) (cid:16) − c nm (cid:17) < /
4. If the inequality is satisfied in the system described bythe CM the information on a ( b ) extracted from a measurement on b ( a ) is sufficient forknowing its state with a precision better than the limit given by the variance for a coherentstate. In turn, EPR correlations are stronger than entanglement [10, 24, 43], i.e. all EPRstates are entangled whereas the converse is not true and there are entangled states violatingIneq. (19).
3. Covariance matrix reconstruction
In this section we describe in some detail the method we have implemented to experi-mentally reconstruct the CM given in Eq. (1) and, thus, to fully characterize a bipartiteGaussian state. As expected, each autocorrelation block, A or B , is retrieved by mea-suring only the single-mode quadratures of the concerned mode a or b . Diagonal termsof A correspond to the variances of x a and y a and are directly available at the output ofthe homodyne detection. Off-diagonal terms are instead obtained by measuring the two9dditional quadratures z a = √ ( x a + y a ), t a = √ ( x a − y a ), and exploiting the relation σ = σ = ( h z a i − h t a i ) − h x a ih y a i [15]. The block B is reconstructed in the same wayfrom the quadratures of b . Elements of block C involve the products of quadrature of modes a and b and cannot be obtained by measuring individually the two modes. Instead they areobtained by homodyning the auxiliary modes c = √ ( a + b ), d = √ ( a − b ), e = √ ( ia + b ),and f = √ ( ia − b ) and by making use of the following relations: σ = 12 ( h x c i − h x d i ) − h x a ih x b i ,σ = 12 ( h y e i − h y f i ) − h x a ih y b i ,σ = 12 ( h x f i − h x e i ) − h y a ih x b i ,σ = 12 ( h y c i − h y d i ) − h y a ih y b i It is worth to note that since h x f i = h x b i + h y a i − h x e i and h y f i = h x a i + h y b i − h y e i , themeasurement of the f -quadratures is not mandatory.As we will see in the following, our experimental setup allows one to mix the modes a and b , say the signal and idler, thank to the polarization systems at the OPO output. Atthe same time, the quadratures x = x , y = x π/ , z = x π/ and t = x − π/ required for theentanglement measurement and for the reconstruction of the CM can be easily and reliablyreconstructed from the pattern function tomography applied to data collected in a 2 π scanof the homodyne detector.
4. Experimentals
The experimental setup is schematically depicted in Fig. 1. It is based on a CW internallyfrequency doubled Nd:YAG laser (Innolight Diabolo) whose outputs @532nm and @1064nmare respectively used as the pump for a non degenerate optical parametric oscillator (OPO)and the local oscillator (LO) for the homodyne detector. The OPO is set to work below theoscillation threshold and it provides at its output two entangled thermal states (the signal, a and the idler b ): aim of the work is indeed to measure the covariance matrix of these twobeams.The OPO is based on an α -cut periodically poled KTP non linear crystal (PPKTP, Raicol Crystals Ltd . on custom design) which allows for implementing a type II phasematching with frequency degenerate and cross polarised signal and idler beams, for a crystaltemperature of ≈ ◦ C. The transmittivity of the cavity output mirror, T out , is chosen in10rder to guarantee, together with crystal losses ( κ ) and other losses mechanisms ( T in ), anoutput coupling parameter η out = T out / ( T in + κ ) @1064 nm of ≈ ≈
60% of the threshold power.The signal and idler modes are then sent to the covariance matrix measurement set-up: this consists in a preliminary polarisation system, that allows choosing the beam tobe detected and a standard homodyne detector. The polarisation system is made of anhalf-wave plate ( λ/
2) followed by a polarising beam splitter (PBS); the different wave-plateorientations allow choosing the beam to be transmitted by the PBS: the signal ( a ), the idler( b ) or their combinations c and d . The other auxiliary modes e and f may be obtained byinserting before the PBS an additional quarter wave plate ( λ/
4) [15]. Acquisition times areconsiderably short thank to pc-driven mechanical actuators that allow setting the λ/ λ/ θ is spanned thanks to a piezo-mounted mirror, linearlydriven by a ramp generator which is, in turn, adjusted to obtain a 2 π variation in 200 ms.The homodyne photodiodes (PDs, model Epitaxx ETX300 ) have both nominal quantumefficiencies of ≈ > few KHz)amplifier. The difference photocurrent is eventually further amplified by a low noise highgain amplifier ( MITEQ AU 1442 ).In order to avoid low frequency noise, the photocurrent is demodulated with a sinusoidalsignal of frequency Ω=3 MHz and low-pass filtered ( B =300 KHz), to be sent to a PCIacquisition board (Gage 14100) that samples it with a frequency of 10 pts/run, with 14-bit11esolution. The total electronic noise power of the acquisition chain is 16 dBm below theshot noise level, corresponding to the a SNR ≈
5. Gaussianity tests
Since the covariance matrix contains the full information only for Gaussian states, a prelim-inary check on the Gaussian hypothesis is necessary in order to validate the entire approach.At first, in order to asses the Gaussianity of our data set we have evaluated the
Kurtosisexcess (or Fisher’s index) is calculated. Then, once Gaussianity is proved, a more sophis-ticate test is used to check the statistical quality of the collected data. In particular, the
Shapiro–Wilk [46] test checks whether the collected data come from a truly random nor-mal distribution, i.e. whether or not the data ensemble is a faithful replica of a Gaussianstatistical population. We underline the importance of Gaussianity tests, which is usuallyassumed rather than actually verified experimentally on the basis of analysis of OPO data.The Kurtosis is the distribution fourth order moment, and can be seen as a sort of“peakedness” measurement of a random probability distribution. Compared to the Gaussianvalue of 3 σ (where σ is the standard deviation) the Kurtosis–excess γ is defined as γ = P ni =1 ( x i − x ) p i P ni =1 ( x i − x ) p i − x is the mean of the data and p i is the probability of the i –th outcome. A γ = 0distribution is Gaussian. As a matter of fact γ gives an immediate check on the Gaussian-ity of the data ensemble, whereas it cannot say anything about accidental (or systematic)internal correlation between data. Overall, the use of the Kurtosis test only may not leadto a conclusive assessment of Gaussianity.For this purpose we adopt the Shapiro-Wilk (SW) tests, which is suitable to test thedeparture of a data sample from normality. SW tests whether a data sample { x , . . . , x n } of n observations comes from a normally random distributed population. The so-called W SW -statistic is the ratio of two estimates of the variance of a normal distribution based on thedata sample. In formula: W SW = (cid:2)P nh =1 a h x ( h ) (cid:3) P nh =1 ( x h − x ) , (20)where x ( h ) are the ordered sample values ( x ( h ) is the h -th smallest value) and a h are weightsgiven by [46]: ( a , ...a n ) = m T V − ( m T V − V − m ) m T the expected values of the order statistics of random variables sampled from thestandard normal distribution, and V is the covariance matrix of the order statistics. Froma mere statistical point of view, W SW is an approximation of the straightness of the normalquantile-quantile probability plot, that is a graphical technique for determining if two datasets come from populations with a common distribution. Notice that W SW ∈ [0 , .
05, if p - W SW ≤ . p - W SW is the p -value of W SW i.e. the probability of obtaining a result at least asextreme as the one that was actually observed, given that the Gaussian hypothesis is true.The two tests verify two complementary aspects. Even if the SW one is considereda faithful Gaussianity test it can fail either for a non-Gaussian or for non truly randomdistributions. Once the Gaussianity of the data is proved, by means of the Kurtosis excess γ , the SW test is used as a test for the randomness of the data ensemble.We have applied the above statistical analysis to our homodyne data distribution dividedinto 104 discrete phase bins (each bin correspond to a θ variation of ≈
60 mrad). As anexample of Gaussianity test, in Fig. 2 we show two typical experimental homodyne tracesfor modes b and d (plots on the left) as well as the corresponding p -value of the Shapiro-Wilktest (plots on the right). As it is apparent from the plots, the mode b is excited in a thermalstate, while the mode c is squeezed with quadratures noise reduction, corrected for non-unitefficiency, of about 2 . p -value p > .
05 (the dashed line in the plots) for all the data set we canconclude that our data are normally distributed and that the signals arriving at the detectorare Gaussian states.
6. Tomographic reconstruction
As already mentioned in Section 4 our setup is suitable to measure all the field quadrature x θ = x cos θ + y sin θ of any input mode by scanning over the phase of the homodyne localoscillator. We exploit this feature twice. On the one hand we use the full homodyne set ofdata to assess Gaussianity of the state and, on the other hand, we may perform full quantumhomodyne tomography to validate results and increase precision for some specific quantities[47].The acquisition of every mode is triggered by the PZT linear ramp: for each value θ , thequadrature x θ = x cos θ + y sin θ of the homodyne input mode is measured, where x and y are13espectively the amplitude and phase field quadratures. Calibration with respect to the shotnoise is obtained by obscuring the OPO output and acquiring the vacuum quadratures. It isworth stressing that experimentally, the acquisition over 2 π intervals presents the advantagethat it does not require sophisticate phase locking set-up to keep θ constant during theacquisition.The collection of homodyne data points, normalised to the shot-noise, is then used toevaluate the bipartite state properties, included the quadratures, x θ , for every θ . The deter-mination of the quadratures mean value, as well as of any other relevant quantity, has beenperformed thank to the so called pattern function tomography. This allows reconstructingthe mean value h ˆ O i of an observable ˆ O as the statistical average of a suitable kernel func-tion R [ ˆ O ] over the ensemble of homodyne data ( x i , θ i ) [48]. By taking into account thenon-unitary detection efficiency η , h ˆ O i is indeed retrieved as: h ˆ O i = R [ ˆ O ] = 1 N N X i =1 R η [ ˆ O ] ( x i ; θ i )where N is the total number of samples. Every datum ( x i , θ i ) individually contributes to theaverage, so that the operator mean value is gradually built up, till statistical confidence inthe sampled quantity is sufficient. Although the method is very general, and can be appliedto any operator, in the following we will only report the kernels for the quantities we areinterested in in this paper. For η > .
5, the following kernels can be calculated (we omitthe dependence of R η [ ˆ O ]( x ; θ ) on x and θ ) R η (cid:2) a † a (cid:3) = 2 x − η , R η h(cid:0) a † a (cid:1) i = 83 x − x R η [ x φ ] = 2 x cos ( φ − θ ) R η (cid:2) x φ (cid:3) = 14 (cid:26) (cid:18) x − η (cid:19) (cid:2) ( φ − θ ) − (cid:3)(cid:27) In principle, a precise knowledge of the h ˆ O i would require an infinite number of measurementson equally prepared states. However, in real experiments the number of data N is of coursefinite, so requiring an errors estimation. Under the hypotheses of the central limit theoremthe confidence interval on the tomographic reconstruction is given as: δ ˆ O = 1 √ N q ∆ R η [ ˆ O ]14here ∆ R η [ ˆ O ] is the kernel variance, say the average over the tomographic data of thequantity R η [ ˆ O ] ( x, φ ) − h ˆ O i . For the particular case of a field quadrature, the confidenceinterval is: ∆ R η [ x θ ] ( x, φ ) = (cid:10) ∆ x θ (cid:11) + 12 h n i + 2 − η η where h n i is the mean photon number of the field under scrutiny.
7. Experimental results
The first step is the Gaussianity test for the each data set which consists, for each acquisition,of a collection of eight homodyne traces: one for the shot-noise (vacuum), one for theelectronic noise and six corresponding to the six homodyne modes { a, b, c, d, e, f } Thenwe check the consistency of the vacuum (shot noise) CM, namely, σ = Diag(1 ,
1) withinthe experimental errors. After this the thermal character of a and b , as expected for a belowthreshold OPO, is verified and their mean photon number as well as A and B CM blocksare retrieved. Then, modes c , d and e , f , are analysed in view of their squeezed thermalnature, with squeezing appearing on the x , y and t , z quadratures respectively. The varianceof x c (squeezed), y c (anti–squeezed), x d (anti–squeezed), y d (squeezed), x e , y e , x f , and y f ,are finally used to retrieve the CM C block.Since modes a and b are both, phase independent, thermal states, the determination oftheir quadrature variances are highly robust against homodyne phase fluctuations. Accord-ingly, the error on blocks A and B elements is obtained by propagating the relative tomo-graphic error. On the other hand, when dealing with, phase dependent, squeezed states, asmall uncertainty in setting the LO phase θ can result in a non negligible indeterminacy onthe quadrature variance used to reconstruct the relative σ element. As a consequence, whenevaluating the errors on the elements of the block C , one must take into account the noiseproperties of the involved modes and critically compare the tomographic error with the errordue to the finite accuracy on θ . σ and σ are obtained as combinations of squeezed/anti–squeezed variances, which, are stationary points of the variance as function of θ , thus theyare quite insensitive to θ fluctuations; accordingly the overall tomographic error can be re-liably used in this case. On the contrary σ and σ depend on the determination of x e,f and y e,f . These quadrature variances are extremely sensible to phase fluctuations being thevariance derivative, in θ , maximum for this values. In this case the error correspond to thedeviation between the variances at x π and at x π ± δθ (or x − π and x − π ± δθ ) with δθ ≃ σ does not correspond to a physical state.A typical matrix is given by σ = . (21)It corresponds to an entangled state that satisfies the Duan criterion ( β D = 0 .
93) andthe Simon criterion ( ˜ d − = 0 . E ( σ ) = 0 .
12) while it it does not show EPR correlations( β EP R = 0 . ≈ S ( ̺ ) = 2 . S (1 |
2) = 0 . S (2 |
1) = 0 .
720 and the quantum mutual information by I ( σ ) = 0 . σ = (22)In this case the corresponding state, whose total energy is n tot ≈ .
9, is both entangled andEPR correlated ( β D = 0 .
64, ˜ d − = 0 . E ( σ ) = 1 .
12, and β EP R = 0 . I ( σ ) = 1 .
633 carried by t he state. Notice that thisstate suffer from non–zero entries on the anti–diagonal elements of the CM. This is due to anon–perfect alignment of the non–linear crystal that give raise to a projection of a residualcomponent of the field polarized along a onto the orthogonal polarization (say along b ),thus leading to a mixing among the modes [45]. This effect is the well known polarizationcross-talk. 16ndeed, in the ideal case the OPO output is in a twin-beam state S ( ζ ) | i , S ( ζ ) =exp { ζ a † b † − ¯ ζ ab } being the entangling two-mode squeezing operator: the correspondingCM has diagonal blocks A , B , C with the two diagonal elements of each block equal inabsolute value. In realistic OPOs, cavity and crystal losses lead to a mixed state, i.e. toan effective thermal contribution. In addition, spurious nonlinear processes, not perfectlysuppressed by the phase matching, may combine to the down conversion, contributing withlocal squeezings. Finally, due to small misalignments of the nonlinear crystal, a residualcomponent of the field polarized along a may project onto the orthogonal polarization (sayalong b ), thus leading to a mixing among the modes [45]. Overall, the state at the out-put is expected to be a zero amplitude Gaussian entangled state, whose general form maybe written as ̺ = U ( β ) S ( ζ ) LS ( ξ , ξ ) T LS † ( ξ , ξ ) S † ( ζ ) U † ( β ), where T = τ ⊗ τ , with τ k = (1 + ¯ n k ) − [¯ n k / (1 + ¯ n k )] a † a denotes a two-mode thermal state with ¯ n k average photonsper mode, LS ( ξ , ξ ) = S ( ξ ) ⊗ S ( ξ ), S ( ξ k ) = exp { ( ξ k a † − ¯ ξ k a ) } denotes local squeezingand U ( β ) = exp { βa † b − ¯ βab † } a mixing operator, ζ , ξ k and β being complex numbers.For our configuration, besides a thermal contribution due to internal and coupling losses,we expect a relevant entangling contribution with a small residual local squeezing and, asmentioned above, a possible mixing among the modes.Given the CM it is also possible to retrieve the corresponding joint photon number dis-tribution p ( n, m ) by using the relation [7]: p ( n, m ) = Z C d λ d λ π χ ( λ , λ ) χ n ( − λ ) χ m ( − λ ) , (23)where χ ( λ , λ ) is the characteristic of the reconstructed two-mode state, that actuallydepends only on σ , and χ n ( λ k ) denotes the characteristic function of the projector | h ih h | , χ n ( λ ) = h n | D ( λ ) | n i = exp {− | λ | } L n ( | λ | ), where L n ( x ) is the n -th Laguerre polynomials.In Fig. 3 we report the joint photon number distribution p ( n, m ) derived from the CM (22)and the single-mode photon distributions (either from data or from the single-mode CM)for modes b and d (the same modes considered in Fig. 2): as one may expect, the photonnumber distribution b is thermal, whereas the statistics of mode d correctly reproduces theeven-odd oscillations expected for squeezed thermal states.The mutual information I ( σ ) of Eq. (9) measures the amount of information one can geton one of the two subsystems by measuring the other one. In turn, it is a measure of thedegree of correlation between the two modes. On the other hand, equally entangled states17ay show different I ( σ ) and the difference appears to be dependent on the total numberof photons. In Tab. 1 we report the mutual information I ( σ ), the total number of photons n tot , the Duan and EPR factors β D and β E , and the symplectic eigenvalue ˜ d − for differentacquisitions. All the states are non–separable and not EPR-correlated; they have differentnumber of photons and, correspondingly, different quantum mutual information.
8. Conclusions
Gaussian states of bipartite continuous variable optical systems are basic tools to imple-ment quantum information protocols and their complete characterization, obtained by re-constructing the corresponding covariance matrix, is a pillar for the development of quantumtechnology. As a matter of fact, much theoretical attention have been devoted to continuousvariable systems and to the characterization of Gaussian states via the CM. On the otherhand, only a few experimental reconstructions of CM have been so far reported due to thedifficulties connected to this measurement.We have developed and demonstrated a reliable and robust approach, based on the use ofa single homodyne detector, which have been tested on the bipartite states at the output ofa sub-threshold type–II OPO producing thermal cross–polarized entangled CW frequencydegenerate beams. The method provides a reliable reconstruction of the covariance matrixand allows one to retrieve all the physical information about the state under investigation.These include observable quantities, as energy and squeezing, as well as non observable onesas purity, entropy and entanglement. Our procedure also includes advanced tests for theGaussianity of the state and, overall, represents a powerful tool to study bipartite Gaussianstates from the generation stage to the detection one.
Acknowledgments
The authors thank S. Solimeno for encouragement and support. SO and MGAP thankM. G. Genoni for useful discussions. This work has been partially supported by the CNR-CNISM agreement.
References
1. M. G. A. Paris, and J. ˇReh`aˇcek (Eds.),
Quantum State Estimation , Lect. Not. Phys. (Springer, Berlin, 2004).2. D.-G. Welsch, W. Vogle, and T. Opatrn´y, “Homodyne Detection and Quantum StateReconstruction”, in Progr. Opt.
XXXIX , E. Wolf Ed., pp. 63–214 (1999).18. G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, “Quantum Tomography”, Adv. Im.Elect. Phys. , 205–309 (2003).4. A. I. Lvovsky, and M. G. Raymer, “Continuous–variable optical quantum–statetomography”, Rev. Mod. Phys. , 299–332 (2009).5. J. Eisert, and M. B. Plenio, “Introduction to the basics of entanglement theory incontinuous–variable systems”, Int. J. Quant. Inf. , 479–506 (2003).6. B.-G. Englert, and K. Wodkiewicz, “Tutorial Notes on One–Party and Two–PartyGaussian States”, Int. J. Quant. Inf. , 153–188 (2003).7. A. Ferraro, S. Olivares, and M. G. A. Paris, Gaussian States in Quantum Information ,(Bibliopolis, Napoli, 2005).8. F. Dell’Anno, S. De Siena, and F. Illuminati, “Multiphoton quantum optics and quan-tum state engineering”, Phys. Rep. , 53–168 (2006).9. S. L. Braunstein, and P. van Loock, “Quantum information with continuous variables”,Rev. Mod. Phys , 513–577 (2005).10. P. D. Drummond, and M. D. Reid, “Correlations in nondegenerate parametric oscilla-tion. II. Below threshold results”, Phys. Rev. A , 3930–3949 (1990).11. Yun Zhang, Hong Su, Changde Xie, and Kunchi Peng, “Quantum variances and squeez-ing of output field from NOPA”, Phys. Lett. A , 171–177 (1999).12. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables”, Phys. Rev. Lett. , 3663–3666(1992).13. Hai Wang, Yun Zhang, Qing Pan, Hong Su, A. Porzio, Changde Xie, and KunchiPeng, “Experimental Realization of a Quantum Measurement for Intensity DifferenceFluctuation Using a Beam Splitter” Phys. Rev. Lett. , 1414–1417, (1999).14. N. Treps and C. Fabre, “Criteria of quantum correlation in the measurement of con-tinuous variables in optics”, Laser Physics , 187–194 (2005).15. V. D’Auria, A. Porzio, S. Solimeno, S. Olivares and M. G. A. Paris, “Characterizationof bipartite states using a single homodyne detector”, J. Opt. B: Quantum Semiclass.Opt. , S750–S753 (2005).16. A. Porzio, V. D’Auria, S. Solimeno, S. Olivares, and M. G. A. Paris, “HomodyneCharacterization of continuous variable bipartite states”, Int. J. Quant. Inf. , 63–68(2007). 197. V. D’Auria, S. Fornaro, A. Porzio, S. Solimeno, S. Olivares, and M. G. A. Paris, “Fullcharacterization of Gaussian bipartite entangled states by a single homodyne detector”,Phys. Rev. Lett. , 020502(4) (2009).18. G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi, “On parametric approximation inQuantum Optics”, Nuovo Cimento B , 339–354 (1999).19. V. D’Auria, A. Chiummo, M. De Laurentis, A. Porzio, S. Solimeno, and M. G. A. Paris,“Tomographic characterization of OPO sources close to threshold”, Optics Express, ,948–956, (2005).20. V. D’Auria, C. de Lisio, A. Porzio, S. Solimeno, J. Anwar, and M. G. A. Paris, “Non–Gaussian states produced by close–to–threshold optical parametric oscillators: role ofclassical and quantum fluctuations”, arXiv:0907.3825v3 (in print on Phys. Rev. A).21. A. Franzen, B. Hage, J. DiGuglielmo, J. Fiur´asˇek, and R. Schnabel, “ExperimentalDemonstration of Continuous Variable Purification of Squeezed States”, Phys. Rev.Lett. , 150505 (2006).22. B. Hage, A. Samblowski, J. DiGuglielmo, A. Franzen, J. Fiur´asˇek, and R. Schnabel,“Preparation of distilled and purified continuous–variable entangled states”, NaturePhys. , 915–918 (2008).23. W. P. Bowen, R. Schnabel, and P. K. Lam, “Experimental Investigation of Criteria forContinuous Variable Entanglement”, Phys. Rev. Lett. , 043601(4) (2003).24. W. P. Bowen, R. Schnabel, and P. K. Lam, “Experimental characterization ofcontinuous-variable entanglement”, Phys. Rev. A , 012304(17) (2004).25. J. Wenger, A. Ourjoumtsev, R. Tualle–Brouri, and P. Grangier, “Time-resolved homo-dyne characterization of individual quadrature-entangled pulses”, Eur. Phys. J. D ,391–396 (2004).26. J. Laurat, G. Keller, J. A. Oliveira–Huguenin, C. Fabre, T. Coudreau, A. Serafini,G. Adesso, and F. Illuminati, “Entanglement of two-mode Gaussian states: charac-terization and experimental production and manipulation”, J. Opt. B , S577–S587(2005).27. J. ˇReh`aˇcek, S. Olivares, D. Mogilevtsev, Z. Hradil, M. G. A. Paris, S. Fornaro,V. D’Auria, A. Porzio, and S. Solimeno, “Effective method to estimate multidimen-sional Gaussian states”, Phys. Rev A , 032111(7) (2009).28. R. Simon, “Peres-Horodecki Separability Criterion for Continuous Variable Systems”,20hys. Rev. Lett. , 2726–2729 (2000).29. Lu–Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability Criterion forContinuous Variable Systems”, Phys. Rev. Lett. , 2722–2725 (2000).30. A. Serafini, F. Illuminati, and S. De Siena, “Symplectic invariants, entropic measuresand correlations of Gaussian states”, J. Phys. B , L21–L28 (2004).31. M. G. A. Paris, F. Illuminati, A. Serafini, and S. De Siena, “Purity of Gaussianstates: measurement schemes and time–evolution in noisy channels”, Phys. Rev. A , 012314(9) (2003).32. G. S. Agarwal, “Entropy, the Wigner Distribution Function, and the Approach toEquilibrium of a System of Coupled Harmonic Oscillators”, Phys. Rev. A :828–831(1971).33. D. Slepian, and J. K. Wolf, “Noiseless coding of correlated information sources”, IEEETrans. Inf. Theory, , 471–480 (1973).34. M. Horodecki, J. Oppenheim, and A. Winter, “Partial quantum information”, Nature , 673–676 (2005).35. M. G. Genoni, M. G. A. Paris, and K. Banaszek, “Quantifying the non-Gaussian char-acter of a quantum state by quantum relative entropy”, Phys. Rev. A , 060303(R)(2008).36. A. Serafini, S. De Siena, F. Illuminati, and M. G. A. Paris, “Minimum decoherencecat-like states in Gaussian noisy channels”, J. Opt. B. , S591 (2004).37. A. Serafini, F. Illuminati, M. G. A. Paris, and S. De Siena, “Entanglement and purityof two-mode Gaussian states in noisy channels”, Phys. Rev A , 022318(10) (2004).38. S. Maniscalco, S. Olivares, and M. G. A. Paris, “Entanglement oscillations in non-Markovian quantum channels”, Phys. Rev. A , 062119(5) (2007).39. R. Vasile, S. Olivares, M. G. A. Paris, and S. Maniscalco, “Continuous-variable-entanglement dynamics in structured reservoirs”, Phys. Rev. A , 062324(11) (2009).40. G. Vidal and R. F. Werner, “Computable measure of entanglement”, Phys. Rev. A ,032314(11) (2002).41. N. J. Cerf and C. Adami, “Quantum extension of conditional probability”, Phys. Rev.A , 893–897 (1999).42. S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, “Entangling Macroscopic Os-cillators Exploiting Radiation Pressure”, Phys. Rev. Lett. , 120401 (2002).213. M. D. Reid, and P. D. Drummond, “Quantum Correlations of Phase in NondegenerateParametric Oscillation”, Phys. Rev. Lett. , 2731–2733 (1989).44. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, andH. Ward, “Laser phase and frequency stabilization using an optical resonator”, Appl.Phys. B , 97-105, (1983).45. V. D’Auria, S. Fornaro, A. Porzio, E. A. Sete, and S. Solimeno, “Fine tuning of a tripleresonant OPO for generating frequency degenerate CV entangled beams at low pumppowers”, Appl. Phys. B , 309-314 (2008).46. S. Shapiro, and M. Wilk, “An Analysis of Variance Test for Normality (CompleteSamples)”, Biometrika , 591–611 (1965).47. Actually quantum tomography is generally more noisy than other less universal esti-mation techniques which incorporate some a priori information about the state (theGaussian character in our case). On the other hand, the intrinsic noise of the recon-struction scheme may be overcome by the statistical noise reduction: this is indeed ourcase, where the reconstruction of the CM is obtained with a small subset of the entirehomodyne sample used in the tomographic reconstruction.48. G. M. D’Ariano, L. Maccone, and M. G. A. Paris, “Quorom of observables for universalquantum estimation”, J. Phys. A , 93–103 (2001).22 ist of captions • Fig. 1. Experimental setup: A type-II OPO containing a periodically poled crystal(PPKTP) is pumped by the second harmonic of a Nd:YAG laser. At the OPO output,a half-wave plate ( λ/ out ), a quarter-wave plate ( λ/ out ) and a PBS out select the modefor homodyning. The resulting electronic signal is acquired via a PC module. • Fig. 2. (Left): from top to bottom, two typical experimental homodyne traces ofmodes b and d (similar results are obtained for the other modes). (Right): p -valueof the Shapiro-Wilk normality test as a function of the bin number (see the text fordetails). Since we have p -value ≥ .
05 (the dashed line in the plots), we can concludethat our data are normally distributed. θ is the relative phase between the signal andthe local oscillator. Kurtosis excess γ for these data is 0 within experimental error. • Fig. 3. (Left): Joint photon number distribution p ( n, m ) for the entangled state ofmodes a and b at the output of the OPO. (Right): single-mode photon distributions p ( n ) for modes b (top right) and d (bottom right). The single-mode distributions ofmode b is thermal and corresponds to the marginals of p ( n, m ). The distributions formodes d is that of squeezed thermal state.23 igures Fig. 1. Experimental setup: A type-II OPO containing a periodically poled crystal (PP-KTP) is pumped by the second harmonic of a Nd:YAG laser. At the OPO output, ahalf-wave plate ( λ/ out ), a quarter-wave plate ( λ/ out ) and a PBS out select the mode forhomodyning. The resulting electronic signal is acquired via a PC module.24ig. 2. (Left): from top to bottom, two typical experimental homodyne traces of modes b and d (similar results are obtained for the other modes). (Right): p -value of the Shapiro-Wilk normality test as a function of the bin number (see the text for details). Since we have p -value ≥ .
05 (the dashed line in the plots), we can conclude that our data are normallydistributed. θ is the relative phase between the signal and the local oscillator. Kurtosisexcess γ for these data is 0 within experimental error.25ig. 3. (Left): Joint photon number distribution p ( n, m ) for the entangled state of modes a and b at the output of the OPO. (Right): single-mode photon distributions p ( n ) for modes b (top right) and d (bottom right). The single-mode distributions of mode b is thermal andcorresponds to the marginals of p ( n, m ). The distributions for modes d is that of squeezedthermal state. Tables I ( σ ) n tot β D ˜ d − β E Table 1. The quantum mutual information for acquisition with different average photonnumbers together with the Duan and EPR factors β D and β E , and the symplectic eigenvalue˜ d − . All the states are non–separable and not EPR-correlated. They have different numberof photons and, correspondingly, different quantum mutual information. I ( σσ