A Piccialli
email: arianna.piccialli@aeronomie.be

A C Vandaele

L Trompet

L Neary

S Viscardy

F Daerden

S Robert

S Aoki

Impact of gradients at the Martian terminator on the retrieval of ozone from SPICAM/MEx

Introduction

Rapid variations in species concentration at the terminator have the potential to cause asymmetries in the species distributions along the line of sight (LOS) of a solar occultation experiment. Ozone, in particular, displays steep gradients across the terminator of Mars due to photolysis [START_REF] Lefèvre | Heterogeneous chemistry in the atmosphere of Mars[END_REF]. Nowadays, most of the retrieval algorithms for solar and stellar occultations rely on the assumption of a spherically symmetrical atmosphere. However, photochemically induced variations near sunrise/sunset conditions need to be taken into account in the retrieval process in order to prevent inaccuracies.

Here, we investigated the impact of gradients along the LOS of the solar occultation experiment SPICAM for the retrieval of ozone under sunrise/sunset conditions. We used the diurnal variations in the ozone concentration obtained from photochemical model calculations together with an adapted radiative transfer code.

SPICAM solar occultations

SPICAM (SPectroscopie pour l'Investigation des

Caractéristiques Atmosphériques de Mars), on board the ESA's spacecraft Mars Express, is a remote sensing spectrometer observing in the ultraviolet (118-320 nm) and in the near infrared (1-17 μm) [START_REF] Bertaux | SPICAM on Mars Express: Observing modes and overview of UV spectrometer data and scientific results[END_REF]. In the solar occultation mode, the UV sensor is particularly well suited to measure the vertical profiles of O 3 and aerosols of the Martian atmosphere [START_REF] Määttänen | A complete climatology of the aerosol vertical distribution on Mars from MEx/SPICAM UV solar occultations[END_REF]. Figure 1 displays transmission spectra obtained at different altitudes for the observation 00633A02. We followed the same method described in [START_REF] Trompet | Improved algorithm for the transmittance estimation of spectra obtained with SOIR/Venus Express[END_REF] to check that the spectra are correctly calibrated and accurately normalized to the solar spectrum.

Retrieval technique

SPICAM-UV spectra are simulated using the line-byline radiative transfer code ASIMUT-ALVL developed at IASB-BIRA [START_REF] Vandaele | Composition of the Venus mesosphere measured by SOIR on board Venus Express[END_REF]. ASIMUT has been modified in order to take into account the atmospheric composition and structure at the daynight terminator. Three different gradients along the LOS can be considered: temperature, total density gradients and the variations of the concentration of specific species. As input for ASIMUT, we used gradients predicted by the 3D GEM-Mars v4 Global Circulation Model (GCM) [START_REF] Daerden | A solar escalator on Mars: Self-lifting of dust layers by radiative heating[END_REF][START_REF] Neary | The GEM-Mars general circulation model for Mars: Description and evaluation[END_REF]. Figure 2

Preliminary results

As preliminary study, we selected four occultations at sunrise and at sunset each. As first step, we retrieved O 3 profiles without taking in account gradients and we obtained results in agreement with previous studies. Then, we investigated the effects of ozone density gradients on the retrieval of ozone. The retrieved ozone profiles are lower compared to retrievals without gradients, even if differences are within the error bars. These effects will be analysed in more detail.

Future work

We will extend analysis to the whole SPICAM-UV solar occultation dataset. The main objective is to investigate fully the impact of these gradients on ozone retrievals. We will compare our retrievals with those from a similar study carried out at LATMOS within the UPWARDS project. Results of this study will then be used for the analysis of the data expected from the NOMAD instrument on the ExoMars 2016 Trace Gas Orbiter.

Figure 1 :

 1 Figure 1: Example of SPICAM transmission spectra at different altitudes: (blue) low altitudes; (red) high altitudes.

  shows the diurnal cycle of ozone derived by GEM-Mars. Ozone EPSC Abstracts Vol. 12, EPSC2018-434, 2018 European Planetary Science Congress 2018 c Author(s) 2018 EPSC European Planetary Science Congress is more abundant during the night time, especially above 40-50 km. As the Sun rises, the destruction of O 3 , although stronger in the high atmosphere, is observed at all altitudes.

Figure 2 :

 2 Figure 2: Diurnal cycle of ozone predicted by the 3D GEM-Mars v4 GCM.

Acknowledgements

The research leading to these results has received funding from the European Union's Horizon 2020 Programme (H2020-Compet-08-2014) under grant agreement UPWARDS-633127. SA has been supported by the FNRS "CRAMIC" project under grant agreement n° T.0171.16.


Marina Korotina

Stanislav Aranovskiy

Rosane Ushirobira

Denis Efimov

Jian Wang
email: wangjian119@hdu.edu.cn

A note on fixed-and discrete-time estimation via the DREM method

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

In this note, we consider the parameter estimation problem for a discrete-time linear regression equation (LRE)

y k = φ k θ + v k , k ∈ N, (1)
where y k , v k ∈ R are the output and the bounded measurement noise, φ k , θ = (θ 1 . . . θ n ) ∈ R n are the bounded regressor and the unknown parameter vectors. The goal is to estimate θ given the online measurements y k and φ k .

The parameter estimation problem for dynamic or static systems is an essential issue in many scientific disciplines, and many well-matured approaches to solving this problem are available [START_REF] Ljung | System Identification: Theory for the User[END_REF], [START_REF] Yan | Linear Regression Analysis: Theory and Computing[END_REF]. Despite various batch methods based on post-processing of collected measurements, two popular iterative parameter estimation methods are the recursive leastsquares and gradient-descent estimators [START_REF] Landau | Adaptive control: algorithms, analysis and applications[END_REF]. The fundamental result in parameter estimation claims that if the regressor φ is persistently excited (PE), that is uniformly excited in time (see Section III for a formal definition of this property), then the recursive least-squares and gradient-descent iterative estimators converge exponentially. They are also input-to-state stable (ISS) and provide filtering of the noise v k .

Numerous recent studies address a possible relaxation of the PE requirement in both continuous-and discrete-time settings. These results include the composite [START_REF] Chowdhary | Concurrent learning adaptive control of linear systems with exponentially convergent bounds[END_REF] and concurrent [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF] learning, directional forgetting [START_REF] Goel | Recursive least squares with variable-direction forgetting: Compensating for the loss of persistency [lecture notes[END_REF], fixed-time estimation [START_REF] Wang | Fixed-time estimation of parameters for non-persistent excitation[END_REF], and the recent regularization-based results [START_REF] Marino | On exponentially convergent parameter estimation with lack of persistency of excitation[END_REF], [START_REF] Bin | Generalized recursive least squares: Stability, robustness, and excitation[END_REF]. However, it is widely understood that if the regressor φ is not uniformly excited in time but only on a particular interval (interval excitation), then the additive noise cannot be efficiently filtered, and the PE-relaxation methods typically consider the noisefree scenario, v ≡ 0. In this note, due to the presence of noise vk in [START_REF] Ljung | System Identification: Theory for the User[END_REF], we are focused on the uniform-in-time excitation, i.e., when the regressor φ is PE.

Standard iterative estimators also have some known shortages, such as the difficulty of accelerating the convergence process (this property is mainly predefined by the level of excitation in the regressor) and the non-monotonicity of the convergence [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF]- [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF]. The latter means that despite a decreasing norm of the parameter vector estimation error, the respective discrepancies for each parameter may demonstrate a complex oscillatory behavior before settling down in a vicinity of the true value, which may postpone a real-time utilization of the obtained estimates. That is why a solution to these drawbacks recently proposed by the dynamical regressor extension and mixing (DREM) method [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] has quickly gained popularity [START_REF] Barroso | Model-based adaptive filtering of harmonic perturbations applied to high-frequency noninvasive valvometry[END_REF]- [START_REF] Schiffer | Online estimation of power system inertia using dynamic regressor extension and mixing[END_REF].

The DREM procedure consists of two steps: extension and mixing. In the first step, the model ( 1) is extended to get a new linear regression with a square n × n regressor matrix. Then, at the mixing step, the single extended LRE for a vector of unknown parameters is transformed into a set of n scalar LREs for each element of the vector θ independently. Then an iterative estimator, e.g., the gradient-descent one, is applied to each of the new scalar LREs separately, yielding the elementwise improved transients with simple and transparent estimator tuning rules.

In this note, we argue that using iterative estimators in the DREM procedure is more a custom than a need. We show that a proper choice of an LRE extension at the first step of the DREM procedure ensures that the novel regressor is strictly positive, allowing for pointwise (algebraic) estimation. A pointwise estimator can be of limited practical interest due to its noise sensitivity, even if it provides the fixed-time (independent in initial conditions) convergence. To this end, we study the worst-case bound on the parameter estimation error and propose tuning guidelines minimizing this bound.

To summarize, the contribution of this paper is as follows:

• we show that a particular choice of filters for an LRE extension ensures strict positiveness of the novel regressor; • we analyze noise propagation for the pointwise (algebraic) estimator and derive the worst-case bound on estimation error.

Finally, we address the argument that standard iterative estimators provide a good trade-off between the transient time and the steady-state noise attenuation (filtering). We show that we obtain faster transients with a comparable steadystate performance by putting the pointwise estimator in a serial connection with a (non-)linear filter. Together, these contributions support our claim that, under the PE condition, the DREM procedure can function as a fixed-time estimator without requiring subsequent gradient-type algorithms.

The rest of the paper is organized as follows. In Section II we briefly describe the DREM procedure. In Section III, we discuss how the initial excitation can be preserved by the DREM procedure. Section IV presents this note's main result, and Section V contains simulations illustrating our results. Finally, Section VI contains some concluding remarks.

Notation

• The sets of nonnegative real and integer numbers are denoted by R + and N, respectively. Also, N * := N \ {0}.

The set of real n × m-matrices is denoted by R n×m . The n-identity matrix is denoted by I n . • For a vector x ∈ R n , x denotes its Euclidean norm.

• The rounding function to the greatest integer smaller than s ∈ R is denoted by s = floor(s).

II. DYNAMICAL REGRESSOR EXTENSION AND MIXING

The original DREM procedure in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] starts with a dynamics extension step. It consists in finding a (dynamic) transformation that applies to the original LRE (1) and leads to a new extended regression

Y k = Φ k θ + V k , V k = V k + k , k ∈ N, (2)
where Y k ∈ R n and Φ k ∈ R n×n are measured signals generated by the extension of the dynamics, V k ∈ R n results from the noise v k propagation, and k ∈ R n is an exponentially decaying term arising due to the initialization of the dynamics extension. Such an extension can be performed, for instance, via delay operators or linear filters, whose auxiliary role is also to filter the noise, see [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF]. A particular choice of the dynamics extension used in this paper is presented in Section III.

For the extended regression (2), we define

φ k = det(Φ k ) , y k = adj(Φ k ) Y k , v k = adj(Φ k ) V k (3)
where adj(M ) denotes the adjugate matrix of a matrix M . This yields the element-wise scalar linear regression:

y k,i = φ k θ i + v k,i , k ∈ N, i ∈ {1, . . . , n}, (4)
where y k,i ∈ R and φ k ∈ R are known signals, θ i ∈ R is the unknown constant parameter to be estimated, and v k,i ∈ R is an unknown bounded measurement distortion. Note that the estimation of each θ i is now explicitly independent of other components of θ.

Let θ k ∈ R n denote an estimate of θ at time k ∈ N. Define e k := y k -φ k θ k = φ k θ k + v k
as the measured regression error, where θ k := θ -θ k is the parameter estimation error. It remains to design an algorithm to compute θ k , which is a simpler problem than (1), since in (4) the estimation of each θ i can be treated separately. Moreover, in scalar linear regression, the convergence of θ k is element-wise monotone and can be accelerated by tuning the estimation procedures [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF].

One of the main issues of the DREM methodology is the excitation of the common regressor φ k , and its relation with the excitation of φ k in the original problem statement, which is obviously predefined by the choice of the dynamics extension generating Φ k . Let us present a solution to this problem.

III. KREISSELMEIER'S REGRESSOR EXTENSION

Let us show how the DREM procedure can keep the excitation level of the original regression for (4). To this end, let us characterize the admissible excitation levels [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF]:

Definition 1 (PE). Let ∈ N * and µ > 0. A bounded signal φ : N → R n is called ( , µ)-persistently excited (PE) if -1 i=0 φ k+i φ k+i ≥ µI n , ∀k ∈ N.
It is said to be intervally excited (IE) if the above inequality is satisfied for k = 0 only.

It is worth noting that in the noise-free case, interval excitation can be related with the conventional identifiability/observability condition for θ in [START_REF] Ljung | System Identification: Theory for the User[END_REF].

To keep the excitation, we proceed as follows. Instead of applying a dynamic extension directly to (1), consider the auxiliary regression problem

φ k y k = φ k φ k θ + φ k v k
and apply an auxiliary filter H(z) = 1-λ z -1 -λ , where λ ∈ (0, 1). Then for

Y k = H(z)φ k y k , Φ k = H(z)φ k φ k , V k = H(z)φ k v k ,
we obtain the extended LRE (2). The state-space implementation of this extension is known as Kreisselmeier's dynamic regressor extension [START_REF] Kreisselmeier | Adaptive observers with exponential rate of convergence[END_REF] and is given by

Φ k+1 = λΦ k + (1 -λ) φ k φ k , (5)
Y k+1 = λY k + (1 -λ) φ k y k , where Y 0 ∈ R n and Φ 0 ∈ R n×n , Φ 0 = Φ 0 ≥ 0.
After the mixing step, it yields the decoupled linear regression equation (4). Specifically, the choice Φ 0 = 0 and Y 0 = 0 guarantees that k = 0 in (2) for all k ∈ N. Kreisselmeier's extension was first used within the DREM context in [START_REF] Gerasimov | On key properties of the lion's and kreisselmeier's adaptation algorithms[END_REF] under the name of memory regressor extension. It is known that (under a proper choice of Φ 0 ), the matrix Φ is semi-positive definite, and Φ is positive definite if φ is PE, see [START_REF] Ioannou | Robust Adaptive Control[END_REF]. The following Lemma further extends this result, providing a lower bound on the excitation characteristics of the signal φ defined in (3). Lemma 1. Consider a ( , µ)-PE signal φ and the Kreisselmeier's dynamic extension [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF].

Then the signal φ defined in (3) is ( , α)-PE, with α = µ (1 -λ) λ -1 n . ( 6
)
Moreover,

φ k ≥ α, ∀k ≥ ; lim inf k→∞ φ k ≥ α 1 1 -λ n .
Proof: The solution of ( 5) is

Φ k = λ k Φ 0 + (1 -λ) k-1 i=0 ψ(k, i), (7)
with ψ(k, i) = λ k-1-i φ i φ i . Set q = k and v = k -q , so k = q + v. Then for all k ≥ k-1 i=0 ψ(k, i) = q j=1   k-1-(j-1) i=k-j ψ(k, i)   + v-1 i=0 ψ(k, i).
Since ψ(k, i) ≥ 0 for all k, i ∈ N and Φ 0 ≥ 0, it follows from ( 7) that for all k ≥

Φ k ≥ (1 -λ) q j=1   k-1-(j-1) i=k-j ψ(k, i)   .
Recalling that λ < 1, for all k ≥ and j ∈ {1, . . . q}, it holds

k-1-(j-1) i=k-j ψ(k, i) ≥λ j -1 k-1-(j-1) i=k-j φ i φ i = λ j -1 -1 i=0 φ k-j +i φ k-j +i ,
and using that φ is ( , µ)-PE, we obtain

k-1-(j-1) i=k-j ψ(k, i) ≥ λ j -1 µI n . Thus Φ k ≥ µ(1 -λ) q j=1 λ j -1 I n = µ (1 -λ) λ -1 1 -λ q 1 -λ I n .
Then for φ k = det(Φ k ), we obtain for all k ≥ (and since q ≥ 1)

φ k ≥ µ (1 -λ) λ -1 n = α.
Moreover, λ q → 0 as k → ∞, and consequently

lim inf k→∞ φ k ≥ µ (1 -λ) λ -1 1 -λ n .
In the continuous-time setting, a similar result on the choice of filters is given in [START_REF] Aranovskiy | On preserving-excitation properties of kreisselmeiers regressor extension scheme[END_REF]. Of course, another filter selection is possible, providing the excitation properties of (1) in (4).

Remark 1. From Lemma 1, it follows that if φ in (1) is PE, then φ is strictly positive for all k ≥ . Moreover, if φ in (1) is IE, then φ resulting from (3), ( 5) cannot be identically zero. In both cases, some k ∈ N exists when φ k = 0. In the following, we consider only such steps and assume that φ k = 0. Otherwise, the linear regression (4) reads y k = v k , and there is no reason to update the estimate for such samples, i.e., θ k = θ k-1 for φ k = 0 (recall that the signal φ is known).

IV. NOISE PROPAGATION

In the noise-free case, after applying the DREM method and obtaining the decoupled linear regression (4), the estimation problem has a trivial pointwise solution

θ k = y k φ k , (8)
which can be unconditionally applied as soon as φ k = 0 for some k ∈ N. Such a condition is always verified with an IEsignal φ.

In the presence of noise, the noise propagation can be studied using the established properties of Kreisselmeier's filters [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF]. Lemma 1 helps to prove the following simple result constituting the main outcome of this note: Theorem 1. Assume that the regressor signal φ in (1) is ( , µ)-PE and Kreisselmeier's dynamic extension (5) is applied to yield (2). Then the estimate θ k can be computed by (3), [START_REF] Marino | On exponentially convergent parameter estimation with lack of persistency of excitation[END_REF], for all k ≥ , and

sup k≥ θ k ≤ sup k≥ v k α , lim sup k→+∞ θ k ≤ 1 -λ n lim sup k→+∞ v k α
.

Proof: According to Lemma 1, for all k ≥ , φ k ≥ α = µ (1 -λ) λ -1 n , hence the division in ( 8) is well defined. By definition,

θ k = θ - y k φ k = v k φ k
giving the required estimates by the properties of φ k . Thus, in the noise-free case, the DREM-based algorithm ( 5), ( 3), (8) guarantees a fixed-time convergence of the estimate θ k to the true value θ, while in the presence of noises, the estimation error will be settled at a vicinity of the origin, whose size is proportional to the noise v amplitude. Note that the convergence (settling) time ≥ n is independent of initial conditions in both cases.

Remark 2. According to the estimates given in Theorem 1, the choice λ * = -1 minimizes the parameter estimation error gain α -1 with respect to the noise, i.e., it maximizes α given in [START_REF] Goel | Recursive least squares with variable-direction forgetting: Compensating for the loss of persistency [lecture notes[END_REF]. Augmenting the value of λ can improve the asymptotic precision of the proposed algorithm,i.e., for k → +∞.

Several final comments are in order: 1) Roughly speaking, the DREM method provides a direct measurement of the unknown parameter vector θ for k ≥ :

Y k = θ + V k , with Y k = φ -1 k y k , V k = φ -1 k v k .
Note also that (3), ( 5), (8) implicitly realize exponentially-weighted least-squares estimation with a forgetting factor λ.

2) Any additional noise filtering can be applied to Y k attenuating the influence of V k . In the existing literature, gradient algorithms are frequently used for (4) [START_REF] Wang | Fixed-time estimation of parameters for non-persistent excitation[END_REF], [START_REF] Vediakova | Finite time frequency estimation for multi-sinusoidal signals[END_REF], [START_REF] Aranovskiy | Parameter identification of linear time-invariant systems using dynamic regressor extension and mixing[END_REF], [START_REF] Ortega | On modified parameter estimators for identification and adaptive control. a unified framework and some new schemes[END_REF], which serve merely for implicit noisecanceling, being probably not the best choice for such a purpose. Indeed, the conventional selection is the leastsquares algorithm:

θ k = θ k-1 + γ k φ k (y k -φ k θ k-1 ), γ k ∈ (0, 2φ -2 k ). For γ k = φ -2
k , it is reduced to (8), while for γ k = φ -2 k with ∈ (0, 1), to a low-pass filtering of [START_REF] Marino | On exponentially convergent parameter estimation with lack of persistency of excitation[END_REF]. In general, many more options are available, as the moving-average filter or nonlinear designs that cannot be directly applied to the linear regression ( 4), e.g., the median filter. See Section V for an illustrative comparison of selected methods. 3) Note that Kreisselmeier's dynamic extension [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF] filters the noise v k already, and other preliminary noise filters can be applied to get [START_REF] Ljung | System Identification: Theory for the User[END_REF]. The DREM method was proposed to process regressor matrices that can be singular (or close to it): multiplying by the adjugate matrix is a viable operation, and the zero (or arbitrarily small) determinant is addressed by a (gradient) estimator. If the original regressor is persistently excited, handling it with DREM after Kreisselmeier's extension, we know that the matrix is not singular after the first samples. Then, applying DREM and dividing by the determinant is the same as multiplying by the inverse. We choose an augmented computational complexity of the DREM framework due to all its advantages, e.g., the possibility of estimating a single element of the vector of unknown parameters with a controlled performance. Another message of this note is that the noise attenuation of the DREM method can be further improved by complementing it with a suitable filtering solution.

V. SIMULATIONS A. General description

We consider a standard example where we estimate parameters of a sinusoidal signal:

y k = A 0 + A sin (ωk + ψ) + v k , k ∈ N, (9)
which can be written as [START_REF] Ljung | System Identification: Theory for the User[END_REF] with

φ k = 1 sin(ωk) cos(ωk) , θ = A 0 A cos(ψ) A sin(ψ) ,
where ω = 2π T , T ∈ N, T ≥ 3, is the known period, and

A 0 ∈ R, A ∈ R + , ψ ∈ [0, 2π
) are unknown constant parameters. For the conciseness of simulations, we focus below on estimating the parameter θ 1 = A 0 only.

The DREM procedure [START_REF] Pan | Efficient learning from adaptive control under sufficient excitation[END_REF] with zero initial conditions yields the element-wise scalar linear regression (4), where we are interested in the equation corresponding to i = 1 only, namely

y k,1 = φ k θ 1 + v k,1 . (10)
Note that the DREM procedure allows for the single parameter estimation; otherwise, the whole vector θ has to be estimated. It is straightforward to show that the regressor φ is ( , µ)-PE with = T and µ = T 2 , and Lemma 1 applies. The signal φ is thus also PE and it is strictly separated from zero for k ≥ T . Then the direct estimation ( 8) is applied. To summarize, the DREM procedure, taking the measurements φ k and y k as inputs and producing the direct estimate of the first element of the vector θ as an output, is given by

Φ k+1 = λΦ k + (1 -λ) φ k φ k , Φ 0 = 0, Y k+1 = λY k + (1 -λ) φ k y k , Y 0 = 0, φ k = det(Φ k ) , y k = adj(Φ k ) Y k , θdir k,1 = y k,1 max {φ k , ε φ } , (11)
where ε φ > 0 is a small constant introduced to ensure the feasibility of implementing the direct estimation for the initial transients of (5).

In the sequel, we consider various filtering techniques applied to the direct estimate [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF] and compare them with estimators used in the element-wise linear regression model [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF]. The results illustrate that fixed-time direct estimation followed by filtering performs better than estimators with asymptotic convergence.

B. Used approaches

By abuse of notation, we use y k , θ, and θk to denote y k,1 , θ 1 , and θk,1 , resp., since we are interested only in θ1 .

For the element-wise linear regression [START_REF] Narendra | Persistent excitation in adaptive systems[END_REF], we consider the following two estimators:

• the gradient estimator given by

θgr k = θgr k-1 + γ 0 1 + γ 0 φ 2 k y k -φ k θgr k-1 , (12)
where γ 0 > 0 is the tuning coefficient; • the least-squares estimator with forgetting given by

p k = p k-1 λ ls + φ 2 k p k-1 , θls k = θls k-1 + p k φ k y k -φ k θls k-1 , (13)
with p 0 > 0 and λ ls ∈ (0, 1) is the forgetting factor. For the direct estimate, we consider the following filtering methods applied to θdir k given in ( 11): • the low-pass filtering given by θlow

k = θlow k-1 + (1 -) θdir k , (14)
where ∈ (0, 1) is the tuning coefficient; • the moving-average filtering given by

θma k = 1 N N -1 i=0 θdir k-i , (15)
where N = min {k, N ma }, and N ma is the length of the averaging window; • the median filter given by

θmed k = median θdir k-N +1 , . . . , θdir k , (16)
where N = min {k, N med }, and N med is the number of samples used in the median computation, • the Kalman filter given by

θkal k = θkal k-1 + P k-1 + Q P k-1 + Q + R k y k φ k -θkal k-1 , P k = R k P k-1 + Q + R k (P k-1 + Q) , R k = R 0 max φ 2 k , ε 2 φ , (17)
where Q ≥ 0 and R 0 > 0 are the process and observation noise covariances. When Q = 0, the case of time-varying parameters can be treated.

C. Simulation results

Consider the signal [START_REF] Bin | Generalized recursive least squares: Stability, robustness, and excitation[END_REF], where A 0 = -3, A = 5, ψ = π 3 , and T = 117. The measurement noise v is chosen as a uniform random variable, v ∼ U (-1, 1). Following Remark 2, the value λ in ( 5) is chosen as λ = T -1

T and the initial conditions in (5) are zero. Then the lower bound α defined in Lemma 1 is α = 0.0063 and the asymptotic lower bound of φ is 0.025.

For the direct estimation, the value ε φ in ( 11) is chosen as ε φ = 10 -8 . The initial values in the estimators [START_REF] Efimov | Robust stability under relaxed persistent excitation conditions[END_REF], [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] and filters ( 14), [START_REF] Schiffer | Online estimation of power system inertia using dynamic regressor extension and mixing[END_REF] are zero; the initial values of p 0 in [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF] and P 0 in ( 17) equal to one. The parameters of the estimators

, ( 13) and filters ( 14), ( 15), ( 17) are chosen to provide equal transient time in the ideal noise-free scenario, where the transient time is measured as the step number when the estimation errors decays to 5% of the initial value. The values are γ 0 = 10 3 , λ ls = e -3 28 , = e -3 73 , N ma = 76, and Q = 2.5 • 10 -6 , R 0 = 1.75 • 10 -4 . For the median filter [START_REF] Vediakova | Finite time frequency estimation for multi-sinusoidal signals[END_REF], the number of used samples is equal to the length of the averaging window in [START_REF] Shao | Data-driven immersion and invariance adaptive attitude control for rigid bodies with double-level state constraints[END_REF], N med = N ma .

Fig. 1 depicts the estimation error transients of the considered methods in the ideal noise-free scenario. Denote the step number when the value φ k overcomes the threshold ε φ as T fix , where T fix = 14 in the considered example. The direct estimate [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF] converges in the fixed time T fix . The median filter also has the fixed-time convergence property with the convergence time min 2T fix , T fix + N med 2 . The movingaverage filter converges in the fixed time T fix + N ma -1, and other estimates converge asymptotically.

Fig. 2 depicts the estimation error transients of the considered methods for noisy measurements. The direct estimate exhibits sensitivity to the noise in the initial transient when the value of φ is small; this sensitivity can be reduced by increasing ε φ . Nevertheless, all considered methods perform well in alleviating the noise, where filtering of the direct estimate converges to a vicinity of the true value faster than the estimators applied to the linear regression model.

The steady-state performance of the considered methods is summarized in Table I, where the mean squared error (MSE) and the mean absolute error (MAE) are given; these values are computed over 10 6 samples after the transients.

To summarize, under equivalent tuning, the direct estimate (8) provides fixed-time convergence, and followed by a filter Estimation error transients for noised measurements: the direct estimation θdir [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF], the estimator θgr (12), the least-squares estimator θls (13), the low-pass filtering θlow (14), the moving-average filtering θma (15), the median filtering θmed (16), the Kalman filter θkal [START_REF] Schiffer | Online estimation of power system inertia using dynamic regressor extension and mixing[END_REF].

provides parameter estimation with an improvement in the noise attenuation, and it should be considered as a reasonable alternative to the standard parameter estimation approaches applied to the element-wise scalar linear regression (4).

VI. CONCLUSION

This work has established that the DREM procedure provides a direct parameter estimation by transforming the initial vector linear regression into one with unknown parameters measured with noise. We have also proved that Kreisselmeier's filters preserve the initial excitation of the regressor in the DREM method. Noise filters can be effectively coupled with DREM. In the future, we shall investigate other estimation algorithms with accelerated convergence rates in discrete-time, especially when it is impossible to obtain a sign-definite regressor after DREM or to improve robustness to measurement noise.

Fig. 1 .

 1 Fig.1. Estimation error transients in the noise-free case: the direct estimation θdir[START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF], the gradient estimator θgr (12), the least-squares estimator θls (13), the low-pass filtering θlow (14), the moving-average filtering θma (15), the median filtering θmed (16), the Kalman filter θkal[START_REF] Schiffer | Online estimation of power system inertia using dynamic regressor extension and mixing[END_REF]. The dashed line corresponds to 5% of the initial error value; the estimators are tuned to have approximately equal transient time. For the median filter, the number of samples is the same as for the moving average.

  Fig. 2.Estimation error transients for noised measurements: the direct estimation θdir[START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF], the estimator θgr (12), the least-squares estimator θls (13), the low-pass filtering θlow (14), the moving-average filtering θma[START_REF] Shao | Data-driven immersion and invariance adaptive attitude control for rigid bodies with double-level state constraints[END_REF], the median filtering θmed (16), the Kalman filter θkal[START_REF] Schiffer | Online estimation of power system inertia using dynamic regressor extension and mixing[END_REF].

TABLE I MSE

 I AND MAE OF THE SELECTED METHODS.

		θgr θls θlow θma θmed θkal
	MSE • 10 3	1.4 1.3 1.2 1.2	1.2	1.1
	MAE • 10 2	3.0 3.0 2.7 2.7	2.7	2.6

supported by the China 111 Project D17019.


P Milman

A Keller

E Charron

O Atabek

Molecular orientation entanglement and temporal Bell-type inequalities

Keywords: numbers: 03.65.Ud, 03.67.-a, 33.20.Sn

We detail and extend the results of [Milman et al. , Phys. Rev. Lett. 99, 130405 (2007)] on Bell-type inequalities based on correlations between measurements of continuous observables performed on trapped molecular systems. We show that, in general, when an observable has a continuous spectrum which is bounded, one is able to construct non-locality tests sharing common properties with those for two-level systems. The specific observable studied here is molecular spatial orientation, and it can be experimentally measured for single molecules, as required in our protocol. We also provide some useful general properties of the derived inequalities and study their robustness to noise. Finally, we detail possible experimental scenarii and analyse the role played by different experimental parameters.

I. INTRODUCTION

Quantum mechanics allows for the existence of states without any classical correspondence. Examples of such states are entangled states, that may appear when describing the total state of a many particle system, or when describing different degrees of freedom of a single particle. Entangled states cause debate because they can present properties which contradict our classical intuition. In addition, some of these properties can be use to increase the security of quantum communication and the efficiency of algorithmic protocols when compared with classical techniques. This is why so much attention has been payed to problems belonging to the foundations of quantum mechanics, and in particular to quantum entanglement, which is believed to be one of its main traits [2]. One of the most important discussions on entanglement and its conflicts with classical physics concerns the realism and (non-)locality of quantum physics. As pointed out by Einstein, Podolski and Rosen in the so called EPR [1] paradox, entangled states are closely connected to apparent contradictions between quantum mechanics and fundamental physical assumptions. Even if experimental evidence has been obtained to support quantum mechanics against the EPR criticism [5], it is still a matter of debate if such experiments were realised in the ideal conditions, closing all the loopholes, so as a definitive conclusion can be reached. At the same time, from the fundamental point of view, identifying precisely which essential quantum properties are involved in local realism violation is still an open problem. This is why extending local realism tests to different physical systems and different physical scenarii still presents so much interest.

A common property of non-local states is entanglement. In spite of its importance and consequences in different physical contexts, it remains an open question to determine whether a general quantum system is entangled or not and to quantify the degree if entanglement of a given state. The problem has been solved for some particular cases, as for a general bipartite system of dimension H 2 ⊗ H 2 and H 2 ⊗ H 3 [7], where H d is the one particle Hilbert space of dimension d. In such cases, necessary and sufficient conditions for telling whether a given state is entangled or not exists. In other particular cases, or subspaces, one can also find necessary and sufficient conditions. For instance, entanglement in bipartite pure states can always be recognised and quantified irrespectively of each parties' dimension. However, when dealing with arbitrary states, including the more realistic mixed ones, only necessary conditions for separability (non-entanglement) can be provided. A notion that will be useful in the following of this paper is the one of entanglement witnesses, defined as an operator Ŵ for which the expectation value Ŵ ≤ S for all separable states. This ensures that the state is entangled if Ŵ > S. On the contrary, the case Ŵ ≤ S [6] is not conclusive. Examples of entanglement witnesses that are also useful for fundamental tests of quantum mechanics are Bell-type inequalities [12], which are the main scope of this paper.

Bell-type inequalities are composed of combinations of observables that, when measured, allow for setting a board between crucial aspects of quantum theory and the classical one. They were formulated by J. S. Bell [3] as a late reply to the EPR criticism to quantum physics [1].

Bell inequalities aim at answering the question: is quantum mechanics a local and realistic theory? In order to do so, they combine correlations between local measurements realised in a multiparty system. Since their first formulation, several other inequalities have been proposed, some studying the same type of problem as Bell, other generalising locality and realism tests to many observers and many possible observables. There are a number of Bell type inequalities, and classifying all of them is a work by itself [8].

The inequalities derived here follow the original formulation of Clauser, Horne, Shimony and Holt (CHSH) [4], and deal with the following scenario: two observers A and B perform local measurements on a bipartite system. Each observer can chose among two experimental set-ups (a and a ′ for A and b and b ′ for B). Each measurement performed by A and B can give only two outcomes. At its origin, CHSH inequalities have been formulated for a pair of spin 1/2 particles or equivalent two-level systems. In the framework of local hidden variable (LHV) theories the measurement outcomes correlation statistics which must fulfil:

| σ a σ b + σ a σ b ′ + σ a ′ σ b -σ a ′ σ b ′ | ≤ 2, (1)
where σ α is the Pauli matrix in the α direction. Briefly, in a LHV theory, one assumes that measurements performed by each observer are independent and their outcomes have a probability distribution which is a product of independent probabilities for each subsystem. Such probabilities can also depend on some random local variable. Details are discussed in many works, as [26], for instance. It can be shown that some entangled states can violate (1), and this experimental violation was observed using photon pairs entangled in polarization [5].

In this case, directions a, a ′ , b and b ′ refer to different orientations of polarizers placed before the detectors, determining the direction of the Pauli matrix that is measured. It was shown by Cirelson [29] that the maximum value of ( 1) is 2 √ 2, and it is easy to verify that this maximal violation can be obtained with maximally entangled states.

Inequalities as (1) have proved to hold for two-level systems or equivalent ones. By "equivalent ones", we also include Bell type inequalities involving a continuum of possible measurement outcomes that are dichotomised and transformed into a two outcomes measurement setup. Dichotomization works as follows: one splits in two classes the range of possible measurement outcomes. All results obtained lying in one of the classes is identified to a given value (+ or -) and the results obtained in the complementary space are associated to the opposite sign. Some examples of systems where this can be sucesfully done are optical fields [9,10]. An interesting problem is to derive Bell-type inequalities for continuous variables without appealing to dichotomisation. Cavalcanti et al. found a way out by using second moment correlations instead of first moment ones, as done in CHSH-type inequalities as (1) [20]. Here, we deal with this problem in a different way: by using bounded observables, one can still use CHSH-type inequalities to detect non-local properties. In this case, a Cirelson bound depending on the norm of the measured observable can also be derived, even if, at least for the specific case treated in this paper, we have not shown yet that it can be attained. An interesting property of the inequality discussed in this work is that it can be used not only for infinite dimensional systems, but also in N levels systems, where N is a finite number. In this case, the numerical value of the bound splitting between a local theory and non-local one depends on the maximal eigenvalue of the measured observable.

Up to now, several studies have been made on Bell-type inequalities in different contexts. The general conclusion is that the subject still presents several open ques-tions and no general theory is available. In particular, a number of intriguing features coming out from such studies somewhat contradict our acquired quantum "intuition": in [25] it is numerically shown that bipartite multidimensional states may violate locality tests more than two qubits. For two qubits, it has been proven that a maximal violation exists, and it is given by the Cirelson bound [29], as will be discussed hereafter. Acin, Gill and Gisin showed, some years later, also using numerical tools, that the maximal violation for pure bipartite multidimensional systems is not obtained for maximally entangled sates [24]. General rules relating local realism violation and entanglement have not yet been found, even if it has been proven that all non-local states are entangled in some way. This fact can be easily understood with the help of entanglement witnesses. It has been shown in [12] that Bell-type inequalities are entanglement witnesses, and all non-local states are entangled.

The inequalities studied in this paper are based on molecular spatial orientation correlations measurements [22] instead of spin-like observables. However, the same type of idea can be generalised to other continuous bounded observable. We have shown that they can be implemented using time delayed measurements of correlations between the spatial orientations of two molecules. As in usual Bell inequalities scenarios, the proposed nonlocality tests rely on measurements performed independently on each molecule by observers placed far apart enough, so that no communication between them is possible during the realization of the protocol. We have shown that the proposed inequalities can be violated by a set of entangled states. An interesting point is that the inequalities derived in the present paper can also be used as entanglement witnesses, as shown, for general CHSHtye inequalities in [6]. In this situation, one can loosen measurement conditions, since we are interested only in detecting a particular quantum correlation, and not a fundamental aspect of quantum physics.

From the experimental point of view, motivations for this work are the recent advances in single molecule manipulation and detection with entanglement creation [13] and quantum information purposes [14,16,17,18]. In particular, trapped cold polar molecules are promissing candidates for quantum information processing based on the manipulation of their rotational levels [27]. Rotational states, which are also involved in the Bell-type inequalities studied in the present paper, are relatively long lived, allowing for short quantum gate implementation times: one can hope to perform about 10 4 gate operations before decoherence takes place. This excellent performance when compared to cold collision based quantum gates [19] is due to the strength of molecular interactions, based on dipolar forces.

The present paper is organised as follows: in section II we discuss some general properties of Bell-type inequalities that are useful in the context of the non-locality tests we propose. In section III we explicit the inequalities based on molecular orientation and show how it can be used for non-locality tests. We exploit its performance and study some entangled states that violate it. We then discuss some constraints involved in an eventual physical implementation of our ideas in section IV, ending up with a concluding note in section V

II. GENERAL PROPERTIES

We describe now some general properties of the inequalities studied in the present paper. Our inequalities involve four observables that can be combined as follows:

B = O 1 (φ 1 ) ⊗ O 2 (φ 2 ) + O 1 (φ 1 ) ⊗ O 2 (φ ′ 2 ) + (2) O 1 (φ ′ 1 ) ⊗ O 2 (φ 2 ) -O 1 (φ ′ 1 ) ⊗ O 2 (φ ′ 2 ).
The operators appearing in the equation above are defined as follows: operators O i (φ i ), i = 1, 2 are local observables chosen by observers 1 and 2. The variables φ i also depend on local properties only, and

O i (φ i ) = U (φ i )O(0)U † (φ i ),
| B | ≤ S, (3)
where S is a number, representing the maximal allowed value for a local theory to hold whenever measurements of correlations between observables are compared as in (2).

The basic assumption to derive (3) is that correlations can be described by probability distributions which are independent for each party (1 and 2) and depend only on local parameters and that the spectrum of observables O i (φ i ) is bounded. The same inequality can be derived by assuming that the average of B is taken with respect to a separable (non-entangled) states.

For simplifying reasons, we focus on the specific case in which the norm of O i (φ i ) is the same for each subsystem and is equal to λ max . In this case, the numerical value of S is S = 2λ 2 max . Quantum mechanics can violate inequality (3), but in order to do so, observables O i and the state considered should be judiciously chosen. In order to check whether an inequality of the type of (3) allows for a non-locality test, one should maximize the left side of (3) for an arbitrary bipartite quantum state. If the maximum obtained value is greater than S, all states violating (3) are nonlocal.

Before proceeding on testing the power of inequalities of the type of (3) for non-locality tests using the proposed molecular orientation-related observables, we demonstrate some simple general properties of (2) which are independent of the observables O i . Such properties are useful since they help to simplify the numerical optimization while giving some physical insight. We show that, thanks to these properties, the number of degrees of freedom to be considered in order to evaluate the maximal value of the violation is decreased. We start by decomposing operators O i (φ i ) in terms of unitary transformations. Using the group property U (φ i + φ j ) = U (φ i )U (φ j ) we can write:

B = (U 1 (φ 1 ) ⊗ U 2 (φ 2 ))(O 1 (0) ⊗ O 2 (0)) + (4) O 1 (0) ⊗ O 2 (φ ′ 2 -φ 2 ) + O 1 (φ ′ 1 -φ 1 ) ⊗ O 2 (0) - O 1 (φ ′ 1 -φ 1 ) ⊗ O 2 (φ ′ 2 -φ 2 )(U † 1 (φ 1 ) ⊗ U † 2 (φ 2 )),
or, more succinctly,

B = (U 1 (φ 1 ) ⊗ U 2 (φ 2 )) (5)
B(0, 0, φ 1 -φ ′ 1 , φ 2 -φ ′ 2 )(U † 1 (φ 1 ) ⊗ U † 2 (φ 2 )), with B(0, 0, φ 1 -φ ′ 1 , φ 2 -φ ′ 2 ) ≡ O 1 (0) ⊗ O 2 (0) + (6) O 1 (0) ⊗ O 2 (φ ′ 2 -φ 2 ) + O 1 (φ ′ 1 -φ 1 ) ⊗ O 2 (0) - O 1 (φ ′ 1 -φ 1 ) ⊗ O 2 (φ ′ 2 -φ 2 ),
The interest of such decomposition is that operator

B(0, 0, φ 1 -φ ′ 1 , φ 2 -φ ′ 2
) depends only on two variables and possesses the same spectrum as B. Thus, B(0, 0,

φ 1 - φ ′ 1 , φ 2 -φ ′ 2
) and B share the same entanglement witnessing properties. Of course, B and B(0, 0,

φ 1 -φ ′ 1 , φ 2 -φ ′ 2 )
do not detect the same entangled states. Nevertheless, they are connected by a local unitary transformation.

The first question we wish to answer concerns the usefulness of orientation correlation measurement for nonlocality tests. In order to answer that, it is enough to determine the norm of B(0, 0,

φ 1 -φ ′ 1 , φ 2 -φ ′ 2
). In the next section, we will study a specific example of an operator O 1,2 (φ 1,2 ), and the physical meaning of the previous results will be made explicit.

III. ORIENTATION BASED NON-LOCALITY TEST A. Molecular orientation and correlation

We introduce now an essential ingredient for the inequalities studied in the present paper, which is molecular spatial orientation. Molecular orientation is an observable, and its classical correspondent is the spatial orientation of the molecular inter-atomic axis with respect to some reference frame. By supposing that the molecules are addressed and manipulated by a linearly polarized laser field, we can define the laser's polarization axis as z, and take it as a reference for molecular orientation. In this case, the orientation of a given molecular state can be defined as the average value of operator cos θ, where θ is the angle relative to the z axis. Note that here, cos( θ) is taken as an operator, in perfect analogy to the position operator and related functions. Orientation as defined above can be experimentally measured, as discussed below.

Since we are interested on probing properties related to a two party system, our system is composed by two molecules. We suppose that they behave like rigid rotors that can freely evolve. Their state depends thus on the time t. The Hamiltonian describing each individual molecule's free evolution is

H i = J 2 i / 2
, where i = 1, 2. It is expressed in units of the rotational energy. J i is the angular momentum operator and therefore the associated evolution operator is given by U i (t) = e -iπHit , where time t is written in units of the rotational period. U i (t) is therefore time-periodic with period 1.

For each molecule of the bipartite set, the orientation at time t is defined as the expectation value of the Ôi (t

) = U -1 2 (t) ⊗ U -1 1 (t) cos( θi )U 1 (t) ⊗ U 2 (t) operator, Ôi (t) = ψ o | Ôi (t)|ψ o
, where |ψ o is the initial state of the system. Orientation correlations between two particles are given by the average value of

C(t 1 , t 2 ) = O 1 (t 1 ) ⊗ O 2 (t 2 ) ,
and this quantity can be measured at different times t 1 and t 2 for each molecule. Operator cos θ is useful for entanglement detection since it "mixes" different values of j, without affecting their projection m. Previous works have considered correlations between different values of the projection m of a given (fixed) value of j [11,31].

With an arbitrary accuracy, each molecule's state (subscripts have been omitted) can be considered to reside in a finite dimensional Hilbert space H generated by the basis set {|j, m ; 0 ≤ j ≤ j max , |m| ≤ j}, where |j, m are the eigenstates of J 2 and J z . Note that H has dimension (j max + 1) 2 . The corresponding wavefunctions are θ, ϕ|j, m = Y jm (θ, ϕ), the spherical harmonics. In the finite space H (jmax) , the cos θ operator is characterized by a discrete, non degenerate spectrum of eigenvalues λ n , with corresponding eigenvectors |λ n , also called orientation eigenstates. The two maximally oriented states |+ and |-are the two eigenstates corresponding to the extreme eigenvalues ±λ max , where λ max ≡ Max n (λ n ). In the particular case of j max = 1 and m = 0, maximally oriented states can be written in the basis of the angular momentum eigenstates |0, 0 and |1, 0 as

|+ = 1/2(|1, 0 +|0, 0 ) and |-= 1/2(|1, 0 -|0, 0 ).
In this particular case, cos θ = 1/3σ x and we recover the results of a two-level system. In this context, the free evolution operator U i (t) changes the orientation of a state, since orientation eigenstates are not eigenstates of the free Hamiltonian. Time evolution creates superpositions of orientation eigenstates, exactly as it happens when one projects photon's polarization with polarizers.

B. The inequalities

We can now combine all the ingredients to build the orientation based Bell-type inequalities. We start with a system composed by two molecules, and measurements independently performed in each one of them should be combined in order to tell whether the total molecular state violates or not local realism. The two molecule bipartite state |ψ o has been created at a given time t o after which it freely evolves. We suppose that, after t o , there is no interaction between molecules. The total molecular state is thus given by the wavefunction ψ(θ 1 , θ 2 , ϕ 1 , ϕ 2 , t) ≡ θ 1 , θ 2 , ϕ 1 , ϕ 2 |ψ(t) , where θ i and ϕ i denote here the polar and azimuthal spherical coordinates in the laboratory frame.

This state is the one whose non-local properties are to be checked. We can use it to compute the average values of Ĉ(t 1 , t 2 ) and combine such correlations taken at different times in a way analog to Eq.( 1) and ( 2), defining the operator

B 1 (t 1 , t 2 , t ′ 1 , t ′ 2 ) ≡ (7) C(t 1 , t 2 ) + C(t 1 , t ′ 2 ) + C(t ′ 1 , t 2 ) -C(t ′ 1 , t ′ 2 ).
For a local theory (LT), it obeys an inequality similar to Eq. ( 3):

| B 1 LT | ≤ 2(λ max ) 2 ; ∀(t i , t ′ i ) ∈ R 2 . ( 8
)
Without loss of generality, we have assumed that each particle state resides in the same finite dimensional Hilbert space H (jmax) . Notice that, in Equation ( 7), time plays the role of polarisers in Bell inequalities based in the photonic polarisation. Other CHSH inequalities using the time evolution instead of polarizers were studied in the literature in very different contexts: in [26,32,33] they allow the detection of entanglement between products of decaying mesons. In [34,35], they reveal quantum properties of single particles. We note that ( 8) is valid for all possible values of j max , and that it can, in particular, be extended to the limit j max → +∞, in which case the spectrum of cos θ forms a continuum. An interesting characteristic of the separability threshold ( 8) is its dependence on λ max .

We will see in the next subsection how the general properties of CHSH-type inequalities can be used in the specific case discussed in this paper to show that the studied inequalities can be violated.

C. Reference Frame and Temporal Origin

A straightforward application of the results derived in section III consists of showing that, in order to study the spectrum of B 1 , we can, without loss of generality, numerically diagonalize it in the specific case of t ′ 1 = t ′ 2 = 0. This result is due to the fact that both operators are related by a local unitary transformation. Using the notation of Section II, we have here that

t 1 = φ ′ A -φ A and t 2 = φ ′ B -φ B .
Another interesting application of the general results of Section II consists of showing that the inequalities discussed here allow for local realism violation even in the case where observers A and B have different time origins. The same happens for the spatial reference frame: violation is independent of any previous agreement between observers. However, different temporal origins and reference frames correspond to different non-local detected states, that are related to each other by local unitary transformations.

We start by discussing in more details the time origin chosen by both observers. Usually, the orientation Belltype inequalities depend on four times of measurements, as defined in Eq. (7). However, as pointed out previously, different times correspond to the application of different unitary transformations. We can thus apply the results of section III, identifying the general operator Ô to the operator cos θ. This leads to the inequalities:

B 1 (t 1 , t 2 , t ′ 1 , t ′ 2 ) = B 1 (0, 0, t ′ 1 -t 1 , t ′ 2 -t 2 ) ≤ 2λ 2 max , (9)
The first identity shows that states maximally violating the Bell-type inequality for t 1 , t 2 , t ′ 1 , t ′ 2 , defined as |s max (t 1 , t 2 , t ′ 1 , t ′ 2 ) can be obtained by the one maximally violating it for 0, 0, t ′ 1 -t 1 , t ′ 2 -t 2 , that we will call |s max (0, 0, t ′ 1 -t 1 , t ′ 2 -t 2 ) by the application of the transformation U (t 1 ) ⊗ U (t 2 )|s max (0, 0, t ′ 1 -t 1 , t ′ 2 -t 2 ) . Temporal uncertainties of τ 1 , τ 2 for each one of the molecules can always be translated as the application of the unitary operator U (τ 1 ) ⊗ U (τ 2 ), so that their only effect is to change the eigenstates of operators B i by the same transformation. Violation can thus still be observed, and the subspace of states violating local realism are obtained by a simple unitary transformation on the original subspace.

The same type of argument can be used for uncertainties of the reference frame for each observer. All reference frames are connected by local unitary transformations describing rotations about some direction of space, and results for different references frames are connected by these same unitary transformations.

We apply these results to simplify the investigation of possible violations of inequality (8).

D. Results

For a given value of j max , we have numerically diagonalized operator B 1 (0, 0, t 1 , t 2 ), and obtained for each (t 1 , t 2 ), its highest eigenvalue β 1 (t 1 , t 2 ), which gives the maximal value of B 1 (maximal violation of ( 8)). This quantity depends on the dimensionality of the system, and we compare the amplitude of the violation when different values of j max are used by defining the relative violation

b 1 (t 1 , t 2 ) ≡ β 1 (t 1 , t 2 ) -2(λ max ) 2 2(λ max ) 2 . ( 10
)
Results for different values of j max and m = 0 are shown in Figure 1. We can see that the proposed inequalities are violated for a significant region of parameters t 1 and t 2 . Figure 1 also calls one's attention because of its symmetries. The central symmetry with respect to the point t 1 = t 2 = 0.5 corresponds to the time reversal symmetry. One can also easily understand the mirror symmetry about the t 1 = t 2 line with the help of the particle exchange symmetry of operator B 1 .

We have shown in [22] that Eq. ( 8) can be violated by a number of pure states, and numerically studied the relative violation (10) with increasing dimension. The results, shown in Figure (2) were obtained in a particular case, where both molecules had a vanishing angular momentum z axis projection (m = 0 for both molecules). The effects of considering different values of m will be discussed in the following. We focus here in the m = 0 case in order to discuss the behaviour of violation of (8) with respect to the dimensionality of the system. We see that the maximal relative violation b 1 (t 1 , t 2 ) decreases with the dimensionality of the system for low dimensions and then starts smoothly growing starting from j max = 4 (blue bars, Figure (2)). Up to now, we have not found an asymtoptical numerical behaviour for j max → ∞. We have shown that the maximal violation is bounded by 3. However, it is still unknown if this value can be reached. Since the exact behaviour of the violation of inequality (8) in infinite dimension is still not known, one pertinent question is whether the proposed inequality is still violated in the limit j max → ∞ for physically acceptable states. We can see that it is indeed the case by calculating the violation relative to the classical bound obtained for an infinite dimensional system. In this case, the classical threshold separating local theory from nonlocal ones is 2, since λ max = 1. Figure (2) shows the results of such violation (pink bars). We see that states violating the infinite dimensional classical threshold can be found when one considers subspaces with dimensions higher than N = j max + 1 ≥ 5 for m = 0. In this subspace, we can see that the violation relative to the infinite dimensional threshold is still small. However, by increasing the size of the subspace and going to j max = 10, we can see that this relative violation increases, and it is not negligible when compared to the two dimensional case, for instance. Notice that the relative violation with respect to the infinite dimensional subspace and the one relative to the restricted subspace approach with increasing dimension, as one should expect. Violating the inequalities (8) in the infinite dimensional limit with entangled states of low dimension is a surprising result that proves that (8) can be violated for all possible values of λ max .

A natural question is what states maximally violate (8). Since we are dealing here with correlations in orientation, one could expect that maximally oriented entangled states of the form 1 √ 2 (|λ max , λ max ± |-λ max , -λ max ) are those that maximally violate them. However, this is not the case, and such state, except for the case j max = 1 and m = 0 do not violate (8) at all. Nevertheless, states maximally violating ( 8) have most of their population in highly oriented states. For j max = 5, states |λ max , λ max and |-λ max , -λ max carry, each one of them, 36% of the population. In addition, states maximally violating (8) are not maximally entangled (except in the case j max = 1, a result which is not surprising and has been observed for other Bell-type inequalities, as in ??, for example. Using the maximally violating state, we have computed the reduced density matrix entropy S = -T r[ρ i log ρ i ], where ρ i is the reduced density matrix with respect to one of the two entangled molecules. In order to compare S for different values of j max , we have normalised it with respect to log (j max + 1), which is the maximal value of the entropy in a subspace of dimension j max + 1. The reduced density matrix entropy is a measure of entanglement for pure states, as the ones 3), together with a plot of log 2/ log (j max + 1). We see that the two plots almost completely match, showing that maximally violating states are very close to entangled states involving only two orthogonal states of each molecule. We have checked that the two states involved are the maximally oriented states. The results presented above were obtained for the specific case of m = 0 for both molecules. Allowing m to take a value different from m = 0 does not bring qualitative changes to our results: the classical threshold will still be given by the case m = 0, since it corresponds to the maximal value of B 1 for a local state. However, the dimension of the subspace considered depends on m. For a given fixed m, it is given by j max -|m| + 1. The value of the maximal orientation is thus determined not only by the dimension of the subspace, but also on the specific value of m. As a consequence, the maximal value of B 1 is determined by the same parameters. For instance, if the considered state is a superposition state of different m's, each one of this subspaces will lead to different contributions when computing B 1 . In particular, those for which j max = |m| give a null contribution, and the maximal possible contribution decreases as |m| increases for each molecule. Physically, this is related to the fact that states with high values of the angular momentum projection |m| are less oriented than those for which j ≫ |m|. As a conclusion, by considering different m's, it is still possible to violate (8), even if fulfilling the required conditions for it becomes harder.

E. Non-pure states and the effect of noise

We now formalize the conditions a non-pure state should satisfy in order to violate Eq. ( 8). This is useful to estimate the effect of noise in our system. A non-pure bipartite state is described by the density matrix ρ 1,2 , where the subscripts 1 and 2 refer to each one of the observers. The average values of ( 7) can be obtained from ρ 1,2 by using

C(t 1 , t 2 ) = O 1 (t 1 ) ⊗ O 2 (t 2 ) = Tr[ρ 1,2 O 1 (t 1 ) ⊗ O 2 (t 2 )
]. Defining the B 1 eigenstates, by {|s i }, with i ranging form 0 to (j max +1) 2 if one assumes that each particle's subspace have the same dimension and m = 0, it is clear that the density matrix ρ 1,2 can be expressed in this basis. By doing so, we have that

Tr[B 1 ρ 1,2 ] = N 2 i p i s i , (11)
where s i is the eigenvalue associated to the eigenstate |s i of B 1 , p i = s i |ρ 1,2 |s i the statistical weight of each one of such eigenstates and N = j max + 1 is the dimensionality of each molecule's subspace. This means, that for a local realistic theory, one should obey

N 2 i=1 p i s i ≤ 2λ 2 max . (12)
An example of non-pure state is given by

ρ 1,2 = P N 1 1 N 2 + (1 -P N )|s max s max |, (13)
where 1 1 is the N 2 × N 2 identity matrix, |s max is the eigenstate of B 1 with maximal eigenvalue s max and P N is the probability of the state to be in a complete mixture. This type of state has been studied in [25] and can illustrate the presence of noise in the preparation of a state |s max maximally violating a Bell-type inequality. They defined that the robustness of a Bell-type test with respect to noise is measured by the maximal allowed value of P N still leading to locality violation. In the notation of Eq. ( 12), s max |ρ|s max = p max = P N /N 2 + (1 -P N ) and s i = s max |ρ|s i = s max = p i = P N /N 2 . By calculating the expectation value of B 1 using (13) one gets that local realism should obey:

P N N 2 i=1 s i N 2 + (1 -P N )s max ≤ 2λ 2 max . (14)
Since B 1 is traceless, i s i = 0, so the inequality is violated for P N < 1 -2λ 2 max /s max . A plot of P N as a function of the considered subspace is given in Figure (4). Notice that, when a two-level system is considered, the known result of P 2 = 1 -1 √ 2 is recovered. Contrary to other Bell-type inequalities for which P N optimised [25], the maximal allowed value of noise in our system decreases with dimensionality up to j max = 5 and then starts increasing again, but doesn't change significantly in the range of dimensions that we have calculated. This result shows that the non-local properties of the maximally violating state are quite robust with respect to noise. We have shown that the operator defined by Eq. ( 7) allows not only for the realization of Bell-type tests in finite angular momentum subspaces, but also when the size of the subspace is not a priori known. This proves that, even in the case where we consider an infinite dimensional space, our inequalities can be violated for some states. However, as shown previously, the contrast of the maximal violation depends on the value of j max . With dichotomizing procedure, a high dimensional system is transformed into an effective two level one by splitting in two the set of measurement outcomes, and the contrast is kept constant, equal to the one for a two-level system, irrespectively of j max . In our case, we can dichotomise as follows: states |ϕ , for which cos θ ϕ > 0, are said to be positively oriented, while those for which cos θ ϕ ≤ 0 are considered as negatively oriented. We define the associated projectors: Π ± = λ± |λ ± λ ± | which project states in the subspace of positive (negative) orientation. λ ± are the positive (negative) cos θ eigenvalues in the given finite dimensional subspace and |λ ± are the corresponding eigenstates. The measured observable for each molecule i is then Π i = Π + -Π -. For a given singlemolecule state |ϕ = λ+ c λ+ |λ + + λ-c λ-|λ -, Π ϕ can take any value in the interval [+1, -1]. The total two molecule observable is Π = Π 1 ⊗ Π 2 . We refer, as before, to two molecule correlation measurements realized at two different times, using Π(t

1 , t 2 ) = Π 1 (t 1 ) ⊗ Π 2 (t 2 ) where Π i (t i ) = U -1 i (t i )Π i U i (t i ).
In analogy with Eq. 7, we now define the operator

B 2 = Π(t 1 , t 2 ) + Π(t 1 , t ′ 2 ) + Π(t ′ 1 , t 2 ) -Π(t ′ 1 , t ′ 2 ). ( 15
)
Since Π(t i , t j ) 2 = 1, one can show that the highest value B 2 can reach is given by the Cirel'son bound 2 √ 2 [29]. Also, for a local theory, we have 10)). The interest of dichotomisation is that the maximal value of the relative violation is always given by b 2 (t 1 , t 2 ) = √ 2 -1, independently of the dimension of the subspace. Fig. 5 shows the maximum value β 2 (t 1 , t 2 ) of B 2 (t 1 , t 2 ) as a function of t 1 and t 2 for j max = 5, while for j max = 1 we obtain trivially the same result as with b 1 (t 1 , t 2 ) (Fig. 1.a right z-axis). We see in Fig. 5 that the dichotomisation procedure not only keeps the contrast of the maximal violation constant, but also allows for high violation for a wider range of values of t 1 and t 2 .

| B 2 LT | ≤ 2; ∀(t 1 , t 2 ) ∈ R 2 . ( 16
)

IV. DISCUSSION ON POSSIBLE EXPERIMENTAL IMPLEMENTATIONS

Orientation entanglement between two molecules can be created, for instance, with the help of the dipolar interaction [14]. A possible experimental scenario consists of two trapped molecules that can be submitted to spatial displacements, as it is currently done with atoms [28]. By putting molecules close enough to each other, and with the help of laser manipulation, one can tailor rotational entangled states. In order to realise Belltype tests, molecules would have to be separated and measurements performed independently in each one of them. The proposed scenario is realistic and there has been rapid progress on cooling, trapping and manipulating polar molecules, especially those formed with two alcali atoms. We believe that such manipulations are going to be shown in laboratories in a near future. With respect to the tests we wish to perform in the present paper, there are some crucial points to analyse for the experimental implementation of the proposed Bell-type inequality test. One of them concerns trapping conditions. Traps should allow molecules to make spatial displacements so that they are sufficiently far apart and so that signaling of one observer's results to the other is not possible during at least one molecular rotational period. This condition is attained for an inter-molecular distance L > cT , where c is the speed of light and T is the molecular rotational period. To put some num-bers on it, the molecular period is usually of the order of 10 -12 s, giving a lower bound to L ≈ 10 -6 m. Even if at this distance dipolar interaction between molecules is not completely negligible, measurements on both molecules are performed during the same rotational period, which is much faster than the characteristic interaction time related to dipolar coupling at this distance (see [14], for example, for a detailed discussion on the relevant time scales). Optical traps seem promising candidates to trap and to displace molecules. Optical tweezers [36], optical lattices and optical conveyor belts [23] are some examples of dipole force based optical traps allowing for atomic trapping and coherence preserving displacements [23,28,37]. In the case of optical tweezers, highly focused beams are used to trap individual atoms [36], and we can, in principle, load two atoms, each one of them in an individual trap [38]. In optical conveyor belts, a stationary field made of two counter propagating beams is used to trap atoms, that can be displaced over distances of ≈ 10nm by changing the relative detuning between the laser beams [23]. Going from atoms to molecules in this type of experimental system could in principle be done by the usual photo association techniques, already demonstrated in the context of optical lattices [15], where diatomic molecules of two alcali atoms of the same specie have been produced. Once molecules are created and trapped, they interact with each other by dipole interaction and a non-local state may be created, as discussed in the previous section. Molecules are thus separated until dipole interaction is negligible, and measurements can be performed. One natural question concerns the effect of the trapping potential itself to the orientation of the molecular state. Optical traps are based on an applied non resonant electric field that interacts with rotational levels. We suppose that molecules in our setup are created in their ground electronic level. The optical trap is created by coupling non-resonantly this electronic level to the first excited one. The detuning between laser and the electronic transition is δ. In usual experiments on optical tweezers, for instance,, δ can vary vary in the range ≈ 10-10 4 GHz, providing photon emission rates from the excited electronic state in the range of 0.1 -100 MHz. These figures are of the same order of magnitude of coherence lifetime for rotational levels, while the range of variation of δ is also of the same order of magnitude of the frequency difference between neighboring rotational levels. In order to estimate various effects of trapping lasers, we will put ourselves in a close to realistic configuration in which the rotational frequency is ≈ 10GHz and δ ≈ 10 4 GHz. Each rotational level has a different energy, making the effective detuning j dependent. This corresponds to adding a factor δ j = Bj(j + 1)/ to the electronic detuning δ, taken with respect to the rotational ground state. Because each rotational level has a detuning which depends on the value of j, there will be a phase difference between each rotational level due to the non resonant coupling to light. In the case of a harmonic trapping potential, this phase is given by e -i2Ω 2 /(δ+δj )t if only one electronic transition is considered. This corresponds to a local action on each molecule that does not play a significant role on non-locality tests, since it maps one state to another one with the same degree of entanglement. As shown before, by properly choosing the time where measurements are performed, such effect can be compensated. This means that the parameters for which maximal violation occurs may be modified. As it can be shown, if a given state violates the inequalities of the type of ( 8) and ( 16), a collection of states connected to it by local unitary transformations will also violate the same inequalities, even if for different parameters. Also, it is important to notice that such dephasing effects can be completely neglected depending on the precise circumstances under which the experiment is performed. This happens because dephasing occurs on a time scale much longer than the rotational period, so that if measurements are performed in this interval, it will not affect the expected results.

Finally, we address the question of orientation measurement.

A possibility is to use Coulomb explosion, a destructive techniques already employed for molecular orientation measurements [30].

After a quasi-instantaneous (when compared to the molecular rotational period), laser induced dissociative multiionisation, the molecular fragments are recorded at different directions of space, in the case of the orientation Bell-type inequalities, or at different hemispheres, in the case of the dichotomized inequalities. This technique is still experimentally challenging since it demands highly efficient single atom detectors. Usually this technique is employed for molecular ensembles, and the detection efficiency is less determinant than in our case. Another possibility is to detect the orientation of single molecules optically, as realized in [21]. Fluorescence intensity when a single molecule is excited by a linearly polarized laser beam that can change polarization in time, can reveal molecular orientation since the scalar product between the field polarization and the molecular dipole depends on its orientation. The phase of the fluorescence intensity with respect to the exciting laser beam when its polarization is turned in time, depends on the molecular orientation, as observed in [21].

V. CONCLUSION

We have extensively discussed some important properties of a recently proposed Bell-type inequality based on the measurement of a continuous, bounded, observable, which is molecular orientation. We have discussed some important properties of it, as its symmetries and its non-dependency on a specific choice of common temporal origin or reference frame. The role of noise in our system, that would transform pure states into statistical mixtures was also analysed, as well as how violation depends on the amount of noise. We have also discussed some important conditions that an eventual experimental set-up should satisfy and the influence of physical parameters as light forces and residual interaction between molecules on the proposed measurements. It seems, from our analysis, that once the necessary degree of advancement is attained by experimental set-ups, our proposal is realistic and should not present major difficulties. Our results open the perspective of entanglement detection and non-locality tests for high angular momentum systems in atomic and molecular physics.

One interesting aspect of the derived inequalities is the fact that they assume the simple form of CHSH inequalities and deal with continuous variables at the same time. This happens because the observable that is measured for each particle has a bounded spectrum, naturally limiting the locality threshold and the norm of the Bell operator. Notice that, in spite of having discussed here the specific case of molecular orientation, the same type of inequalities could be derived for other continuous bounded observables, easily measurable in other physical contexts.

1 FIG. 1 :

 11 FIG. 1: (Color online) Maximal value of B1 as a function of t1 and t2 in units of the rotational period. Left z-axis: highest eigenvalue β1. Right z-axis: relative violation b1 defined by Eq. (10). (a): jmax = 1, m = 0, (b): jmax = 5, m = 0.

FIG. 3 :

 3 FIG.3: Normalised entropy of the maximally violating state as a function of the subspace dimensionalty jmax (dots). Continuous (green) line represents log 2/ log (jmax + 1).

FIG. 4 :

 4 FIG. 4: Maximal value of PN for which our inequalities are violated as a function of jmax = N -1

  with U (φ i ) a one-parameter group of unitary operators with periodicity 2π. If the operator O

i (φ i ) is bounded, we can show that, under the assumption of local realism, (2) satisfies:

  Relative violation of the inequality (8) as a function of the dimension of the subspace for m = 0. Blue: relative violation with respect to the locality threshold when only the finite dimension subspace is considered. Red: relative violation when the infinite dimensional threshold of 2 is considered.

			1	2	3	4	5	6	7	8	9	10
			0.41
		0.4
				0.33
	Relative violation	0.2 0.3			0.3	0.3	0.29 0.13	0.29 0.16	0.3 0.19	0.3 0.23	0.31 0.24	0.31 0.25
		0.1				0.07
		0.0	1	2	3	4	5	6	7	8	9	10
					0.03		N 1
	FIG. 2:
				0.2
			0.5

  As in the case of B 1 , we can define the relative violation b 2 (t 1 , t 2 ) as a function of the maximal eigenvalueβ 2 (t 1 , t 2 ) of B 2 (t 1 , t 2 ) (see Equation (

							1
							0.45
		2.9					0.35
		2.7					0.25
	β 2 β 2	2.5					0.15	b 2
		2.3					0.05
		2.1					1
		0	0.2				0.6	0.8
			0.4 t 1	0.6	0.8	0.2	0.4	t 2
		FIG. 5: (Color online) Same as Fig. 1.b, but for B2

Acknowledgments

This work was partially supported by ANR (Agence Nationale de la Recherche), project ImageFemto number ANR-07-BLAN-0162-02


P Milman

A Keller

E Charron

O Atabek

Bell-type inequalities for cold heteronuclear molecules

Keywords: numbers: 03.65.Ud, 03.67.-a, 33.20.Sn

We introduce Bell-type inequalities allowing for non-locality and entanglement tests with two cold heteronuclear molecules. The proposed inequalities are based on correlations between each molecule spatial orientation, an observable which can be experimentally measured with present day technology. Orientation measurements are performed on each subsystem at different times. These times play the role of the polarizer angles in Bell tests realized with photons. We discuss the experimental implementations of the proposed tests, which could also be adapted to other high dimensional quantum angular momenta systems.

Entanglement and non-locality are two related peculiarities of quantum theory. Historical debates [1] and the advent of quantum information theory, to which entanglement is an essential ingredient, increased the interest paid to those concepts considered to be among the main traits of quantum mechanics [2]. The experimental realization of Bell [3] and Bell-type inequality tests [4] is of extreme importance to confirm the validity of quantum theory in a loophole free context [5]. Since entanglement and non-locality are related (non-locality implies entanglement but not all entangled states are non-local), such tests may equally serve as entanglement witnesses [6]. The interest of such approach also lies in the fact that for arbitrary high dimensional systems one does not dispose of necessary and sufficient conditions for entanglement characterization. However, detecting quantum properties of high dimensional systems is of clear interest in atomic, molecular and optical physics.

A number of experimental proposals are aiming to throw some light to these questions by means of Belltype tests performed in different continuous or multidimensional systems: it was shown recently that correlated atoms originated from the dissociation of a molecular Bose-Einstein condensate can be used to test the Einstein-Podolski-Rosen paradox as originally formulated [3,7]. The dichotomization of measurement results performed in a continuous system can also lead to maximal violation of Bell-type inequalities in phase space [8]. Finally, discrete multi-dimensional systems also allow for non-locality tests [9], and recent experiments show the violation of Bell-type inequalities with two effective spin-1 particles [10]. In parallel, recent advances in single molecule manipulation and detection open the way to the controlled creation of molecular entanglement [11]. Moreover, cold polar molecules confined in optical or magnetic traps have been recognized as a promissing candidate for quantum information processing [START_REF]Special issue of[END_REF], especially when their rotational levels are used as qubits [13,14]. Rotational states are relatively long lived, allowing for short quantum gate implementation times: one can perform about 10 4 gate operations before decoherence takes place. This excellent performance, when compared to cold collision based quantum gates [15], originates from the strength of dipolar interactions.

In the present paper, we propose realistic non-locality and entanglement tests for a system composed of two cold and trapped heteronuclear diatomic molecules. As in usual Bell tests scenarios, measurements are performed independently on each molecule by observers placed far apart, so that no comunication between them is possible during the realization of the protocol. We show in the following that inequalities built from local realism assumptions can be violated by a set of entangled states using measurements of correlations between the two molecules spatial orientation. These inequalities can also be used as entanglement witnesses [6] when the measurement conditions are loosen.

The inequalities derived here follow the original formulation of Clauser, Horne, Shimony and Holt (CHSH) [4]. For a pair of spin-1/2 particles or equivalent two-level systems, it can be shown using local theories that

| σ a σ b + σ a σ b ′ + σ a ′ σ b -σ a ′ σ b ′ | 2, ( 1
)
where σ α the Pauli matrix in the α direction. a and a ′ refer to the first particle while b and b ′ refer to the second one. In order to derive the bound appearing in Eq. ( 1) one assumes that the measurements performed by each observer are independent and their results have a probability distribution which is a product of independent probabilities for each subsystem. Such probabilities can also depend on some random local variable. Details are discussed in many works, as [16] for instance. It can be shown that a collection of entangled states can violate Eq. ( 1), and this experimental violation was observed using photon pairs entangled in polarization [5]. In this case, the directions a, a ′ , b and b ′ refer to different orientations of polarizers placed before the detectors. The value of the maximal possible violation, 2 √ 2, was first derived by Cirel'son [17].

The Bell inequalities we propose here have a structure similar to Eq. (1). They are based on molecular orientation correlation measurements instead of spin-like observables. We show that they are experimentally implementable using time delayed measurements.

Molecular spatial orientation : We suppose that the two linear molecules that compose our system behave like rigid rotors described by their angular properties at a given time t. An arbitrary two-molecule bipartite state |ψ 0 has been created at time t = 0, after which it freely evolves. This state can be entangled using techniques as the ones proposed in [13,14]. No interaction between molecules is allowed for t > 0. The Hamiltonian describing each individual molecule's free evolution is Ĥi = Ĵ2 i / 2 , where i = 1, 2. It is expressed in units of the rotational energy. Ĵi is the angular momentum operator and therefore the associated evolution operator is given by Ûi (t) = e -iπ Ĥit/ , where time t is written in units of the rotational period τ . Ûi (t) is therefore time-periodic with period 1. The total molecular state is thus given by the wavefunction ψ(θ 1 , θ 2 , φ 1 , φ 2 , t) ≡ θ 1 , θ 2 , φ 1 , φ 2 |ψ(t) . θ i and φ i denote here the polar and azimuthal spherical coordinates of both molecules in the laboratory frame. For each subsystem of the molecular bipartite set, the orientation at time t is defined as the average value

ψ 0 |O i (t)|ψ 0 of the orientation operator Ôi (t) = Û -1 2 (t) ⊗ Û -1 1 (t) cos(θ i ) Û1 (t) ⊗ Û2 (t).
Note that here cos(θ i ) is taken as an operator, in perfect analogy to the position operator and related functions. Orientation as defined above can be experimentally measured by recording each molecule's fragments angular distribution following a quasi-instantaneous molecular dissociation induced by ultrafast Coulomb explosion [18].

With an arbitrary accuracy, each molecule's state (subscripts will be omitted) can be considered to reside in a finite dimensional Hilbert space H generated by the basis set {|j, m ; 0 j j max , |m| j}, where |j, m are the eigenstates of Ĵ2 and Ĵz . Since it is a priori possible to prepare entangled bipartite molecular states with m = 0 using ultra-cold atomic photoassociation or Feshbach resonances followed by a dipolar interaction scheme [START_REF]Special issue of[END_REF]13], we have considered here, for simplicity, that the angular momentum projection m is fixed at 0 for each molecule. We have checked that our conclusions remain valid even if m = 0. The dimension of each molecule's Hilbert space H is then j max + 1, and the states |j, m = 0 will now simply be written as |j . The corresponding wavefunctions are the spherical harmonics θ|j = Y j0 (θ). In the finite Hilbert space H, the cos θ operator is characterized by a discrete, non-degenerate spectrum of eigenvalues λ n (0 n j max ), with corresponding eigenvectors |λ n , also called orientation eigenstates. The two maximally oriented states |+ and |-are the two eigenstates corresponding to the extreme eigenvalues ±λ max , where λ max = Max(λ n ). In the particular case of j max = 1, the maximally oriented states can be simply written in the basis of the angular momentum eigenstates |0 and |1 as |+ ∝ |0 +|1 and |-∝ |0 -|1 . In this particular case, cos θ ∝ σx , and we can of course recover the well-know results of a two-level system.

We now describe how orientation measurements performed locally on each subsystem can be used for nonlocality tests through inequalities analogous to Eq. ( 1).

Temporal inequalities : The orientation correlation between the two molecules is now defined as the average value of the two-particle operator C(t 1 , t 2 ) = O 1 (t 1 ) ⊗ O 2 (t 2 ), measured at times t 1 and t 2 . Note that the operator cos θ is useful here for entanglement detection since it "mixes" different values of j without affecting their projection m. Previous works have considered correlations between different values of the projection m for a given (fixed) value of j in other physical contexts [9,19].

The rotation of the molecular axis is described by the free evolution operators Ûi (t). As coherent superpositions of a finite numbers of |j , the orientation eigenstates are not stationnary states of the free Hamiltonian. Time evolution therefore modifies these quantum superpositions, thus changing the molecular orientation, in exact analogy to the usual projection of the photon polarization with polarizers. We would like to stress here that other CHSH inequalities using free evolution instead of polarizers were studied in different contexts: in Ref. [16,20], they allow the detection of entanglement between products of decaying mesons. In Ref. [21], they reveal quantum properties of single particles.

By combining measurements realized at different times, one can define, in analogy to Eq. ( 1), the operator

B 1 (t) = C(t, t) + C(t, 0) + C(0, t) -C(0, 0). ( 2
)
For simplicity, we have used a single time variable t here in comparison with the four angles of Eq. ( 1). We have verified that this limitation does not affect the generality of our results, since introducing four measurement times t i only increases the number of entangled states detected.

In the framework of a local theory (LT), the operator B 1 (t) obeys an inequality similar to Eq. ( 1):

| B 1 (t) LT | 2λ 2 max , ∀t > 0. ( 3
)
Without loss of generality, we have also assumed that each particle state resides in the same finite dimensional Hilbert space H. We note that Eq. ( 3) is valid for all possible values of j max , and that it can, in particular, be extended to the limit j max → +∞, in which case the spectrum of cos θ forms a continuum. An interesting characteristic of the separability threshold (3) is its dependence on λ max . In order to study the spectrum of B 1 (t), we can numerically diagonalize this operator. We obtain, for each time t, its highest eigenvalue, which gives the maximal value of B 1 (t) , and therefore the maximal violation of Eq. ( 3). This quantity depends on the dimensionality of the system, and we compare the amplitude of the violation for different values of j max in Fig. 1. While j max = 1 corresponds to a two-level system, j = 2 and j max = 5 are two examples of a higher dimensional system where local realism can, in principle, be violated.

A first motivating result is that for a broad range of times, Max[ B 1 (t) ] > 2λ 2 max , thus violating Eq. ( 3). It proves that B 1 (t) is useful for non-locality tests. In particular, in the two dimensional case j max = 1, Max[ B 1 (t) ] reaches its maximal value of 2 √ 2 λ 2 max , in a similar fashion as for CHSH inequalities with two level systems. However, with increasing dimensionality, the maximum relative violation decreases. This effect is shown in Fig. 1: the maximal relative violation of Eq.( 3) decreases from 41% to 29% when j max varies from 1 to 5. In addition, the highest possible separability threshold given by Eq. ( 3) is obtained in a true infinite dimensional space, since in this case λ max → 1. The cos θ eigenvalues then form a continuum, and Eq. ( 3) becomes a CHSH inequality for continuous measurement values. In a real experiment, one does sometimes not control the dimension of the subspace where entanglement is created, and the maximum threshold 2λ 2 max → 2 should then be considered. Fig. 1 shows that already for low values of j max it is possible to violate this general threshold. Indeed, for j max = 5, the maximum value of B 1 (t) is 9/4, and for all values j max 5, one can find entangled states violating Eq. ( 3). This result can be very useful in the realization of an experiment, and it is a consequence of two facts: high orientation (high values of λ max ) can be obtained in reduced angular momentum subspaces [22] and entanglement enhances two particle orientation correlations [13].

We have shown that the operator defined by Eq. ( 2) allows not only for the realization of Bell- finite angular momentum subspaces, but also when the size of the subspace is not a priori known. However, the maximal value of the violation decreases with dimensionality. This may render the proposed test more difficult in very high dimensions. We can overcome this by a dichotomizing procedure, in which a high dimensional system is transformed into an effective two level one. The dichotomization is defined as follows: the states |λ n for which λ n = λ n | cos θ |λ n > 0 are said to be positively oriented, while those for which λ n 0 are considered as negatively oriented. The orientation eigenstates are thus separated in two classes, |λ + and |λ -, depending on the sign of their associated eigenvalue. We now define, for each molecule, the projectors on the subspaces of positive or negative orientation: Π ± = λ± |λ ± λ ± |. The measured observable for the molecule i is then Π i = Π + -Π -. This measurement consists in counting the asymmetry of the molecular angular distribution. For a given single-molecule state |ϕ = λ+ c λ+ |λ + + λ-c λ-|λ -, ϕ|Π i |ϕ can take any value in the interval [+1, -1]. The total two-molecule observable is defined as Π = Π 1 ⊗ Π 2 . We refer, as before, to two-molecule correlation measurements realized at two different times, using Π(t

1 , t 2 ) = Π 1 (t 1 ) ⊗ Π 2 (t 2 ) where Π i (t i ) = Û -1 i (t i )Π i Ûi (t i ).
In analogy to Eq. ( 2), we now define the operator

B 2 (t) = Π(t, t) + Π(t, 0) + Π(0, t) -Π(0, 0). ( 4
)
Since Π(t i , t j ) 2 = 1, one can show that the highest value B 2 (t) can reach is given by the Cirel'son bound 2 √ 2 [17]. In addition, in the framework of a local theory, we have

| B 2 (t) LT | 2 , ∀t > 0.
(5) Fig. 2 shows the maximum value of B 2 (t) as a function of time for j max = 1, 2 and 5. For j max = 1 we obtain trivially the same result as with B 1 (with a simple scaling factor). However, for j max > 1, measuring B 2 has some advantages over B 1 . Cirel'son and locality bounds do not depend on dimensionality anymore, remaining valid even in the continuous case. Furthermore, Fig. 1 and2 show that the range of time when B 2 violates locality is much larger than the one of B 1 . We now illustrate the principles of an ideal experiment leading to non-locality (and entanglement) tests using Eq. (4). Our scenario consists of two molecules, created in a bipartite state |ψ 0 whose non-local properties one wishes to test. The molecules are spatially separated and evolve freely so that observers A and B can independently measure their orientation following the principles of [18]. Measuring the average value of the dichotomized orientation (4) requires only two detectors for each molecule, one placed in each hemisphere, determining whether molecules are positively or negatively oriented. Observers A and B perform such measurements at different times t i . To ensure non-locality tests are realized, we impose that the measurements performed by each observer cannot causally affect the other's. For that, we suppose that A and B are separated by a distance ℓ such as ℓ > c τ , where c is the speed of light and τ the molecule's rotational period. For typical values of τ , this means that the observers should be separated by a distance greater than a few hundred µm. A possible experimental setup allowing for such molecular separation and manipulation are optical tweezers, currently used as individual atoms traps [23]. The protocol is thus the following: after collecting orientation measurements for different times t i , observers A and B compare their results and construct the quantity (4). Using measurements lying in the interval determined by a rotational period τ , they can experimentally infer the violation of (4). In a simpler approach, we can also consider the situation where ℓ < c τ . In this case, correlations between measurements realized at different times and the proposed inequalities can serve as entanglement witnesses. This version of the protocol find direct applications for molecules trapped in optical lattices [24].

In conclusion, we have introduced two experimentally realistic distinct high dimensional Bell-type inequalities based on the measurement of a quantity which is usually considered as being "almost classical": the molecular orientation. Both can be violated not only in the case of a restricted angular momentum subspace but also in the continuous limit. The proposed inequalities also present another original feature: orientation measurements are realized at different times for each particle. We have shown that dichotomization into two classes, namely positive and negative orientation, allows for a simple experi-mental measurement procedure that can be implemented with present day technologies using molecules in optical traps. This second approach also increases the number of detected entangled states. Our results open the perspective of entanglement detection and non-locality tests for high angular momentum systems in atomic and molecular physics.

During the realization of this work, P.M. was financially supported by the ACI "Simulations Moléculaires". This work was partially supported by the CEA, under contract No. LRC-DSM-0533.

FIG. 1 :

 1 FIG.1:(Color online) Maximal value of B1(t) in the subspace H with j jmax as a function of t in units of the rotational period τ . The black, red and blue curves correspond to jmax = 1, 2 and 5 respectively. The associated separability thresholds are represented by dashed horizontal lines. The continuous case associated with the limit jmax → +∞ is also shown as a dotted horizontal line.

  type tests in Color online) Maximal value of B2(t) as a function of t in units of the rotational period τ . The solid, dashed and dotted curves correspond to jmax = 1, 2 and 5 respectively. The separability and Cirel'son bounds are shown as horizontal lines.

		2.8		j max = 5		2 √  2
		2.6
	ˆ B2(t)	2.4		j max = 2
	Max	2.2
				j max = 1
		2.0	separability threshold
	00 FIG. 2: (	.25	0.5 t	0.75	1