Aurore Lomet
email: aurore.lomet@utc.fr

Gérard Govaert
email: gerard.govaert@utc.fr

Yves Grandvalet
email: yves.grandvalet@utc.fr

Model Selection in Block Clustering by the Integrated Classification Likelihood

Keywords: Block clustering, integrated classification likelihood, model selection, Gaussian data

Block clustering (or co-clustering) aims at simultaneously partitioning the rows and columns of a data table to reveal homogeneous block structures. This structure can stem from the latent block model which provides a probabilistic modeling of data tables whose block pattern is defined from the row and column classes. For continuous data, each table entry is typically assumed to follow a Gaussian distribution. For a given data table, several candidate models are usually examined: they may differ in the numbers of clusters or in the number of free parameters. Model selection then becomes a critical issue, for which the tools that have been derived for model-based one-way clustering need to be adapted. In one-way clustering, most selection criteria are based on asymptotical considerations that are difficult to render in block clustering due to dual nature of rows and columns. We circumvent this problem by developing a non-asymptotic criterion based on the Integrated Classification Likelihood. This criterion can be computed in closed form once a proper prior distribution has been defined on the parameters. The experimental results show steady performances for medium to large data tables with well-separated and moderately-separated clusters.

Introduction

These last years have seen an increased interest in research and use of co-clustering of data tables, in domains of machine learning, statistics, data mining [START_REF] Berkhin | A survey of clustering data mining techniques[END_REF] or genomics [START_REF] Jagalur | Analyzing in situ gene expression in the mouse brain with image registration, feature extraction and block clustering[END_REF]. We consider block clustering, whose objective is to organize a data table in homogeneous blocks (clusters). Latent block models [START_REF] Govaert | Clustering with block mixture models[END_REF] enable to define a series of solutions to the block clustering problem, which may differ in their numbers of clusters or in the number of free parameters. Then, model selection [START_REF] Fraley | How many clusters? Which clustering method? Answers via model-based cluster analysis[END_REF], becomes a "critical point" to obtain a meaningful classification. a mixture of block components pertaining to the latent row and column classes: p(X) = (z,w)∈Z×W p(z, w) p(X|z, w) .

The latent block model is then derived from the two following assumptions:

Assumption 1.
Conditionally on the row and column labels, all table entries are independent, and all the entries belonging to the same block are identically distributed:

p(X|z, w, α) = i,j p(x ij |α z i ,w j ) ,
where α is a shorthand notation for all α z i ,w j parameters that characterize the distribution of x ij given (z i , w j ). For Gaussian data, α k is the mean µ k and the variance σ 2 k of block (k, ). Assumption 2. All latent variables are independent. The row labels are independent from the column labels: p(z, w) = p(z) p(w) .

The row labels are independent and identically distributed, and correspondingly, the column labels are independent and identically distributed: where π = (π 1 , . . . π g ) is the prior probability of row labels and ρ = (ρ 1 , . . . , ρ m ) is the prior probability of column labels.

Under these assumptions, the probability distribution of the model is:

p(X|θ) = (z,w)∈Z×W p(z|π) p(w|ρ) p(X|z, w, α) , = (z,w)∈Z×W i,j π z i ρ w j p(x ij |α z i ,w j ) ,
where θ = (π, ρ, α, n, d) is a shorthand notation for all parameters.

Model Types

A family of models can be defined by considering several numbers of classes in rows and columns. We additionally consider the options where the class priors (π, ρ) are free or equal and where the variance parameters σ 2 are free or equal for all clusters . Thus, four models are defined for fixed numbers of clusters by combining these two options.

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Complete Data

The latent block model considers that the distribution of the observed data table X depends on the unobserved latent class variables (z, w). The complete data refers thus to the triplet (X, z, w), and the complete data log-likelihood for the latent block model that assumes that (X, z, w) is observed is then:

L c (X, z, w|θ) = i,k z ik log π k + j, w j log ρ + i,j,k, z ik w j log p(x ij |α k, ) ,
where, slightly abusing notations, the double subscript class indicators z ik and w j are such that

z ik = 1 if z i = k and z ik = 0 otherwise.
The complete data likelihood plays a central role in a variational expectation-maximization (VEM) algorithm [START_REF] Nadif | Algorithms for Model-based Block Gaussian Clustering[END_REF] which is used to the estimate model parameters. As the regular EM algorithm, VEM maximizes the log-likelihood iteratively by maximizing the conditional expectation of the complete data log-likelihood given a current estimate of θ and the observed X. This maximization relies on the computation of p(z, w|X, θ) at the E-step. Without further assumptions, this computation is intractable for the latent block model, requiring n g × d m operations. This problem is circumvented thanks to a variational approximation assuming the conditional independence of z and w: p(z, w|X) = p(z|X)p(w|X). Both terms of the factorized distribution are easily dealt with, requiring respectively n × g and d × m operations. The M-step then proceeds as for the regular EM algorithm. At convergence, this algorithm returns an approximate log-likelihood L, its maximizer θ, and the estimation of the corresponding most probable row and column labeling (ẑ, ŵ). An approximate classification likelihood L c (X, ẑ, ŵ| θ) can also be computed.

ICL for the Gaussian Latent Block Model

The objective of model selection is to choose the "best model". In the Bayesian framework, the best model may be defined as the most probable one given the observed data and a prior distribution over models. When this prior is flat on the finite set of models under consideration, the best model maximizes the integrated likelihood:

p(X|M ) = T p(X|θ, M ) p(θ|M )dθ ,
where T is the parameter space and p(θ|M ) is the prior distribution over parameter θ for model M . This integrated likelihood is commonly approximated by the Bayesian information criterion (BIC). Nevertheless, this approximation is not valid for block clustering due to the inter-dependence between rows and columns and the violation of the necessary regularity conditions [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF].

The integrated classification likelihood (ICL) [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF] is constructed on the complete data to overcome the problems pertaining to regularity; it then relies on the ability of clustering models to provide evidence with regards to the unobserved data:

p(X, z, w|M ) = T p(X, z, w|θ, M ) p(θ|M )dθ ;
for the latent block model described in Section 3, it also solves the issues raised by the dependencies among table entries.

For convenience, the ICL criterion evaluates the logarithm of the integrated classification likelihood:

ICL(M ) = log p(X, z, w|M ) = log p(X|z, w, M ) + log p(z|M ) + log p(w|M ) , (1)
where the decomposition follows from Assumption 2, supposing the independence of priors p(θ) = p(α) p(π) p(ρ).

Then, ICL(M ) can be expanded in several forms according to the model type, as listed in Section 3. For the first term, assuming that the variances differ, we choose a Gaussian prior distribution of parameters (µ 0 , σ 2 k /κ 0 ) for the conditional mean of a block µ k |σ 2 k and a scaled inverse χ 2 prior distribution of parameters (ν 0 , σ 2 0 ) is then appropriate for the variance parameter σ 2 k . The first term of (1) then becomes (see appendix):

log p(X|z, w, M ) = - nd 2 log π + gmν 0 2 log(ν 0 σ 2 0 ) -gm log Γ( ν 0 2 ) + gm 2 log κ 0 + k, log Γ( ν 0 + n k d 2 ) - 1 2 log(κ 0 + n k d ) - ν 0 + n k d 2 log ν 0 σ 2 0 + (n k d -1)s 2 k + κ 0 n k d κ 0 + n k d (x k -µ 0 ) 2 , (2)
where xk = 1 2 , n k and d are respectively the mean, the unbiased sample variance, the number of rows and the number of columns of the block (k, ).

n k d i,j z ik w j x ij , s 2 k = 1 n k d -1 i,j z ik w j (x ij -xk )
When the variances are assumed to be equal, a Gaussian prior distribution of parameters (µ 0 , σ 2 /κ 0 ) may be postulated for µ|σ 2 and a scaled inverse χ 2 prior distribution of parameters (ν 0 , σ 2 0 ) for the variance parameter σ 2 . The first term of (1) is then:

log p(X|z, w, M ) = - nd 2 log π + ν 0 2 log(ν 0 σ 2 0 ) -log Γ( ν 0 2 ) + gm 2 log κ 0 + log Γ( ν 0 + nd 2 ) - 1 2 k, log(κ 0 + n k d ) - ν 0 + nd 2 log ν 0 σ 2 0 + (nd -gm)s 2 w + k, κ 0 n k d κ 0 + n k d (x k -µ 0 ) 2 , ( 3
)
where

s 2 w = 1 nd-gm k (n k d -1)s 2
k is the unbiased within-cluster sample variance. When the proportions (π, ρ) are assumed to differ, a symmetric Dirichlet prior distribution of parameters (δ 0 , ..., δ 0 ) is postulated, and the last two terms of (1) respectively become:

log p(z|M ) = log Γ(gδ 0 ) + k log Γ(n k + δ 0 ) -g log Γ(δ 0 ) -log Γ(n + gδ 0 ) , (4)
log p(w|M

) = log Γ(mδ 0 ) + log Γ(d + δ 0 ) -m log Γ(δ 0 ) -log Γ(d + mδ 0 ) . (5)
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Parameters ICL(M ) expansion Variances Proportions log p(X|ẑ, ŵ, M ) log p(ẑ|M ) log p( ŵ|M )

σ 2 π, ρ (3)
(6) (7) σ 2 k π k , ρ (2)
(4) (5) σ 2 π k , ρ (3)
(4) (5) σ 2 k π, ρ (2) (6) (7)
Table 1: Variants of the ICL(M ) expansion, for a single or multiple variance parameters, and for free or fixed proportion parameters.

When the parameters (π, ρ) are fixed (and equal), p(z|M ) and p(w|M ) are multinomial variables (of parameter 1/g and 1/m respectively), and the last two terms of (1) respectively become:

log p(z|M ) = n log g , (6)
log p(w|M

) = d log m . (7)
As the latent variables z and w are unobserved, we replace them by their most probable inferred values, ẑ and ŵ, as estimated by the VEM algorithm [START_REF] Nadif | Algorithms for Model-based Block Gaussian Clustering[END_REF]. Thus, we obtain four variations of ICL(M ), whose lengthy expressions are summarized in Table 1.

Numerical Experiments

In this section, the performances of ICL are studied on simulated data, which were generated from the latent block model defined in Section 3, so as to have a known ground truth to assess model selection. We use block models with 3 row and 3 column clusters from two model types, either with free or fixed proportions, and both with multiple variances parameters. These two intermediate model types allow for under and over-estimation of the number of free parameters. To avoid unbalanced errors between clusters, the row and column means are equidistant. We consider three degrees of cluster overlap (5%, 12% and 20% of classification error according to conditional Bayes risk) and four data sizes (50 × 50, 200 × 200, 500 × 500 and 1000 × 1000) (see [START_REF] Lomet | Design of Artificial Data Tables for Co-Clustering Analysis[END_REF] for thorough details). For each setup, twenty data sets are simulated, and for each data set, 64 candidate latent block models are estimated by the VEM algorithm: the numbers of row and column clusters are varied from 1 to 4, and for each such number, the four model types are considered. Note that these 64 models define the set of possible models over which a flat prior will be assumed to compute the ICL criterion.

Our ICL criterion requires specifying the five hyperparameters (δ 0 , µ 0 , κ 0 , ν 0 , σ 2 0 ). For the Dirichlet distribution on the proportions parameters, we chose to use a non-informative prior (δ 0 = 1 2 ). For the prior on block means, µ 0 is set to the overall mean of the data table entries. The hyperparameter κ 0 is set to nd/gm: its increase with the number of entries in each cluster encodes a prior assumption on the difficulty of the problem, and the chosen rate reflects our observations on the decrease error rates as table size increases. For the prior on the block variances, the hyperparameters ν 0 and σ 2 0 are defined such as the mode (

ν 0 σ 2 0 ν 0 +2
) is equal to the overall variance. The hyperparameter ν 0 is set to 10 -1 in order to allow for large deviations from this modal value. Therefore, σ 2 0 is equal to 20s 2 where s 2 is the unbiased sample variance. This type of empirical Bayes method is commonly used to set the value of some of the hyperparameters, such as the ones related to the mean distribution [START_REF] Wyse | Block clustering with collapsed latent block models[END_REF]. They may even be employed in more elaborate construction of priors, such as in hierarchical models [START_REF] Richardson | On Bayesian analysis of mixtures with an unknown number of components (with discussion)[END_REF] . That being said, the mathematical properties of this approach are little known and have been little investigated in general and in co-clustering in particular. In this paper, we fix the values of the hyperparameters as stated above and study their sensitivity to evaluate their effects on model selection.

The ICL criterion is compared to the extension of Akaike Information Criterion 3 proposed by [START_REF] Van Dijk | A Bayesian approach to two-mode clustering[END_REF]:

AIC 3 (M ) = -2 L(X| θ, M ) + 3 × (η α + n(g -1) + d(m -1) + g + m -2) ,
where L is the variational approximation of the likelihood, θ is the set of parameters estimated by VEM and η α denotes the number of free parameters in α. This criterion is adapted to the four variants of the latent block model by adjusting η α and dropping the last three terms when proportions are fixed.

The competing criteria aim to retrieve the correct model, which was used for generating the data table. This model is defined by two arguments: its number of row and column clusters and the model type (with single or multiple variance parameters and free or fixed proportions). We tested the ability to select the correct number of clusters knowing model type, the model type knowing the correct number of clusters and to retrieve both model arguments.

Tables 2 and3 report the number of successes among the 20 repetitions for each setup (respectively for free and fixed proportions). For each criterion three types of selection are studied: the correct number of clusters knowing the type of model, the correct type of model knowing the number of clusters and the correct number of clusters and type of model. Overall, ICL performs better than AIC 3 , though AIC 3 is usually slightly better at recovering the model type for known number of clusters. The main weakness of AIC 3 is its inability to retrieve the correct number of clusters even in the most favorable cases (with well-separated clusters and large data tables). This behavior contrasts with the performances of ICL that improve when table size increases and when the degree of overlap decreases, eventually selecting systematically both model types and number of clusters in the most favorable cases tested here. When these criteria fail to select the correct model, AIC 3 and ICL underestimate the number of clusters, while they pick a more complex model type (with free proportions for example).

An empirical study of the sensitivity of the hyperparameters highlights that wrong choices may badly influence the results of the model selection realized by the ICL criterion, but that @ COMPSTAT 2012 a wide range of values perform about equally well in this regard. Figure 1 shows the influence of each hyperparameter on the number of correct selections (clusters and models). Each graph represents this number versus one hyperparameter (in log-scale), while the other ones are fixed as stated above. For the hyperparameter µ 0 , different values were tested and revealed that the mean of the whole data table is a well-advised choice. The other hyperparameters behave well, with wide plateaus of nearly optimal values. Note that this univariate analysis only provide a partial picture of sensitivity, as we observed a strong coupling between some of them, like σ 2 0 and ν 0 which govern the distribution of the prior on the variance parameter(s) of the model.

Conclusion

The Latent Block Model is a probabilistic model that enables to organize a data table in homogeneous blocks. A critical point of the overall fitting procedure is to select a correct model, in order to obtain a meaningful co-clustering. We proposed a model selection strategy based on the adaptation of the exact integrated classification likelihood criterion to the Gaussian latent block model. This non-asymptotic criterion allows to avoid asymptotical problems due to dual nature of rows and columns in block clustering. Moreover, ICL does not require costly additional calculations. Indeed, once the clusters have been estimated, this criterion can be computed in closed form once a proper prior distribution has been defined on the parameters.

The experiments on simulated data targeted two types of model selection, either related to the number of row and column clusters, or to the model type, or both. Model type is defined from parameter restraints, such as considering equal variances or proportions for all clusters. Our results show that AIC 3 is unable to select the correct number of clusters and that it is less successful than ICL regarding model type. The latter perform well when the overlap between clusters is small to moderate, and its performances improve with the size of the data table. The study of the sensitivity of hyperparameters highlights that adjusting the values of some of them allows to improve the performances of ICL.

Beyond the Gaussian latent block model, this approach can be used for other model types such as the latent block model for binary or contingency data, by modifying appropriately the prior of the conditional distribution on the data. For the terms related to the proportions, when the proportions are free, we assume a symmetric Dirichlet prior distribution of parameters (δ 0 , ..., δ 0 ) for the row and column parameters (π, ρ), so that: Details can be found in [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF].

(σ 2 ) -1 2 exp - κ 0 + n k d 2σ 2 µ k - κ 0 µ 0 + n k d xk κ 0 + n k d 2 - (κ 0 n k d )(x k -µ 0 ) 2 2σ 2 (κ 0 + n k d ) = k, (σ 2 ) -1 2 exp - κ 0 + n k d 2σ 2 µ k - κ 0 µ 0 + n k d xk κ 0 + n k d
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  p(z|π) = i p(z i |π) , p(w|ρ) = j p(w j |ρ) ,

Figure 1 : 2 ×(σ 2 ) -nd 2 exp - 1 2σ 2 i 2 ,= (σ 2 ) -( ν 0 +nd 2 +1) exp - 1 2σ 2 (ν 0 σ 2 0

 12222222 Figure 1: Number of correct selections (clusters and model) according to variations in hyperparameter values. Here, the data table size is 500 × 500, the latent block model has different proportions and variances and the clusters are moderately-separated (Bayes' error at 12% for 3 × 3 clusters).

2 ×(σ 2 ) -( ν 0 +nd 2 +1) exp - ν 0 σ 2 0 2 .

 22222 + (nd -gm)s 2 w + k, κ 0 n k d κ 0 +n k d (x k -µ 0 ) 2 2σThis probability can be factorized:p(µ, σ 2 |X) = p(µ|σ 2 , X)p(σ 2 |X)Thus, the posterior distribution is defined (assuming the posterior independence of µ k ):p(α|X, z, w, M ) = k, N µ k ; κ 0 µ 0 + n k d xk κ 0 + n k d , σ 2 κ 0 + n k d × Inv-χ 2 σ 2 ; ν 0 + nd, ν 0 σ 2 0 + (nd -gm)s 2 w + k, n k d κ 0 n k d +κ 0 (x k -µ 0 ) 2 ν 0 + nd .

1 k

 1 0 )...Γ(δ 0 ) π k =1 dπ , = Γ(gδ 0 ) Γ(δ 0 ) g Γ(δ 0 + n 1 )...Γ(δ 0 + n g ) Γ(n + gδ 0 ) .

Table 2 :

 2 Number of correct selections over 20 trials for different table sizes and cluster overlaps.The simulated model has multiple variances and free proportions.

		Table Size		50 × 50		200 × 200	500 × 500	1000 × 1000
		Bayes' risk	5% 12% 20% 5% 12% 20% 5% 12% 20% 5% 12% 20%
	3 AIC	Cluster Model Type Cluster and Type	0 8 0	0 8 0	0 15 0	0 19 20 20 0 0 0 0 0	0 20 20 20 0 0 0 0 0	0 20 20 20 0 0 0 0 0
	ICL	Cluster Model Type	9 5	0 5	0 15	18 2 17 20 20 0	20 17 20 20 20 0	20 20 20 20 20 0
		Cluster and Type	2	0	0	16 2	0	20 17	0	20 20	0

Table 3 :

 3 Number of correct selections over 20 trials for different table sizes and cluster overlaps. The simulated model has multiple variances and fixed proportions.

		Table Size	50 × 50		200 × 200	500 × 500	1000 × 1000
		Bayes' risk	5% 12% 20% 5% 12% 20% 5% 12% 20% 5% 12% 20%
	3 AIC	Cluster Model Type Cluster and Type	0 19 13 16 0 0 0 0 0	0 16 13 13 0 0 0 0 0	0 20 20 17 0 0 0 0 0	0 20 20 19 0 0 0 0 0
	ICL	Cluster Model Type	13 1 20 17 16 0	16 13 16 13	2 4	20 20 14 20 20 15	20 20 19 20 20 16
		Cluster and Type 12 0	0	16 9	1	20 20	0	20 20	1

Acknowledgment This research was partially supported by the French National Research Agency (ANR) under grant ClasSel ANR-08-EMER-002, the PASCAL2 Network of Excellence, and the European ICT FP7 under grant No 247022 -MASH. We would also like to thank the reviewers for their constructive and detailed comments.

Appendix: Derivation of ICL

The first term of the expansion (1) is rewritten using the following decomposition:

where p(α|M ) and p(α|X, z, w, M ) are respectively the prior and posterior distributions of α.

For the latent block model with different variances, given the row and column labels, the entries x ij of each block are independent and identically distributed. We thus apply the standard @ COMPSTAT 2012

Model Selection in Block Clustering by the Integrated Classification Likelihood results for Gaussian samples [START_REF] Gelman | Bayesian data analysis[END_REF], where the distributions are defined by:

Using the definitions of these distributions, the first term of the expansion (1), log p(X|z, w, M ) = log p(X|z, w, α, M ) + log p(α|M ) -log p(α|X, z, w, M ) , is identified, after some calculations, as [START_REF] Biernacki | Assessing a mixture model for clustering with the integrated completed likelihood[END_REF].

For the latent block model with equal variances, the standard results need to be adapted to account for the shared parameter σ 2 . The prior distributions are now defined as follows: p(X|z, w, α, M ) = i,j,k, N (x ij ; µ k , σ 2 ) z ik w j , p(α|M ) = k, N (µ k ; µ 0 , σ 2 κ 0 ) × Inv-χ 2 (σ 2 ; ν 0 , σ 2 0 ) .


F Kaiser

A Issautier

L A Ngah

O Dȃnilȃ

H Herrmann

W Sohler

A Martin

S Tanzilli

High-quality polarization entanglement state preparation and manipulation in standard telecommunication channels

Keywords: numbers: 03.65.Ud, 03.67.Bg, 03.67.Hk, 03.67.Mn, 42.50.Dv, 42.65.Lm, 42.65.Wi

We report a simple and practical approach for generating high-quality polarization entanglement in a fully guided-wave fashion. Both deterministic pair separation into two adjacent telecommunication channels and the paired photons' temporal walk-off compensation are achieved using standard fiber components. Twophoton interference experiments are performed, both for quantitatively demonstrating the relevance of our approach, and for manipulating the produced state between bosonic and fermionic symmetries. The compactness, versatility, and reliability of this configuration makes it a potential candidate for quantum communication applications.

Introduction

Photonic entanglement is a key resource for quantum communication protocols [START_REF] Tittel | Photonic entanglement for fundamental tests and quantum communication Quant[END_REF], among which today's main application is quantum key distribution [START_REF] Gisin | Quantum cryptography[END_REF][START_REF] Scarani | The security of practical quantum key distribution[END_REF]. Producing entangled photons at telecom C-band wavelengths (1530-1565 nm) enables implementing relatively long-distance quantum links, thanks to low-loss optical fibers and high performance standard components. One of the most natural and accessible entanglement observables is the polarization state of the photons. Many experiments have been performed in this field, often taking advantage of the so-called type-II spontaneous parametric down conversion (SPDC) process in suitable non-linear bulk [START_REF] Kwiat | New High-Intensity source of Polarization-Entangled photon pairs[END_REF][START_REF] Piro | An entangled photon source for resonant single-photon-single-atom interaction[END_REF] or waveguide [START_REF] Zhong | High-quality fiberoptic polarization entanglement distribution at 1.3 µm telecom wavelength[END_REF][START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF] crystals. Moreover, integrated optics in LiNbO 3 is a powerful resource for engineering quantum circuits for both generating and manipulating entangled photons [START_REF] Tanzilli | On the genesis and evolution of integrated quantum optics Laser[END_REF]. Strong nonlinear optical effects as well as single spatial mode operation are indeed key ingredients for future developments of quantum information technology [START_REF] Martin | A quantum relay chip based on telecommunication integrated optics technology[END_REF].

In this paper we present a simple and practical approach for generating highquality polarization entanglement in a fully guided-wave fashion. Both deterministic pair separation in two adjacent telecommunication channels and walk-off compensation are achieved using standard fiber components. Two-photon interference experiments are carried out both for precisely determining the fiber compensation stage, and for manipulating the produced state between bosonic (|Ψ + ) and fermionic (|Ψ -) symmetries [START_REF] Matthews | Simulating quantum statistics with entangled photons: a continuous transition from bosons to fermions[END_REF]. Eventually, we perform a Bell inequality test on the |Ψ + state to quantitatively demonstrate the relevance of both our approach and results.

In the case of type-II waveguides, paired photons are emitted co-propagatively in the cross-polarization states |H |V . Successful formation of entanglement is then obtained upon probabilistic pair separation at a beam-splitter, at the price of a 50% pair loss. To circumvent this, several strategies have been proposed. They are based on either multi-waveguide configurations [START_REF] Yoshizawa | Generation of polarisation-entangled photon pairs at 1550 nm using two PPLN waveguides[END_REF][START_REF] Jiang | The generation of polarization-entangled photon pairs using periodically poled lithium niobate waveguides in a fibre loop[END_REF][START_REF] Kawashima | Type-I Quasi-Phase-Matched waveguide device for Polarization-Entangled twin photon generation[END_REF], interlaced poling periods in lithium niobate [START_REF] Thyagarajan | Generation of polarization-entangled photons using type-II doubly periodically poled lithium niobate waveguides[END_REF][START_REF] Suhara | Quasi-phase matched waveguide devices for generation of postselection-free polarization-entangled twin photons[END_REF][START_REF] Thomas | Novel source of polarization entangled photon pairs using a PPLN waveguide with interlaced domains 15 th ECIO[END_REF], or concurrent type-0/I SPDC processes in AlGaAs ridge waveguides [START_REF] Kang | Generation of polarization entangled photons using concurrent type-I and type-0 processes in AlGaAs ridge waveguides[END_REF]. Related experiments reported limited entanglement qualities due to technological imperfections [START_REF] Yoshizawa | Generation of polarisation-entangled photon pairs at 1550 nm using two PPLN waveguides[END_REF][START_REF] Jiang | The generation of polarization-entangled photon pairs using periodically poled lithium niobate waveguides in a fibre loop[END_REF][START_REF] Kawashima | Type-I Quasi-Phase-Matched waveguide device for Polarization-Entangled twin photon generation[END_REF][START_REF] Suhara | Quasi-phase matched waveguide devices for generation of postselection-free polarization-entangled twin photons[END_REF][START_REF] Thomas | Novel source of polarization entangled photon pairs using a PPLN waveguide with interlaced domains 15 th ECIO[END_REF]. In addition, the temporal walk-off between the paired photons naturally induced by the waveguide birefringence needs to be compensated. In earlier works, the latter was achieved using bulk optics strategies, reducing both the reliability and the applicability of the source [START_REF] Zhong | High-quality fiberoptic polarization entanglement distribution at 1.3 µm telecom wavelength[END_REF][START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF]. Here, we address these two issues using a low-loss telecom fiber solution.

Principle of the source for deterministic paired-photons separation

The experimental setup of the source is shown in Figure 1. The type-II photon-pair generator is based on a 3.6 cm long, 9.0 µm periodically poled, lithium niobate waveguide structure (PPLN/W) [START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF]. More details of our fabrication processes and numerical simulations carried out for reaching the desired phase-matching are given in [START_REF] Martin | Integrated optical source of polarization entangled photons at 1310 nm[END_REF]. We couple 2.5 mW (nominal power) from a 770 nm diode laser into the waveguide towards creating paired photons around the degenerate wavelength of 1540.2 nm. As shown in Figure 2, the |H and |V emission modes overlap almost perfectly with a spectral bandwidth of about 0.85 nm FWHM.

We address the paired photons deterministic splitting via a pertinent strategy which consists of dividing the emission spectrum of Figure 2 into high (+) and low (-) wavelength windows, so as to create a state of the form

|Ψ(φ) = 1 √ 2 |H + |V -+ e i φ |V + |H -. (1)
To this end, we employ two complementary standard 100 GHz dense wavelength division multiplexers (DWDM). As shown in Figure 2, this set of filters is chosen so as to nearly cover the full emission spectrum, but also to remove the side peaks of the spectrum that provide with a polarization determination as a function of the wavelength [START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF]. This way, almost no useful photons are lost in the spectrum central peak. Note that the actual overlap between the two filters at the center of the spectra is on the order of 1%. Taking into account the relative spectral intensities, this leads to a probability of <0.25% of having the two photons sent to the same wavelength region, i.e. |V + |H + and |V -|H -given to the same user (+ to Alice and -to Bob, respectively). Therefore, additional noise contributions, due to the detection of a non-entangled photon in one wavelength region and a dark-count in the other detector, are negligible.

Temporal walk-off compensation using a fiber approach

To compensate the temporal walk-off between the |H and |V photons induced by the waveguide birefringence, we take advantage of a simple, reliable, and low-loss strategy. As previously used with fiber-based photon-pair sources [START_REF] Li | Optical-Fiber Source of Polarization-Entangled Photons in the 1550 nm Telecom Band[END_REF][START_REF] Li | Storage and long-distance distribution of telecommunications-band polarization entanglement generated in an optical fiber[END_REF], a birefringent polarization maintaining fiber (PMF) is butt-coupled to the waveguide's output and oriented such that the fiber birefringence axis is rotated by 90 • compared to that of the waveguide. To find the correct compensation length of the PMF, we perform a two-photon interference experiment based on the so-called Hong, Ou, and Mandel (HOM) dip setup [START_REF] Hong | Measurement of subpicosecond time intervals between two photons by interference[END_REF] for the polarization observable [START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF], in which the temporal delay between the two photons is scanned. On one hand, this permits inferring the natural temporal walk-off induced by the waveguide birefringence. On the other hand, it allows characterizing the specific walk-off compensation achieved per meter of PMF. The HOMlike experimental apparatus is shown in Figure 3 with the left inset (a) connected to the output of the source. The generated photons are first sent to a tunable polarization dependent delay based on a Michelson interferometer like arrangement, as detailed in [START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF]. One of the mirrors can be displaced using a motor to introduce an artificial delay between the photons. Thereafter, the bi-photon state is rotated by 45 • using a fiber polarization controller (PC) compared to the {H, V } creation basis, and sent to a fiber polarizing beam-splitter (f-PBS). After the f-PBS, two DWDMs are used for selecting high (+) and low (-) wavelength photons. Note that in principle, the actual position of the filtering stage, i.e. before or after the f-PBS, does not matter. This allows considering the state of Eq. 1, with φ=0, at the input of the measurement apparatus of Figure 3(a). In this case, the evolution of the sate reads

|Ψ = 1 √ 2 |H + |V -+ |V + |H - 45 • -→ 1 2 √ 2 |H + |V --|H + |H -+ |V + |V --|V + |H - +|V + |H -+ |V + |V --|H + |H --|H + |V -. (2)
When the delay between the photons is perfectly compensated, the states associated with cross-polarized contributions cancel each other. As a consequence, the quantum state after the action of both the f-PBS and the filters leads to the state

|Ψ HOM = 1 √ 2 |V + |V --|H + |H -. (3)
The coincidence detection rate is then recorded, as a function of the artificial delay, between two InGaAs avalanche photodiodes (APD). One is a passively quenched device (IDQ-220, featuring ∼20% detection efficiency and ∼10 -6 / ns dark-count probability) which triggers, upon detection, the second one operated in gated mode (IDQ-201, featuring ∼25% detection efficiency and ∼10 -5 / ns dark-count probability). As shown in Figure 4, we obtain HOM-dips when performing the experiment with and without the PMF walk-off compensator. The associated raw visibilities are both of 94±5%. Note that no particular efforts have been made towards optimizing these figures. Without the PMF, the dip is centred at a delay of -4.40 ps, which corresponds to the natural walk-off induced by the waveguide birefringence, as predicted by the theory [START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF]. We characterized the specific PMF compensation to be 1.38 ps/m, meaning that a 3.2 m long PMF is necessary in our case. With such a fiber, the dip position is shifted to 0.03 ps, indicating that the walk-off is nearly perfectly compensated.

Manipulation of the entangled state symmetry

The above dips have been obtained with photons at different wavelengths, contrary to what is commonly done in similar HOM measurements [START_REF] Zhong | High-quality fiberoptic polarization entanglement distribution at 1.3 µm telecom wavelength[END_REF][START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF]. This indicates that this quantum effect does not simply originate from a conventional two-photon interference, but rather from an entangled state post-selected by the measurement apparatus (f-PBS+DWDMs). Namely, in standard polarization HOM experiments, i.e. involving wavelength degenerate photons, the dip is observed independently of potential phase fluctuations between the two photons impinging the PBS [START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF]. Entanglement, however, implies a constant phase relation between its two contributions. To show this, we now discuss a second measurement which makes it possible to switch the phase of the entangled state, if any, from 0 to π, corresponding to the |Ψ + and |Ψ -maximally entangled Bell states, respectively. In the case of a |Ψ + state, associated with an even wave-function parity (bosonic symmetry for the polarization observable), the two photons can always be related to an even spatial distribution due to their bosonic character. As a consequence, photon coalescence can be observed upon the projection by a PBS preceded by a 45 • rotation of the bi-photon state compared to the {H, V } basis, as demonstrated above.

If, however, a

|Ψ -= 1 √ 2 |H + |V --|V + |H -state,
showing an odd wavefunction parity (fermionic symmetry), is analyzed with such a setup, then the spatial distribution needs to be associated with an odd parity as well. In this case, after the bi-photon state is rotated by 45 • using a fiber PC compared to the {H, V } creation basis, projection at an f-PBS oriented in the {H, V } basis leaves the state unchanged. This is why an anti-coalescence behaviour is expected, i.e. the paired photons are deterministically separated at the second f-PBS, therefore leading to increased coincidence rates. Figure 3 depicts the experimental setup with the right inset (b) connected, which enables tuning the phase between the two contributions of the initial state |Ψ + . After the polarization dependent delay line the paired photons are separated deterministically regarding their wavelengths using a first DWDM filter. In one arm, a variable delay is employed and in the other arm a phase shift between the contributions |H + and |V + is introduced thanks to a fiber coupled Soleil-Babinet (SB) phase plate. The paired photons are then recombined at a second DWDM, rotated by 45 • using a fiber PC, and sent, eventually, to an f-PBS so as to perform HOM dip/peak measurements. Note that this two-DWDM configuration, which provides deterministic separation and recombination of the photons, has the arrangement of a Mach-Zehnder interferometer. However, it requires no phase stabilization since the two arms are associated with a different wavelength, therefore preventing single-photon interference. As shown in Figure 5, we observe again a clear dip when the phase introduced by the SB is φ = 0 and the birefringence compensation is made. If the phase is chosen to be π, then the maximally entangled state |Ψ -is generated and a coincidence peak is observed. With the same detectors as for Figure 4, both the dip and peak are associated . HOM dip and peak obtained for two different SB settings. For φ = 0 the |Ψ + still holds and a dip is obtained, as was the case in Figure 5. Changing the phase to π turns the state to |Ψ -for which the spatial distribution is expected to have an odd symmetry, therefore leading to a coincidence peak. with raw and net (i.e. after subtraction of events associated with the dark-counts in the detectors) visibilities of 86±5% and 94±6%, respectively. Note that the curves of Figure 5 have been recorded for the same integration time as in Figure 4. The reduced raw visibility is ascribed to an increase of the pump power to compensate the extra losses introduced in the setup of Figure 3(b), since a free-space S/B and additional fiber coupling stages are used.

It is also remarkable that this behaviour is observed independently of the delay introduced between the + and -components. For the peak measurement of Figure 5, the fiber delay was of 22 ns (↔5 m), which is much greater than the ∼5 ps (↔100 GHz) single photons' coherence time travelling through the device. This condition therefore prevents them from temporally overlapping at the second f-PBS. As a conclusion of this analysis, which is similar to energy-time entanglement in a folded Franson configuration [START_REF] Franson | Bell inequality for position and time[END_REF][START_REF] Thew | Experimental investigation of the robustness of partially entangled qubits over 11 km[END_REF], the two properties discussed above can be attributed only to entangled states.

Entanglement quality measurement

To further quantify the generated entanglement quality using the simple setup in Figure 1, we perform a Bell inequality test using the standard polarization settings [START_REF] Zhong | High-quality fiberoptic polarization entanglement distribution at 1.3 µm telecom wavelength[END_REF][START_REF] Martin | A polarization entangled photon-pair source based on a type-II PPLN waveguide emitting at a telecom wavelength[END_REF]. As discussed above, the birefringence compensation is made using a 3.2 m long PMF. Then, the paired photons are deterministically split up and sent to Alice and Bob using two standard 100 GHz DWDMs centred at the channels ITU-46 and ITU-47 (see Figure 2). As the reflection of each DWDM is the complement of it's transmission window, two complementary filters are necessary. The generated entangled state is then analyzed by Alice and Bob, each of them comprising a fiber PC, a half-wave plate (HWP), a polarization beam-splitter (PBS), and a single photon detector. In order to reduce the noise in this measurement, we take advantage of two free-running InGaAs APDs (IDQ-220) featuring ∼20% detection efficiency and ∼10 -6 / ns dark-count probability. In addition, on Alice's side, an SB phase compensator is employed to adjust the phase relation φ between the contributions to the entangled state of Eq. 1. For the following measurements, we set φ = 0 for generating the maximally entangled Bell state |Ψ + = |Ψ(0) . Then Alice fixes her polarization analyser to be subsequently oriented at horizontal (H), vertical (V ), diagonal (D) and anti-diagonal (A). Simultaneously, the coincidence detection rate between Alice and Bob is measured as a function of Bob's HWP angle, which is continuously rotated. As shown in Figure 6, we obtain excellent visibility interference patterns for all settings. The fitting parameters infer average raw and net (i.e. after subtraction of events associated with the dark-counts in the detectors) visibilities of about 97.3±0.6% and 99.5±0.8%, respectively. These visibility figures correspond to net and raw Bell parameters of S net =2.824±0.007 and S raw =2.806±0.005, respectively, and therefore to the violation of the Bell, Clauser, Horne, Shimony, and Holt (B-CHSH) inequalities by more than 100 standard deviations for each value [START_REF] Clauser | Proposed Experiment to Test Local Hidden-Variable Theories[END_REF]. Note that the raw visibility is only limited by the dark-count probabilities in the detectors, in our case ∼ 1 • 10 -6 / ns. The obtained net visibility underlines the high quality entanglement generated by the source itself. Note that similar results have been obtained for the |Ψ -state (curves not represented).

Regarding the other figures of merit of this source, we have reached a coincidence rate of about 1100 s -1 for a waveguide coupled pump power of 2.5 mW. The spectral brightness of the PPLN/W was measured to be ∼ 2 • 10 4 generated pairs per second, per mW of coupled pump power, and per GHz of spectral bandwidth. In addition, the losses experienced by the single photons are of about 3 dB, from the output of the generator to the measurement apparatus. This is an exceptionally good loss figure, and mainly ascribed to the advantageous use of off-the-shelf standard telecom components. Note that the main loss (∼ 2 dB) comes from the butt-coupling between PPLN/W and the PMF, which could be strongly reduced using a fiber pigtail and/or a segmented tapered waveguide structure for enabling a better channel waveguide to fiber mode overlap [START_REF] Castaldini | Soft-Proton-Exchange tapers for low insertion-loss LiNbO 3 devices[END_REF].

Conclusion & perspectives

As a conclusion, we have demonstrated a high quality polarization entanglement source based on a type-II PPLN/W. The source is reliable and efficient thanks to the use of lowloss standard telecom components, such as a polarization maintaining fiber for waveguide birefringence compensation and a set of DWDMs for deterministic pair separation into standard communication channels. The excellent quality entanglement obtained at the output of the source (events due to the dark-counts in the detectors discarded), combined with the simplicity of the setup, make this approach a good candidate for future quantum networking solutions based on photonic entanglement. For instance, the reported source could be used in long-distance quantum key distribution schemes, provided detectors featuring much lower dark-count rates would be employed. Notably, state-of-the-art superconducting devices show dark-count probabilities on the order of 10 -9 / ns [START_REF] Verevkin | Ultrafast superconducting single-photon detectors for near-infrared-wavelength quantum communications[END_REF][START_REF] Engel | Superconducting single-photon detector for the visible and infrared spectral range[END_REF].

Figure 1 .

 1 Figure 1. Setup for generating polarization entangled photons at 1540 nm using a type-II PPLN/W pumped by a 770 nm laser. A PMF is used as birefringence compensator and a set of two DWDMs separates the photons deterministically into the ITU-46/47 channels. A standard Bell inequality apparatus measures the entanglement quality. On Alice's side, a Soleil-Babinet phase compensator (SB) permits adjusting the phase of the entangled state. L: lens; PC: fiber polarization controller; HWP: half wave-plate; PBS: polarising beam-splitter; &: AND-gate.

Figure 2 .

 2 Figure 2. Polarization dependent emission spectrum of the PPLN/W at 110 • C. The two polarization modes show a near-perfect overlap. Filled curves represent the transmission spectra of the two employed DWDM filters. + and -signs indicate the high and low coupled wavelength transmission windows, respectively.

Figure 3 .

 3 Figure 3. Experimental setup for (a) observing the HOM-dips with and without the PMF, and (b) shifting the phase of the entangled state. In the former situation, the photons states are rotated by 45 • thanks to a fiber PC and sent to an f-PBS for state analysis. Wavelength post-selection is made using two DWDMs centred in the ITU-46 and 47 channels. In the latter situation, the paired photons are deterministically separated at a first DWDM (ITU-46). Photons in the -region are subjected to a variable delay, while those in the + region are subjected to a polarization dependent phase shift introduced thanks to an SB plate. Eventually, the paired photons are recombined at a second DWDM (ITU-47), rotated by 45 • (PC), and sent to an f-PBS for state analysis. L: lens; QWP: quarter wave-plate; M: Mirror; &: AND-gate.

Figure 4 .

 4 Figure 4. HOM-dip measurements with and without the PMF. The vertical dashed line symbolizes zero delay.

Figure 5

 5 Figure 5. HOM dip and peak obtained for two different SB settings. For φ = 0 the |Ψ + still holds and a dip is obtained, as was the case in Figure5. Changing the phase to π turns the state to |Ψ -for which the spatial distribution is expected to have an odd symmetry, therefore leading to a coincidence peak.

Figure 6 .

 6 Figure 6. Violation of the B-CHSH inequalities for the standard settings, i.e. horizontal {H}, vertical {V }, diagonal {D}, anti-diagonal {A}. For all the settings, visibilities exceeding 99% (net) and 97% (raw) are obtained, which underlines the quality of our approach.

Acknowledgement

The authors thank O. Alibart, A. Kastberg, M. P. De Micheli and D. B. Ostrowsky for their help and fruitful discussions. Financial support from the CNRS, l'Université de Nice -Sophia Antipolis (UNS), both the European ERA-SPOT program "WASPS" and the ICT-2009.8.0 FET project "QUANTIP" (grant 244026), l'Agence Nationale de la Recherche project "e-QUANET" (grant ANR-09-BLAN-0333-01), le Ministère de l'Enseignement Supérieur et de la Recherche (MESR), la Direction Générale de l'Armement (DGA), the Malaysian government (MARA), la Fondation iXCore pour la Recherche, and le Conseil Régional PACA, is acknowledged.


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