Jonathan Lenglet
email: jonathan.lenglet@agroparistech.fr

Jean-Yves Courtonne
email: jean-yves.courtonne@inria.fr

Sylvain Caurla
email: sylvain.caurla@inra.fr

Material flow analysis of the forest-wood supply chain: a consequential approach for log export policies in France

Part of the French timber transformation industry suffers from difficulties to adapt to recent changes on global markets. This translates into net exports of raw wood and imports of transformed products, detrimental to both the trade balance and the local creation of wealth. At the same time, no consistent and homogeneous accounts for wood product production and consumption exist at this time. This article first aims at objectifying this situation by undertaking the first material flow analysis of the French forestwood supply chain. We then evaluate the potential consequences of various scenarios of raw wood exports reduction policies, namely subsidies for consumption or transformation and taxation of exports, on both economic outcomes for the different actors and material flows. We thus provide an example of coupling material flow analysis with economic modeling in an attempt to move from the diagnostic phase to the assessment of possible actions within a decision-making perspective.

Introduction

In France, the forest cover is expanding rapidly. In 2016, forests and woodlands cover more than one third of the national territory, making France the fourth most forested country in Europe. The current volume of wood in forest stands is estimated to be 2.6 billions cubic meter and represents a significant ressource potential for forests and timber industries [START_REF] Ign | Le memento inventaire forestier. la foret en chiffres et en cartes[END_REF]). Yet, the economic balance-sheet remains relatively modest and the sector still presents an important trade-balance deficit. Relocating sawmill activities on the national territory is more and more presented as a solution to ensure the provision of sawnwood products while reducing trade deficit [START_REF] Alexandre | Rapport de mission de la déléguée interministérielle à la forêt et au bois[END_REF]. However this raises questions such as which sawmill industries should be relocated, to what extent and what would be the consequences over the rest of the wood sector, in terms of competition, trade-offs and/or synergies.

With this in mind, we strongly believe that any recommendation to relocate sawmill activities, either by stimulating private investments or by implementing public policies, should rely on a comprehensive pattern of wood harvests, flows and uses. The primary goal of this paper is therefore to assess wood product flows in a comprehensive manner, following their pathway from the time the wood is harvested in the forest, along the transformation process, and ending with the consumption of transformed products. We particularly focus on outgoing and ingoing flows. This endeavor typically falls in the field of Material Flow Analysis (MFA), that is, a systematic assessment of the flows and stocks of materials within a system defined in space and time [START_REF] Brunner | Practical handbook of material flow analysis[END_REF].

Recent developments in the industrial ecology (IE) research field emphasize the relevance of MFA methods to obtain a precise understanding of systems and sub-systems functioning. The IE approach aims to provide necessary knowledge to support environmental, economic and social progress in an interactive system of processes and industries (for a detailed history of IE, see [START_REF] Erkman | Industrial ecology: an historical view[END_REF]). Even though the main objective is to enhance sectors eco-efficiency (i.e. wastes and emissions reduction, energy and raw material use optimization), benefits can be broader [START_REF] Burström | Municipalities and industrial ecology: reconsidering municipal environmental management[END_REF]. As stressed by [START_REF] Ehrenfled | Industrial ecology: a new field or only a metaphor[END_REF] and [START_REF] Bringezu | Material flow analysis[END_REF], MFA is now recognized as a scientific support tool for decision makers, notably in the field of environmetal policy. However, within this context, MFA can also contribute to economic and societal development by shifting from a linear to a circular approach, leading to an overall 'win-win-win' outcome [START_REF] Gibbs | Industrial ecology and eco-industrial development: A potential paradigm for local and regional development?[END_REF]. The forest-wood sector offers a good example this multifunctionality. It provides employment, partly non-relocatable, recreation and participate in resource management and development of rural areas.

To our knowledge, so far, only [START_REF] Bais | Global patterns and trends of wood harvest and use between 1990 and 2010[END_REF] have extensively quantified wood flows at the global scale, suggesting that there are still considerable uncertainties about the extraction of wood at the global level and especially of woodfuels. Other identified wood flow studies have focused on regional levels [START_REF] Binder | Transition towards improved regional wood flows by integrating material flux analysis and agent analysis: the case of appenzell ausserrhoden, switzerland[END_REF][START_REF] Mantau | Wood flow analysis: quantification of resource potentials, cascades and carbon effects[END_REF], national levels [START_REF] Hekkert | Analysis of the paper and wood flow in the netherlands[END_REF] for the Netherlands, [START_REF] Hashimoto | Data book: Material and carbon flow of harvested wood in japan. center for global environmental research[END_REF] for Japan, [START_REF] Piskur | Roundwood flow analysis in slovenia[END_REF] for Slovenia, [START_REF] Knaggs | Estimated woodflow for the republic of ireland in 2007[END_REF] for Ireland, [START_REF] Cheng | Spatial and temporal flows of china's forest resources: development of a framework for evaluating resource efficiency[END_REF] for China, [START_REF] Metla | Wood flows in finland 2010. finnish statistical yearbook of forestry[END_REF] for Finland, [START_REF] Weimar | Der holzfluss in der bundesrepublik deutschland[END_REF] for Germany, [START_REF] Parobek | Analysis of wood flows in slovakia[END_REF] for Slovakia), as well as on the European Union as a whole [START_REF] Mantau | Wood flows in europe (eu27)[END_REF]. Such studies are currently lacking in France where no consistent and homogeneous accounts for wood product production and consumption exist at this time. Forest management and wood extraction are generally treated separately from wood transformation (which is itself organized into many different federations). Different units are used in official reports (cubic meters of roundwood, cubic meters of product, metric tons, number of units produced) making it difficult to cross-check the data. In addition, large uncertainties prevail in some existing databases, especially regarding woodfuels that are known to be largely under-reported in official statistics [START_REF] Chevallier | Etude de faisabilité pour l'évaluation des flux de matière de biomasse[END_REF]. This article is a first attempt to fill this gap and provide a comprehensive view of the flows in the French forest-wood supply chain. One limitation of MFA that is usually pointed out concerns the lack or the availability of data and its high degree of uncertainty [START_REF] Schwab | A data characterization framework for material flow analysis[END_REF][START_REF] Patricio | Uncertainty in material flow analysis. indicators at different spatial levels[END_REF]. To tackle this issue, we propose to rely on the MFA framework depicted in [START_REF] Courtonne | Downscaling material flow analysis: the case of the cereals supply chain in france[END_REF], which uses the wood fiber equivalent (presented in Section 2) as a basic accounting unit.

The second goal of this paper is to assess the impacts of the relocation of sawmill activities on other forest sectors such as the fuelwood and pulp sectors, as well as on the economic outcomes for the sector agents. To address this issue, we couple the MFA with a partial equilibrium forest sector model, the French Forest Sector Model [START_REF] Lobianco | The importance of introducing spatial heterogeneity in bio-economic forest models: Insights gleaned from ffsm++[END_REF]Caurla et al., 2013b,a;[START_REF] Lecocq | Paying for forest carbon or stimulating fuelwood demand? insights from the french forest sector model[END_REF] and we explore the economic consequences of three policy options that reduce roundwood export.

By combining a MFA and a partial equilibrium framework, this paper makes it possible to bridge the gap between the diagnosis phase and the recommendation phase for policy makers through a multi-scenarios analysis. Indeed, as [START_REF] Binder | From material flow analysis to material flow management part i: social sciences modeling approaches coupled to mfa[END_REF] and [START_REF] Binder | Implementing the results of material flow analysis[END_REF] recall, the impact of MFA on policy making is not clear since most studies stop at the diagnostic phase. We propose to go beyond the diagnosis in order to provide policy relevant information by using a partial equilibrium model that projects the evolution of the forest sector at the same scale and for the same products as those described in the MFA. Compared to [START_REF] Bouman | Material flows and economic models: an analytical comparison of sfa, lca and partial equilibrium models[END_REF] who focused on how different approaches (LCA, MFA and partial equilibrium model) independently contribute to elucidating a research question, we undertake the coupling of MFA with economic modeling. Note that in the title, we refer to this as a consequential approach in reference to consequential Life Cycle Assessment (LCA) that, contrary to attributional LCA, models economic links between processes and is therefore able to account for the indirect effects of a change. In practice, this generally translates into nonlinearities, because of the existence of thresholds or scaling effects.

The article is organized as follows. The first section is dedicated to the presentation of the data sources, hypotheses and methodologies used, both for the MFA and the parts dealing with economic prospectives. Results are presented and discussed in a second section before concluding on the main findings of the study and on future research perspectives within the context of a transition to sustainability.

Materials and methods

Material Flow Analysis (MFA) 2.1.1. Boundaries of the study

We focus on the upstream part of the forest-wood supply chain, which is the most relevant to our research questions. Firstly, this makes it possible to analyze to what extent the country is exporting raw products and importing products of first transformation. Secondly, it provides information on log outlets, a key question, for example, being: is there a reason to worry that firewood is cannibalizing log resources at the expense of the lumber industry? As a consequence, the model implemented does not aim to represent some second or third transformation industries such as the furniture industry. More precisely, our MFA covers all log products (distinguishing hardwood from softwood), wooden pallets, sawnwood, veneer, plywood and wood panels, as well as energy wood, sawmill residues, pulp and paper products and niche products such as wood staves. A full list of the products considered is presented in Figure 6 in the supplementary material. The geographic boundary of the study is metropolitan France.

Reference unit

To make the MFA consistent we need to define a homogeneous unit for the entire wood chain. This is all the more important given that wood products data are reported in various units. [START_REF] Bösch | Physical input-output accounting of the wood and paper flow in germany[END_REF] note that when quantities of wooden products are converted into the same unit in order to compare them, the cubic meter of roundwood equivalent is often used. It is defined as the volume of roundwood necessary to make one unit of processed product. However, we share their opinion that it cannot be directly used in MFA studies since it leads to double-counting of by-products such as sawmill residues. Using cubic meters or tonnes also raises consistency issues since other constitutive materials such as adhesives in panels or additives in paper products are then considered to be wood, whereas, volume changes due to drying would not be accounted for.

We therefore use the wood fiber equivalent as the reference unit. The unit is used in comparable studies in other european countries [START_REF] Mantau | Wood flows in europe (eu27)[END_REF]Bosch, 2015 ;Weimar, 2011). The name "m 3 of fiber equivalent" is taken from [START_REF] Weimar | Der holzfluss in der bundesrepublik deutschland[END_REF] and [START_REF] Bösch | Physical input-output accounting of the wood and paper flow in germany[END_REF], while [START_REF] Mantau | Euwood -real potential for changes in growth and use of eu forests[END_REF] and the forest department of UNECE/FAO (UNECE and FAO, 2010) use the term "solid wood equivalent" to refer to the same thing. The main advantage of using m 3 of fiber equivalent compared to mass unit is that it is equivalent to m 3 of roundwood in the case of greenwood (e.g, wood increment, harvest)1 .

The wood fiber equivalent (refered to as m 3 [f] in the text) is the volume of the wood fibers that are contained in the product at the fiber saturation point [START_REF] Weimar | Der holzfluss in der bundesrepublik deutschland[END_REF]. For each wood-based product (e.g., fiber board, plywood, pulp), a specific conversion factor is calculated depending on its primary measurement units. A section of the Supplementary Material is dedicated to a presentation of the formula and hypotheses used for this unit, with Figure 6 providing conversion factors for each product studied. Although similar in principle, our theoretical formula slightly differs from the one provided by [START_REF] Weimar | Der holzfluss in der bundesrepublik deutschland[END_REF] in order to actually provide results expressed at the fiber saturation point. However, in practice, the lack of data (particularly concerning moisture rates) leads us to use the same factors for many products.

Data sources

Table 2.1.3 summarizes the sources used to build the MFA model. Data is taken from a number of different sources, which makes it very unlikely that it will "fit" the mass conservation law. However, to take advantage of all of the information available, we used a data reconciliation technique inspired by [START_REF] Cencic | Material flow analysis with software stan[END_REF].

Uncertainty of input data and period of study

Data quality is assessed either by directly retrieving precision indicators from the statistic sources, or, in most of the cases, on expert judgments based on the way data is built (source, type of survey or poll, scope of the study). Note that what matters the most for MFA results is the relative difference between data sources (e.g. source a is twice as precise as source b). Estimated 95% confidence intervals range from 1.4% (forest primary production) to 50% (wood logs used for individual heating, in official circuits).

Additionally, we chose to study the annual average between the years 2009, 2010, 2011, 2012 and 2013. This presents at least two advantages: first, it reduces the uncertainty (of data from polls), and second, it allows us to neglect stock variations, which would be very difficult to quantify otherwise. Indeed, in this type of capital-intensive industry, stocks of roundwood and final products are minimized to avoid high additional storage costs. Due to the high heterogeneity in timber sales practices (private and public, contacts and auctions) and the significant rate of informal transactions and re-selling of roundwood, it is not feasible to get information on stocks on any scale [START_REF] Marty | A socio-economic analysis of french public timber sales[END_REF]. In this study we consider that a fiveyears average is the best compromise for reducing the bias while remaining representative of the situation over a relatively short period of time.

Item

Supply and use tables and data reconciliation

Physical Supply and Uses Tables (SUT) have a long tradition of being used in the fields of ecological economics and industrial ecology OECD (2008). They consist of product-industry matrices indicating:

• which local industries and/or foreign territories are supplying the product to the studied system (Supply Table ),

• which local industries and/or foreign territories are using the product available in the studied system (Use Table ).

Data reconciliation is used to phase out inconsistencies. The model imposes mass conservation laws, which implies constraints on the supply-use tables. In particular, supply equals use constraints must be verified for each product and for each transformation industry (the materials that enter an industry must also exit this industry, regardless of the products they are embodied in) as well. Equations of the constraints are provided in the supplementary material. As we explained before, these constraints are unlikely to be met by input data so we resort to data reconciliation. While details are provided in supplementary material, the principle can be summarized as follow: in order for the constraints to be fulfilled, output data (results) are allowed to differ from input data but the less uncertainty there is about the input data the smaller the distance between output and input data will be. SUTs at the level of France are provided in the supplementary material both before and after the data reconciliation process (Figures 5 and 6 in supplementary material).

Uncertainty propagation

We use Monte Carlo simulations to evaluate uncertainties of output results. As stated before, we assessed uncertainty of each input data. We use a Gaussian distribution hypothesis and expressed it in the form of 95% confidence interval (i.e. 2 standard deviations)2 .

• Step 1: for each input variable (e.g., harvest of hard roundwood, production of hard sawnwood) we draw a realization based its Gaussian distribution3 . Given i input data, we therefore have a random set of i realizations.

•

Step 2: this random set is used as input of the reconciliation process and leads to an output set, for instance composed of j variables.

• Step 3 : steps 1 and 2 are repeated a sufficient number of times n (typically in our case, about 1000 times4 ).

•

Step 4: for each of the j output variables we analyze the output distribution (n realizations): output distributions are Gaussian as well and we use the standard deviation to estimate the 95% confidence interval of the output variable (2 standard deviations).

French Forest Sector Model (FFSM)

2.2.1. An overview of the model FFSM is a bio-economic recursive model of the French forest sector. The version used in this article is FFSM 1.0 (Caurla et al., 2013b,a; Lecocq et al., 2011), 5 . This version is composed of two modules: an inventory-based dynamics forest module (FD) and a partial equilibrium market module (MK). At each period (year), the MK module computes all market equilibria in the forest sector, from which it deducts the annual harvest. Harvest is then entered into the FD module, which computes available timber resources at year t + 1. In turn, these results are then entered into the MK module, and so on (see Fig. 1). To be consistent with our MFA, the base year is 2009 and the model runs until 2020. (Caurla et al., 2013b). The time horizon is 2020.

FD module

The FD module represents the dynamics of the French forests that accounts for natural growth and mortality [START_REF] Colin | Rapport technique module biologique LEF[END_REF] and wood removals from human harvest. It simulates timber stock dynamics using a diameter-class approach. Since French forests are very diverse in terms of climate, soils, species and types of management, the FD module breaks down the timber resource into 1716 cells that differ by region (22 administrative regions), type of management (high forests, coppice, mixed), species (coniferous and broadleaved) and diameter classes (a total of 13). Resource dynamics in each cell are calibrated using data from the 20052007 French forest inventories [START_REF] Colin | Rapport technique module biologique LEF[END_REF].

Between two successive years, the dynamics of wood volume for a specific cell follows the equation:

V u,t+1 = V u,t × (1 -tp u -m u -h u,t ) + V u-1,t × β u-1 (1)
where V is the volume of wood, u is the diameter class, t is the time period (year), tp u is the time of passage for a tree in diameter class u, m u is the rate of natural mortality in diameter class u, h u,t is the harvest rate (derived from the economic module of the FFSM), and β u-1 is a coefficient that accounts for the height growth of trees when upgrading from class u -1 to class u. . . .

MK module

Armington, 1969

French administrative French administrative

Figure 2: The economic module of the FFSM represents three levels of the forest sector and distinguishes nine products (Caurla et al., 2013b) The MK module solves, for each French administrative region, a partial equilibrium problem according to [START_REF] Samuelson | Spatial price equilibrium and linear programming[END_REF] spatial price equilibrium framework. To do so, it annually computes (1) the optimal demand of processed products p: sawnwood (hard and softwood, respectively referred to as hsw and ssw), plywood (pw), fuelwood (fw), pulp, panels and (2) the optimal supply of raw products w: roundwood (hard and softwood, respectively referred to as HRW and SRW) and industrial wood (IW) and (3) the optimal prices associated to these quantities. This equilibrium explicitely accounts for wood availabilities in forest, for the costs of transformation in processing-indsutries and for the costs of transportation from one region to another into account following [START_REF] Samuelson | Spatial price equilibrium and linear programming[END_REF] theory. In addition, international trade is computed through an Armington framework assuming that domestic and foreign products are not fully substitutable. The international trade is modeled using exogenous international prices derived from the FAO (2013) and elasticities of substitution between local and international products specifically estimated for FFSM in [START_REF] Sauquet | Estimating armington elasticities for sawnwood and application to the french forest sector model[END_REF].

The regional supply S of primary forest products w in region i takes the form of Eq. 2, where w is the elasticity of supply to price P w,i,t and β w is the elasticity of supply to available resources in forest. Both elasticities are derived from Buongiorno et al. (2003) and are exogenous in the context of this study.

S w,i,t = S w,i,t-1 P w,i,t P w,i,t-1 w F w,i,t F w,i,t-1 β w (2)
Likewise, in each region i, the demand curves D of processed products p use exogenous elasticities σ p of demand to price (specifically estimated for FFSM in [START_REF] Sauquet | Estimating armington elasticities for sawnwood and application to the french forest sector model[END_REF].

D p,i,t = D p,i,t-1 P p,i,t P p,i,t-1 σ p (3)
Eventually, a Leontief function represents the transformation of raw products into processed products6 (see Table 4 in Appendix). Compared to a single Leontieff matrix, FFSM makes thus it possible to deal with nonlinear relationships between the supply and demand levels and the prices of wood products.

Consistency with MFA and input-output calibration in FFSM

The number of products represented in FFSM is lower than in MFA analysis. To make the modeling framework consistent, results from MFA are aggregated to match the definition of products in FFSM. Inputoutput coefficients used in the Leontief function of FFSM are then calibrated using MFA results for each group of products. They are presented in Table 4 in the Supplementary Material.

Prospective: simulating policies to relocate sawmill activities

Several papers assess the impacts of export restrictions on global and national forest sectors. Using a global log-lumber trade model, Van [START_REF] Van Kooten | Global impacts of russian log export restrictions and the canada-u.s. lumber dispute: modeling trade in logs and lumber[END_REF] show that moving an export tax at the national level (Canada or Russia) has a small impact on the global forest sector welfare. Using data for the State of Washington, [START_REF] Johnson | Expanding u.s. log export restrictions: impacts on state revenue and policy implications[END_REF] suggest that domestic mills and consumers clearly benefit from the ban while the welfare of Japanese consumers and the export industry is reduced. The same conclusion holds in [START_REF] Van Kooten | Global impacts of russian log export restrictions and the canada-u.s. lumber dispute: modeling trade in logs and lumber[END_REF] for British Columbia (BC). They report that, while free trade of logs might maximize global wellbeing, it might also reduce the domestic welfare of economic forest sector agents. While these papers clearly suggest that export bans are beneficial to the domestic forest sector, neoclassical economy judges them inefficient at the global scale [START_REF] Goodland | If tropical log export bans are so perverse, why are there so many?[END_REF]. However, these studies focus on the impacts of a restriction without analyzing the different design of policy options to reduce export and their implications on forest sectors. This is precisely the aim of the prospective part of this paper. To do this, we tested three alternative policies. Two of them consist in subsidizing transformed products made of roundwood: Sub C is a subsidy to domestic sawnwood and plywood consumption, while Sub T is a subsidy to sawnwood and plywood transformation. The other policy, referred to as Exp, is an extra-cost to roundwood export. Sub C and Sub T differ from their transmission channels: while Sub C acts through processed product consumers, Sub T directly irrigates the upstream and the downstream of the roundwood sector through the reduction of transformation costs.

More precisely, Sub C is introduced in Eq. 3 such as:

D p,i,t = D p,i,t-1 P p,i,t × S ub C p,i,t P p,i,t-1 × S ub C p,i,t-1 σ p (4)
Sub T is introduced in the transformation agents surplus, in the objective function of FFSM, decreasing C p,w , the cost of transforming products w in products p.

Exp acts differently and models an extra cost for plant-health control and preventive treatment. Such a policy already exists for wood packaging material [START_REF] Strutt | Evaluating the impacts of an international phytosanitary standard for wood packaging material: Global and united states trade implications[END_REF] as a tool to limit the diffusion of pests and diseases. We assume that this policy costs 10 e/m 3 which is close to the export tax equivalent imposed by Canada on its exports to the U.S. (Van Kooten and Johnston, 2014) and which is considered as plausible by the French government. In the model, this policy is introduced indirectly by increasing international prices by 10 e/m 3 . This value leads to a reduction of roundwood log exports of 20% regarding a baseline scenario without policy. To make the comparison between policies meaningful, we based them on the same level of export reduction derived from the Exp policy result (-20% for roundwood, either hardwood or softwood). We calibrated the levels of subsidies Sub C and Sub T consistent with this objective by a trial and error process (55% and 110%, respectively).

Results

Wood flow analysis

MFA results are presented in two figures: Figure 3 provides a global view of the upstream part of the supply chain (forest stock, increment, harvest, self-consumption of households, etc.) and Figure 4 focuses on the transformation of lumber and industrial and pulpwood 7 . Every flow comes with a figure that specifies the quantity of wood in m 3 (f). Besides, following [START_REF] Schmidt | The sankey diagram in energy and material flow management[END_REF] we specify the volumes of import and export for each product. They are represented as vertical input (import) and output (export) flows on the Sankey diagrams. Confidence intervals are also represented as well as some quantities in a more intelligible unit, e.g., paper production is expressed in tonnes.

Six main outcomes stem from the diagrams. First, Figure 3 shows that, at the national level, the stock of standing wood is increasing since the total of harvested wood (including self-consumption) represents about half of the annual increment. Although this result is in line with figures published elsewhere [START_REF] Colin | Disponibilités forestières pour l'énergie et les matériaux à l'horizon 2035[END_REF], it must be qualified since all biological production at the national level cannot be harvested. Indeed, some species or diameters do not have a commercial outlet and some areas, such as mountainous ones, are inaccessible. This can explain the paradoxical situation encountered in many French regions where forest stocks increase while wood harvest seems to reach a ceiling. In addition, as suggested by its wide confidence interval, the production of energy wood remains difficult to assess. The majority of energy wood flowing in anthropogenic activities is composed of self-consumption energy wood. These self-consumption volumes are estimated by [START_REF] Pouet | Etude sur le chauffage domestique au bois : marché et approvisionnement[END_REF] based on a poll taken on some 1,500 households. Even if the total household consumption probably gives a fair idea of the situation, quantifying the share of the different firewood supply chains is more difficult since some suppliers may have an interest 7 The global input flow in Figure 4 corresponds to the combination of all roundwood and pulpwood/industrial wood flows on Figure 3. in under-reporting their sales. In addition, the core business of energy wood suppliers is often logging or sawmilling, which makes the identification of energy wood flows even more complex (INSEE, 2014). Second, although the standing wood resource at national level is mostly composed of broadleaves (about 63% [START_REF] Fcba | Mémento[END_REF]), it appears that HRW represents 9.5% of log volumes, wheareas SRW accounts for 28%. This result is explained by the higher adequacy of softwood with the process industry requirements. Indeed, sawmills consume about four times more SRW than HRW, and the panel and paper pulp industries nearly twice as much.

Third, France is a net exporter of logs. This is all the more true for hardwood. Exports represent 22% of total HRW and 7% of total SRW. France is also a net exporter of pulpwood since about 15% of the pulpwood harvest is exported.

Fourth, Figure 4 suggests that ssw represents the largest production in volume closely followed by panels production. Half of the sawmill residues are used in the pulp and panel industries as raw material, which reveals a well-established recovery chain and potential competition with new additional fuelwood demand. The pulp and paper sectors have a large recycling chain: more than 80% of papers consumed are sorted to be recycled, representing about 55% of the resources of the paper industry (expressed in volume of wood fiber equivalent). These sectors determine the economic value of the wood sector in France. It can be noted that France also exports large quantities of paper wastes that are therefore not valued in the country.

Fifth, Figure 7 in the Supplementary Material shows that niche products such as stave woods and plywood, even if negligible from a volume point of view, remain important from a value-added perspective. Conversely, the weight of sawmill residues is strongly reduced. Note that these niche products are mainly produced from HRW and could become interesting outlets from the point of view of export reduction.

Sixth, from a trade balance perspective, we can notice that panels and paperboards are largely exported (resp. 60% and 54% of the total production is exported). Hardwood sawmilling is also a net exporter, which shows a possible commercial outlet for producing hsw in France for export. On the contrary, 28% of total ssw consumed in France is imported which shows an inadequacy between local supply and demand. Imported products are likely to be species and/or diameters that are not produced in France (even if data cannot corroborate this assumption).

These results mask important regional differences. For example, about half of the wood harvest takes place in only five regions (of the 22 French regions). These regions are not analyzed in this article, which focuses on diagnosis and prospective for the national forest-wood supply chain, although they constitute major leads for future work. We will come back to this question in the conclusion.

Prospective

We assessed the impacts of the policies described above on two groups of economic indicators: (i) the variations in prices and quantities produced, consumed, imported and exported, and (ii) the budgetary cost and the variations in economic agent surpluses. Variations are given for the year 2020. Sankey diagrams outlining the potential impact of each policy on material flows are proposed in the Supplementary Material (Figures 8, 9 and 10).

Impacts on quantities

Table 2 shows the impacts of the three policies on production and consumption of wood products in the sector. Three outcomes stem from these results. First, while Sub C and Sub T largely modify production and consumption patterns, Exp does not have a major impact on domestic wood flows. It only slightly reduces HRW and SRW production. This is because Exp acts via an extra cost on export without modifying domestic prices and domestic agents' behaviors. As a consequence, total roundwood production decreases because of export reduction but this does not lead to a rebound effect on the domestic market. Second, while Sub C and Sub T have almost the same impact on the production of raw products upstream, their impacts on consumption patterns differ. In fact, the impacts of Sub C mostly depend on the price elasticity of the transformed product demand and the price elasticity of the primary product supply. Since the demand of hsw (-0.5) is more elastic than that of ssw (-0.4) and pw (-0.3) and the supply of HRW is more elastic than that of SRW (1.2 vs. 0.9), Sub C has a relatively bigger impact on hsw than ssw. For its acts, Sub T plays via the costs of transformation so that the higher the cost of transformation is, the higher the impact of Sub T will be. This explains the higher impact on plywood (cost of transformation: 534 e/m 3 ) than on hardwood sawnwood (268 e/m 3 ) and on softwood sawnwood (48 e/m 3 ). Third, although Sub C and Sub T are subsidies of the sawnwood and plywood sectors, they also impact fuelwood and, to a lesser extent, the pulp and panel sectors via the increase in production of sawmill residues. This implies a crowding-out effect with IW, which results in a reduction of IW production (-20% and -18%, respectively) that is not observed with Exp policy.

Regarding impacts on exports and imports, Table 3 shows that Sub C increases imports of hsw, ssw and pw in a proportion depending on the price elasticity of the demand. Sub C therefore creates an important windfall effect. On the other hand, Sub T decreases imports of ssw and pw and does not affect imports of hsw. This is because Sub T only acts on domestic production costs, which makes domestic products more competitive than foreign substitutes.

Variation of surpluses and budgetary costs

Table 4 shows that while the two subsidies increase all surpluses except the surplus related to IW production Exp reduces surplus related to HRW/SRW productions, slightly increases surplus related to IW production and has no significant impact on surpluses related to consumption activities. In the case of Sub T, the driver is the reduction of transformation costs that reduces the consumer price and increases the producer price and, as a consequence, increases both the quantities produced and consumed at the national scale. This results in overall gains in economic agents' surplus. In the Sub C case, the consumer-perceived price is reduced which leads to a higher demand and, therefore, an increase in production upstream. In that case, however, part of the additional demand is fulfilled by imports, as shown in the previous section, which do not increase domestic agents' surplus. Both subsidies have a negative impact over industrial wood activities since they lead to an additional production of sawmill residues, leading to a crowding-out effect with industrial wood upstream.

Table 5 shows that while Sub C is the subsidy leading to a higher welfare benefits, it is also the most expensive. When looking at the social costs, Sub T performs better than Sub C since Sub T does not increase imports and does not not imply a windfall effect. Budgetary costs of Exp are 0 since FFSM does not take transaction costs into account.

Discussion about the results

Results show there is no first best policy option. From a policy-making perspective, the implementation of a policy to reduce roundwood export must consider the different levels of impacts, namely the welfare variations for each economic segment, the total welfare costs, and the total budgetary costs. Sub C leads to the highest welfare gains for economic agents (except IW suppliers) but is the most costly for the government budget because of the windfall effect. Sub T follows a comparable surplus gains pattern while being half as expensive. Exp policy is not beneficial to wood sector surplus but is the least costly if we assume that the transaction and functioning costs are covered by the payment made by roundwood exporters. In addition, it is the only policy in which the IW sector remains unchanged. According to these results, the only possible solution for (1) reducing the crowding out effect between sawmill residues and industrial wood competition with the pulpwood sector, and (2) increasing welfare surplus in the forest sector while (3) preventing a windfall effect would be to combine Exp with Sub T. Figure 11 in the Supplementary Material shows the effect of a range of combinations between these two policies on social cost. However, as outlined before, social cost is not the only criterion to be considered. The choice of a policy mix should imply a debate among all stakeholders impacted by the policy (e.g., including citizens paying for the potential subsidies). We believe that they alone can express which criteria matter the most and which efforts they are willing to make.

In addition, since FFSM has no endogenous growth module, it cannot represent the impacts of subsidies on capital accumulation. Yet, it is very likely that, in reality, when looking at the long term, the net social costs of policies would diminish over time as capital investments increase. At one point, the net social costs may even become negative.

Conclusion

Our paper fills two important gaps. It is the first in the literature to draw a comprehensive picture of wood flows in the upstream part of the wood sector in France that takes all available data sources into account. To do this, we built an MFA model using a reconciliation process and a homogeneous unit to make the different sources consistent and comparable while quantifying uncertainty. Second, it assesses the economic impacts of three policy alternatives to reduce exports of logs by using FFSM, a partial equilibrium model. Impacts on material flows are evaluated as well. By combining an MFA and an economic modeling framework, our analysis goes beyond diagnosis and proposes insights into policy-makers' recommendations.

We demonstrated the relevance of MFA to reconcile highly heterogeneous datasets. Since no common database exists in the forest-wood sector, these results offer a solid base for analysis and opportunities for development prospects. However, attention must be paid to the temporal aspects of the study. Our MFA provides a snapshot of the situation for a specific reference year and does not capture trends. For instance, concerns on raw wood leaks are not only due to current export levels but also, and maybe mostly, to increasing exports trends and sawmill shut-downs. Thus, it would be relevant to repeat the study according to a predetermined time-step (e.g., 5 years using either a fixed or sliding average), using the same methodology. Comparison across time of wood flow dynamics may be encouraged as a tool to study the wood-forest sector's health and as a powerful monitoring instrument for policy makers. The systematic framework implemented here will hopefully foster this endeavor.

MFA results suggest that while broadleaves dominate conifers in French forests, production of HRW is three times lower than that of SRW. Meanwhile, exports represent 22% of the total HRW production but only 7% of SRW production. This leads to an apparent paradox: the resource is abundant and potentially underharvested but the sector is unprofitable since part of the value added is created abroad. Nevertheless, the MFA suggests that both hsw production and niche products such as stave wood are a potential commercial outlet for domestic HRW. Yet, as long as the international demand for logs and especially broadleaf products is the driving force behind exports, production of HRW will be oriented toward exportation and investments in the sector will be postponed, exacerbating the trade balance deficit. With this in mind, as the second step of our analysis, we tested three policy options to reduce log exports: a subsidy to consumers, a subsidy to process industries and an additional costs to export due to phytosanitary control, which acts as a ban in the model. The potential effects of these policies help to specify several criteria of interest to aid the decision process: budget costs, welfare variation in the forest-wood sector and redistribution of this welfare among agents. These indicators could also serve to assess potential impacts on local employment given that labor intensiveness differs between sectors. Moreover, a healthier, more dynamic supply chain would arguably make it easier to reach national climate mitigation goals. These last two points would require more in-depth studies but seem crucial if we are to have a holistic view of the sector. While mono-criterion optimization approaches (e.g., optimizing cost-effectiveness) are often used to make policy recommendations, we believe that multi-criteria analysis would be better suited to study possible alternatives. This approach would go hand-in-hand with a decision process involving all of the stakeholders, that is, ideally, anyone affected by the decision, where they are asked what they value most and what compromises they are willing to accept and support.

Although this was not the purpose of this article, we know that the forest-wood sector is highly geographically heterogeneous. Therefore, similar studies at regional levels would be beneficial. Rather than a series of individual regional assessments, we would argue for a coherent multi-regional assessment, that is, disaggregating the national MFA and economic model to provide evaluations in all regions. Regional strategies, e.g., specialization strategies, could then be revisited both in terms of their internal effects and consequences on other regions. Interestingly, there is no guarantee that the same policy would be preferred, regardless of whether the focus is put on the national level or on the regional level. We think that a multi-regional approach will therefore be a major direction for future research. Nevertheless, one limitation that should be kept in mind when reproducing such a study is the possible bias induced by the scale effect. Indeed, working at a national scale and on a five-years average reduces the impact of potential extreme regional events (such as forest fires, storm or pest outbreaks). This homogeneizing effect would not be relevant at a lower level, for example when downscaling the computation at regional level and may lead to faulty conclusions due to misinterpretation. It is thus necessary to put the MFA results in a meaningful and broader context when looking at local flows arrangement. This should stand for any local MFA. Table 7: Input-output coefficients used in the Leontief function of FFSM, calibrated using MFA results. These coefficients represent the quantity of raw products (HRW, SRW, IW) required to produce one unit of transformed product (fw, hsw, ssw, pw, panels, pulp). Some industries are assumed to produce no waste residues (fw, panels) according to MFA. Plywood is assumed to be made of either HRW or SRW.

Equations of the Material flow analysis

This formalism was already presented in [START_REF] Courtonne | Downscaling material flow analysis: the case of the cereals supply chain in france[END_REF]. The optimization function has however been adapted for the present study as we explain below. As shown above, the supply table comprises the supply matrix V and the imports vector I whereas the use table comprises the use matrix U, the exports vector E and the (final) consumption vector C. For instance U i j refers to the quantity of product i that is used by sector j and C i refers to the quantity of product i that is consumed by end-users on the territory. These tables are subject to constraints (we use hats above letters to refer to data resulting from the reconciliation process, original data are represented without a hat):

• Constraint 1: mass conservation of each product (note that stocks are neglected because we are working on a five-years average): j Vi j + Îi = j Ûi j + Êi + Ĉi i = 1..n (5)

• Constraint 2: mass conservation in every transforming industry:

i Vi j = i Ûi j j = 1..m and j is a transforming sector (6)

• Constraint 3: Technical conversion factors between products (e.g. rate of sawmill residues),

• Constraint 4: All flows have positive values,

• Constraint 5: Matrices V and Û are sparse (mostly empty), for instance the wood panel industry cannot produce nor consume sawnwood.

Note that constraint 1 does not apply to standing wood in forest to allow the increase of final stock in forest and natural mortality.

Our goal is to minimize the discrepancy between original data and estimated/output data while respecting the constraints. This problem can be expressed in many ways depending on the expression of the distance between original and final data. We first used a classic weighted least square optimization: ) and monetary flows on the right (in Me). Monetary flows are estimated based on average unit prices for each product category. We can see niche products as wood stave and plywood mobilize little material flows but generate significant added value. Conversely, products of lesser quality such as pulpwood and sawmill residues appear a lot less significant in monetary terms.

Figure 11: Evolution of social cost depending on the Exp -Sub T policy mix. The graph represents social costs for 11 combinations that all lead to the same decrease in export of 20 %. Only Exp value is presented on the X-basis, the level of the Sub T is then chosen by a trial and error process. Results shows a U-shaped curve with a minimum for Exp= 5 e/m3 and Sub T=80%. The net social cost is then 7 Me, which is smaller than the Exp policy alone and which makes this combination potentially interesting from a cost-efficiency perspective.

Figure 1 :

 1 Figure1: The FFSM is based on a recursive and modular framework(Caurla et al., 2013b). The time horizon is 2020.

Figure 3 :

 3 Figure3: Sankey diagram of the upstream part of the forest-wood MFA. Flows are expressed in Mm 3 (f) and correspond to a one-year average during the period 2009-2013. Wood stock is not represented at the same scale as the other flows. Besides, initial stock and its associated uncertainty is not an output from the model (that does not handle stocks) but is directly retrieved from IGN.

Figure 4 :

 4 Figure 4: Sankey diagram of the downstream part of the forest-wood MFA. Flows are expressed in 1000 m 3 (f) and correspond to a one-year average during the period

Figure 5 :

 5 Figure 5: Input SUT for the wood MFA (France, expressed in 1000 m 3 (f).)

Figure 7 :

 7 Figure 7: Comparison of material flows on the left (in 1000 m 3 ) and monetary flows on the right (in Me). Monetary flows are estimated based on average unit prices for each

Table 1 :

 1 Data sources used for the MFA on wood products.

		Source	Comments
	Biological increment / Total	French Geographic Institute	Year average for the 2009-2013 pe-
	harvest / Mortality	(IGN)	riod. Available for each department
			(nuts 3 level)
	Harvest (commercial) and	Annual agricultural statistics	Available for each department and
	harvest residues	(SSP)	for groups of species (coniferous
			and broadleaves) and groups of uses
			(roundwood, pulpwood and energy
			wood)
	Firewood self-consumption	Ademe (2013)	Survey on household wood con-
			sumption
	Production of sawnwood	Annual agricultural statistics	Available for each department and
	and residues	(SSP)	for groups of species (coniferous
			and broadleaves)
	Production of other trans-	Annual reports from federa-	Available at the country level, quan-
	formed products	tions (e.g. FCBA)	tities reported in specific units
	Consumption of paper / pa-	Federation of paper produc-	Available at the country level
	per recycling rate	ers
	Outcome of wood wastes	INSEE	The order of magnitude is highly
			variable among available reports
			underlining the fact that these quan-
			tities are not known with precision.
			For this study, we chose the official
			Insee statistics.
	International trade	SitraM database (based on	Available for 382 positions
		French customs)	(NST 2007),	department of
			origin/destination,	country of
			destination/origin.

Table 2 :

 2 Quantities produced (P) or consumed (C) in 2020 compared (variations compared to the baseline scenario) in Mm 3 [f].

		HRW	SRW	IW (E)	fw (I)	hsw (I)	ssw (I)	pw (I)	pulps	panels
		(E)	(E)						(I)	(I)
	Baseline 1.13	1.06	1.61	0	0.16	2.45	0.32	3.77	2.02
	Sub C	1.26	1.04	0.13	0	0.05	1.09	0.15	1.06	0.91
		(-20%)	(-20%)	(-3%)	(-2%)	(+83%)	(+42%)	(+29%)	(-1%)	(0%)
	Exp	0.85	0.85	0.17	0 (0%)	0.15	2.12	0.3	1.14	0.94
		(-20%)	(-20%)	(0%)		(0%)	(0%)	(0%)	(0%)	(0%)
	Sub T	1.23	1.02	0.14	0	0.06	1.55	0.06	1.08	0.92
		(-20%)	(-20%)	(-3%)	(-2%)	(0%)	(-7%)	(-24%)	(-1%)	(0%)

Table 3 :

 3 Exports of raw products and imports of transformed products in 2020 (variations regarding baseline without policy) in Mm 3 [f].

		HRW	SRW	IW (P)	fw (C)	hsw	ssw (C) pw (C)	pulps	panels
		(P)	(P)			(C)			(C)	(C)
	Sub C	+93%	+41%	-30%	+7%	+47%	+35%	+46%	+2%	+1%
	Exp	-10%	-6%	+1%	0%	0%	0%	0%	0%	0%
	Sub T	+88%	+36%	-25%	+6%	+34%	+13%	+34%	+1%	+1%

Table 4 :

 4 Variations in surpluses (C: consumers surpluses; P: producers surpluses) regarding baseline scenario without policy in 2020.

		Total welfare vari-	Budgetary	costs	Social costs (bud-
		ations compared to	(Me)		getary costs -Total
		baseline (Me) (+			welfare variations,
		means a net gain, -			Me)
		means a net loss)
	Sub C	+2248	+3000		+752
	Exp	-33	0		+33
	Sub T	+1448	+1653		+205

Table 5 :

 5 Impacts on welfare, budgetary costs and social costs.

Table 6 :

 6 This work was supported by the French Ministry of Agriculture (Programme 215 du MAAF). The authors thank Mélanie Lucas from DRAAF for her valuable comments. The authors thank Alexandra Niedzwiedz from AgroParisTech for her support in data management. Conversion factors (to wood fiber equivalent (m 3 (f))) used in the study. Products marked with a * are taken fromWeimar (2011), the others are provided by the authors.

	Product	Original unit	Conversion factor (m 3 (f) / original unit)
	Standing timber	m 3	1
	Lumber	m 3	1
	Roundwood (imported or	Tonnes	softwood: 1.05, hardwood: 1.12, hy-
	exported)		pothesis: greenwood density
	Pulpwood	m 3	1
	Fire wood	m 3	1
	Forest wood chips	m 3	1
	Ties	m 3	1
	Staves	m 3	1
	Sawnwood	m 3	1
	Sawnwood (imported or ex-	Tonnes	softwood: 1.36, hardwood: 1.75, hy-
	ported)		pothesis: 15% moisture
	Sawmill residues	Tonnes	1.1
	Sawmill woodchips	Tonnes	1.1
	Veneer	m 3	1
	Particle board *	m 3	1.25
	Fibreboard *	m 3	1.47
	MDF board *	m 3	1.39
	Plywood *	m 3	0.96
	Wood floor	m 2	0.015
	Pallets and wood packaging Tonnes	1.1
	Mechanical paper pulp *	Tonnes	2.22
	Chemical paper pulp *	Tonnes	2.13
	Old papers *	Tonnes	1.54
	Paper, cardboards *	Tonnes	1.54
	Wood wastes	Tonnes	1.1

In addition, results are more easily conveyed to field experts and practitioners since they are used to work with this unit.

The Gaussian hypothesis is the most commonly used in MFA data reconciliation processes (e.,g.[START_REF] Cencic | Material flow analysis with software stan[END_REF].

In fact, we add a subtlety by not allowing realizations outside 3 standard deviations (99% of the distribution) in order to prevent extreme values that would not make sense (the Gaussian distribution itself otherwise stretches from -infinity to +infinity).

We can plot the evolution of uncertainty estimation depending on the number of simulations to see when it has stabilized. 6

The most recent version of the model (FFSM++;[START_REF] Lobianco | The importance of introducing spatial heterogeneity in bio-economic forest models: Insights gleaned from ffsm++[END_REF][START_REF] Lobianco | Accounting for active management and risk attitude in forest sector models. an impact study on french forests[END_REF]) involves a third module, referred to as the microbased Area-Allocation module (AA) making it possible to project over longer horizons, and accounting for forest resource heterogeneity and the degree of forest owners expectations as well as their degree of risk-aversion.

In addition, the model considers sawmills and harvest residues as by-products of primary activities.

Availability Availability

Resource dynamics Resource dynamics

Supplementary material

The Wood fiber unit Definitions and abbreviations M : mass, V : volume, µ : volumicmass, p : product, w : water, o : other components, τ : moisture rate,

τ saturation ≈ 30%. Water can be present in wood in two forms: water in wood fibers and free water. Free water appears above the saturation point and has no impact on the volume. Below the saturation point, wood fibers start to lose water and the volume decreases.

w f : wood f iber at the saturation point

The product is made of a single wood specie and no other components

Note that if τ p = τ saturation (which is the default assumption when no other information is available), then V w f = V p . In the model, this is for instance the case for the production of sawnwood. Furthermore, given a coefficient of retraction f (τ p ), defined by V p = f (τ p )V w f , we have:

and:

The product is made of a single wood specie and other components

The product is made of different wood species and other components Let α i be the fraction of wood specie i in the product mix. We have:

Ci subject to the set of constraints 1 to 5 (7)

In equation 7, σ refers to the standard deviation of the data. They make it possible to treat data sources differently depending on the assumed (or possibly measured, or constrained) uncertainties. We however changed the distance used for minimization we because we found weighted least squares could lead to extreme allocations in specific cases. The minimization implemented is the following:

subject to the set of constraints 1 to 5 (8)

We divide each term by σ instead of dividing it by σ 2 . We consider this distance a good balance between penalizing absolute and relative differences. Testing the impact of different plausible distances on results is part of our next research leads.


Mihai Dascalu
email: mihai.dascalu@cs.pub.ro

Gabriel S Gutu
email: gabriel.gutu@cs.pub.ro

Cristian Paraschiv

Stefan S Ruseti
email: stefan.ruseti@cs.pub.ro

Philippe Dessus
email: philippe.dessus@univ-grenoble-alpes.fr

Danielle A Mcnamara
email: dsmcnama@asu.edu

Scott A Crossley
email: scrossley@gsu.edu

Ionut Cristian Paraschiv
email: ionut.paraschiv@cs.pub.ro

Danielle S Mcnamara

Stefan Trausan-Matu

Cohesion-Centered Analysis of CSCL Environments using ReaderBench

Keywords: CSCL, Cohesion Network Analysis, Textual Cohesion, Natural Language Processing, Virtual Communities, Massive Open Online Course

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Teachers and tutors have a limited amount of time to manually assess and grade student output. Moreover, monitoring and scoring student activities using indicators reflective of their performance in terms of participation or collaboration with peers is a cumbersome processes. Hence, there is necessity for automated analyses, which led us to develop our Cohesion Network Analysis (CNA) approach and integrate this approach within the ReaderBench framework available at http://readerbench.com. ReaderBench [START_REF] Dascalu | ReaderBench -Automated Evaluation of Collaboration based on Cohesion and Dialogism[END_REF][START_REF] Dascalu | Cohesion Network Analysis of CSCL Participation[END_REF] is a fully functional open-source framework centered on discourse analysis that consists of various Natural Language Processing (NLP) techniques designed to support students and teachers in their educational activities. This paper presents a brief overview of Computer Supported Collaborative Learning (CSCL) experiments performed within ReaderBench which help to validate the framework.

Three experiments in different CSCL environments have been conducted. These experiments focused on chat conversations, virtual communities, and a Massive Open Online Course (MOOC). In ReaderBench, CNA is defined by cohesive links spanning throughout the discourse and is tightly coupled with dialogism and polyphony which define the theoretical framing of CSCL. Moreover, CNA closely resembles Social Network Analysis by relying on equivalent indices to quantify participation within the generated sociograms [START_REF] Dascalu | ReaderBench -Automated Evaluation of Collaboration based on Cohesion and Dialogism[END_REF][START_REF] Dascalu | Cohesion Network Analysis of CSCL Participation[END_REF].

The first CNA evaluated the involvement of participants in CSCL chats [START_REF] Dascalu | Cohesion Network Analysis of CSCL Participation[END_REF]. The statistical analysis, run on a corpus of 10 multi-participant academic conversations, showed that cumulative utterance scores and outdegree were significant predictors of participation, while the cumulative utterance scores highlighted different participant interactions. The second CNA generated significant, moderate to strong, and positive correlations with human dialogue assessments while analyzing academic Virtual Communities of Practice [START_REF] Nistor | Finding student-centered open learning environments on the internet: Automated dialogue assessment in academic virtual communities of practice[END_REF]. This analysis showed that dialogue quality is higher for full time faculty members as they sustain a more elaborated dialogue and initiate longer discussion threads than part time members of the community (i.e., students). Third, CNA was used to predict students' completion rates in the context of the Big Data in Education MOOC [START_REF] Crossley | Predicting Success in Massive Open Online Courses (MOOC) Using Cohesion Network Analysis[END_REF]. Based on a cohesion graph and a longitudinal analysis, 56 indices were computed for each of the 300 students; twenty-six indices were statistically relevant and subsequently used in a Discriminant Function Analysis. Our method predicted the completion of the course with 76% accuracy, showing that students that are active on forums are more likely to finish the course.

These experiments demonstrate the capability of our ReaderBench framework to analyze a wide range of CSCL environments and to perform in-depth, cohesioncentered analyses of conversations' characteristics with a high accuracy.

As a continuation of the interactive events at AIED 2013, LAK 2015, AIED 2015 and EC-TEL 2015, this demo is focused on the latest experiments centered on CSCL performed with the ReaderBench framework. All functionalities of ReaderBench (https://git.readerbench.com/ReaderBench/ReaderBench) are wrapped into dedicated packages and most of them can be tested freely online at http://readerbench.com. A demo developed for CSCL analysis is available at http://readerbench.com/demo/cscl.

Equipment provided by us: 1 Laptop (Mac Pro 15"), no additional hardware required.

Installed environment: Java JDK 1.8 Internet connection required to access ReaderBench website, in addition to the remote UPB server exposing the REST APIs. For demonstration purposes, serialized outputs will be used in order to speed up processing.

Acknowledgments. This research was partially supported by the FP7 2008-212578 LTfLL project, by the 644187 EC H2020 RAGE project, as well as by the "Excellence Research Grants" Program UPB-GEX 12/26.09.2016.


Lionel Tenemeza Kenfack
email: kenfacklionel300@gmail.com

Martin Tchoffo

Cornelius Lukong

Fai

detrimental effects on the evolution of entanglement using a system of three-qubit as probe

Keywords: Classical noise, Non-Gaussian noise, Entanglement, Qubit

HAL is

Introduction

Quantum entanglement is one of the most remarkable properties of quantum systems. It has been defined in 1935 by Erwin Schrdinger [1] as one of the most peculiar phenomenon of quantum mechanics in which the global states of a compound quantum system cannot be written as product of the states of individual subsystems [2]. It has been demonstrates that quantum entanglement is a fundamental an indispensable resource for many potential applications in the field of quantum information theory such as for instance the quantum cryptography [3,4] and information processing [START_REF] Nielsen | Quantum computation and quantum information[END_REF][START_REF] Bennett | [END_REF][7]. Unfortunately, real quantum system cannot refrain from interacting with its external environment (the so-called decoherence) which introduces noise in the system and which, in turn, results into the destruction of the quantum properties of the system and consequently make its unusable. However, it has been recognized that decoherence may play a nondetrimental role on the evolution of entanglement existing among different marginal parts of a quantum system [8,9].

The interaction of a quantum system with its external environment may be simulated either classically or quantum mechanically. In the former, the system and the environment is look as a single quantum entity governed by an overall unitary operator; it is worth nothing that this description is more realistic because it includes the transfer of information and it is also related to the idea that the phenomenon of decoherence is strongly connected to the entanglement between the quantum system and its external environment. On the other hand, in the classical approach (also known as random classical force (RCF) approach), the system and the environment is considered as a quantum system coupled to random classical forces. Contrary to the first description, in the RCF approach, the system is not entangled with the environment. Furthermore, it is important to note that when the environment is very complex (with many degree of freedom); the quantum approach becomes very challenging. In such a situation the RCF approach represents a valid and reliable alternative to overcome the challenge. Nevertheless, for certain system-environment (SE) interactions, it has been pointed out that it is possible to obtain a classical simulation that is completely equivalent to the quantum simulation [10][11][12][13][14][15].

Motivated by the desire to minimize the detrimental effects of decoherence on the properties of quantum systems (e.g., quantum entanglement), the effects of different classical environments characterized either by Gaussian or non-Gaussian distributed noise on the evolution of many quantum systems (e.g., small dimensional systems or qubits) have been intensely investigated in recent years both theoretically [16][START_REF] Benedetti | Time-evolution of entanglement and quantum discord of bipartite systems subject to 1/f α noise[END_REF][START_REF] Benedetti | [END_REF][19][20][21][22][23][24][25][26][27][28][29][30] and experimentally [31,32]. So far, interesting results have been obtained. For instance, it has been proved in [20][21][22][23][24][25][26] that the indirect interaction between the subsystems of a composite system due to their coupling to a common classical environmental noise (CEN) can enable to preserves their entanglement indefinitely.

Besides, the effects of the input configuration of the system as well as the QNC configuration on the evolution of entanglement have been investigated both for two-and three-qubit systems [16][START_REF] Benedetti | Time-evolution of entanglement and quantum discord of bipartite systems subject to 1/f α noise[END_REF][START_REF] Benedetti | [END_REF][19][20][21][22][23][24][25][26]. On the other hand, the joint effects of several kinds of CENs on the evolution of entanglement have also been investigated both for two-and three-qubit system [22,23]. While most of these studies are devoted to the possibility of protecting entanglement via the QNC configuration and the initial input state, little attention has been paid to classify different kinds of CENs with respect to their detrimental effects on the evolution of entanglement.

For example is has been shown that when the subsystem of a composite system are coupled to a CEN in a common environment, entanglement is better preserves than when there are coupled either in mixed or independent environments [16][START_REF] Benedetti | Time-evolution of entanglement and quantum discord of bipartite systems subject to 1/f α noise[END_REF][START_REF] Benedetti | [END_REF][19][20][21][22][23][24][25][26]. In other words, even if the effects of many kinds of CENs on the evolution of entanglement have been widely investigated in the literature, the classification of these CENs with respect to their harmful impact on the evolution of entanglement has not yet been investigated.

In this paper, we intend to use a three-qubit system coupled in independent environments and subjected to different CENs, as a quantum probe to classify these CENs with respect to their detrimental effects on the evolution of entanglement of the probe. In particular, three different kinds of CENs namely the RTN, SN and CN are considered. In the case of RTN, two particular regimes namely the fast or Markov RTN and slow or non-Markov RTN are considered while for CN, we consider both the case of pink and brown noise.

Our approach to classify these CENs involves seven QNC configurations: in the first QNC configuration, we consider that the first qubit of the probe system is coupled with the RTN, the second with the SN and the third with the CN and this is denoted (RTN;SN;CN); the other QNC configurations are obtained from the latter by switching each qubits local noise by the local noises of the remaining two qubits. Concretely speaking, the second and third QNC configuration denoted (SN;SN;CN) and (CN;SN;CN) are obtained from the first one by considering that the first qubit is no longer coupled with the RTN, but rather with either the SN or RTN. In the same way, the fourth and fifth QNC configuration denoted (RTN;RTN;CN) and (RTN;CN;CN) are obtained from the first QNC configuration by switching the SN to which the second qubit of the probe is coupled either by the RTN or the CN. Finally, the sixth and seventh QNC configuration denoted (RTN;SN;RTN) and (RTN;SN;SN) are derived from the first one by switching the CN to which the third qubit of the probe system is coupled either by the RTN or the SN. It is worth recalling that the notation (XX;YY;ZZ) means that the first qubit of the probe system is locally coupled to XX, the second to YY and the third to ZZ. By employing the tripartite negativity as entanglement measure, we compare the dynamics of entanglement in the seven QNC configurations to each other. These comparisons allow us to appreciate the destructive influence of the presence or absence of each kinds of CEN on the evolution of entanglement which thereafter should enable us to classify these noises with respect to their detrimental effects. The probe is initially prepared in the GHZ state and each CEN is introduced by means of a stochastic process. Therefore, the time evolution of the probe system is obtained by performing an ensemble average over all the possible realizations of each stochastic process and the entanglement is quantified by means of tripartite negativity.

The rest of the paper is organized as follows: In Sec. 2, we present the physical model and introduce the tripartite negativity. The analytical results as well as the behaviors of entanglement for the different QNC configurations are presented in Sec. 3. Finally, we summarize our result in Sec. 4.

The physical model

Our model, similar to those considered in [20][21][22][23][24][25][26], consists of three noninteracting qubits coupled in independent environments in which each qubit interacts with a specific CEN. As we have already pointed out, three different kinds of CEN namely RTN, SN and CN are considered. In order to fully classify these CENs with respect to their detrimental effects on the evolution of entanglement of the probe system (threequbit system), seven QNC configurations namely respectively (RTN;SN;CN), (SN;SN;CN), (CN;SN;CN), (RTN;RTN;CN), (RTN;CN;CN), (RTN;SN;RTN) and (RTN;SN;SN) are considered. In these configurations, the dynamics of the system is governed by the following Hamiltonian

H(t) = H 1 (t) ⊗ I 2 ⊗ I 3 + I 1 ⊗ H 2 (t) ⊗ I 3 + I 1 ⊗ I 2 ⊗ H 3 (t), (1)
where, I k , k = 1, 2, 3 denotes the identity operator in the subspace of the qubits k; and H k (t) stands for the single-qubit Hamiltonian which contains a stochastic variable giving rise to the external noise and can be explicitly expressed as

H k (t) =∈ k,0 I k + g k ϑ s k (t)σ x k , (2)
where ∈ k,0 is the degenerated qubits energy in the absence of noise, I k and σ x k are respectively the identity and the spin-flip Pauli matrix acting on the subspace of the qubit k. g k characterizes the strength of interaction between qubit k and its local CEN. ϑ s k (t) (s ∈ {RT N, SN, CN }) stands for a stochastic parameter whose statistic depends upon the kind of CEN to which the qubit k is coupled. More precisely, when the qubit is coupled to the SN, the stochastic parameter ϑ s k (t) ≡ ϑ SN k (t) is assumed to be independent on time and characterized by the following flat probability distribution [33,34]

P ϑ SN k =        1 η m -→ ϑ SN k (t) -η 0 ≤ η m 2 0 -→ otherwise , (3)
where η 0 denotes the average value of the distribution and η m characterizes the degree of disorder of the environment. However, in the case of RTN the random terms ϑ s k (t) ≡ ϑ RT N k (t) behaves as a random bistable fluctuator (RBF) with a constant switching rate γ. Note that a RBF can be any quantity which switched randomly between two discrete values with a certain switching rate γ. In this case ϑ RT N k (t) is characterized by the probability distribution given in Eq. (11). It is worth noting that, depending on the ratio between the switching rate γ and the SE coupling constant g k , two regimes arise: strong or Markovian RTN and weak or non-Markovian RTN regime. In the case of CN or 1/f α noise, the random parameter

ϑ s k (t) ≡ ϑ CN k (t)
also behaves as a RBF. However, the main difference with the RTN case is that the switching rate of the RBF is not known a priori [16,[START_REF] Benedetti | Time-evolution of entanglement and quantum discord of bipartite systems subject to 1/f α noise[END_REF].In point of fact, here, ϑ CN k (t) describes a RBF with a switching rate γ distributed according to a specific probability law, depending on the colored of the noise. More precisely, the colored of the noise depends upon the value of the exponent α and the cases with α = 1 and α = 2 are often called pink and brown noise respectively. The probability distribution of the switching rate γ as a function of the parameter α has been found [16,[START_REF] Benedetti | Time-evolution of entanglement and quantum discord of bipartite systems subject to 1/f α noise[END_REF] and it reads

P α (γ) =              1 γ ln (γ max /γ min ) -→ α = 1 α -1 γ α (γ min γ max ) α γ α-1 max -γ α-1 min -→ 1 < α ≤ 1 , (4)
where γ min and γ max are respectively the minimum and the maximum value of the of the switching rate γ.

Note that, it is also possible to obtain the 1/f α noise by considering a collection of many RBFs [16] but in this work, we will limit ourselves to case of single RBF. As we have already pointed out, since the single qubit Hamiltonian of Eq. ( 2) is stochastic, the time evolution system is obtained by averaging the time-evolved density ρ ({ϑ} , t) = U ({ϑ} , t) ρ(0)U † ({ϑ} , t) both over all the possible realizations of the stochastic ϑ s k (t)

and over all the possible realizations of the switching rate γ. More precisely the dynamics of the probe system at a given time t can be written as

ρ(t) = ρ ({ϑ} , t) {ϑ s k } {γ} = U ({ϑ} , t) ρ(0)U † ({ϑ} , t) {ϑ s k } {γ} , (5)
where ρ(0) is the input state of the probe system, U ({ϑ}

, t) = U 1 (ϑ s 1 , t) ⊗ U 2 (ϑ s 2 , t)⊗ U 3 (ϑ s 3 , t
) stands for the global unitary time evolution operator of the probe system for a given noise configuration {ϑ} = {ϑ s 1 , ϑ s 2 , ϑ s 3 }.

. . . {ϑ s k } and . . . {γ} denote the average over all the possible realizations the stochastic process ϑ s k and switching rate γ, respectively. Since the qubits are not interacting, the single qubit time evolution operator

U k (ϑ s k , t
) can be written (in the unit of = 1) as

U k (ϑ s k , t) = exp   -ı t 0 H k (t ′ ) dt ′   = e -ı∈ k,0 t cos θ s k (t) ı sin θ s k (t) ı sin θ s k (t) cos θ s k (t) , (6)
where

θ s k (t) = -g k t 0 ϑ s k (t ′
) dt ′ stands for the noise phase picked by qubit k during its evolution. In this work, we assumed that the probe system is initially prepared in the GHZ state, that is, ρ(0) = |GHZ GHZ| with

|GHZ = 1 √ 2 (|0 1 0 2 0 3 + |1 1 1 2 1 3 ).
As pointed out in the introduction, we quantify the entanglement between the qubits of the probe system by means of the tripartite negativity [35], which is a powerful entanglement estimator arising from the positive partial transpose (PPT) criterion [36] defined as

N (3) (ρ) = 3 N 1-23 N 2-13 N 3-12 , (7)
where

N ℓ-mn = ı |λ ı (ρ T ℓ )|-1 [with λ ı (ρ T ℓ
) the eigenvalues of the partial transpose ρ T ℓ of the total density with respect to the qubit ℓ] denotes the bipartite negativity between the subsystem ℓ and the compound system mn with ℓ ∈ {1, 2, 3} and mn ∈ {12, 13, 23}. Note that for symmetrical tripartite systems (systems that remain unchanged under the permutation of its parts) the tripartite negativity reduces to the bipartite negativity of any bipartition of the system.

Results and discussions

In this section, we deal with analytical and numerical results of the evolution of entanglement in all the above mentioned QNC configurations. The entanglement is access by recourse to the tripartite negativity and the probe system is initially prepared in the GHZ state.

First QNC configuration: (RTN;SN;CN)

For this configuration, the dynamics of the system as given in Eq. ( 5) can be written in its more explicit form as

ρ c1 (t) = γmax γmin dγ 3 P α (γ 3 )ρ(γ 3 , t), (8)
with

ρ(γ 3 , t) = τ 2 τ 1 dθ SN 2 P θ SN 2 ρ θ RT N 1 , θ SN 2 , θ CN 3 θ RT N 1 θ CN 3 , (9)
where

τ 1 = η 0 - η m 2 , τ 2 = η 0 + η m 2 and . . . {θ s k } = dθ s k (. . .) P (θ s k ) (10)
In Eq. ( 10), P (θ s k ), s =RTN(CN) is the probability distribution of the noise phase θ s k whose explicit form can be written as [37-39]

P(θ s k , t) = 1 2 e -γ k t × δ(θ s k -g k t) + δ(θ s k + g k t) + γ k g k Θ(θ s k + g k t)+ + Θ(θ s k -g k t) × I 1 θ s k t 1 -(θ s k /g k t) 2 1 -(θ s k /g k t) 2 + I 0 γ k t 1 -(θ s k /g k t) 2 , (11)
where δ(x) is the Dirac delta function, I k (x) is the modified Bessel function and Θ(x) is the Heaviside step function. Let recall that in the case of RTN the switching γ k is constant, that is, γ k ≡ γ whereas in the case of CN, γ k is no longer constant and is distributed according to the probability law of Eq. ( 3). Once the calculations are performed, we obtain the following matrix

ρ c1 (t) = Y c1 (t) |000 + |111 000| + 111| + K c1 (t) |001 + |110 × × 001| + 110| + L c1 (t) |010 + |101 010| + 101| + + F c1 (t) |100 + |011 100| + 011| , (12)
where

Y c1 (t) = 1 8 (Φ(t)Ω(t) + Φ(t)Λ(t) + Ω(t)Λ(t) + 1) , K c1 (t) = - 1 8 (Φ(t)Ω(t) + Φ(t)Λ(t) -Ω(t)Λ(t) + 1) , L c1 (t) = 1 8 (Φ(t)Ω(t) -Φ(t)Λ(t) -Ω(t)Λ(t) + 1) , and
F c1 (t) = - 1 8 (Φ(t)Ω(t) -Φ(t)Λ(t) + Ω(t)Λ(t) + 1) .
The time dependent functions Φ(t), Ω(t) and Λ(t) are the decoherence factors induced by the RTN, SN and CN, respectively. They are defined as follows

Ω(t) =        e -γt cosh(δt) + γ δ sinh(δt) → γ > 2g, δ = γ 2 -4g 2 e -γt cos(δt) + γ δ sin(δt) → γ < 2g. δ = 4g 2 -γ 2 . ( 13
For this QNC configuration, the analytical expression of the tripartite negativity obtained from Eq. ( 7) can be written as follows

N (3) c1 (t) = 3 1 8 3 µ=1 ϕ µ (t) (µ,ν) ϕ µ (t) + ϕ ν (t) + ϕ µ (t) -ϕ ν (t) , (15)
with (µ, ν) = {(1, 2), (1, 3), (2, 3)}, ϕ 1 (t) = Ω(t), ϕ 2 (t) = Λ(t) and ϕ 3 (t) = Φ(t). In Fig. 1, we report the evolution of the tripartite negativity as a function of the scaled time τ = gt, in the case of Markov (weak coupling regime) and non-Markov (strong coupling regime) RTN and both for the case of pink (α = 1) and brown (α = 2) noise. Note that the integral Λ(t) in the analytical expression of the tripartite negativity has been computed numerically in the scaled range [γ min , γ max ] /g = 10 -2 , 10 2 . We see that the evolution of entanglement of the probe system decays with damped oscillations to zero. On the other hand, we see that the evolution of entanglement is strongly affected not only by the colored of the noise (pink or brown) but also by the Markov or non-Markov character of the RTN. In point of fact, we observe from both figures (Markov and non-Markov regime) that for 0 ≤ τ ≤ 1, both pink and brown noises have almost the same destructive influence on the entanglement of the probe system. However, for τ > 1 we observe that the pink noise has more destructive influence on the evolution of system with respect to the brown noise. In other words, the pink noise is more fatal to the survival of entanglement than the brown noise, regardless of the Markov or non-Markov character of the RTN. Moreover, we observe both in the case of pink and brown noise that the residual amount of entanglement in the system seems to higher in the Markov regime than in the non-Markov one.

Second and third QNC configuration: (SN;SN;CN), (CN;SN;CN)

Here, we investigate how entanglement is affected when the RTN is switched either by the SN or CN.

This should allow us to evaluate the destructive influence of the presence of the SN as well as CN by comparing the evolution of the probe system entanglement obtained in both configurations with the one of the first configuration. For both configurations, the dynamics of the system obtained by averaging the time-evolved density matrix of the system both over all the possible realizations of the stochastic noise phase and switching rate can be written as follows

ρ c2 (t) = γmax γmin dγ 3 P α (γ 3 )ρ(γ 3 , t), (16)
with

ρ(γ 3 , t) = τ 2 τ 1 τ 2 τ 1 dθ SN 1 dθ SN 2 P θ SN 1 P θ SN 2 ρ θ SN 1 , θ SN 2 , θ CN 3 θ CN 3 , (17)
and

ρ c3 (t) = γmax γmin γmax γmin dγ 1 P α (γ 1 )dγ 3 P α (γ 3 )ρ(γ 1 , γ 3 , t), (18)
with

ρ(γ 1 , γ 3 , t) = τ 2 τ 1 dθ SN 2 P θ SN 2 ρ θ RT N 1 , θ SN 2 , θ CN 3 θ CN 1 θ CN 3 . (19)
Once the calculations are performed, we find that the density matrices ρ c2 (t) and ρ c3 (t) can be written in the three qubits computational basis as

ρ c2 (t) = Y c2 (t) |000 + |111 000| + 111| + K c2 (t) |001 + |110 × × 001| + 110| + L c2 (t) |010 + |101 010| + 101| + + K c2 (t) |100 + |011 100| + 011| , (20)
and

ρ c3 (t) = Y c3 (t) |000 + |111 000| + 111| + K c3 (t) |001 + |110 × × 001| + 110| + L c3 (t) |010 + |101 010| + 101| + + L c3 (t) |100 + |011 100| + 011| , (21)
where

Y c2 (t) = 1 8 Φ 2 (t) + 2Φ(t)Λ(t) + 1 , K c2 (t) = 1 8 1 -Φ 2 (t) , L c2 (t) = Y c2 (t)    Φ=-Φ Y c3 = 1 8 Λ 2 (t) + 2Φ(t)Λ(t) + 1 , K c3 (t) = Y c3 (t)    Λ=-Λ , L c3 (t) = 1 8 1 -Λ 2 (t) .
with Φ(t) and Λ(t) given in Eq. (14). From these density matrices, we can compute from Eq. ( 7) the tripartite negativities as

N (3) c2 (t) = Φ(t) 3 Λ(t)   Φ(t) + Λ(t) + Φ(t) -Λ(t) 2   2 , (22)
and

N (3) c3 (t) = Λ(t) 3 Φ(t)   Φ(t) + Λ(t) + Φ(t) -Λ(t) 2   2 . ( 23
)
Note unlike the first configuration case, these expressions depend only on Φ(t) and Λ(t) which are the decoherence factors due to the SN and CN, respectively. It is worth nothing that in the limit of sufficiently long but finite times, both Φ(t) and Λ(t) tend to zero and, as a consequence, the entanglement initially present in the probe system is completely suppressed. In figure 2, we plot the evolution of N immediately that when the RTN is switched by the SN (second QNC configuration), the rate of degradation of entanglement is increases with respect to case of the first QNC configuration (Fig. 1), regardless of the Markov or non-Markov regime. This clearly indicate that the SN has more destructive effects on the probe system than the RTN. Another information that one can observe in this figure is that the CN, independently on the colore considered, is less detrimental than the SN. Indeed, we observe from Fig 1 that whatever the colore of the CN is, the curves obtained for the second QNC configuration are either slightly or completely below those of the third QNC configuration. Overall, Fig. 2 clearly shows that the SN is more fatal to the probe system than the RTN and CN. However, for RTN and CN no conclusion can be draw yet.

Fourth and fifth QNC configuration: (RTN;RTN;CN), (RTN;CN;CN)

Here, we analyze the situation in which the coupling of the second qubit of the probe system is switched from SN to either RTN (fourth QNC configuration) or CN (fifth QNC configuration). Both configurations shall enable us to illustrate how the RTN and CN affect the probe system. For both configurations, the evolution of the system is described by the following density matrices

ρ c4 (t) = γmax γmin dγ 3 P α (γ 3 )ρ(γ 3 , t), (24)
with

ρ(γ 3 , t) = ρ θ RT N 1 , θ RT N 2 , θ CN 3 θ RT N 1 θ RT N 2 θ CN 3 , (25)
and

ρ c5 (t) = γmax γmin γmax γmin dγ 2 P α (γ 2 )dγ 3 P α (γ 3 )ρ(γ 1 , γ 3 , t), (26)
with

ρ(γ 2 , γ 3 , t) = ρ θ RT N 1 , θ CN 2 , θ CN 3 θ RT N 1 θ CN 2 θ CN 3 . (27)
After performing calculations, we find that the density matrix ρ c4 (t) has the same functional expression of ρ c2 (t) (Eq. ( 20) ), but with the following coefficients

Y c2 (t) → Y c4 (t) = 1 8 Ω 2 (t) + 2Ω(t)Λ(t) + 1 , K c2 (t) → K c4 (t) = 1 8 1 -Ω 2 (t) F c2 (t) → F c4 (t) = Y c4 (t)    Ω=-Ω
However, the time-evolved density matrix ρ c5 (t) can be written in the three-qubit computational basis as

ρ c5 (t) = Y c5 (t) |000 + |111 000| + 111| + K c5 (t) |001 + |110 × × 001| + 110| + K c5 (t) |010 + |101 010| + 101| + + F c5 (t) |100 + |011 100| + 011| , (28)
with

Y c5 (t) = 1 8 Λ 2 (t) + 2Ω(t)Λ(t) + 1 , K c5 (t) = 1 8 1 -Λ 2 (t) , F c5 (t) = Y c5 (t)    Λ=-Λ (29)
where Ω(t) and Λ(t) are given in Eq. ( 13) and Eq. ( 14) respectively. From the definition of the tripartite negativity expressed in Eq. ( 7), we obtain

N (3) c4 (t) = lim Φ→Ω N (3) c2 (t) = Ω(t) 3 Λ(t)   Ω(t) + Λ(t) + Ω(t) -Λ(t) 2   2 , (30)
and

N (3) c5 (t) = Λ(t) 3 Ω(t)   Λ(t) + Ω(t) + Ω(t) -Λ(t) 2   2 . (31)
In figure 3, we report the evolution of the tripartite negativities as a function of the dimensionless time τ , for the cases of pink and brown noise and both for Markov and non-Markov RTN. The observation of this figure

clearly confirms what we have already pointed out that the SN is more fatal to the probe system than the CN and RTN, regardless of the Markov or non-Markov character of the RTN as well as the color of the CN.

Indeed, by comparing Fig. 3 with Figs. 1 and2, we observe that when the second qubit of the probe system is no longer affected by the SN but rather by the RTN or CN, the entanglement is less degraded. This figure also reveals that the pink noise is more fatal to the probe system than both Markov and non-Markov RTN.

However, the situation is totally reversed in the case of brown noise. In other words, the brown noise is less fatal to system than the RTN, regardless of the Markov or non-Markov character of the RTN. In point of fact, as it can clearly be seen in Fig. 3, the curves obtained both in the weak and strong coupling regime (Markov and non-Markov RTN) in the fourth QNC configuration are either slightly or completely above (pink noise)/below (brown noise) those obtained in the fifth QNC configuration. So, at this level we can say that we have achieved the main objective of this work, which was to classify the SN, CN and RTN with respect to their destructive influence on a quantum system. Nevertheless, in the following we will investigate two other QNC configurations to reinforce what we have already concluded in both previous subsections.

Sixth and seventh QNC configuration: (RTN;SN;RTN), (RTN;SN;SN)

As we have already mentioned above, if we limit ourselves to previous analyzed QNC we can already classify the SN, CN and RTN with respect to their detrimental effects. However, in order to reinforce what has already been concluded, here we are going to investigate the situation in which the coupling of the third qubit of the probe system is switched from CN to either RTN or SN. In such a situation, the dynamics of the probe system at a given time t can be written as

ρ c6 (t) =

Figure 1 :

 1 Figure 1: (Color online) Evolution of the tripartite negativity as a function of the scaled time τ = gt in the case of Markov (left) and Non-Markov (rigth) RTN, both for pink (black solid line curves) and brown (red solid line curves) noise, with the following parameters: ηm = 10, η 0 = 0, γ/g = 10 (weak coupling regime), γ/g = 0.1 (strong coupling regime) and [γ min , γmax] /g = 10 -2 , 10 2 .

  as a function of the scaled time in the case of pink (left) and brown (right) noise. We observed

Figure 2 :

 2 Figure 2: (Color online) Evolution of the tripartite negativity as a function of the scaled time τ in the case of pink (left) and brown (rigth) noise, both for the second (red solid line curves) and third (black solid line curves) QNC configurations, with the following parameters: ηm = 10 , η 0 = 0 and [γ min , γmax] /g = 10 -2 , 10 2 .

Figure 3 :

 3 Figure 3: (Color online) Upper panels: evolution of the tripartite negativity as a function of the scaled time τ in the case of:(left) pink noise and Markov RTN; (right) pink noise and Non-Markov RTN, both for the fourth (black solid line curves) and fifth (red solid line curves) QNC configuration, with the following parameters: γ/g = 10 (weak coupling regime), γ/g = 0.1 (strong coupling regime) and [γ min , γmax] /g = 10 -2 , 10 2 . Lower panels: Same as in the upper panels but in the case of brown noise.

Acknowledgement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Once performing the calculations, we find again that the density matrices ρ c6 (t) and ρ c7 (t) have the same functional expressions of ρ c3 (t) (Eq. ( 21) ) and ρ c5 (t) (Eq. ( 28) ), respectively, but with the following coefficients

From the definition of the tripartite negativity expressed in Eq. ( 7), we obtain

and

In figure 4, we report the evolution of N the probe system is less influence by the overall external noise when its third qubit is coupled to the RTN than when it is coupled to the SN, demonstrating that the SN is more harmful than the RTN. This result clearly reinforces what has been previously found. On the other hand, by comparing the left panel of Fig. 2 with Fig. 4, we clearly observe that the pink noise is more detrimental than the Markov and non-Markov RTN. Moreover, by comparing the left panel of Fig. 2 with its right panel, we observe (on the whole) that entanglement is less degraded by the Markov RTN with respect to the non-Markov RTN.

Conclusions

In this paper we have investigate the classification of three different classical non-Gaussian noises, namely the static, colored and random telegraph noise, with respect to their negative effects on the evolution of entanglement, using a system of three noninteracting qubits as probe. The probe system has been initially prepared in the GHZ state and seven QNC configurations have been analyzed in detail. Moreover, the probe system has been described by a stochastic Hamiltonian and the dynamics of the system for each QNC configuration has been computed by performing an ensemble average over all the possible realizations of the stochastic processes. Using the tripartite negativity as measure of entanglement, the evolutions of entanglement in these configurations have been compared to each other to evaluate the detrimental effects of the presence/absence of each of kind of noise considered.

In all QNC configurations analyzed, we found that the evolution of entanglement is severely destroyed by the external noise. On the other hand, we found that depending on the QNC configuration as well the noise parameters (Markovian or non-Markovian regime/pink or brown noise) considered, the entanglement decay rate is slowed down or quickened. By comparing the evolutions of entanglement obtained in the first, second and third QNC configuration to each other, we have found that the SN is more fatal to the survival of entanglement than the RTN and CN. In the same way, the evolutions of entanglement in the first, fourth and fifth QNC configuration have been compared to each other and we have found that depending on the noise color considered, the RTN may be more fatal to the probe system than the CN. In point of fact, we found that the RTN is less detrimental to the survival of entanglement of the probe system than the pink noise and that the situation is totally reversed in the case of brown noise. Finally, the evolutions of entanglement in the sixth and seventh QNC configuration have been compared with those in the first QNC configuration and once again we have found that whatever the regime (Markov or non-Markov) considered is, SN is more fatal than the RTN.

Overall, we found that these noises can be classified in descending order (according to their detrimental influence) as: SN>pink noise>RTN (both Markovian and non-Markovian)>brown noise (where ">"means more detrimental than). Hence, we believe that our investigation can be useful for the engineering of classical non-Gaussian distributed environmental noise.