Florin Constantin Mihai
email: mihai.florinconstantin@gmail.com

Waste collection in rural communities: challenges under EU regulations. A Case study of Neamt County, Romania

Keywords: waste collection, waste management, municipal waste, rural areas, EU

The paper aims to examine the changes in the rural waste management sector at regional scale since the Romania adhesion to the EU in 2007. Traditional waste management based on the mixed waste collection and waste disposal often on improper sites prevailed in municipal waste management options of transitional economies across the globe. The lack of formal waste collection services in rural areas has encouraged the open dumping or backyard burning. The paper analyses the improvements and challenges of local authorities in order to fulfill the new EU requirements in this sector supported by data analysis at local administrative unit levels and field observations. Geographical analysis is compulsory in order to reveal the local disparities. The paper performs an assessment of waste collection issues across 78 rural municipalities within Neamt County. This sector is emerging in rural areas of Eastern Europe, but is far from an efficient municipal waste management system based on the waste hierarchy concept.

Introduction

The rural waste management sector is poorly developed in low and middle-income countries compared to urban areas, even a significant share of the population lives in such regions. The rural waste management issues are less debated in the literature than urban areas due to the lack of proper data. There has been little discussion about this issue which focuses on a local scale using the data analysis in a geographical point of view. In this context, the paper seeks to provide a spatial dimension of the waste collection issue across rural areas. Open dumping is a complex environmental threat which is often widespread in rural regions across the developing countries due to the lack of formal waste management services. The illegal dumping of waste has occurred even in countries where waste management systems are better developed and cover almost all population as in Spain [1] or Italy [2]. Rural waste management sector is an emerging issue in developing and transitional countries across the world [3][START_REF] Taboada-Gonzalez | Waste characteristic and waste management perception in rural communities in Mexico: a case study[END_REF][START_REF] Taghipour | Characterizing and quantifying solid waste of rural communities[END_REF]. The landfill is far the main waste management option across new EU member states [START_REF] Dubravská | Recycling of the MSW in the Slovak Republic[END_REF][START_REF] Gabršček | Communal Waste Management: Case study for Slovenia[END_REF]. Reorganization of waste collection services, closure or the upgrade of non-compliant landfills, development of recycling centers are priorities in the case of new EU members [START_REF] Stanic-Maruna | Solid waste management in Croatia in response to the European Landfill Directive[END_REF][START_REF] Szántó | Waste Management Facility Siting and Social Conflictsthe Case of Hungary In[END_REF]. The implementation of Landfill Directive 1999/31 is challenging even for older EU countries such as Greece [START_REF] Lasaridi | Implementing the Landfill Directive in Greece: problems, perspectives and lessons to be learned[END_REF]. The investments in the modernisation process of municipal waste management sector are expensive and CEE countries rely on EU funds [START_REF] Cudecka-Purina | Evaluation of Financial Investment Effectiveness in Latvian Waste Management Regions[END_REF][START_REF] Scheinberg | Multiple modernities: transitional Bulgaria and the ecological modernisation of solid waste management[END_REF]. Extension of waste collection towards less populated areas lead to a reduction of illegal dumping activities [START_REF] Podgaiskyte | Waste management sector value changes in Lithuania along the last decade[END_REF]. The changes of municipal waste composition varied differently among urban and rural households in the last decade which lead to different waste management options [START_REF] Doležalova | The changing character of household waste in the Czech Republic between 1999 and 2009 as a function of home heating methods[END_REF]. Rural areas of Eastern Europe were often ignored by waste management services until the implementation of the EU Landfill Directive. Recent studies pay attention to rural waste management issues from Poland [START_REF] Malinowski | Assessment of segregated waste accumulation efficiency in selected suburban communities[END_REF][START_REF] Steinhoff-Wrześniewska | The pilot study of characteristics of household waste generated in suburban parts of rural areas[END_REF], Romania [START_REF] Mihai | Spatial distribution of rural dumpsites parameters in Romania[END_REF] or from EU candidate countries such as Serbia [START_REF] Nenković-Riznić | Socio-cultural models as an important element of the site selection process in rural waste management[END_REF] concerning illegal waste disposal practices, poor waste management facilities and future perspectives related to EU waste policy. The intermunicipal cooperation should be developed in order to combat the administrative and logistical inefficiency of rural areas with poor results in terms of separate collection and recycling activities [START_REF] Poldnurk | Optimisation of the economic, environmental and administrative efficiency of the municipal waste management model in rural areas[END_REF]. Romania must upgrade the poor waste management facilities across rural municipalities. The paper examines the progress and the gaps of a new EU member in rural solid waste management sector.

3

Material and Methods

Study area

Neamt County is located in Nort-East Region of Romania, the Eastern border of EU. The population is 470776 inhabitants according to the last Population Census (2011), of which 301167 lives in rural areas and only 169599 people are residents in urban areas. The administrative territorial unit of this county includes 5 cities (Piatra Neamtcounty capital, Roman, Targu Neamt, Bicaz, and Roznov) and 78 communes as shown in Fig. 1.

The commune is the basic administrative unit for Romanian rural areas which may include one or more villages and where a single village play the role of the local administrative center. The landscape varies from mountain region in the western half to subCarpathian depression and hills in the center to corridor valleys (Moldova and Siret rivers) and the plateau region in the southeastern part of the county.

2 Data source and the geographic analysis

Raw data of waste streams (eg. amounts of mixed household waste collected, amounts of separately waste collected breakdown per waste fractions, number inhabitants served by waste of collection, waste collection facilities, frequency, and type of waste collection, sanitation fees) were collected from local authorities, waste operators and Environmental Protection Agency of Neamt County in order to calculate at commune level the main waste indicators. The analysis of quantitative data related to local geographical areas, demographic features and between rural municipalities is a difficult task. The paper analyzes the annual and monthly variations of the household waste stream in case of the five communes in order to outline the seasonal variations. The comparative analysis regarding the amounts of waste collected (2011-2012) reveals some oscillations within a commune or between rural localities served by the same waste operator. The accurate records of waste streams (mixed/separately collected, recovered and disposed) owned by the local authorities or waste operators are crucial in the analysis of key performance indicators, especially in a geographical context. The responses received from local authorities vary from case to case, some of them the are incomplete, inaccurate or local authorities have no such data. Most of the waste collection services are delegated to private operators and they also have no concrete data at the commune level because the data are mixed with other rural municipalities. Some of the waste operators refuse to share the waste statistics data. Therefore, the quality of the data varies from the one commune to another or between waste operators. The majority of waste fractions data are volumetric estimations based on bin/containers volumes or according to the garbage compactor truck capacity.

There are no weighbridge systems at Roman and Targu Neamt landfills where most of the localities disposed their waste. Data submitted in tons are obtained based on volumetric estimations (the waste operator using a specific density of 0.4t/m 3 ). The household waste is transported via a waste compactor truck (including waste fractions collected from population & economic agents). The quantitative and qualitative data are displayed by thematic maps which reveal on the one hand, the coverage rates of the rural population access to waste collection services and on the other hand, the current waste collection methods used by rural communities.

3 Governance of waste management sector

The closure of non-compliant urban landfill sites according to the calendar stipulated into Government Decision nr. In 2014, the new regional sanitary landfill located in the Girov commune should be operational until the end of the year. In this context, the paper examines the garbage crisis (2014-2015) due to the legal issues debated in the regional mass media. Local details are provided for some communes in order to reveal the concrete difficulties encountering by local authorities in providing basic waste collection services.

3.Results and Discussion

The closure of wild dumps and non-compliant urban landfills

Local dumpsites, river dumping, and open burning were the main options for rural waste disposal until 16 July (2) the local authorities are obliged to provide regular waste collection services across their administrative areas and to transport the wastes collected to urban landfills since the above deadline;

(3) the non-compliant urban landfills (where these wastes could be disposed within the county) should be operational until 16 July 2009 (Bicaz) and 16 July 2012 (Roman & Targu Neamt);

(4) poor budgets allocated to waste management sector, particularly in the case of rural communities.

The gap between the deadlines mentioned above and the implementation status of the new integrated solid waste management system at the county level led to serious issues in waste management sector during July 2012 and July 2015 (subsection 3.9).

The expansion of waste collection services (WCS) in rural areas

Rural waste management sector is in a full expansion process at national, regional and local levels in order to comply the targets assumed by Romania to EU. The waste collection companies prefer to serve dense urban centers with high population density and avoid sparsely populated rural areas with a lower-income population

Alternatives for non-compliant landffills

Piatra Neamt, the capital city of Neamt county, has a modern waste management system supported by ISPA funds since 2007, but the sanitary landfill must accept only the garbage generated within the city. In this context, the periurban communes (Dumbrava Rosie, Girov, Garcina, Alexandru cel Bun) were forced to look for other alternatives with supplementary costs due to the transport. The investments achieved with pre-accession funds (PHARE-CES) had been implemented and new facilities (1 transfer station, 2 sorting plants) have been operational since 2011 in the county, serving the surrounding communes of Bicaz, Targu Neamt & Roznov cities. In order to avoid to the transportation of waste in other counties at higher costs, two temporary sites (dumps) were designed in the proximity of old landfills (Roman -25 000 m 3 , Targu Neamt -13 000 m 3 ) in the summer of 2012. This temporary solution was an environmental compromise of local authorities because the new regional sanitary landfill was under construction and the old landfills should be closed until 16 July 2012. The residual waste collected (mixed fraction) from Roznov, Bicaz and surroundings communes were transported to Roman city landfill site. Furthermore, these two temporary sites are non-compliant with EU standards, pollute the surroundings and these has a limited capacity, serving three cities and dozens of communes. Basically, all the wastes disposed in these sites should be transported to the new regional sanitary landfill located in the Girov commune. This fact will imply supplementary costs for waste operators and local councils.

Lack or poor waste collection facilities

Seven communes (Doljeşti, Dulcești, Icusești, Pancesti, Poienari, Pastraveni, Bargaoani) were not covered by WCS in 2012 as shown in Fig. 1, the wastes generated are disposed on improper sites (roadsides, riverbanks, pastures, creeks) or burnt by the inhabitants. The field observations revealed that waste dumping is still present and poor waste management services are implemented across several communes due to the lack of funds. Some inhabitants still refuse to pay the sanitation fees because the uncontrolled waste disposal is a cheap and convenient bad practice.

Backyard burning or river dumping still occurs in the proximity of households, particularly in the mountain or sub-Carpathian sector of the county. Lack of appropriate infrastructure and improper location of collection points within a commune favor this bad practice. In this context, waste collection through "door to door " system may be more suitable. New regulations stipulate that the frequency of waste collection from community points should not exceed the following deadlines [23]: (1) in the warm season (April 1 to 30 September) daily from the central areas, catering establishments, health facilities with beds, kindergartens, and nurseries and at most two days in other areas for biodegradable and residual fractions;

(2) in the cold season (October 1 to 31 March ) within 3 days for all areas;

(3) once of 3 days in case of dry recyclables Furthermore, waste collection through "door to door" system must be performed once per week for biodegradable and residual fractions and once per two weeks in case of dry recyclables. This system is commonly encountered in the rural areas and households districts of urban areas. The waste collection infrastructure varies from one community to another, depending on financial resources. For instance, in the case of the Trifești commune, the waste collection is carried from door to door in trash bags (household waste is taken by teams formed by individuals who benefit from the minimum income guarantee under Law 416/2001), transported by their own means or by tractor with trailer at the special collection point provided with containers. At this collection point, household wastes are transported by the waste operator (SC Rossal Roman SA) to the Roman city landfill. Some localities (villages) are inaccessible and private waste operators do not consider economically profitable to serve all the villages within a commune, therefore, such local authorities need to provide the primary waste collection services. Thus, the household waste collected is transported to the collection center (administrative village) then carried out by the waste operator to the urban landfill site. In other cases, containers are spread across the commune where certain public locations are established (based on population density and households).

Population performs the primary collection of mixed household waste which are further transported by the waste operator to the urban landfill site.

Separate waste collection services

The poor infrastructure and the few separate collection facilities across Romania reveal the early stage of waste management system in rural areas. The plastics and paper fractions are expected to increase in Romanian rural areas in the following years. This scenario is valid for other new EU members where source-separate collection must be improved in rural areas [START_REF] Doležalova | The changing character of household waste in the Czech Republic between 1999 and 2009 as a function of home heating methods[END_REF]. In 2012, the mixed waste collection through "door to door" system prevailed in rural communities [START_REF] Mihai | Assessment of biowaste losses through unsound waste management practices in ruralareas and the role of home composting[END_REF] of Neamt County, followed by special collection points [START_REF] Podgaiskyte | Waste management sector value changes in Lithuania along the last decade[END_REF] or a mixed situation between these two options [START_REF] Cudecka-Purina | Evaluation of Financial Investment Effectiveness in Latvian Waste Management Regions[END_REF]. Basically, there is no separate collection at source, no reuse or recycling facilities in these 48 communities (61.5 % of total) according to the Fig. 4. The biowaste fraction of household waste is diverted from wild dumpsites via home composting (open piles) and animal feeding. The improvement of home composting procedure in an organized manner will have a better result in terms of compost quality and net GHG's emissions if 70 % of biowaste is processed as a realistic scenario [START_REF] Mihai | Assessment of biowaste losses through unsound waste management practices in ruralareas and the role of home composting[END_REF]. On the other side, the pre-accession funds helped the other rural communities to provide separate collection services. The construction of sorting station in Tasca commune (2400 t/yr) has introduced regular waste collection services in the mountain region of Bicaz Valley covering one city (Bicaz) and six communes as follows: Tarcău, Tașca, Bicazu Ardelean, Bicaz Chei and Dămuc.

Such services are provided by SC Ave Huron SRL which also operates the sorting plant. Waste collection from households is done either by "door to door" method using bags or bins or by collection points (containers -1,1 mc). The collection of recyclables (from the commercial sector and/or households) was performed in other several communes in 2011-2012 as follows:

• Gherăeşti commune: PET/plastics are separately collected in special containers as 5.72 t in 2011 and 13.89 t in 2012.

• Farcașa commune: 124 m 3 of plastics, 50 m 3 of paper/cardboard; 12 m 3 of wood in 2011; 136 m 3 of PET/plastics, 54 m 3 of paper/cardboard, 14 m 3 wood in 2012; waste collection service is provided by SC Agmady SRL.

• Tupilați commune: 80 t of waste collected (in 2011) of which 0.8 tons of paper/cardboard, 0.5 t PET/plastics were separately collected; in 2012, 93.5 t of waste collected of which 1.33 t of paper/cardboard and 2.8 t PET/plastics were separately collected • Tazlău commune (in 2012): 1.09 t of PET (six containers); 0,455 t of paper/cardboard; 0,003 t of metal (Al); 5 t of WEEE The SC Ecorec Recycling SRL recovers these recyclables These recyclables are sent to recovery units (companies) by waste operators. Waste management services has experienced an emergent process between 2010-2012 when all municipalities (urban and rural) signed the intermunicipal association agreement "ECONEAMŢ" under the patronage of Neamt County Council which runs the project "Integrated Waste Management System in Neamt County".

Rural areas of Neamt county will be provided with waste collection platforms with residual (mixed fraction) and source-separated containers which are further discussed in subsection 3.8.

Waste collection costs

The costs of sanitation services (2009-2012) were supported by the local budget or by the population through annual or monthly fees. For instance, according to the Bahna Hall, the expenses of sanitation services (established by contract with SC ROSSAL SRL Roman) was carried from local budget as follows: 44679 lei in 2010 ( 44000 LEI = 10 000 EUR), 44407 in 2011 and 54066 lei in 2012 varying according to the amounts of waste collected (volumetric estimations) such as: 667/663/806 m 3 , with no separate accounts for population and economic agents.

The taxation system and sanitation fees vary from case to case (depending on the collection method, infrastructure, waste operator, transport, waste disposal site) for population/households and businesses as follows (source of data: Partial coverage of waste management services implies a share of the population who do not pay these services. On the other hand, although the full population may be served, there are households who do not pay the charges for services received and wastes are uncontrolled disposed into the surroundings. The awareness of population to such services plays a key role for the economic sustainability of this activity and local environmental protection.

Seasonal variation of household waste collected

Rural waste generation rates are much lower than in urban areas due to: the lower socio-economic conditions, poorer consumption power, a significant share of biodegradable wastes (such as food waste) may be recovered in households as compost or for livestock feed. The recyclables (paper /cardboard, plastic, metal, wood, glass) textiles or inert fractions (construction, demolition) are usually discharged in bags or containers. However, some organic fraction from gardens, agricultural sources or manure may be encountered in the mixed containers.

Per capita waste generation rates is frequently smaller than 0.3 kg.inhab.day -1 , but significant fluctutions may appear across several communes in the county 0,01 -0,9 kg.inhab.day -1 [START_REF] Mihai | Rural waste generation : a geographical survey at local scale[END_REF]. The most populated commune of the county, Sabaoani commune (9901 inhab in 2011), has a waste generation rate of 0.63 kg.inhab.day -1 in 2009.

The waste management plans have a default value set to 0.4 kg.inhab.day -1 for rural areas and of Romania and 0.9 kg.inhab.day -1 for urban areas. Such fluctuations are also encountered in rural areas of emerging economies. As an example, in China, waste generation rates range from 0.25 to 2.1 kg.inhab.day -1 [3]. Geographic and socioeconomical features of municipalities influence the waste production per capita. For instance, in Greece, per capita municipal waste generation varies from 0.23 kg.inhab.day -1 in East Zagorio (Epirus) to 3.59 in Vari (Attica) and the smaller rates (< 0.8 kg.inhab.day -1 ) characterized the small (1000-3000 inhabitants) and rural municipalities, particularly those located in mountain regions [START_REF] Lasaridi | Waste management costs in Greece: spatial patterns and causal factors[END_REF].

The comparative analysis (annual and monthly) regarding the amounts of waste collected reveals frequent oscillations within a commune or between rural localities served by the same waste operator (as shown in Figures 5 and6.) There is a downward trend of household waste collected in the cold season (November-December-January-February) and a progressive trend in April-August. Monthly changes in 2011 are stronger than in 2012, but the latter case reflects a more homogeneous annual trend. Also, the amounts of waste collected in 2011 are lower with a maximum in July and August, in contrast to 2012 when the peak is from May to June. The oscillation of the amounts of waste collected within a commune is derived from the data accuracy, particularly in 2011. Major differences between communes may be explained by the demographic factor and by the coverage rate of waste collection services.

In the case of Brusturi commune, only 611 persons are reported to be served by the waste operator in 2011 and 624 in 2012, although the total population was 3852 inhabitants according to the Population Census in 2011. According to this data, only 50 % of the main village (Brusturi) is served by the waste operator in 2012 compared to full coverage (100 %) in the case of Draganesti (1382 inhab.), Baltatesti (4090 inhab.) and Ghindaoani (1807 inhab) communes. There is a significant increase of household waste collected in 2012 compared to 2011 in case of Brusturi commune (193 t.yr -1 to 353 t.yr -1 ) and Baltatesti commune (480 t.yr -1 to 600 t.yr -1 ), a relatively constant rate for Ghindaoani (170 to 176 t.yr -1 ) and a decrease in case of Drăgăneşti (264 t.yr -1 to 234 t.yr -1 ). The monthly data of 2011 is not available for Urecheni commune, the amounts of waste collected in 2012 is 359 t.yr -1 . This variation also depends on the accuracy of data registered by the waste operator. The per capita waste generate rate range between 0.26 to 1.54 kg.inhab.day -1 according to the Table 1. The total household waste collected (HSW) by the waste operator is divided by the number of population served by WCS which reflect the higher rates of Brusturi and Urecheni communes.

If the full population of Brusturi commune is served by WCS, the per capita generation rate will decrease to 0.253 kg.inhab.day-1 in 2012 and 0.138 in 2011 based on actual data of HSW. The waste operator collected larger amounts of household waste than those generated by 624 persons reported, or such amounts are overestimated. The proper assessment of population served by WCS and the amounts of household waste collected is crucial in order to obtain reliable waste generation rates at commune level. Despite the fact the Baltatesti is the most populated commune among these 5 localities and on the other hand, it is a national touristic spa resort, the per capita generation rate is much lower than Brusturi commune. The tourism input could represent over 5 % of total waste generation rate within the commune [27]. There is a clear evidence that data reported for Brusturi commune are questionable.

The calculation and mapping of rural waste generation rates across the communes of a county have several challenges because the lack of basic waste statistics database broken down per local administrative levels [START_REF] Mihai | Rural waste generation : a geographical survey at local scale[END_REF]. Such statistics must be supported by experimental studies across rural municipalities.

Regional integrated waste management system

This major infrastructure project "Integrated Waste Management System in Neamt County" aims the full coverage of county population to proper waste management services and it has several objectives such as (Source: http://cjneamt.ro/smid/):

(1) Full coverage of rural population to residual waste collection via 3024 metallic containers (1.1 m 3 )

(2) The expansion of source-separated waste collection schemes in urban and rural areas through collection points (plastic containers 1.1 m 3 ) : 1 container for papers/cardboard; 1plastics and metals; 1-glass);

(3) Separate collection of biowaste in urban areas (households areas) through plastic containers (240 l) (4) 52000 of individual composting units (240 l) for biodegradable waste covering 50 % of the rural population;

(5) Building of 2900 waste manages platforms in rural areas where containers ( residual / dry recyclables) will be located (other 154 in urban areas) [START_REF] Dubravská | Recycling of the MSW in the Slovak Republic[END_REF] sorting station Cordun commune (in the vicinity of Roman city)sorting station has a designed capacity 17000 t.yr -1 for source-separated waste fraction such as paper, plastics, glass and metals . This station will serve Roman and 26 communes from south-east of county. [START_REF] Gabršček | Communal Waste Management: Case study for Slovenia[END_REF] transfer station with compaction at Cordun -designed capacity 45 000 t.yr -1 , the residual waste generated by Roman and 26 communes will be transported to the regional sanitary landfill site [START_REF] Stanic-Maruna | Solid waste management in Croatia in response to the European Landfill Directive[END_REF] transfer station at Tasca commune (in the proximity of Bicaz city) -9000 t.yr -1 will serve Bicaz city and rural communities from western part of the county (9) transfer station of Targu Neamt city -serving the urban area and surrounding communes (10) regional landfill site (located in the Girov commune -27 ha) which will serve all localities in the county except Piatra Neamt city where a sanitary landfill is operational until 2017. The total capacity is 9 million of m 3 shared by 3 cells , first -980 000m 3 . [START_REF] Cudecka-Purina | Evaluation of Financial Investment Effectiveness in Latvian Waste Management Regions[END_REF] the full closure of non-compliant landfills from Bicaz, Targu Neamt and Roman cities.

The project costs is estimated to be 176.992.016 lei. The financial source is covered by European Regional Development Fund (80 %) Government Funds (18%) and 2 % (local budget). The project is implemented during February 2011 and June 2016. The county waste management system will integrate the previous waste management facilities implemented through Phare-CES such as sorting stations (Tasca commune and Targu Neamt city), transfer station (Roznov city) or ISPA funds (composting and sorting plants -Piatra Neamt city).

The local garbage crisis

The closure of non-compliant landfills with EU Directive 1999/31, according to the calendar established by the G.D nr. 345/2005, has created several dysfunctionalities of municipal waste management services across urban and rural municipalities during 2009-2014. Despite the fact the new regional sanitary landfill was finished in 2014, the tender process, which delegates the site to a waste operator, was contested by one participant to the court of justice, therefore, the integrated waste management system could not be operational. In this context, the County Council had to pay from the public budget the conservation and security of this facility until a legal waste operator is finally nominated to manage the site. In the meantime, the temporary sites were almost full of waste and further garbages generated by cities and rural communities have no place where to be disposed. This situation has severe implications for the local environment. Mass media reveals that some rural localities dispose their wastes on improper sites, even they have been fined by the local environmental guard. The transport of wastes is too expensive for local budgets of rural councils in order to dispose the garbage to landfills located in other counties. In March 2015, County Council decided to start the procedure for a public administration of Girov sanitary landfill as a temporary solution until a waste operator is nominated by the tender procedure. The landfill is operating from July 2015. This crisis highlights the gaps between the EU deadlines concerning the closure of non-compliant landfill sites (July 2012) and the construction of new waste management facilities in the field (November 2014) with no suitable alternatives in this period than two temporary dumps. Moreover, a court case prolonged this garbage crisis across 2015. In this context, the most of the municipal waste generated in the county (except Piatra Neamt city) have been disposed in the non-compliant sites during these 3 years. The lack of a coherent waste management policy and the bureaucracy between the EU Commission, Government, and local authorities lead to such situation.

Conclusions

This paper performs a depth analysis of rural waste management transition toward a sustainable system in the context of EU region (NUTS3) and it points out the local governance issues related to this sector. (subsections 3.1-3.3 ; 3.9). Most of rural municipalities have a waste collection coverage over 70 %, but there are 7 communes with no such basic services in 2012. Mixed waste collection prevails in rural areas where "door to door" collection system is frequently used. There is no separate collection at source, no reuse or recycling facilities in 48 rural municipalities. Some municipalities have implemented separate collection schemes via pre-accession funds ( eg. Roznov city plus Borlesti, Piatra Soimului, and Rediu communes). Taxation system and sanitation fees vary from one municipality to another and private sector has expanded the waste management services towards rural areas since 2009. Seasonal variation of household waste generation (5 communes) reveals the main peak during the warm season (May-July) and the second one in October. The per-capita waste generation rates range between 0.26 -0. 6 kg.inhab.day -1 , but an overestimation is observed in case of Brusturi commune due to the questionable raw data. The improvement of waste statistics (based on weighings) at commune level is imperative for future studies. The local garbage crisis led to disposal of household waste in two temporary dumps (Roman, Targu Neamt ) and favored the illegal dumping practices across rural areas. The new regional integrated waste management system aims to cover all rural municipalities to proper waste management services. The regional sanitary landfill ( Girov commune) , transfer stations (Tasca, Cordun, Tirgu Neamt and Roznov) will provide the disposal solutions for residual household waste collected among rural localities via waste collection platforms .

Sorting stations (Piatra Neamt, Cordun, Tasca, Targu Neamt) , composting plants ( Piatra Neamt city) and individual composting facilities will increase the recycling and reuse rate of biowaste and dry recyclables across the county. The rural solid waste management sector must be further analyzed in a geographical context in order to provide a holistic approach and to help the decision-makers to implement viable waste management policies related to local, regional and national features.

  345/2005 [20] which transposed the EU Landfill Directive 1998/31 led to serious governance issues associated to waste management sector. The papers examines the implications of the landfills closure deadlines on rural municipalities of Neamt county such as: (i)16 July 2009 -the closure of local dumps from rural areas and Bicaz city landfill (ii) 16 July 2012 -the closure of not-compliant landfills of Roman and Targu Neamt cities;

  2009. A little attention to this issue has been paid by rural municipalities till then, but environmental authorities have begun a better monitoring of the law enforcement. The calendar of urban landfills closure stipulated by G.D. nr.345/2005 [20] created several difficulties at the county level such as: (1) the obligation of local authorities to close and rehabilitate the rural dumpsites by 16 July 2009;

  [21]. In Neamt county, most of the rural municipalities have a coverage rate over 70 % to waste collection services in 2012 according to Figure 2. The map also reveals the lack of such services in 7 communes and poor coverages (< 50 %) in Brusturi and Negresti communes. Waste operators have expanded their services towards rural areas after the closure of rural dumpsites in July 2009 compared to previous years [22]. This fact is visible year by year since 2009 due to the expansion of the private sector. In the first phase, urban operators have extended their activity in surrounding rural areas (SC Rossal SA Roman, SC Bratner Ecological Services Piatra Neamț, SC Romprest Service SA) then, the new waste operators came into the local market (SC Ave Huron SRL, SC Lemar Industries SRL, SC DDD Service SRL, SC Eurosal Trade SRL, SC Diana Service SRL). There are long distances between the collection point and landfill site, for example, wastes collected across Tazlău & Cândeşti communes are transported to Târgu Neamt city landfill. In other cases, the wastes collected from communes in the vicinity of Bacău County (Români, Podoleni) are disposed in the urban landfills of Buhuşi or Bacău cities (Nicolaie Bălcescu-sanitary landfill) which are located in the Bacau county. The coverage area of certain waste operators can be locally (e.g. SC Huron Ave-Bicaz Valley) or to overlap with a heterogeneous geographic region. For instance, SC Agmady SRL Durău collects the wastes from communes located in a mountain area (Ceahlău, Grinties, Farcașa, Hangu) or Moldavian Plateau (Bozieni, Oniceni). These wastes are collected from economic agents, local institutions or households. Most of the municipal waste collected across the county, during July 2009 -July 2012 were transported to old urban landfills of Târgu Neamț (see Figure 3) and Roman cities.

  Recyclables (paper/cardboard, plastic/PET, metal) are processed in sorting plant, the residual & household wastes are transported to the Roman city landfill. Before the sorting station, most of the rural communities were not served by an organized waste collection service favoring the waste dumping into Bicaz river and its tributaries. The Bicaz landfill had to be closed in July 2009 which complicated the situation in the region. The implementation of sorting station in Targu Neamt city has introduced the separate containers in the urban area and waste collection services in surrounding localities (the city plus Băltatesti, Ghindaoani, Petricani, Brusturi, Draganesti & Urecheni communes).The sorting station proceses the paper/cardboard, PET bottles and plastics films collected from the population (almost 300 tons per year). The sorting band has a magnet for the metal fraction, and wastes are sorted manually by the workers. The processed recyclables are further sold to economic agents. The transfer station of Roznov city and separate waste collection facilities (plus Piatra Soimului, Borlesti & Rediu communes) are operational in the southwestern part of the county. Special collection points are placed across villages 4 containers ( 1-paper/cardbord; 1-PET/plastics, 1-biowaste, 1-residual).

  local authorities)  Brusturi -4.11 lei inhab.month (4,4 lei = 1EUR), 12.48 lei for bin of 240 l /month (companies) in 2012 ;  Drăgănești -51 lei/m 3 (2012);  Urecheni -2,4 lei.inhab.month ; 45 lei / month (economic agents) in 2012  Alexandru cel Bun-27.92 lei /household/month, 55,85 lei per economic agent/month (2012)  Agapia -5 lei / family / month (2009)  Pângărați -25 lei/household/bin of 240 l/month, 1,6 lei/inhab/month per eurocontainer 1,1 m 3 (2010). Timişeşti -5 lei / family / month(15. 09.2009)

  The deadlines stipulated by G.D nr. 345/2005 for closure of wild dumps (16 July 2009) and non-compliant urban landfills (Targu Neamt, Roman -16 July 2012; Bicaz -16 July 2009) led to serious challenges in terms of waste collection , transportation and disposal of household waste across Neamt Conty which are higlighted in the paper

8 Fig. 2

 82 Fig.2Coverage rates of waste collection services (reported & estimated data) at commune level in 2012. The numbers refer to the communes which are discussed in the paper and cover different geographical areas; not all communes can be labeled because will affect the map visibility.

Fig. 3 Fig. 4 Fig. 5 Fig. 6

 3456 Fig.3 Mixed waste collection system in Petricani commune (2011)


Philippe Woloszyn

Béatrice Quenault

Guillaume Faburel

Guillaume Faburel Territorial

Philippe Woloszyn -Béatrice Quenault

Territorial intelligence of vulnerability systems -A transitional viewpoint of sustainable development

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

Introduction

If we represent the three interlinked dimensions of territorial vulnerability (socioeconomical, econological, socioecological), and their dynamic interdependencies within a triangle (see following Figure 1), the first side illustrates the dependence of the vulnerability system on the use of resources that could become scarce as a consequence of the occurrence of hazards and the induced variation of the system. This interaction dimension, formalized as econological vulnerability, represents failures of the global transition towards sustainable systems through environmental degradation due to the materialist-consumerist paradigm, i.e. economic growth. This degradation is mainly combined with the diminution of social capital.

On a complementary side, socio-ecological vulnerability clearly refers to the fact that natural resources are overused to ensure a basic standard of living, thus placing a great deal of pressure on the ecological capital. To overcome these ecological limits of unsustainable paths of development, the system should rather move towards a transitional perspective in rupture with this current global dynamics. Territorial development would therefore not only depend on vital supplies, but also on education, income levels and amenities access.

At the last side of the following sustainability trade-off interaction triangle, socioeconomic breaking point can be detected where the cost of the system stability exceeds both the value of the property threatened and the financial means of the various local authorities. Thus, socioeconomic vulnerability can be read as a function of territorial needs to be provided to ensure resilience of the sustainability system. Moreover, vulnerability studies should simultaneously imply economical trading functions, human societies and ecological systems, through their corresponding (social /economical /environmental-) capitals like socioeconomic pillar, lifestyle preferences, consumption behaviour and policy rules, underlined by biodiversity and cultural diversity similarities.

While adopting a holistic systemic viewpoint, this transitional conception of development may be evaluated by indicators of sustainability 'state', relatively to ecological and socioeconomical systems.

In this prospect, corresponding decision and regulation instruments have also to converge, aiming to create adaptive conditions through sustainable development cognitive and sociopolitical territorial expertise. Here, socio-political dimension of resilience, implying not only public policy, but also others stakeholders behaviors or decisions, has mainly to ensure the capacity to restore the trust of populations. Territorial intelligence of vulnerability systems will therefore depends on local capacities to empower people in order to facilitate their involvement in decision making process linked to socio-ecological transition goals.

Thus, through studying system response or adaptive capacity to answer to the pressure of major effects of the changes in its environment as well as inequalities access to environmental amenities [START_REF] Faburel | Vécu environnemental et qualité de vie en région Ile-de-France. Une approche interdisciplinaire d'évaluation des disparités, rapport de recherche intermédiaire pour le MEDD[END_REF], we could identify the benefits of cooperative structures and decision-making at local levels, and therefore measure the consequent benefits within the sustainability global system [START_REF] Eso | Avancement du rapport quadriennal d'activité de l[END_REF].

Econological dimension of vulnerability: environmental alea in Europe and its economic treatment: environmental economics behavioral laws, the case of ecological monetization (right to pollute)

The adoption of new environmental regulations such as the implementation of tradeable pollution permits which intends to create incentives to pollute less in order to alleviate econological vulnerability can have the mere opposite effect. This type of market-based regulation may be an opportunity for private actors to register new financial gains without having any significant results on the pollution reduction. In the same way, the implementation of dikes to prevent the risks of floods may result in an increased future vulnerability if it should lead to new constructions in areas liable to flooding [START_REF] Quenault | Vulnérabilité et résilience au changement climatique en milieu urbain : vers des stratégies de développement urbain durable ?[END_REF].

Prospect theory asserts two fundamental ways in which people perceive options differently than the rational actor model would predict. First, people value a loss of a certain amount more negatively than the positive value they associate with a gain of the same amount (MiniMax principle, [START_REF] Moles | Théorie des actes, vers une écologie des actions[END_REF]. Second, people tend to overweight low probabilities and to underweight moderate and high probabilities, with the latter effect being more pronounced than the former. (Kahneman and Tversky, 2000). Development of this principle underlines the importance of this environmental economics behavioral laws.

Socio-ecological vulnerability: the example of climate change impacts on cities

Cities, which represent complex socio-ecological systems, are coevolving with the climate of the earth system. Urban areas are approximately responsible globally for 75% of greenhouse gas (GHG) emissions (UN, 2007) as well as being hotspots of vulnerability to climate change impacts, in particular extreme meteolorogical events such as flooding, drought and heatwaves, heavy rains, storms and hurricanes, leading to catastrophic functional disruptions in the territorial systems.

Thus, cities are both drivers of climate change and foci of climate impacts. In this respect, the city level must be a priority to implement not only mitigation strategies but also adaptation strategies to climate change. In order to reduce vulnerability and to better prevent potential disruptions in case of extreme event by improving the resilience, adaptation strategies to be developed in the urban area have to adress simultaneously its various component: exposure, sensibility and adaptive capacity to climate impacts, with analyzing the key interactions and dependencies, linkages and feedbacks between its territorial system components (individuals, institutions, infrastructure) [START_REF] Quenault | Vulnérabilité et résilience au changement climatique en milieu urbain : vers des stratégies de développement urbain durable ?[END_REF].

This example of socio-ecological system interactions illustrates how a multidimensional appraisal of vulnerability can constitute a break with traditional approaches of risk management focused on hazard impacts assessment and technological solutions; instead, it invites to promote an alternative management of territorial vulnerability situations with a greater attention paid on adaptive response, leading to social protection innovations as well as new governance skills [START_REF] Woloszyn | Towards an interaction evaluation between dimensions and objectives of sustainable development at territorial levels[END_REF]. This paradigmatic change constitutes the basic principle of socio-ecological transition movement.

Socio-economical vulnerability to conjonctural stress: the two dimensions of resilience

Along with the elaboration of sustainable policy sequences, emerging concepts and paradigms about socio-economical dimension of vulnerability mainly underlines the rule of societal resilience. Resilience is a way to account for the adaptive capacity of societies to pursue their development pathways while being under the pressure of the major unavoidable effects of the changes in their environment. Resilience (from the Latin etymology resiliare, to rebound) is literally the act or action of springing back. This notion of resilience has then been elaborated in different domains. In child psychology and psychiatry [START_REF] Engle | Child development: vulnerability and resilience[END_REF], it is referring to living and developing successfully when facing adversity; in ecology [START_REF] Holling | Resilience and stability of ecological systems[END_REF], referring to moving from a stability domain to another one under the influence of disturbances; in business [START_REF] Hamel | The quest for resilience[END_REF], referring to the capacity to reinvent a business model before circumstances force to; in industrial safety [START_REF] Hollnagel | Resilience Engineering Precepts[END_REF], referring to anticipating risk changes before damage occurrence.

The environmental concern we aim to develop allows the sustainability trade-off system to retain resilience, with implying a reduction in its self-organization to minimize risk of instability. Resilience theory claims also that entropy evaluates adaptive systems [START_REF] Bailey | Social Entropy Theory[END_REF] defining to what constraint quality and level the environmental system is able of selfrepairing, through multidimensional sustainability converging [START_REF] Woloszyn | Platos poor city and rich city: Towards an inequality multidimensional evaluation[END_REF].

When resilience theory claims that perturbated systems soon returns to their dominant stable equilibrium, it describes both economic and social systems: as economic system resilience is defined by the ability to allocate resources efficiently in the face of major shocks, social systems resilience traduces the capacity of human societies to face conjunctural stress (Munasinghe, 1994, Costanza andalii. 1997).

Moreover, socio-economical systems vulnerability impacts interaction processes between human facts and economic environment can lead, for example, to crime [START_REF] Hillier | Can streets be made safe?[END_REF]; social segregation [START_REF] Vaughan | Space and Exclusion: Does urban morphology play a part in social deprivation?[END_REF][START_REF] Legeby | Does the urban structure of Swedish cities inhibit the sharing of public space?[END_REF], or accessibility differentials [START_REF] Ståhle | Compact sprawl: Exploring public open space and contradictions in urban density[END_REF]. In that frame, resilience studies investigate the influence of socioeconomics in environmental protection, as well as territorial well-being.

Territorial resilience as a response to vulnerability: the role of politics (r)evolution and public involvement

So, human and society resilience has to converge, thanks to their corresponding decision and regulation instruments, aiming to create both cultural and technological adaptative conditions through sustainable development cognitive and socio-political territorial expertize.

Here, social dimension of resilience, implying both state and public policy, but also stakeholders, ensures the capacity to restore the social system well-functioning through the trust of populations. Thus, state securisation, as a necessary condition of resilience processing, could not constitute the only response to socio-economical vulnerability. In this way, resilience could constitute an objective (i.e. for state administration), but is mainly attended as a state (in the systemic point of view), i. e. a guarantee of the social system structural stability.

Hence, following the new goal of territorial resilience as a response to vulnerability, local or metrolitan governances has also newly (mainly?) to deal differently with population's attempts and demands, so to deal with involvements wishes, participation appraisal, even with inhabitants empowerments. That is a remarked fact throughout the world: decisions making processes and regulation tools have been progressively adapted to environmental and urban planning risks of disputes. Stakeholder's scenes have nowadays to facilitate the participation informal actors, for instance community groups, but also power coalition (Sabatier, 1988) gathering other types of expertises (and experts too).

That is also a former condition in order to negotiate new balances between needs, resources and territorial development, so to find vector equilibrium between econological, socioecological and socioeconomical vulnerabilities, pursuing. In a tension context, requiring new bargaining between economic, ecological and socio-cultural capital, places attachments, lifestyles, wellbeing appear as new resources and knowledge's themes. For instance, local habits of production could feed new economical model for transition cities.

Here, admittedly, the required adaptive capacity of societies for territorial resilience passes by new political roles, regulation tools efficiency, and other democratic processes (participative ones). Mainly, it deals with recognition people capabilities (Sen, 2001), and democratic innovations (for instance controversies methods, in Callon, Lascoumes and Barthe, 2001).

That the raison why capitals should recover people capacities to directly cooperate to transition model, and the vector equilibrium should also put capabilities ands governance processes of empowerment at cornerstone.

In that way, nevertheless, vulnerability could become differently economical (i.e. participation inequalities, environmental iniquities and injustice… Faburel, 2012) and, at least, differently politic, being vigilant to inequal capacities to participate at strategic scale and step in decision-making processes for socio-ecological transition of large territories. According to us, that is a new essential stake for sustainable planning in resilience necessity paradigm: how to differently manage with local singularities in a general perspective of transition model of development. Therefore, convergence of adaptive actions could strengthen the resilience of the territories and most disadvantaged groups as parts of a coherent strategy for social cohesion, with both introducing political priorities for local action and adapt public action goals to the socio-ecological new i-deal.

Improving the visibility of the resilience of territories and populations: sustainability indicators

To achieve this goal, vulnerability, and therefore, resilience evaluation should qualify interaction laws between economical system, human societies and ecological systems through transverse dimensions, involving non-commensurable variables and objectives for multicriteria analysis.

With adopting a holistic systemic viewpoint, indicators of system 'state' relatively to the durability of ecological and socio-economic dimensions, may help to realize this interactional measurement. Therefore, the self-organizing and internal structure map of ecological and socioeconomical system can proceed to cooperative structures and decision-making processes identification.

Here, according to the necessity of empowerment, undoubtedly, well being and quality of life should have to be put at political assessment agenda, as "new" sustainable indicators. For instance, on well-being, the New Economics Foundation develops the SWEMWBS » scale (ShortWarwick-EdinburghMental-WellbeingScale) at european level (Michaelson, 2011). Or the European Council (Palatin, 2011) developps the Societal Progress Indicators and Responsibilities for ALL methods by focus groups, deling with well-being and on social cohésion,

Conclusion

Improving our understanding of the interactions among social, economic and ecological systems should allow a better appreciation of vulnerability management. To achieve this goal, associated risk and uncertainty (entropy) will also be parametered by the use of decision analysis tools, with using specified indicators which have to answer the following questions:

What kind of territorial indicators to valuate well-being and quality of life within sustainable development pillar convergence ?

What articulation between new acknowledgement fields and stakeholders practice for those convergence indicators ?

To what territorial action objects shall they refeer ?

What new governance practices shall they imply to improve territorial resilience ?

Answering those questions will constitute a major key for trans-actional analysis of the sustainable development objectives, in order to define the cognitive frames, the policicocultural coding as well as the scientifical expertise performativity for territorial action evaluation.

Figure 1 :

 1 Figure 1: Sustainable-weighted structuration of vulnerability within Territorial Intelligence trade-off.


Emmanuel Jeandel
email: emmanuel.jeandel@loria.fr

Simon Perdrix
email: simon.perdrix@loria.fr

Renaud Vilmart
email: renaud.vilmart@loria.fr

Quanlong Wang
email: quanlong.wang@cs.ox.ac.uk

ZX-Calculus: Cyclotomic Supplementarity and Incompleteness for Clifford+T quantum mechanics

The ZX-Calculus is a powerful graphical language for quantum mechanics and quantum information processing. The completeness of the language -i.e. the ability to derive any true equation -is a crucial question. In the quest of a complete ZX-calculus, supplementarity has been recently proved to be necessary for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity consists in merging two subdiagrams when they are parameterized by antipodal angles. We introduce a generalised supplementarity -called cyclotomic supplementarity -which consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.

We exhibit another new simple axiom that cannot be derived from the existing rules of the ZX-Calculus, implying in particular the incompleteness of the language for the so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of an extended ZX-Calculus, including an axiom schema for the cyclotomic supplementarity.

Introduction

The ZX-Calculus is a powerful diagrammatic language for reasoning in quantum mechanics introduced by Coecke and Duncan [START_REF] Coecke | Interacting quantum observables: categorical algebra and diagrammatics[END_REF]. Every diagram is composed of three kinds of vertices: red and green dots which are parametrised by an angle, and a yellow box; and each diagram represents a matrix thanks to the so-called standard interpretation. Moreover, any quantum transformation can be expressed using ZX-diagrams, meaning they are universal. For instance, some particular states can be represented as evidenced by [START_REF] Coecke | Three qubit entanglement within graphical z/x-calculus[END_REF]. The language initially describes pure quantum state transformations, though some work has been made to adapt it to non pure evolutions [START_REF] Coecke | Axiomatic description of mixed states from selinger's cpm-construction[END_REF][START_REF] Coecke | Environment and classical channels in categorical quantum mechanics[END_REF] and measurement-based quantum computing [START_REF] Duncan | Rewriting measurement-based quantum computations with generalised flow[END_REF].

Unlike quantum circuits, the ZX-Calculus comes with a set of equalities between diagrams that preserve the matrix that is represented. Hence, using a succession of locally applied such equalities, one can prove that two diagrams represent the same matrix, for the language is sound i.e. all the equalities do indeed preserve the matrix.

The converse of soundness is called completeness. Here, it amounts to being able to transform any diagram into another one, as long as both represent the same matrix. Hence, the concept of completeness is here totally defined by one particular interpretation, the standard interpretation, unlike other definitions of completeness such as in [START_REF] Selinger | Finite dimensional hilbert spaces are complete for dagger compact closed categories[END_REF] in which it is related to a whole family of interpretations.

It has been proven that the ZX-Calculus is in general not complete [START_REF] Schröder | The zx-calculus is incomplete for quantum mechanics[END_REF]. Yet, some restrictions have been proven to be complete. The π 2 -fragment -the language restricted to angles that are multiples of π 2 , which represents the stabiliser quantum mechanics -is complete [START_REF] Backens | The zx-calculus is complete for stabilizer quantum mechanics[END_REF], its pseudonormal form using graph states introduced in the case of the ZX-Calculus in [START_REF] Duncan | Graphs states and the necessity of euler decomposition[END_REF]. Moreover, this proof can be adapted to show the completeness of a ZX-like calculi used for graphically representing Spekken's toy model [START_REF] Backens | A complete graphical calculus for spekkens' toy bit theory[END_REF][START_REF] Spekkens | Evidence for the epistemic view of quantum states: A toy theory[END_REF] or for graphically representing the real matrices [START_REF] Jeandel | Y-calculus: A language for real matrices derived from the zx-calculus[END_REF]. The π-fragment -representing the real stabiliser quantum mechanics -is also complete, with a slightly adapted set of rules [START_REF] Duncan | Pivoting makes the zx-calculus complete for real stabilizers[END_REF].

A fragment is approximately universal when any quantum transformation can be approached with arbitrarily great precision using only the angles in the fragment. Sadly, the π 2 -fragment is not approximately universal, but the π 4 -fragment is [START_REF] Selinger | Quantum circuits of t-depth one[END_REF]. It is called the Clifford+T quantum mechanics. Completeness for this fragment was an open question, one of the main ones in the fields of categorical quantum mechanics [20] -even though a partial answer has been given for the fragment composed of path diagrams involving angles multiple of π 4 [START_REF] Backens | The zx-calculus is complete for the single-qubit clifford+t group[END_REF]. In this paper, we show that in the ZX-Calculus, the π 4 -fragment is not complete, showing that a scalar equality is derivable using matrices, but not diagrammatically. We propose to replace the "inverse rule" by this equality, and show that it can prove the former one as well as a third one: the "zero rule". Notice that this axiomatisation has been recently turned into complete axiomatisation of the ZX-calculus for this fragment [START_REF] Jeandel | A complete axiomatisation of the zx-calculus for clifford+ t quantum mechanics[END_REF].

We also show that an infinite number of fragments are also incomplete, by proving that a generalised form of the "supplementarity rule" [START_REF] Perdrix | Supplementarity is necessary for quantum diagram reasoning[END_REF] cannot be derived in them. Supplementarity, which has been proved to be necessary, consists in merging two subdiagrams when they are parameterized by antipodal angles. The generalised supplementarity called cyclotomic supplementarity consists in merging n subdiagrams at once, when the n angles divide the circle into equal parts. We show that when n is an odd prime number, the cyclotomic supplementarity cannot be derived, leading to a countable family of new axioms for diagrammatic quantum reasoning.

Finally, we propose to add the new scalar equation, as well as the cyclotomic supplementarity to the set of rules, and to get rid of the now obsolete "inverse" and "zero" rules. We address the question of the incompleteness of the -new -general ZX-calculus with a modified version of the proof by Zamdzhiev and Schröder de Witt [START_REF] Schröder | The zx-calculus is incomplete for quantum mechanics[END_REF], for theirs does not stand any more because of the introduction of the cyclotomic supplementarity.

We present the ZX-Calculus in section 2, prove the incompleteness of the π 4 -fragment and give a new scalar rule in section 3, and in section 4 we show how to generalise the supplementarity rule, discuss the way some are derivable from others, present the altered set of rules and give a new proof of incompleteness of the general ZX-Calculus.

ZX-Calculus

Diagrams and standard interpretation

A ZX-diagram D : k → l is an open graph with k inputs and l outputs -read from top to bottom -and is generated by: R

(n,m) Z (α) : n → m α • • • • • • n m R (n,m) X (α) : n → m α • • • • • • n m H : 1 → 1 e : 0 → 0 I : 1 → 1 σ : 2 → 2 : 2 → 0 η : 0 → 2
where n, m ∈ N and α ∈ R and the two compositions: The standard interpretation of the stabiliser ZX-diagrams associates to any diagram D : n → m a linear map D : C 2 n → C 2 m inductively defined as follows:

•
D 1 ⊗ D 2 := D 1 ⊗ D 2 D 2 • D 1 := D 2 • D 1 := 1 := 1 0 0 1 := 1 √ 2 1 1 1 -1 :=     1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1     := 1 0 0 1 :=     1 0 0 1     α := 1 + e iα α • • • • • • n m := 2 m              2 n        1 0 • • • 0 0 0 0 • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • 0 0 0 0 • • • 0 e iα        n + m > 0 For any n, m ≥ 0 and α ∈ R, α • • • • • • n m = ⊗m • α • • • • • • n m • ⊗n where M ⊗0 = 1 and M ⊗k = M ⊗ M ⊗k-1 for any k ∈ N * := N \ {0} .
To simplify, the red and green nodes will be represented empty when holding a 0 angle:

• • • • • • 0 • • • • • • := and • • • • • • 0 • • • • • • :=
Also in order to make the diagrams a little less heavy, when n copies of the same subdiagram occur, we will use the notation (.) ⊗n as defined above.

With the general calculus -with angles being in R -we can represent any matrix of size a power of 2 i.e. ZX-Diagrams are universal:

∀A ∈ C 2 n × C 2 m , ∃D, D = A
This requires dealing with an uncountable set of angles, so it is generally preferred to work with approximate universality -the ability to approximate any linear map with arbitrary accuracy -in which only a finite set of angles is involved. The π 4 -fragment -ZX-diagrams where all angles are multiples of π 4 -is one such approximately universal fragment, whereas the π 2 -fragment is not.

Calculus

The diagrammatic representation of a matrix is not unique in the ZX-Calculus. Hence, a set of equalities has been proposed to axiomatise the language. This set is summed up in Figure 1.

The initial set of axioms [START_REF] Coecke | Interacting quantum observables: categorical algebra and diagrammatics[END_REF] included the rules (S1), (S2), (S3), (B1), (B2), (K1), (K2) and (H). The rule (EU) has been proven to be necessary in [START_REF] Duncan | Graphs states and the necessity of euler decomposition[END_REF] and the rules (ZO) and (IV) result from [START_REF] Backens | Making the stabilizer zx-calculus complete for scalars[END_REF][START_REF] Backens | A simplified stabilizer zx-calculus[END_REF]. Finally, the rule (SUP) has been added in [START_REF] Perdrix | Supplementarity is necessary for quantum diagram reasoning[END_REF].

• • • = α+β β • • • • • • • • • • • • α • • • • • • (S1) = (S2) = (S3) = (IV) = (B1) = (B2) = π π π • • • • • • (K1) = π α -α π α π (K2) π 2 π 2 -π 2 = (EU) α • • • = α • • • • • • • • • (H) π = π (ZO) α α+π 2α+π = (SUP)
Figure 1: Rules for the ZX-calculus with scalars, augmented with the supplementarity rule [START_REF] Perdrix | Supplementarity is necessary for quantum diagram reasoning[END_REF]. All of these rules also hold when flipped upside-down, or with the colours red and green swapped.

The right-hand side of (IV) is an empty diagram. (• • • ) denote zero or more wires, while ( • • • ) denote one or more wires.

When we can show that a diagram D 1 is equal to another one, D 2 , using a succession of equalities of this set, we write ZX D 1 = D 2 . Given that the rules are sound, this implies that D 1 = D 2 . The rules can be applied to any subdiagram, meaning, for any diagram D:

(ZX D 1 = D 2 ) ⇒ (ZX D 1 • D = D 2 • D) ∧ (ZX D • D 1 = D • D 2 ) (ZX D 1 ⊗ D = D 2 ⊗ D) ∧ (ZX D ⊗ D 1 = D ⊗ D 2 )
Scalars: We will identify diagrams with 0 input and 0 output -hence representing a 1 × 1 matrix -with scalars. We will not ignore them in this paper, while in some versions of the ZX-Calculus, the global phase or even all the scalars are ignored. Ignoring them would imply taking the risk of ignoring a zero scalar, which can lead to false statements -if ZX 0 × D 1 = 0 × D 2 , we can not say that ZX D 1 = D 2 . The first rules to palliate it appeared in [START_REF] Backens | Making the stabilizer zx-calculus complete for scalars[END_REF] and were simplified in [START_REF] Backens | A simplified stabilizer zx-calculus[END_REF].

Only Topology Matters is a paradigm -to be taken as a rule -stating that any wire of a ZX-diagram can be bent at will, without changing its semantics:

= = = = = α = α α = α 3 The π 4 -Fragment is not Complete
In this section, we identify the following simple equation (E), which is sound -both sides of the equation represent the scalar 1 -but which cannot be derived from the rules of the ZX-Calculus expressed in Figure 1.

-π 4

π 4 = (E)
Since equation (E) only involves angles multiple of π 4 , it implies the incompleteness of the π 4fragment of the ZX-Calculus. In the following, we exhibit a simple invariant of the ZX-Calculus to prove that (E) is not derivable, and then we show that (E) subsumes two existing rules of the ZX-Calculus -namely (IV) and (ZO) -, leading to a simpler -(IV) and (ZO) rules are replaced by (E) -but more expressive ZX-Calculus that we call ZX E .

A Graphical Invariant for the ZX-Calculus

We introduce a simple graphical quantity for ZX-diagrams, the parity of the number of odd-degree red dots plus the number of H-dots (yellow squares), formally defined as follows:

Definition 1. Given a ZX-diagram D : n → m, let D ∈ {0, 1} be inductively defined as

D 1 ⊗ D 2 = D 1 • D 2 = D 1 + D 2 mod 2, α • • • • • • n m = n + m mod 2,
= 1, and . = 0 for all the other generators.

One can define similarly . as the parity of the number of odd-degree green dots plus the number of H-dots. Notice that for any scalar D : 0 → 0, D + D = 0 mod 2, thanks to the well known degree sum formula which implies that the sum of the degree of the vertices of a graph is even. More generally, for any D : n → m, D + D = n + m mod 2, which is clearly an invariant of the ZX-calculus since all the rules preserve the number of inputs/outputs. As a consequence, a rule preserves . if and only if it preserves . .

Lemma 1 (Invariant). For any ZX-diagram D 1 , if D 1 = 0 and ZX D 1 = D 2 , then D 1 = D 2 .
Proof. Notice that all the rules in Figure 1, but (ZO), preserve . . Since D 1 = 0, the scalar π cannot appear in any derivation transforming D 1 into D 2 , thus (ZO) is not applied, as a consequence

D 1 = D 2 .
Proposition 1. Equation (E) is not derivable using the rules in Figure 1: ZX (E).

Proof. The two diagrams of equation (E) are non zero, and they differ for . , so according to lemma 1, ZX (E).

Since the diagrams of equation (E) are in the π 4 -fragment, it implies that the π 4 -fragment of the ZX-calculus is not complete.

Remark 1. -π 4 π 4 = π 4 -π 4
the "doubled" version of (E), contrary to it, is derivable in the ZX-Calculus.

By completeness of the π 2 -fragment, for any ZX-diagrams D 1 and D 2 in this particular fragment, if D 1 = D 2 = 0, then D 1 = D 2 . This property is obviously not true in the π 4 -fragment, equation (E) being a counter example. However, this property is also satisfied by other, a priori not complete, fragments: Proposition 2. For any k = 0 mod 4 and any two diagrams D 1 , D 2 with angles multiple of π k , if

D 1 = D 2 = 0 then D 1 = D 2 .
Proof. Given k > 0 and D a ZX-diagram with angles multiple of π k , one can show, by induction on D, that (

√ 2 D ) D is a matrix whose entries are in Q[e iπ k ]
, the smallest subfield of C which contains e iπ k . Since there is a non-zero entry in D 1 , there exist

q 1 , q 2 ∈ Q[e iπ k ] such that √ 2 D1 q 1 = √ 2 D2 q 2 = 0, so √ 2 ( D1 -D2 ) ∈ Q[e iπ k ]. Suppose √ 2 ∈ Q[e iπ k ]: i. If k = 2 mod 4, then i = e iπ 2 ∈ Q[e iπ k ], so e -iπ 4 = √ 2 1-i 2 ∈ Q[e iπ k ]. Therefore e iπ 2k = (e iπ k ) k+2 4 × e -iπ 4 ∈ Q[e iπ k ]. This implies Q[e iπ 2k ] = Q[e iπ k
] which is not possible as they are vector spaces over Q of dimension respectively ϕ(4k) = 2ϕ(2k) and ϕ(2k) where ϕ is Euler's totient function, and ϕ(2k

) = 0. ii. If k is odd then 2k = 2 mod 4. Moreover, since Q[e i π k ] ⊆ Q[e i π 2k ], √ 2 ∈ Q[e i π 2k
] which is impossible according to the previous case (i).

Thus √ 2 / ∈ Q[e iπ k ] when k = 0 mod 4, so D 1 = D 2 .

A Simpler and More Expressive ZX-calculus

Equation (E) cannot be derived in the ZX-calculus (proposition 1), as a consequence we propose to add this equation (E) as a rule of the language to make it more expressive. We show in the following that the introduction of this new rule makes the two scalar rules (ZO) and (IV) obsolete, leading to a language with less rules than the one define in Figure 1.

Let us define ZX E = {(E)} ∪ ZX \ {(IV), (ZO)}. First, notice that, thanks to [START_REF] Backens | A simplified stabilizer zx-calculus[END_REF], the so-called Hopf law is derivable from ZX \ {(IV), (ZO)}, and hence from ZX E : Proof. In appendix at page 13.

Hence ZX E (IV), (ZO).

Remark 2. The other rules of the language remain, a priori, necessary in the presence of equation (E). In particular the supplementarity which has been recently proved to be necessary in ZX [START_REF] Perdrix | Supplementarity is necessary for quantum diagram reasoning[END_REF] is necessary in ZX E : one can prove using the interpretation . k,l -defined in [START_REF] Perdrix | Supplementarity is necessary for quantum diagram reasoning[END_REF] -with k = 3 and l = 8 that ZX E \ {(SUP)} (SUP).

Remark 3. One may want to generalise equation (E), replacing the particular angles ± π 4 by some generic angle α. Proposition 2 is a strong evidence that such a generalisation is not possible and that the language requires at least one rule which is specific to the π 4 angle.

4 Cyclotomic Supplementarity

Generalisation of (SUP)

The concept of supplementarity in quantum diagram reasoning has been first introduced by Coecke and Edwards [START_REF] Coecke | Three qubit entanglement within graphical z/x-calculus[END_REF], turned into a simple but necessary rule (SUP in Figure 1) in [START_REF] Perdrix | Supplementarity is necessary for quantum diagram reasoning[END_REF]. Roughly speaking, supplementarity consists in merging two dots sharing the same neighbour when the difference of their angles is π, i.e. when the two angles are antipodal. We generalize this concept to cyclotomic supplementarity as follows: for any n ∈ N * , n dots sharing the same neighbour can be merged when their angles divide the circle into equal parts (cyclotomy), i.e. when their angles are of the form α + 2kπ n for k ∈ 0; n -1 :

= α+ 2π n α+ n-1 n 2π α • • • nα+ (n-1)π • • • (SUP n )
Notice that there are n green dots in the left diagram, and n parallel wires in the right diagram. Any of these equations is valid for the standard interpretation of ZX-diagrams:

Proposition 5. (SUP n ) is sound.
Proof. In appendix at page 14.

Cyclotomic supplementarity has a generalisation: the green dots can be merged not only when they share the same neighbour, but also when they share the same neighbourhood. It leads to the notion of cyclotomic twins, which generalise the notion of antiphase twins [START_REF] Jeandel | A complete axiomatisation of the zx-calculus for clifford+ t quantum mechanics[END_REF]: Definition 2 (Cyclotomic Twins). n dots in a ZX-diagram are cyclotomic twins if:

• they have the same colour

• their angles divide the circle into equal parts α + 2kπ n for k ∈ 0; n -1 • they have the same neighbourhood: for any vertex, the number of wires connecting it to any of the twins is the same Proposition 6 (Cyclotomic Twins and Supplementarity). With ZX ∪ {(SUP n )} n∈N , cyclotomic twins can be merged.

Proof. In appendix at page 14.

The rest of the section is dedicated to the structures of this family of equations: we show that (SUP n ) is necessary when n is an odd prime number and that (SUP n ) can be derived when n is not prime. As a consequence, we exhibit a countable family of equations that cannot be derived in the ZX-calculus.

The Set of Supplementarity Rules for Prime Numbers

It is not necessary to define the supplementarity rules for all numbers n ∈ N as axioms. For instance, we will prove that their restriction to the set of prime numbers is enough to show all the others.

Let P be the set of prime numbers.

Theorem 7.

• ∀p, q ∈ N * , ZX E ∪ {(SUP p ), (SUP q )} (SUP pq )

• ∀n ∈ N * , ZX E ∪ {(SUP p )} p∈P (SUP n ) • ∀p ∈ P, p ≥ 3, ZX E ∪ {(SUP q )} q∈P\{p} (SUP p )
Proof.

First statement: If n is not prime, its supplementarity can be derived. Indeed, suppose n can be decomposed in two numbers p and q (n = pq), for which we know the supplementarity rule.

q q q pq = α+ 2π pq α+ p-1 pq 2π α • • • • • • α+ 2π q α+ pq-1 pq 2π α+ 2π q • • • α α+ q-1 q 2π • • • α+ pq-1 pq 2π α+ 2π pq α+ 2π pq + 2π q α + 2π pq + q-1 q 2π • • • qα+ (q-1)π qα+ 2π p +(q-1)π • • • qα+ p-1 p 2π +(q-1)π = = pqα+ (pq-1)π
with p-ticked edge representing p parallel wires. The first equality is just a rearranging of the branches, the second uses (SUP q ) p times and the last one exploits Proposition 6 with p(qα + (q -1)π) + (p -1)π = pqα + (pq -1)π.

Second Statement: As a direct consequence of the previous statement, since (SUP 1 ) is trivial, the supplementarity rules for prime numbers are enough to derive all the others. Third Statement: Let p ∈ P and p ≥ 3. Let us consider the interpretation . p 2 which amounts to multiplying all the angles of a diagram by p 2 .

• The interpretation . p 2 coincides with the interpretation . 1,p 2 -1 defined in [START_REF] Perdrix | Supplementarity is necessary for quantum diagram reasoning[END_REF]. As stated in this article, since the first parameter is odd and the second one is even, all the rules of ZX\{(SUP 2 )} hold.

• The rule (E) also holds. Indeed, p is odd, and whether p mod 8 is 1, 3, 5 or 7, p 2 mod 8 = 1, so p 2 π 4 = π 4 mod 2π. • The rule (SUP q ) when q ∈ P, q = p holds, since gcd(p 2 , q) = 1: = α+ 2π q α+ q-1 q 2π α ...

p 2 qα+ (q-1)π ... → ← p 2 α+ 2p 2 π q p 2 α ... p 2 α+ q-1 q 2p 2 π p 2 α+ 2π q p 2 α ... p 2 α+ q-1 q 2π = ... qα+ (q-1)π
• The rule (SUP p ) does not hold:

= α+ 2π p α+ p-1 p 2π α ... p 3 α ... → ← ... pα p 2 α ... p 2 α p 2 α
Indeed, when α = 0, for instance, using on the left side p -1 times (B1) and (IV), and on the right side (IV) and p-1 2 times the Hopf law (HL), since p ≥ 3:

...

= = = ... = ⊗(p-1) ( ) ⊗(p-1) ⊗(p-1) ⊗(p-1)
Every rule but the p-supplementarity (with p ∈ P and p ≥ 3) holds with this interpretation, so it cannot be derived from the others:

∀p ∈ P, p ≥ 3, ZX E ∪ {(SUP n )} n∈P \ {(SUP p )} (SUP p )
Corollary 1. For any n ≥ 3 odd, the π 2n -fragment of the ZX E -Calculus is incomplete. Proof. Let p be an odd prime factor of n. Theorem 7 proves that ZX E (SU P p ), and notice that all the angles involved in the rule are multiples of π 2p , hence in the π 2n -fragment.

Remark 4. We can also notice that all the rules (SUP n ) respect the quantity . , so that the rule (E) remains necessary.

Discussion on the Supplementarity's Derivability Structure

Let p and q be two natural numbers. We have previously shown ZX E ∪ {(SUP p ), (SUP q )} (SUP pq ). In other words, (SUP p ) can be deduced from the supplementarity of the dividers of p. Now, can we deduce this same equality from the supplementarity of some of its multiples?

The first result comes when p is odd:

Proposition 8.
∀p, q ∈ N * , (p = 1 mod 2) ⇒ {(HL), (IV), (SUP p ), (SUP pq )} (SUP q )

Proof. In appendix at page 15.

There exists another -weaker -derivation when p is even:

Proposition 9.
∀p, q ∈ N * , {(HL), (IV), (SUP p ), (SUP p 2 q )} (SUP pq )

Proof. In appendix at page 15.

Remark 5. In the last two propositions, we require that the ZX be "general" i.e. with angles either real or a rational multiple of π because we need α/p to be in the fragment in both cases. Though, the result can be expanded to any fragment for some α provided α/p be in the fragment.

To sum up:

ZX E ∪ {(SUP p ), (SUP q )} (SUP pq ) ZX E ∪ {(SUP p ), (SUP p 2 q )} (SUP pq ) (p = 1 mod 2) ⇒ ZX E ∪ {(SUP p ), (SUP pq )} (SUP q ) • • • = α+β β • • • • • • • • • • • • α • • • • • • (S1) = (S2) = (S3) -π 4 π 4 = (E) = (B1) = (B2) = π α -α π α π (K) α • • • = α • • • • • • • • • (H) π 2 π 2 -π 2 = (EU) = α+ 2π n α+ n-1 n 2π α • • • nα+ (n-1)π • • • (SUP n ) n ∈ N * or n ∈ P

Updated Set of Rules

We propose to add the generalisation of the supplementarity rule to the set of rules of the ZX-Calculus, and to restrict to the set necessary when dealing with particular fragments. We notice that the rule (K1) is derivable from the others [START_REF] Backens | A simplified stabilizer zx-calculus[END_REF] so we can get rid of it, and the new set of rules of the ZX-Calculus is shown in figure 2.

Remark 6. We can prove that (SUP 2 ) is not derivable from ZX E \ {(SUP 2 )}, using the interpretation . k,l defined in [START_REF] Perdrix | Supplementarity is necessary for quantum diagram reasoning[END_REF] with k = 3 and l = 8. However, it is important to notice that we have not proven that (SUP 2 ) can not be derived from the rest once the cyclotomic supplementarity is added. Indeed, the family of interpretations used in the proof of Theorem 7 only works when p is odd, and the one used previously, . k,l , does not hold for many supplementarity rules.

The rule (SUP 2 ) is all the more peculiar as, due to the Hopf law (HL), it is the only supplementarity rule that creates a non-trivial scalar -except for the supplementarity rules for even numbers, which anyway derive from (SUP 2 ). For instance it may create a scalar worth 0 -when applied with α = 0 -which is the first step towards proving the rule (ZO) in the π 4 -fragment. Moreover, it is the only supplementarity that can not be proven to be necessary by simply multiplying the angles by a constant.

The General ZX-Calculus is still Not Complete

The argument given by Schröder de Witt and Zamdzhiev [START_REF] Schröder | The zx-calculus is incomplete for quantum mechanics[END_REF] to show the incompleteness of the general ZX-Calculus is not valid anymore -when multiplying the angles by any integer, there is at least one supplementarity that does not hold. But we can patch the demonstration to make it valid again.

Theorem 10. The general ZX-Calculus is incomplete with the set of rules in figure 2.

Proof. We will make the proof using a combination of ZX-rules and matrix calculus on the interpretations of the diagrams. Consider the following diagrams: When applying the π green state at the top and the 0 green state at the bottom of both D 1 and D 2 , we end up with:

D 1 : π 2 π 4 4 π = π π 4 -π 2 (B1) (S1) (K1) D 2 : α α π 3 π θ π = α π π 3 θ α+π 2α+π θ π 3 π = (S1) (SUP2) (HL)
In order for their interpretations to be equal, we need: , thus α = ± π 2 ± arccos 2 3 mod 2π. α is not a rational multiple of π: One can check that e iα0 is a root of the polynomial 3X 4 + 2X 2 + 3 which is irreducible in Z (since 30203 is a prime number, thanks to Cohn's irreducibility criterion, 3X 4 + 2X 2 + 3 is irreducible in Z). The polynomial is not cyclotomic because its coefficient of higher degree is not 1, hence e iα0 is not a root of unity, i.e. α 0 is not a rational multiple of π. As a consequence, none of the

e i π 4 √ 2e -i π 4 =
± π 2 ± arccos 2 3
are rational multiples of π. Now, let us put back all the pieces together. Assume ZX D 1 = D 2 for some α and θ. Then there exists a finite sequence of rules of the ZX that transforms D 2 into D 1 . We define q ∈ N * such that for any (SUP p ) in the sequence, p ≤ q, and S = {k(q + 4)! + 1 | k ∈ N}. For all q ∈ S and for -.

q the interpretation that multiplies the angles by q , the rules of the ZX are preserved, -D 1 q = D 1 , and -D 2 q is in the form D 2 . Indeed:

• (q + 4)! is clearly a multiple of 8 so q π 4 = π 4 mod 2π, so all the rules but the supplementarity rules hold, and -D 1 q = D 1 since it is in the π 4 -fragment.

• for any p ∈ P such that p ≤ q, then (q + 4)! = 0 mod p so gcd(p, q ) = 1, which implies that (SUP p ) also holds.

• (q + 4)! is a multiple of 6 so q π

3 = π 3 mod 2π hence - D 2 - q is in the form D 2 .
Then, ZX D 1 = -D 2 q . D 1 has a finite number of decompositions in the form D 2 , but -D 2 q | q ∈ S is infinite -since α is an irrational multiple of π -and all these diagrams are decompositions of D 1 in the form D 2 , hence we end up with a contradiction. So ZX D 1 = D 2 , which proves the incompleteness.

A Appendix

A.1 (ZO) is Derivable in ZX E

The following scalar lemmas are provable in ZX [START_REF] Backens | Making the stabilizer zx-calculus complete for scalars[END_REF][START_REF] Backens | A simplified stabilizer zx-calculus[END_REF]:

Lemma 3. = Lemma 4. α =
Proof of Proposition 4. First, using (S2), (HL), (SUP), (B1) and (S1), and then applying a red state at the top:

π = π π = = π π = ⇒ π = π
Then, using (IV), (S1), the previous result and lemma 4:

= π π α = α α π π = = π
Notice that so far we have not used (E). It becomes necessary in the following, since . does not hold there. Using (E), the previous results, and lemmas 4 and 3:

A.2 Cyclotomic Supplementarity

In order to prove the soundness of (SUP n ), let us first define the equality:

α+ n-1 n 2π α • • • α+ 2π n = nα+ (n-1)π (1) Lemma 5. (SUP n ) is sound ∀α ∈ R ⇔ (1) is sound ∀α ∈ R.
Proof. Notice that π , form a basis of C 2 . Hence, using (K1) and (B1): Hence ( 1) is sound which implies (SUP n ) is sound.

(SUP n ) is sound ∀α ∈ R ⇔ kπ α+ n-1 n 2π = nα+ (n-1)π • • • α • • • α+ 2π n kπ is sound ∀α ∈ R, ∀k ∈ {0, 1} ⇔ α+ n-1 n 2π = nα+ (n-1)π • • • α • • • α+ 2π n kπ kπ kπ kπ kπ kπ is sound ∀α ∈ R, ∀k ∈ {0, 1} ⇔ = n(α+kπ)+ (n-1)π • • • α+kπ + n-1 n 2π α+kπ + 2π n α+kπ is sound ∀α ∈ R, ∀k ∈ {0, 1} ⇔ (1
Proof of Proposition 6. Consider n green cyclotomic twins for n ∈ N * . Notice that since the dots are twins, they are either an independent set or a clique if two of them are connected, they are all connected. If another green dot is part of the neighbourhood, or if the twins are connected, they can be merged using (S1). Otherwise, up to (HL), the neighbourhood is composed of only m red dots connected with exactly 1 wire to any of the twins. If m = 0, the equation becomes (1), proven in lemma 5. If m = 1, then the direct use of (SUP n ) gives the result. If m ≥ 1, we have to use the generalised bialgebra rule proven in [START_REF] Duncan | Graphs states and the necessity of euler decomposition[END_REF]:

• • • = • • • α α+ n-1 n 2π • • • m m = m • • • • • • nα+ (n-1)π = • • • m nα+ (n-1)π n • • • n • • • • • • n = • • • α m α+ n-1 n 2π • • • α α+ n-1 n 2π = • • • m • • • nα+ (n-1)π ⊗(n-1)(m-1) ⊗(n-1)(m-1)
The proof is the same for red cyclotomic twins.

A.3 Derivations of the Supplementarity

Proof of Proposition 8. With (HL) the Hopf Law and (IV) the inverse rule, which are both derivable from ZX E : p p p pq q α α+ 2π q α+ q-1 q 2π

. . .

= α+

q-1 q 2π α . . . With p-ticked edge representing p parallel wires. Those are created -always by multiples of two -using the Hopf Law (HL). The 3 rd diagram is obtained by rearranging the branches so that we can use (SUP pq ). For the last two diagrams, notice that q is odd so (q -1)π = 0 mod 2π.

Proof of Proposition 9. If p is odd, then the previous proposition implies the wanted result, taking q := pq. Now, if p = 0 mod 2: The second call to (SUP p ) makes use of the equality (1) of lemma 5. The last equality is just a rearranging of the branches.

  Spacial Composition: for any D 1 : a → b and D 2 : c → d, D 1 ⊗ D 2 : a + c → b + d consists in placing D 1 and D 2 side by side, D 2 on the right of D 1 . • Sequential Composition: for any D 1 : a → b and D 2 : b → c, D 2 • D 1 : a → c consists in placing D 1 on the top of D 2 , connecting the outputs of D 1 to the inputs of D 2 .

Proposition 3 . 4 =

 34 (IV) is derivable from ZX E . Proof. Using (E), (B1), (HL), (S2) and (S1): Proposition 4. (ZO) is derivable from ZX E .

Figure 2 :

 2 Figure 2: New set of rules for the ZX-calculus with scalars. All of these rules also hold when flipped upside-down, or with the colours red and green swapped. The right-hand side of (E) is an empty diagram. (• • • ) in (S1) and (H) denote 0 or more wires, while ( • • • ) denote 1 or more wires.

6 .

 6 We will try to express D 1 in the form D 2 . One can notice that D 1 = D 2 when α 0 = π 2 -We can even show: No more than four values for α are possible when decomposing D 1 in the form D 2 :

1 k=0( 1 +

 11 ) is sound ∀α ∈ R Proof of Proposition 5. According to the previous lemma, (SUP n ) is sound ∀α ∈ R if and only if (1) is sound ∀α ∈ R. The standard interpretation of the right part of the equality 1 yields 1 + e i(nα+(n-1)π) , and, using the fact thatn k=0 (X -e 2ikπ n ) = X n -1 (the e 2ikπ nare the n th roots of unity), the interpretation of the left part amounts to:ne i(α+ 2kπ n ) ) = e in(α+π) n-1 k=0 (e -i(α+π) -e2ikπn ) = e in(α+π) (e -in(α+π) -1)= 1 + e i(nα+(n-1)π)