Nutrient dynamics under different ocean acidification scenarios in a low nutrient low chlorophyll system: the Northwestern Mediterranean Sea

Keywords: ocean acidification, plankton community, mesocosm experiments, nutrient dynamics, nitrogen, phosphorus, iron, stoichiometry, Oligotrophy, Mediterranean Sea

Highlights:

-Mesocosm experiments performed in two Mediterranean sites during two seasons -Contrasted nutrient stoichiometry in surface waters in summer and winter -Dissolved organic pool was a large stable fraction of N and P in summer and winter -CO 2 had no effect on nutrient dynamics that was mostly biologically controlled

Introduction

Since the beginning of the industrial era, the atmospheric carbon dioxide (CO 2 ) level has increased by nearly 40% (from ~280 to 390 ppm in 2011;Hartmann et al., 2013) and may reach between ~450 and ~1000 ppm by 2100 depending on the considered future emission scenario [START_REF] Collins | Long-term Climate Change: Projections, Commitments and Irreversibility[END_REF]. As the ocean absorbs about a quarter of total anthropogenic CO 2 emissions (Le [START_REF] Quéré | Global carbon budget 2013[END_REF], the increase in CO 2 dissolved in seawater leads to an increase in seawater acidity (decrease in pH). Surface ocean pH has already declined by approximately 0.1 since the beginning of the industrial era [START_REF] Orr | Anthropogenic ocean Acidification over the twenty-first century and its impact on calcifying organisms[END_REF] and is expected to decrease an additional 0.06 to 0.32 by the end of this century depending on the considered CO 2 emission scenario [START_REF] Ciais | Carbon and Other Biogeochemical Cycles[END_REF]. Although the chemical forms of compounds such as macro-and micro-nutrients in the ocean are controlled by multiple environmental factors including pH, only a restricted number of studies have addressed the impact of ocean acidification on nutrient dynamics and its consequences on biological activity. ) and an alteration of the NH 4 + / NH 3 equilibrium. Therefore, ocean acidification could directly affect the chemical speciation of phosphorus and nitrogen with impacts on their bioavailability and stoichiometry.

In addition to pH equilibrium reactions, ocean acidification can impact biological processes and indirectly shift nutrient cycles. In the case of nitrogen, an increase in the partial pressure of CO 2 (pCO 2 ) could (1) enhance nitrogen fixation (e.g. [START_REF] Hutchins | CO 2 control of Trichodesmium N 2 Fixation, photosynthesis, growth rates, and elemental ratios: implications for past, present, and future ocean biogeochemistry[END_REF] and

(2) reduce the rate of marine nitrification (e.g. [START_REF] Beman | Global declines in oceanic nitrification rates as a consequence of ocean acidification[END_REF]. The alteration of the NH 4 + / NO 3 -ratio and the increasing trend toward P-limitation could ultimately impact primary [START_REF] Ciesm | Biogeochemical impacts of ocean acidification-emphasis on carbonate production and dissolution[END_REF][START_REF] Hutchins | Nutrient cycles and marine microbes in a CO 2 -enriched ocean[END_REF].

For example, [START_REF] Romero | Dynamic forcing of coastal plankton by nutrient imbalances and match-mismatch between nutrients and turbulence[END_REF] showed that waters rich in NH 4 + favor the growth of small autotrophic species and heterotrophic bacteria while NO 3 -mostly favor diatom growth.

The organic fraction of dissolved nutrients (dissolved organic nitrogen: DON and dissolved organic phosphorus: DOP) can represent a large source of P and N for microorganisms in oligotrophic areas (e.g. [START_REF] Jackson | Importance of dissolved organic nitrogen and phosphorus to biological nutrient cycling ». Deep Sea Research Part A[END_REF][START_REF] Karl | Dymanics of DOP[END_REF][START_REF] Bronk | DON as a source of bioavailable nitrogen for phytoplankton[END_REF]. Regarding DOP, most of its compounds cannot be used by phytoplankton directly, unless specific enzymes (e.g. alkaline phosphatase) are produced to hydrolyze DOP into PO 4 3- (Holland et al., 2005;[START_REF] Dyhrman | Presence and regulation of alkaline phosphatase activity in eukaryotic phytoplankton from the coastal ocean: Implications for dissolved organic phosphorus remineralization[END_REF]. These hydrolysis reactions are sensitive to pH with the consequence that a change in pH could alter the bioavailability of DOP in seawater [START_REF] Doney | Ocean acidification: the other CO 2 problem[END_REF].

Furthermore, decreases in ocean pH theoretically increase the proportion of free dissolved trace metals and increase metal solubility [START_REF] Millero | Solubility of Fe (III) in seawater[END_REF]. This would lead to substantial increases in the total bioavailable fraction of many trace elements and result in toxic concentrations for marine microorganisms [START_REF] Raven | Ocean Acidification due to Increasing Atmospheric Carbon dioxide[END_REF]. Several studies have

shown that a decrease in ocean pH enhances the solubility of Fe(III), a form of iron that is bioavailable for phytoplankton (e.g. [START_REF] Liu | The solubility of iron in seawater[END_REF][START_REF] Millero | Effect of ocean acidification on the speciation of metals in seawater[END_REF]. Iron speciation is largely controlled by organic complexation and photochemical processes (Breitbarth et al., 2010a and ref within). A decrease of pH below 8 favors the iron complexation with organic ligands against the precipitated form Fe(OH) 3 [START_REF] De Rijck | Cationic speciation in nutrient solutions as a function of pH[END_REF]. The effect of ocean acidification on Fe(III)-organic complexation needs further investigations (Hoffmann et al., 2012). [START_REF] Shi | Effect of ocean acidification on iron availability to marine phytoplankton[END_REF] have shown that depending on the nature of the chelating agent, a pH decrease could enhance the solubility of Fe by by organic compounds rather than its oxy-hydroxide form (Fe(OH) 3 ).

Ligand production is facilitated by phytoplankton dissolved organic matter (DOM)

production. Enhanced release of DOM with increasing pCO 2 (i.e. [START_REF] Engel | Direct relationship between CO 2 uptake and transparent exopolymer particles production in natural phytoplankton[END_REF] could, as a feedback, increase iron binding and make it more bioavailable for primary production. This has recently been shown in in situ mesocosm experiments conducted as part of the Pelagic Ecosystem CO 2 Enrichment project (PeECE) in Bergen, Norway (Breitbarth et al., 2010b).

Most studies investigating the impact of ocean acidification on pelagic ecosystems have been performed in laboratories but, during the last decade, several mesocosm experiments have been conducted, for instance as part of the PeECE project and the European Project on Ocean Acidification (EPOCA; i.e. [START_REF] Engel | Testing the direct effect of CO 2 concentration on a bloom of the coccolithophorid Emiliania huxleyi in mesocosm experiments[END_REF]Grossart et al., 2006;[START_REF] Schulz | Build-up and decline of organic matter during PeECE III[END_REF][START_REF] Riebesell | Technical note: a mobile sea-going mesocosm system -new opportunities for ocean change research[END_REF]. Mesocosms offer the advantage to follow the response of pelagic communities to an environmental driver under close-to-natural environmental conditions. The above-mentioned mesocosm experiments have been mostly performed in nutrient-replete cold waters. However, a large portion of the open ocean is characterized by low-nutrient, low-chlorophyll (LNLC) conditions. To our knowledge, no ocean acidification mesocosm study has been conducted under these limiting conditions as observed in the Mediterranean Sea.

The Mediterranean Sea is one of the most nutrient-poor waters in the world with a trophic status varying from oligotrophic-mesotrophic in the Northwestern Mediterranean to extremely oligotrophic in the eastern basin (Moutin and Rimbault, 2002;[START_REF] Mermex Group | Corrigendum to "Marine ecosystems' responses to climatic and anthropogenic forcings in the Mediterranean[END_REF] and references within). The Mediterranean Sea is in general a sink for atmospheric CO 2 [START_REF] D'ortenzio | Satellite-driven modeling of the upper ocean mixed layer and air-sea CO 2 flux in the Mediterranean Sea[END_REF]. As this region is relatively more alkaline than the open ocean, it absorbs relatively more anthropogenic CO 2 and is a significant source of CO 2 for the Atlantic [START_REF] Aït-Ameur | Distribution and transport of natural and anthropogenic CO 2 in the Gulf of Cádiz[END_REF]. Long-term data series of carbonate properties and ocean acidification trends in the Mediterranean Sea are scarce. However, current interpolation suggests that pH may have already declined by 0.15 in some areas of the surface Mediterranean Sea (Touratier and Goyet, 2011).

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The objective of the present study is to assess the effect of ocean acidification on nutrient (Fe, N and P) dynamics in a nutrient-depleted system, limited or co-limited by P and N availability [START_REF] Tanaka | Lack of P-limitation of phytoplankton and heterotrophic prokaryotes in surface waters of three anticyclonic eddies in the stratified Mediterranean Sea[END_REF]: the Mediterranean Sea. (MedSeA; www.medsea-project.eu). In this paper, we report on the evolution of nutrient concentrations, under both dissolved inorganic and organic forms, and discuss the potential impact of ocean acidification on nutrient dynamics in LNLC ecosystems.

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Material and methods

Experimental setup and sampling

The first mesocosm experiment was performed in the Bay of Calvi (BC; Corsica, France) during summer (June-July 2012), and the second experiment was conducted in the Bay of Villefranche (BV; France) during winter (February-March 2013). A complete description of the sites, experimental setup, initial conditions and evolution of the core parameters is given in Gazeau et al. (submitted, this issue-a). Briefly, for each experiment, nine mesocosms (2.5 m in diameter, 12 m depth and ~50 m 3 in total volume) were deployed during 20 (BC) and 11 days (BV). Mesocosm bags were made of plastic material (polyethylene mixed with vinyl acetate) with nylon mesh to allow a maximum resistance and light penetration. In order to avoid atmospheric deposition, the top of the mesocosms were covered with UV-transparent ethylene tetrafluoroethylene (ETFE) roofs, except for periods of sampling during which the roofs were partially opened. Three unperturbed mesocosms were used as controls (C1, C2 and C3) and six mesocosms were enriched with CO 2 to obtain a gradient of pCO 2 levels (P1 to P6). Targeted pCO 2 levels were obtained via addition of 75 to > 300 L of CO 2 -saturated water into mesocoms. A diffusing system was used to ensure a perfect mixing of this CO 2 -saturated seawater inside the mesocosms. In order to minimize the stress induced by the addition of large quantities of acid water, the acidification of the mesocosms was performed over the course of 4 days. Control mesocosms presented initial pCO 2 levels of 450 and 350 µatm in BC and BV, respectively. In BC, the six targeted elevated pCO 2 levels were P1: 550, P2: 650, P3: 750, P4: 850, P5: 1000 and P6: 1250 µatm. In BV, the targeted levels were P1: 450, P2: 550, P3: 750, P4: 850, P5: 1000 and P6: 1250 µatm.

Mesocosms were anchored in clusters of 3, each cluster containing one control mesocosm accompanied by a medium and a high pCO 2 level (cluster 1: C1, P1, P4; cluster 2: C2, P2, P5

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ACCEPTED MANUSCRIPT 8 and cluster 3: C3, P3, P6). On June 24 th 2012 (in BC) and on February 21 st 2013 (in BV), the targeted pCO 2 levels were reached and the experiments started (day 0). Daily depth-integrated sampling (0 -10 m) was performed using 5 L Hydro-Bios integrated water samples in the nine mesocosms and in the external environment (OUT) in order to follow the dynamics of each considered parameter in natural free waters.

Seawater samples for total alkalinity (A T ; 500 mL) measurements were filtered on

Analytical method

Carbonate chemistry, chlorophyll a and heterotrophic prokaryote abundance

A T was determined potentiometrically and C T was determined using an inorganic carbon analyser (AIRICA, Marianda©, Kiel, Germany) coupled to an infrared gas analyser (LI-COR© 6262). More detailed analytical protocols are given in Gazeau et al. (submitted, this issue-a). All parameters of the carbonate chemistry were determined from C T , A T , temperature and salinity using the R package seacarb [START_REF] Lavigne | Seacarb: seawater carbonate chemistry with R[END_REF]. Chlorophyll a (Chla) measurements were performed by high performance liquid chromatography (HPLC, following the protocol described in Gazeau et al. (submitted, this issue-b). The abundance of free-living heterotrophic prokaryotes (HP) was estimated by flow cytometry as described in Celussi et al. (in press, this issue).

Dissolved inorganic phosphorus (PO 4 3-

) and dissolved organic phosphorus (DOP)

A Liquid Waveguide Capillary Cell (LWCC), made of quartz capillary tubing, was used to manually analyze PO 4 3-

. The LWCC was connected to a spectrophotometer and measurements were performed in the visible at 710 nm. This spectrometric method is based on Murphy and Riley's molybdenum blue method and uses two reagents: (1) a mixed reagent composed of an ammonium molybdate solution (40 g L -1 in ultrapure water) and an antimony potassium tartrate solution (3 g L -1 in ultrapure water) in 5N sulfuric acid solution, and (2) an ascorbic acid solution (18 g L -1 in 2/3 of ultrapure water and 1/3 of SDS surfactant solution;

Zimmer and Cutter, 2012;Zhang and Chi, 2002). The LWCC was 2.5 m long in order to ]) were calculated using a calibration method in ultrapure water, following the Beer-lambert's law, and validated by a standard addition method (t student = 0.988, n = 25, p value = 0.30, α = 0.05).

To determine DOP concentrations ([DOP] UV ), samples were irradiated by ultra-violet light (UV) using the 705 UV Digester (Metrohm). Samples were transferred into 10 mL (a total of 20 mL was irradiated for each sample) quartz tubes and irradiated during 90 min with a mercury vapor lamp, according to the method UV light photooxidation described by [START_REF] Armstrong | Photochemical combustion of organic matter in sea water, for nitrogen, phosphorus and carbon determination[END_REF]. With the standard molybdenum blue method of Murphy and Riley, the concentration of total dissolved phosphorus [TDP] was measured in irradiated samples.

[DOP] UV was then determined by subtracting [PO 4

3-

] from [TDP].

Dissolved inorganic nitrogen (NO x ) and dissolved organic nitrogen (DON)

The exact same protocol (and equipment) as described by [START_REF] Louis | Dissolved inorganic nitrogen and phosphorus dynamics in seawater following an artificial Saharan dust deposition event[END_REF] was used to determine dissolved inorganic nitrogen concentrations. Briefly, the analysis of NO x (= NO 3 -+ NO 2 -) was performed by a spectrometric method in the visible at 540 nm, with a 1 m LWCC. This method is based on the colored reaction between NO 2 -and 2 reagents (sulfanilamide and NED) to form a pink azo compound. Previously, samples were passed through a copperized cadmium column to reduce NO 3 -to NO 2 -and determine NO x concentrations ([NO x ]) using a calibration curve in ultrapure water. The limit of detection was ~10 nM and the reproducibility was ~6% (tested using a reference solution, CertiPUR, Merck;

Dissolved iron (DFe)

DFe concentrations were measured by flow injection with online preconcentration and chemiluminescence detection using the exact protocol, instrument, and analytical parameters as described by [START_REF] Bonnet | Atmospheric forcing on the annual iron cycle in the Western Mediterranean Sea: a 1-year survey[END_REF]. Briefly, our flow injection analysis and chemiluminescence detection (FIA-CL) instrument is based on the method described by [START_REF] Obata | Automated determination of iron in seawater by chelating resin concentration and chemiluminescence detection[END_REF]. In order to quantify the low DFe concentration corresponds to the difference of pCO 2 level between the treatments and the 3 controls at each time-point. A significant linear trend between MD and pCO 2 was considered when p value < 0.05 (Fisher'test). Above this threshold, we assumed that acidification had no linear effect on nutrient dynamics. All regressions were conducted using the R software.

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Results

Initial conditions

Table 1 presents the initial conditions (day 0) of pCO 2 and pH T (on the total scale) in the nine mesocosms during both experiments. In BC, the initial pCO 2 in control mesocosms was 467 ± 5 µatm corresponding to a pH T of 8.02. At the start of the experiment, perturbed pCO 2 levels were close to the targeted levels and ranged from 609 to 1353 µatm (P1 to P6) corresponding to a pH T gradient from 7.92 to 7.61. In BV, initial pCO 2 levels in control mesocosms were lower than in BC, with an average pCO 2 of 358 ± 17 µatm corresponding to a pH T of 8.12. Initial pCO 2 levels in perturbed mesocosms were close to the targeted levels and ranged from 494 µatm (P1) to 1250 µatm (P6) corresponding to a pH T ranging from 8.0 to 7.63.

Initial nutrient and Chla concentrations in BC and in BV inside control mesocosms and outside are presented in Table 2. In BC, [Chla] was higher outside (0.12 µg L -1 ) than inside mesocosms (0.06 ± 0.01 µg.L -1 ). The same holds true for [PO 4 In BV, [Chla] was much higher than in BC with initial values of 1.15 ± 0.06 µg L -1 inside control mesocosms and 0.95 µg L -1 outside. This was also the case for [DOP] UV with values of 45 ± 1 nM inside control mesocosms and 48 nM outside. Regarding [PO 4

3-], values were lower in BV than in BC and were similar inside and outside control mesocosms (~10 nM).

[DON] UV were also similar outside and inside control mesocoms at 4.0 and 3.6 ± 0.7

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ACCEPTED MANUSCRIPT 14 µM respectively. In contrast, [NO x ] was higher in BV than in BC and a difference between OUT and control mesocosms was observed (1.17 vs 0.13 ± 0.03 µM).

[DFe] was 1.2 nM inside and outside control mesocosms.

Environmental and experimental conditions during the summer experiment in the Bay of Calvi

In BC, in the perturbed mesocosms, as no further addition of CO 2 -saturated seawater was performed, pCO 2 levels slightly decreased after the acidification phase over the course of the experiment, especially for high CO 2 mesocosms (P5 and P6). During the experiment, mean pCO 2 levels were C1: 429, C2: 427, C3: 429, P1: 508, P2: 586, P3: 660, P4: 747, P5:

828, P6: 990 µatm.

[Chla] did not show any strong temporal trend, although levels slightly increased on day 10 to maximal concentrations of 0.09 ± 0.003 µg L -1 on day 14 (Fig. 1).

With regard to the heterotrophic prokaryotes, the initial average abundance (day 0) in all mesocosms was 5.0 ± 0.3 x 10 5 cell mL -1 . Their density remained relatively constant during the experiment, and ranged from 4.0 x 10 5 to 6.1 x 10 5 cell mL -1 . The evolution of [PO 4

3-],

[NO x ], [DOP] UV and [DON] UV in BC in all mesocosms and outside is shown in Figure 2.

After an initial decrease in [PO 4

3-] inside and outside mesocosms between day 0 and day 1,

[PO 4 3-
] remained low and ranged from 3 to 12 nM inside all mesocosms, and from 7 to 13 nM outside until day 20. Regarding DOP, a decrease was also observed between day 0 and day 1 outside mesocosms and in control mesocosms (no data available for perturbed mesocosms at day 0 except for P2). From day 1 to day 13, [DOP] UV varied between 5 and 20 nM with no significant trend, except for P1 where an increase from 8 to 16 nM (linear regression: r² = 0.89, n = 5, p value = 0.02) was observed.

[NO x ] increased from 47 ± 14 nM to 97 ± 22 nM in all mesocosms between day 0 and day 1, followed by a decrease in both inside and outside mesocosms until day 8 (linear A slight decrease between day 1 and day 13 could be observed in all mesocosms (linear regression: r² = 0.83, n = 6, p value = 0.01).

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Environmental and experimental conditions during the winter experiment in the

Bay of Villefranche

In BV, drops in pCO 2 levels were more important than in BC due to strong winds with occurred between day 3 and day 9 (linear regression: r² = 0.95, n = 6, p value < 0.001) with the largest rise observed in P4, P2 and C2. [NO x ] in those mesocosms was the highest measured during the BV experiment (Fig. 4). The lowest [NO x ] was measured in P1 and the standard deviation was the lowest in P6 indicating that [NO x ] in that mesocosm was more stable than elsewhere (Fig. 4).

With respect to DON, concentrations were around 3.5 ± 0.7 µM inside and outside mesocosms and no clear trend could be observed, except for C1 and outside mesocosms

Representativeness of the environmental settings

The mesocosm experiments presented here focused on the impact of ocean acidification on plankton communities and are the first of such experiments to have been conducted in a low nutrient, low chlorophyll system. Our experimental protocol allowed for quantification of both inorganic and organic nutrient concentrations at nanomolar levels, and therefore allowed us to assess, in detail, temporal nutrient variability and contribution to the functioning of the community. Furthermore, this study provided the opportunity to compare nutrient levels measured at the two sites, with data compiled from the literature and obtained using both conventional (micromolar level) and nanomolar techniques (see Table 3). Finally, this compilation allowed us to put our results in a broader context of seasonal and spatial variability of nutrient levels in the Mediterranean Sea.

In the Bay of Calvi, the low initial in situ [NO x ] (50 nM) was in agreement with [NO x ] < 30 nM measured during a mesocosm experiment conducted in summer 2008 and 2010 in coastal waters of Corsica [START_REF] Ridame | Contrasted saharan dust events in LNLC environments: impact on nutrient dynamics and primary production[END_REF]. The low initial in situ [PO 4

3-] (35 nM) was consistent with data reported in previous studies conducted in the same season (Table 3; 5-20 nM in surface waters). These nitrate and phosphate depleted conditions, observed at the start of this experiment, are representative of summer oligotrophic conditions in the Mediterranean Sea.

In the Bay of Villefranche, the initial in situ [NO x ] (1.17 µM) was very close to the concentration (1.25 µM) measured during the same week at a sampling site located at the entrance of the Bay (Service d'Observation en Milieu LITtoral; http://somlit.epoc.u-bordeaux1.fr/fr/). The same range [NO x = 0.7-2.3 µM] was also found at that site in surface waters in February 2012 [START_REF] Thyssen | Onset of the spring bloom in the Northwestern Mediterranean Sea: influence of environmental pulse events on the in situ hourly-scale dynamics of the phytoplankton community structure[END_REF]. [PO 4

3-] can be variable in winter in surface

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waters: ~10 nM in this study (but ~50 nM from SOMLIT, close to the detection limit using the conventional analytical method), from 25 to 92 nM using a conventional analytical method [START_REF] Thyssen | Onset of the spring bloom in the Northwestern Mediterranean Sea: influence of environmental pulse events on the in situ hourly-scale dynamics of the phytoplankton community structure[END_REF] and 7 nM using nanomolar techniques [START_REF] Louis | Dissolved inorganic nitrogen and phosphorus dynamics in seawater following an artificial Saharan dust deposition event[END_REF].

Those concentrations were low compared to the ones measured at the open ocean DYFAMED site (43°25'N, 7°52'E): 200-300 nM during the winter-mixing period (January-February)

followed by a decrease from 200 to 30 nM during the winter-spring transition [START_REF] Pulido-Villena | Transient fertilizing effect of dust in Pdeficient LNLC surface ocean: fertilizing effect of dust in LNLC ocean[END_REF]. While a NO x -replete system outside mesocosms was in agreement with conditions usually reported for the Mediterranean Sea in winter, it must be stressed that this was no longer the case at the start of the experiment inside all mesocosms. Instead, a rapid NO x uptake by phytoplankton during the acidification phase (4 days before day 0; Gazeau et al., submitted, this issue-a) was concomitant with the increase in biomass in all mesocosms (Figs. 1 and3). This led to a sharp drop in [NOx] from 0.96 to 0.13 µM. Although [NO x ] outside mesocosms also slightly declined (from 1.6 to 1.2 µM) during that acidification period, [NO x ] was maintained at a higher level due to a winter-mixing situation (the vertical profile of temperature from SOMLIT indicated a homogeneous water column over 200 m) and/or due to external input via atmospheric wet depositions (Migon, pers. comm.).

Therefore, the conditions encountered at the start of the experimental phase were not fully representative of winter conditions for that area.

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In surface waters, reported ratios are much more variable: from 6 to 68 in the 0-150 m layer (Table 3). This reflects the difficulty to accurately measure nutrients with conventional methods in severely nutrient depleted waters, with concentrations often below DL during the stratification period (Pasqueron de [START_REF] Pasqueron De Fommervault | Temporal variability of nutrient concentrations in the Northwestern Mediterranean Sea (DYFAMED time-series station)[END_REF]. Estimates of surface water ratios are thus questionable in terms of quality. In surface waters of the Bay of Calvi, using nanomolar techniques, we found an initial in situ [NO x ] / [PO 4

3-] of 1.7 ( ] at the start of the experiment. As mentioned above, NO x uptake occurred during the acidification phase, causing [NO x ] to decrease to 129 ± 30 nM and leading to an average [NO x ] / [PO 4

3-] = 13 in all mesocosms. These conditions suggest a potential N and P co-limitation for the plankton community.

During both summer and winter experiments, the contribution of dissolved organic nutrients to the total dissolved phosphate and nitrogen pool was high. This is in agreement with previous studies in the Mediterranean Sea ([DON] / [TDN] = 60-100% and [DOP] /

[TDP] = 30-100%; Table 3). Considering that organic forms of nutrients follow the same dynamics as DOM, which are known to accumulate in surface waters in the Northwestern Mediterranean Sea during the stratification period (Copin-Montegut and Avril, 1993), [DON] and [DOP] should peak in summer. In contrast, in our study, initial in situ [DON] UV and

[DOP] UV were higher in winter than in summer. The large labile fraction of DOP (60-80%; Aminot and Kerouel, 2004;Lønborg and Alvarez-Salgado, 2012) and the oligotrophic nature of the Mediterranean Sea could explain a high turnover rate of DOP, thus preventing its accumulation in surface waters. Previous studies in the Sargasso Sea [START_REF] Lomas | Sargasso Sea phosphorus biogeochemistry: an important role for dissolved organic phosphorus (DOP)[END_REF], as

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21 well as in an oligotrophic subtropical gyre [START_REF] Steinberg | Overview of the US JGOFS Bermuda Atlantic Time-series Study (BATS): a decade-scale look at ocean biology and biogeochemistry[END_REF], showed that [DOP] peaked during the winter/spring bloom and declined slowly throughout the summer resulting in enhanced P-stress. This was concomitant with an increase in the alkaline phosphatase activity (APA). Similarly, in the present study, APA was higher in summer than in winter (Celussi et al., in press, this issue), possibly explaining lower [DOP] in summer than in winter. At the start of the experiment in the Bay of Villefranche, in situ [DON] UV and [DOP] UV were close to reported concentrations for the Mediterranean Sea in winter (Table 2; [START_REF] Raimbault | Wet-oxidation and automated colorimetry for simultaneous determination of organic carbon, nitrogen and phosphorus dissolved in seawater[END_REF]Santinelli et al., 2002). In the Bay of Calvi, in situ [DOP] UV was in the range (0.01 -0.10 µM) of reported data in summer in surface waters of the Mediterranean Sea (Table 3).

With respect to in situ [DON] UV, the observed value of 2 µM was lower than concentrations reported previously for the western basin in winter (from 4.5 to 5.5 µM; [START_REF] Raimbault | Wet-oxidation and automated colorimetry for simultaneous determination of organic carbon, nitrogen and phosphorus dissolved in seawater[END_REF]; Table 3).

In the Bay of Villefranche, initial in situ [DFe] (1.2 nM; Table 1) was lower than values reported in the Bay of Villefranche at the same season by [START_REF] Bressac | Post-depositional processes: what really happens to new atmospheric iron in the ocean's surface?[END_REF] 3.8 nM at 5 m) and, instead, more similar to winter offshore concentrations (DYFAMED site, 0.81 ± 0.07 nM; [START_REF] Bonnet | Atmospheric forcing on the annual iron cycle in the Western Mediterranean Sea: a 1-year survey[END_REF]. The annual [DFe] dynamics in surface waters in the Northwestern Mediterranean Sea are controlled by several factors such as atmospheric deposition, winter mixing, phytoplankton uptake, and particle scavenging. Over an annual time scale, [DFe] ranged from 0.2 to 1.2 nM at DYFAMED site [START_REF] Bonnet | Atmospheric forcing on the annual iron cycle in the Western Mediterranean Sea: a 1-year survey[END_REF].

Minimal values were measured when [Chla] was the highest, as a consequence of biological uptake and removal by the sinking particles. Under stratified conditions, atmospheric deposition results in an accumulation of [DFe] in surface waters that is isolated from deeper layers [START_REF] Bonnet | Atmospheric forcing on the annual iron cycle in the Western Mediterranean Sea: a 1-year survey[END_REF][START_REF] Pulido-Villena | Transient fertilizing effect of dust in Pdeficient LNLC surface ocean: fertilizing effect of dust in LNLC ocean[END_REF]

Nutrient dynamics: Biological vs. Acidification effect

During both experiments, results showed that none of the parameters investigated in this study were linearly related to CO 2 enrichment (Fig. 5). These two experiments, conducted in waters depleted in PO 4 3-, showed low variations in PO 4 3-over the experimental period and no significant differences in PO 4 3-dynamics over the different pCO 2 conditions. With respect to NO x , concentrations followed the same temporal trend with no significant differences between control and perturbed mesocosms (Figs. 4 and5). In the Bay of Villefranche, during the acidification phase, although targeted pCO 2 conditions were not yet reached, a drop in NO x observed in all mesocosms suggested that NO x uptake by phytoplankton was not impacted by an increase in pCO 2 in the perturbed mesocosms. During the summer and winter experiments, a trend in DON UV and DOP UV was now observed, and concentrations were not affected by acidification (Fig. 5). Although a decrease in pH would favor the solubility of DFe (Breitbarth et al., 2010a), DFe concentrations did not increase with increasing pCO 2 (Fig. 5). As such, it appears that both direct chemical and indirect biological processes (further discussed in this section) that controlled nutrient dynamics during these experiments were independent of pCO 2 conditions.

Previous mesocosm experiments, similar to the ones presented here, focused on the effect of ocean acidification on plankton communities but were performed under nutrient replete conditions using artificial nutrient addition. For instance, in the framework of the Pelagic Ecosystem CO 2 Enrichment (PeECE) project, three land-based mesocosm three experiments, and in agreement with our results, NO x and PO 4 3-uptake were similar in all mesocosms, indicating that phytoplankton nutrient uptake was insensitive to changes in seawater pH [START_REF] Engel | Testing the direct effect of CO 2 concentration on a bloom of the coccolithophorid Emiliania huxleyi in mesocosm experiments[END_REF][START_REF] Ciesm | Biogeochemical impacts of ocean acidification-emphasis on carbonate production and dissolution[END_REF][START_REF] Schulz | Build-up and decline of organic matter during PeECE III[END_REF][START_REF] Bellerby | Marine ecosystem community carbon and nutrient uptake stoichiometry under varying ocean acidification during the PeECE III experiment[END_REF]. As similar increases in DOC, DON and DOP were observed in all PeECE mesocosms, this further suggests that acidification, as observed in our experiments, did not impact the production or consumption of dissolved organic nutrients. In contrast to the lack of pCO 2 effects on inorganic and organic macronutrients, Breitbarth et al. (2010b) have shown, during the PeECE III experiment and in agreement with previous studies (e.g. [START_REF] Liu | The solubility of iron in seawater[END_REF][START_REF] Millero | Effect of ocean acidification on the speciation of metals in seawater[END_REF], that ocean acidification may lead to enhanced Fe-bioavailability due to

(1) an increase in the organically complexed Fe (III) and (2) an increase in Fe(II) concentrations. This is not consistent with data acquired during our experiment in the Bay of Villefranche where [DFe] in high-CO 2 mesocosms were not significantly higher than in low-CO 2 mesocosms.

More recently, a large off-shore mesocosm experiment has been conducted in the Arctic ocean (Svalbard) following a pCO 2 gradient ranging from 185 to 1420 µatm. This study has shown that, before nutrient addition ([NO x ] = 0.1 µM and [PO 4

3-] = 0.06-0.09 µM), in agreement with our results, organic and inorganic nutrient dynamics were similar in all pCO 2 treatments [START_REF] Schulz | Temporal biomass dynamics of an Arctic plankton bloom in response to increasing levels of atmospheric carbon dioxide[END_REF]. During the phytoplankton growth stimulated by nutrient addition (after day 13, [NO x ] d13 = 5.5 µM and [PO 4

3-] d13 = 0.4 µM), while the same

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT
24 dynamics in DON and DOP were observed in all mesocosms, inorganic nutrient uptake (NO x and PO 4 3 ) was faster at higher CO 2 levels indicating a shift in phytoplankton community composition [START_REF] Schulz | Temporal biomass dynamics of an Arctic plankton bloom in response to increasing levels of atmospheric carbon dioxide[END_REF], a shift that was not observed during our experiments (Gazeau et al. submitted, this issue-b).

Among the very few bottle experiments focused on the effect of ocean acidification on planktonic communities under low nutrient conditions, results of Yoshimura et al. (2010;

[NO x ] initial = 0.05 ± 0.02 µM and [PO 4

3-] initial = 0.25 ± 0.01 µM) support our conclusion that under limiting conditions, nutrient dynamics are not dependent on pCO 2 levels.

In our study, nutrient measurements were performed at nanomolar levels providing greater detail to assess temporal dynamics. As nutrients dynamics did not differ between pCO 2 levels, the nine mesocosms can be considered as replicate measurements. As such, daily average nutrient concentrations were calculated (Figs. 6, 7 and 8) providing insight to the relationship between nutrient dynamics and major biological parameters (chlorophyll a, particulate organic carbon, organic matter) and processes (production and degradation, heterotrophic enzymatic activities, nitrification). Such stocks and fluxes, measured during the experiments, are presented and discussed in details in companion papers (Gazeau et al., submitted, this issue-b; Maugendre et al., in press, this issue-a; Celussi et al., in press, this issue).

As a brief summary of the companion papers, during the summer experiment in the Bay of Calvi, CO 2 enrichment did not have any significant effect on primary production (Maugendre et al. in press, this issue-a), abundance and activity of heterotrophic prokaryotes (Celussi et al., in press, this issue), as well as Chla concentrations and phytoplankton community composition (Gazeau et al., submitted, this issue-b). The summer plankton community was mostly composed of small species such as haptophytes and cyanobacteria decreased during the first day and then slightly decreased until day 7 (Fig. 6). In such a nutrient-depleted system, haptophytes, considered as mixotrophic, are known to be good competitors for low nutrient resources [START_REF] Sanders | Mixotrophic protists in marine and freshwater ecosystems[END_REF]. The ratio between uptake rates of NO x (12.8 nmol N L -1 d -1 ) and PO 4 3-(0.5 nmol P L -1 d -1 ), determined from the slope of NO x and PO 4 3-decreases between day 1 and day 7, was around 25. This is close to the typical [NO x ] / [PO 4

3-] in Mediterranean waters below the mixed layer (~22). Despite a slight decrease in

[DON] UV , an uptake of DOP was not observed (except between day 0 and day 1). This was unexpected since the gradual increase of the alkaline phosphatase activity (APA) observed until day 12 (on average 169 nM h -1 ; Celussi et al., in press, this issue) should be concomitant with a decrease in [DOP] (i.e., Ruttenburg and Dyhrman, 2002;Mahaffey et al., 2014;Fig. 7).

It could be due to either the low [DOP] UV during this experiment and/or a rapid turnover of DOP (Benitez-Nelson and Buesseler, 1999). Towards the end of the experiment, [NO x ] raised while [PO 4

3-] remained stable (Fig. 6). Excluding the external inputs in the mesocosms by UV-transparent roofs covering them, the release of NO x would result from bacterial remineralization.

In the Bay of Villefranche, as already mentioned, a rapid nutrient uptake by phytoplankton was observed during the acidification phase, concomitant with the increase in

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT
26 biomass (Fig. 6). After this [Chla] increase (and [NO x ] drop) in all mesocosms, the biomass of the autotrophic community decreased (from 1.1 to 0.9 µg L -1 ) while the abundance of heterotrophic prokaryotes increased in all mesocosms (from 7.3 x 10 5 cell mL -1 to 1.1 x10 6 cell mL -1 ; Fig. 6). From day 3, an increase in [NO x ] occurred in all mesocosms that was likely linked to enhanced heterotrophic prokaryotes activity (Fig. 6). It is possible that the increase in [NO x ] of ~18 nmol N L -1 d -1 was induced by nitrification activity estimated to be ̴ 19 nmol N L -1 d -1 (Al-Moosawi, pers. comm.) although no negative correlation between [NH 4 + ] and

[NO x ] could be observed (Fig. 8b). Nitrification could have been preceded by ammonification of the DON pool produced from the degradation of particulate organic matter by heterotrophic prokaryotes. This could explain the negative correlation between particulate organic carbon concentrations ([POC]) and [NO x ] (Fig. 8c), while [DON] remained stable (Fig. 8a; sampling and analytical protocols for POC are described in Gazeau et al., submitted, this issue-b). The decrease in POC could also be due to the phytoplankton cell lysis under experimental stress (e.g. water column trapped in mesocosm). The large amount of released nutrient-rich organic matter resulted in an increase in bacterial abundance and heterotrophic activity. Ultimately, regenerated inorganic nutrients such as NH 4 + and NO x by nitrification were produced (Al-Moosawi, pers. comm.). While the heterotrophic activity based on the regenerated production was usually higher in summer, the increase in phytoplankton biomass (mostly composed of small species such as cryptophyceae and haptophyceae; Gazeau et al.

submitted, this issue-b) at the start of experiment in the Bay of Villefranche allowed an increase in the abundance and activity of heterotrophic prokaryotes (see Celussi et al., in press, this issue, for further details).

Finally, in the Bay of Villefranche, in relation to the increase in heterotrophic activity and nitrification, [DFe] increased significantly (p value < 0.001) from day 3 to day 7 (Fig. 9a).

This could be the result of either an enhancement of Fe solubility by organic ligands exuded indeed observed between day 3 and day 7 (Fig. 9b). Although [Chla] continued to decline until day 11, [DFe] did not increase after day 7. Finally, other biotic and abiotic processes such as scavenging and/or aggregation [START_REF] Johnson | What controls dissolved iron concentrations in the world ocean?[END_REF], that were not explicitly studied here, could have explained this decrease in [DFe] observed at the end of the experiment. Based on these two experiments, ocean acidification did not affect the dynamics of nutrients both under inorganic and organic forms: (1) the concentration of dissolved organic nutrients remained stable regardless of pCO 2 conditions, (2) the expected increase in iron solubility at high pCO 2 was not observed and (3) the dynamics of NO x and PO 4 3-were controlled by the biological processes independently of pCO 2 conditions.

Although the use of nanomolar techniques during these experiments facilitated documentation of nutrient dynamics with great precision, the direct measurement of the turnover times of P and N would provide a better view on these dynamics resulting from biological uptake. Such measurement would improve the assessment of ocean acidification impacts on plankton nutrient uptake, particularly in summer, when the turnover time of P is very low (1-2 h; Moutin et al., 2002b). As such, we recommend the measurement of nutrient turnover rates over nanomolar measurements in future mesocosm studies conducted in oligotrophic areas.

This absence of pCO 2 effects on the bioavailability of nutrients in low nutrient and low chlorophyll waters suggests that there will be no direct positive or negative feedback by the biological pump in response to the rise in atmospheric CO 2 . However, other changes ), dissolved organic and total nitrogen (DON and TDN, respectively) and dissolved organic and total phosphate (DOP and TDP, respectively) concentrations and their ratio reported in the literature for the Mediterranean Sea.

Following

  established theoretical equilibriums of dissolved inorganic phosphate and ammonium (NH 4 + ) in seawater as a function of pH (i.e. Zeebe and Wolf-Gladrow, 2001), a decrease in surface ocean pH expected to occur this century will lead to a decrease of phosphate

  III) binding and affect the chemical speciation of Fe by favoring Fe chelated

  of detection to 1 nM. The repeatability and reproducibility standard deviations were around 4.8% and 7%, respectively. Phosphate concentrations ([PO 4 3-

-.

  t student = 0.263, n = 7, p value = 0.80, α = 0.05). Similar to DOP, DON concentrations ([DON] UV ) were determined by the irradiation UV method during which DON and all reduced inorganic ions, such as NO 2 -, are oxidized to NO 3 Using the same spectrometric method used to analyze NO x (see section above), the dissolved nitrogen [TDN] was measured, and [DON] UV was calculated by subtracting [NO x ] from [TDN].

  [DFe] in seawater, samples (with pH adjusted to ~5) were passed through MAF-8HQ (8-quinolinol-immobilized fluoride containing metal alkoxide glass) resin column to selectively collect and preconcentrate DFe. After elution of DFe by a HCl solution, the eluent was mixed with a luminol solution (0.74 M), an ammonium hydroxide solution (1 M) and a hydrogen peroxide (H 2 O 2 ) solution (0.7 M). A luminescence reaction occurs between the luminol and H 2 O 2 , catalyzed with DFe in alkaline conditions. Two distinct seawater reference materials were used to control for analytical accuracy: NASS-5 (North Atlantic Surface Seawater) and D1-508 (SAFe reference seawater) with measured [DFe] of 3.74 ± 0.19 nM and 0.65 ± 0.03 nM respectively in good agreement with consensus values (3.763 ± 0.001 and 0.67 ± 0.04 nM respectively). The repeatability and reproducibility were lower than 10%, and the detection limit was ~20 pM. Data of [DFe] are only available for the experiment in BV.2.3. Data availabilityData collected during both experiments are freely available on Pangaea, Bay of Calvi (BC): http://doi.pangaea.de/10.1594/PANGAEA.810331 and Bay of Villefranche (BV): http://doi.pangaea.de/10.1594/PANGAEA.835117. DOP] UV , [DON] UV and [DFe], the measured concentration in each perturbed mesocosm for each sampling date was subtracted from the average of concentrations in the 3 controls at that time-point. The relative mean deviation (MD, expressed as a %) for each treatment and time-point was obtained by dividing that value by the average concentration in the 3 controls. A simple linear regression model was used to test MD variations as a function of delta pCO 2 , which

  ), while lower values were measured outside (~16 nM). DON UV represented the largest dissolved N pool with concentrations of 2 and 3.6 ± 2.1 µM inside and outside control mesocosms respectively, while [NO x ] was lower with no difference between outside and inside control mesocosms (0.05 µM vs 0.06 ± 0.01 µM).

  mesocosms P1 to P4 showing very similar levels by the end of the experiment. Mean pCO 2 levels were C1: 357, C2: 356, C3: 352, P1: 456, P2: 486, P3: 544, P4: 545, P5: 719, P6: 941 µatm.[Chla] averaged 0.98 ± 0.15 µg L -1 in the nine mesocosms along the 12-day experiment. Chlorophyll a remained slightly above levels in the surrounding waters for the entire experimental period, except for the last day (day 12) when concentrations increased abruptly outside the mesocosms. As one can see in Figure1,[Chla] increased during the acidification phase, consuming a large proportion of available nutrients (see below), notably NO x , before the start of the experimental phase (day 0). In all mesocosms, after this initial peak, [Chla] gradually decreased until the end of the experiment. While the abundance of the heterotrophic prokaryotes outside the mesocosms remained constant over the experiment (6.3 ± 0.3 x 10 5 cell mL -1 ), it increased from day 2 to day 12 to reach 1.1 ± 0.1 x 10 6 cell mL -1 M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 16 inside all mesocosms. The initial average density of heterotrophic prokaryotes (day 0) was 7NO x ], [DOP] UV , [DON] UV and DFe evolution in BV in all mesocosms and outside are shown in Figure 3. During the acidification phase, the increase in [Chla] outside and inside mesocosms was concomitant with the decrease in [PO 4 3-] (from 15 ± 2 to 10 ± 2 nM) and [NO x ]. At day 0, while [NO x ] outside mesocosms remained high (~ 1 µM), all mesocosms were depleted in NO x with an average concentration of 129 ± 30 nM. No strong variations for [PO 4 3-] were observed in the mesocosms over the course of the BV experiment (global average: 9 ± 1 nM). [PO 4 3-] was generally higher outside than inside mesocosms and a peak of 23 nM was reached on day 6 followed by a decrease until the end of the experiment with similar inside and outside concentrations (12 nM at day 12). For the strongest perturbations (P4, P5 and P6),[DOP] UV dropped between day 0 and day 1, while[DOP] UV outside and in P2, P3 and control mesocosms tended to increase. [DOP] UV then remained stable (~44 ± 6 nM) in all mesocoms and outside until the end of the experiment.Over the course of the BV experiment, [NO x ] was much higher outside (range 889-1618 nM) than inside mesocosms (range 40-682 nM). Inside all mesocosms, [NO x ] was stable until day 3 with an average concentration of 121 ± 50 nM. A general increase in[NO x ]

  ] UV increased after day 1 until day 10 (linear regression C1: r² = 0.86, n = 6, p value = 0.02; linear regression OUT: r² = 0.82, n = 6, p value = 0.03).In addition to NO x , PO 4 3-, DON and DOP, [DFe] data were documented during the experiment in BV.[DFe] were similar outside and inside mesocosms (2.3 ± 0.8 vs. 2.5 ± 1.2 nM, respectively). No clear trend was observed over the course of the experiment, although[DFe] slightly increased between day 3 and day 7 in all mesocosms. 3.4. Nutrient dynamics vs. pCO 2 Box-plots of inorganic and organic nutrient concentrations during both experiments are shown in Figure 4, and relative MD (see section 2.4) of each perturbed mesocosm during the entire experiments was plotted against delta pCO 2 for each perturbed mesocosm at each time-point, in Figure 5. During both experiments, [PO 4 3-] dynamics were not significantly related to pCO 2 levels. In addition, there was no significant relationship between pCO 2 and relative MD for PO 4 3-(Fig. 5; BC: p value = 0.06; BV: p value = 0.69). The same holds true for [DOP] UV (Fig. 5; BC: p value = 0.97; BV: p value = 0.33) and [DON] UV in BC (Fig. 5; BC: p value = 0.35). In BV, despite the significant relationship between relative MD for [DON] UV and delta pCO 2 (p value of 0.03), this relationship appears weak considering the large observed deviation between the 3 controls (Figs. 4 and 5). In BC, pCO 2 had no impact on [NO x ] (Fig. 5; p value = 0.31). In BV, although box-plots for [NO x ] indicate a difference between controls, P2 and P4on one side and P1, P3, P5 and P6 on the other side (Fig.4), no significant effect of pCO 2 gradient on NO x dynamics was observed (Fig.5; p value = 0.36). Despite the fact that[DFe] was generally lower in CO 2 -enriched mesocosms compared to controls (Fig.4), the pCO 2 gradient did not significantly affect the evolution of [DFe] when considering the full duration of the experiment (Fig.5; BV: p value = 0.91).

  used to depict the P or/and N (co)-limitation(s) for phytoplankton carbon fixation. In the Mediterranean Sea, below the mixed layer, [NO x ] / [PO 4 3-] is close to 22 (e.g. Mc Gill, 1965; Krom et al., 1991; Pujo-Pay et al., 2011) and higher than the Redfield ratio (N / P = 16). In deep waters, estimates are more robust as ratios are calculated based on [PO 4 3-] and [NO x ] well above the analytical detection limit (DL) of conventional analytical method (DL: [PO 4 3-] ≤ 20-50 nM and [NO x ] ≤ 20 nM).

  . Because atmospheric inputs, such as Saharan dust deposition, have high interannual variability (Guieu et al., 2014 and ref within), concentrations in surface waters are likely also highly variable from one year to another. This could explain the different [DFe] observed in the Bay of Villefranche at similar season. At the start of the winter experiment in the Bay of Villefranche, [DFe] in surface waters was reflective of [DFe] found in the open western Mediterranean Sea in winter.

  inBergen (Norway), in 2001[START_REF] Engel | CO 2 increases 14 C primary production in an Arctic plankton community[END_REF] and 2005 (PeECE I, II and III, respectively). In 2001 and 2003, three treatments represented low (190 µatm), present-day (370-410 µatm) and high (700 µatm) pCO 2 conditions. In PeECE III, there was one presentday condition and two high pCO 2 levels (350, 700 and 1050 µatm). After nutrient addition, [NO x ] initial and [PO 4 3-] initial were 15 and 0.5 µM respectively during PeECE I, 9 and 0.4 µM respectively during PeECE II, and 16 and 0.8 µM respectively during PeECE III. During all

  al., submitted, this issue-b), dominated by mixotrophs or heterotrophs. While the carbon biomass of autotrophs was very low (from 0.009 ± 0.003 to 0.022 ± 0.009 µmol C L -1 ; based on compound specific 13 C incorporation; Maugendre et al., in press, this issue-b), mixotrophs represented the most productive group with an increasing biomass over the experimental period (from 0.05 ± 0.01 to 0.08 ± 0.03 µmol C L -1 ). The low biomass of autotrophs determined by Maugendre et al. (in press, this issue-b) was consistent with the low average [Chla] measured in the Bay of Calvi (from 0.06 to 0.08 µg L -1 ; Fig. 6). Despite a low autotrophic production, [NO x ] decreased by 50% throughout the experiment and [

  the release of DFe in seawater after cell death (Witter et al., 2000) while [Chla] decreased. A negative relationship between [DFe] and [Chla] (p value = 0.001) was

  acidification mesocosm experiments conducted in coastal LNLC areas allowed for investigation of the effect of ocean acidification on nutrient dynamics using nanomolar techniques. While the summer experiment was fully representative of oligotrophic summer conditions in the Mediterranean Sea, the winter experiment failed to reproduce the mesotrophic conditions typical of the winter time. Instead post-bloom conditions were also encountered, with high concentrations of phytoplankton exudates and low nutrient concentrations.

Figure 1 :

 1 Figure captions

Figure 2 :

 2 Figure 2: Evolution of the concentrations of inorganic and organic nutrients (nitrate + nitrite:

Figure 3 :

 3 Figure 3: Evolution of the concentrations of inorganic and organic nutrients (dissolved iron:

Figure 4 :

 4 Figure 4: Box-plots of nutrient concentrations (dissolved iron: DFe, nitrate + nitrite: NO x ,

Figure 5 :

 5 Figure 5: Relative mean deviation (MD; in %) of the concentrations of inorganic and organic

Figure 6 :

 6 Figure 6: Average (all mesocosms) concentrations of chlorophyll a (Chla; green line), nitrate

Figure 7 :

 7 Figure 7: Comparison of the evolution of the concentration of dissolved organic phosphorus

Figure 8 :Figure 9 :

 89 Figure 8: Nitrate + nitrite ([NO x ]) vs dissolved organic nitrogen ([DON]) (a), [NO x ] vs

  photo-oxidation UV; b) high temperature combustion; c) persulfate wet oxidation; d) UV-peroxide oxidation.

  Figure 1 :

Figure 9 :

 9 Figure 9 :

Table 3

 3

	), well below

Table 1 .

 1 Initial (day 0) pCO 2 (µatm) and pH T (total scale) conditions inside the nine mesocosms during the experiments in the Bay of Calvi in summer 2012 and in the Bay of Villefranche in winter 2013.

	Mesocosm experiment	C1	C2	C3	P1	P2	P3	P4	P5	P6
	pCO 2	474	465	462	609	731	790	920	1198	1353
	Bay of Calvi
	pH T	8.01	8.02	8.02	7.92	7.85	7.82	7.76	7.66	7.61
	pCO 2	378	347	350	494	622	690	743	932	1250
	Bay of Villefranche
	pH T	8.10	8.13	8.13	8.00	7.91	7.87	7.85	7.76	7.63

Table 2 .

 2 Initial (day 0) nutrients (dissolved iron: DFe, nitrate + nitrite: NO x , phosphate: PO 4 3-, dissolved organic nitrogen: DON and dissolved organic phosphorus: DOP) and chlorophyll a (Chla) concentrations inside the 3 'control' mesocosms (average and standard deviation calculated from the three independent measurements) for the Bay of Calvi (BC) and the Bay of Villefranche (BV) experiments. (*) When data on day 0 were not available, values have been measured on day 1.

		[PO 4	3-]	[NO x ]	[DOP]	[DFe]	[DON]	[Chla]
	Mesocosm Experiment
		nM		µM	nM	nM	µM	µg L -1
	BC	35		0.05	16		2	0.12
	OUT
	BV	10		1.17 *	48	1.2 *	4	0.95
	BC	22 ± 3	0.06 ± 0.01	23 ± 4		3.6 ± 2.1	0.06 ± 0.01
	Mean control
	BV	10 ± 2	0.13 ± 0.03	45 ± 1	1.2 ± 0.1	3.6 ± 0.7	1.15 ± 0.06

Table 3 .

 3 Nitrate + nitrite (NO x ), phosphate(PO 4

	3-

Acknowledgements

This work was funded by the EC FP7 project 'Mediterranean Sea Acidification in a changing climate' (MedSeA; grant agreement 265103), the project European Free Ocean Carbon Enrichment (eFOCE; BNP-Paribas foundation), the MISTRALS-MERMEX program (Institut des Sciences de l'Univers, INSU), the Corsican local authorities and the Rhone-Mediterranean and Corsica Water Agency (http://www.eaurmc.fr). It is a contribution to the Surface Ocean -Lower Atmosphere Study (SOLAS), the Integrated Marine Biogeochemistry and Ecosystem Research (IMBER) and the LOICZ projects. The STARESO marine station in Corsica is gratefully acknowledged for its assistance and boat support carried out within the framework of the STARECAPMED project funded by the Rhone-Mediterranean and Corsica Water Agency. A grant provided by the University of Paris 6 supported J.L. The staff of the Observatoire Océanologique de Villefranche is gratefully acknowledged for their assistance and boat support, colleagues of the Laboratoire d'Océanographie de Villefranche for providing laboratory space. J.-M. Grisoni, A. Sallon, G. Obolensky, S. Alliouane, B. Hesse, D. Luquet, D. Robin, P. Mahacek and E. Cox are acknowledged for assistance with diving operations. L. Kapsenberg is acknowledged for a careful check of English grammar.


Gilles Pütz

Anthony Martin

Nicolas Gisin

Djeylan Aktas

Bruno Fedrici

Sébastien Tanzilli

Quantum nonlocality with arbitrary limited detection efficiency

The demonstration and use of nonlocality, as defined by Bell's theorem, rely strongly on dealing with non-detection events due to losses and detectors' inefficiencies. Otherwise, the so-called detection loophole could be exploited. The only way to avoid this is to have detection efficiencies that are above a certain threshold. We introduce the intermediate assumption of limited detection efficiency, that is, in each run of the experiment, the overall detection efficiency is lower bounded by ηmin > 0. Hence, in an adversarial scenario, the adversaries have arbitrary large but not full control over the inefficiencies. We analyse the set of possible correlations that fulfil Limited Detection Locality (LDL) and show that they necessarily satisfy some linear Bell-like inequalities. We prove that quantum theory predicts the violation of one of these inequalities for all ηmin > 0. Hence, nonlocality can be demonstrated with arbitrarily small limited detection efficiencies. We validate this assumption experimentally via a twin-photon implementation in which two users are provided with one photon each out of a partially entangled pair. We exploit on each side a passive switch followed by two measurement devices with fixed settings. Assuming the switches are not fully controlled by an adversary, nor by hypothetical local variables, we reveal the nonlocality of the established correlations despite a low overall detection efficiency.

Introduction -When studying the discoveries in fundamental physics of the past century, one cannot help but come across Bell's seminal work on the nonlocal nature of quantum theory [1]. It implies that quantum physics can produce correlations which cannot be explained by a common past with local variables propagating contiguously. This has not only proven fascinating from a foundational point of view, but also given rise to applications in device independent quantum information processing (DIQIP) [2], such as quantum key distribution [3][4][5], randomness generation [START_REF] Colbeck | Quantum And Relativistic Protocols For Secure Multi-Party Computation[END_REF][START_REF] Pironio | [END_REF], or entanglement certification [8,9]. For a semi broad-audience presentation of these concepts, see [START_REF] Gisin | Quantum Chance[END_REF].

Let us briefly recall the concept of local and nonlocal correlations. Assume that a source emits pairs of particles that travel to two distant stations, traditionally called Alice and Bob. As depicted in FIG. 1, the two experimentalists perform one out of several possible measurements on the individual particles they each receive and record the associated outcomes. We denote Alice's and Bob's measurement choices by x and y and their recorded outcomes by a and b, respectively. They can then compute the correlation P (ab|xy). Given the setup, it seems natural to think that any correlations that Alice and Bob can observe in this way are due to particles having a common past, as they come from the same source. We refer to this common past by λ. Correlations that can be explained by the existence of such a parameter are called local : P L (ab|xy) = dλρ(λ)P (a|xλ)P (b|yλ).

(1)

Bell's work shows that there are quantum correlations that cannot be reproduced by such a local model, proving Two boxes receive each a particle, emitted by a common source, each. They are given inputs x and y, which we depict here as the setting of an active switch, and return outputs a and b, respectively. There is the possibility for non-detection events, in which case the corresponding output variable takes the value ∅. Since these losses can be seen as happening inside the box, they can depend on the inputs x and y, respectively. We analyze the limited detection local case of this scenario, meaning that a local hidden variable λ not only fully describes the state of the particle, but can also influence whether or not a non-detection event occurs.

that quantum physics is inherently nonlocal. Nonlocality has since been demonstrated many times [2], and finds repercussion in a variety of applications [3,[5][START_REF] Colbeck | Quantum And Relativistic Protocols For Secure Multi-Party Computation[END_REF][START_REF] Pironio | [END_REF][8]. However, when demonstrating quantum nonlocality, several issues have to be addressed. Here, we are specifically interested in one of them: what happens if the particles can be lost on the way to or inside the measurement stations, including the possibility that they reach the detectors but are simply not registered. In this case, we say that a = ∅ or b = ∅. One immediate idea is of course to carefully analyze why the particles get lost and, if the mechanisms are well understood, to simply discard these cases. This means that Alice and Bob postselect on the cases for which they both register a detection event: a = ∅ and b = ∅. Notably, this opens up the possibility that fully local correlations appear nonlocal if our understanding of the cause of the non-detection events is incorrect, a situation that we wish to avoid [START_REF] Branciard | [END_REF]12]. This is especially relevant in the case of active adversaries in DIQIP applications. Another option is to consider the non-detection events as an additional possible outcome and simply check if the resulting correlation is nonlocal. In this case, one will never mistake a local correlation for a nonlocal one. The drawback, however, is that even highly nonlocal distributions may now appear local.

In the end, the only way address this issue, usually called the detection loophole, consists of not only producing highly nonlocal correlations but also having a high enough detection efficiency [2]. If the latter is not satisfied, even a perfect state preparation and perfectly calibrated measurement apparatus do not help and one is left with an inconclusive experiment unless the detection loophole is assumed to be not exploited. It is only recently that all the potential loopholes in a Bell test have been closed in a single experiment for the first time [13]. However, closing the detection loophole in long-range experiments and applications remains difficult. Namely, after a transmission through 10 km of the currently best optical fiber, the losses already exceed the tolerable threshold. We therefore take a different route in this paper.

Here, we introduce the concept of limited detection locality. It consists of an intermediate assumption between neglecting the detection loophole and closing it completely. We show that this assumption, even when arbitrarily weak, allows demonstrating nonlocality by postselection with arbitrarily low overall detection efficiency. In addition, we show that the two previously mentioned methods of dealing with detection inefficiencies (postselection and assignment to an additional outcome) can be seen as a special case of a more general method. We then connect the concepts of limited detection locality and measurement dependent locality [14,15]. Finally, we discuss a dedicated twin-photon experiment that we performed, and show how quantum nonlocality can be revealed despite low detection efficiencies without having to make the fair-sampling assumption.

Limited Detection Locality (LDL)-Assume that there exist a fixed η min and η max with [η min , η max ] [0, 1] such that

η min ≤ P (a = ∅|xλ) ≤ η max , (2)
and similarly for Bob. This corresponds to the assumption that, for any input x and any common local variable λ, there is a probability of at least η min and at most η max of having a detection. Consider for example a world in which the polarization degree of freedom of photons was as of yet undiscovered. It is nowadays well known that the detection efficiency of most photonic detectors is indeed sensitive to the polarization degree of freedom of light. However the detection efficiency never goes up to 1 or down to 0, which corresponds to our assumption of limited detection efficiency with nontrivial η min and η max . We refer to correlations fulfilling conditions (1) and (2) as limited detection local. Note that technically, the case of [η min , η max ] = [0, 1] can still be analyzed with our model, making it possible to retrieve the results discussed in [START_REF] Branciard | [END_REF].

In an experiment, one can additionally determine the actual observed detection efficiencies, which may of course be different for the different sets of inputs. It is necessary that some detections occurred for all possible sets of inputs, which reads

P (a = ∅|x) = η A x > 0. (3)
Since

P (a = ∅|x) = dλρ(λ)P (a = ∅|xλ), (4)
we have that η min ≤ η A x ≤ η max . All of this holds analogously for Bob's side. To ease notation, we define η xy = η A x η B y . We can now focus on the postselected limited detection local distributions given by

P (ab|xy, a = ∅, b = ∅) = P (ab|xy) η xy . (5)
Similarly to local correlations, these postselected limited detection local correlations fulfil certain conditions. More precisely, they form a convex polytope and therefore respect a set of linear Bell-like inequalities. More details on such a polytopal structure is given in the Supplemental Material [16]. Making the additional assumption that η xy = η x y for all x, x , y, y , one of these inequalities is, for example, given by

η 2 min P (00|00, a = ∅, b = ∅) -η min η max P (01|01, a = ∅, b = ∅) -η min η max P (10|10, a = ∅, b = ∅) -η 2 max P (00|11, a = ∅, b = ∅) ≤ 0. (6)
In an experiment with given losses, the experimentalists can check for which values of η min and η max their observed correlations violate this inequality. He can then conclude that no limited detection local model with these parameters could have reproduced them. Interestingly, there are quantum correlations that do not fulfil this inequality independently of the observed detection efficiencies η xy (including the case where the detection efficiency is different for each input pair) and for any upper bound η max as long as η min > 0. In fact, this is achieved by all quantum correlations violating Hardy's paradox [17] and can therefore be realised using any sufficiently pure partially entangled 2qubit state with the right set of projective measurements. This may be quite surprising since it is wellknown that without the assumption of limited detection efficiency (2), a minimal observed detection efficiency of η = xy η xy /4 > 2 3 is required to demonstrate nonlocality for 2 parties using binary inputs and outputs. However, making the arbitrarily weak additional assumption that P (a = ∅|xλ) ≥ η min > 0 allows demonstrating quantum nonlocality despite arbitrarily large losses and detection inefficiencies.

A more general method of dealing with detection inefficiencies-It is possible to impose any desired η min at the price of adding some noise to the system. Assume that Alice and Bob set their detection systems, which we assume to have an efficiency η, such that any time a non-detection event occurs, the system still gives an outcome with the probability η min . In this way, Alice and Bob impose that their detection systems have a limited detection efficiency given by the chosen η min and η max = 1 and they can treat the resulting correlations by the tools presented above. This however comes at the price of adding local noise to their correlations. In fact, assume that Alice and Bob would share the nonlocal correlation P N L if the detectors were perfect and there were no losses (e.g., given by projective measurements on a pure quantum state) and denote by P A N L and P B N L the marginal distributions of Alice and Bob, respectively. In the non-detection cases, the detection systems are set up such that they give with probability η min an outcome given by the local distributions P A L and P B L , respectively. Then, by postselecting on the cases where the detection systems gave an outcome, Alice and Bob share the correlation:

P = η 2 P N L + η(1 -η)η min (P A N L P B L + P A L P B N L ) +(1 -η) 2 η 2 min P A L P B L 1 (η + (1 -η)η min ) 2 . (7)
They can then analyse this correlation using the tools of limited detection efficiency presented above.

As mentioned in the introduction, the possibility of dealing with losses and detectors' inefficiencies by assigning the non-detection events to an additional outcome and treat the resulting correlations using the usual tools of nonlocality. This corresponds exactly to the strategy we just presented with η min = 1. However, our approach is more general, allowing to assign only a fraction of the non-detection events to an outcome and postselecting on the rest. For a fixed detection efficiency η, our method therefore encompasses both of the previous strategies, full postselection and full assignment to an additional outcome, and additionally allows for an arbitrary mixture of the two. It is at this point not obvious to us that for a given experiment (meaning for a given P N L and a given η), all of these strategies would yield the same result. We leave it up for future works to analyse this question in more detail.

Link to measurement dependent locality (MDL)-Another way to counter-fit nonlocal correlations using only local resources is if the common history λ is correlated with the inputs x and y. If the correlation can be arbitrary, then any nonlocal correlation can be counter-fitted in this way, so limitations have to be imposed to be able to make any conclusions. Together with some co-authors, we recently studied the case of measurement dependent local correlations [14,15] that are defined in the following way: P (abxy) = dλρ(λ)P (xy|λ)P (a|xλ)P (b|yλ) (8)

≤ P (xy|λ) ≤ h. (9)
Note that if Alice and Bob each have N inputs, then 0 ≤ ≤ 1 N 2 ≤ h ≤ 1 due to the normalization of probability distributions. Similarly to this paper, we showed that the set of MDL-correlations for fixed and h can be analysed using Bell-like inequalities.

It turns out that there exists a strong link between the concepts of limited detection locality and measurement dependent locality. We make this connection explicit by the following theorem, whose demonstration is detailed in the Supplemental Material [16]:

Theorem: Assume that we have a correlation that can be produced by using a combination of postselected limited detection (2) and measurement dependent ( 9) local (8) resources, with parameters (η min , η max ) and ( , h), respectively. Then, this correlation can also be reproduced using only measurement dependent local resources with =

η 2 min η 2 max and h = η 2 max η 2 min h.
Intuitively, the link comes from the fact that an adversary that is allowed, via postselection, to discard unsatisfying inputs is in fact influencing the choice of inputs. A consequence of this theorem is that whenever a correlation cannot be reproduced by a measurement dependent local model with bounds and h , then it can also not be realized using limited detection efficiencies with

η 2 min η 2 max ≥ N 2 and η 2 max η 2 min ≤ N 2 h
, where N is the number of inputs for each of the two parties. This allows us to use any result derived for the MDL-scenario and apply them to LDL-correlations. Even more interestingly, we are now able to deal with the problems of losses and measurement dependence in a straightforward way since we can simply focus exclusively on measurement dependence.

Experiment: We now discuss the twin-photon based experimental realization whose principle is depicted in FIG. 2. Each user's station comprises 2 measurement devices with fixed settings, i.e., 4 single-photon detectors (8 in total). The detection events at Alice's are spacelike separated from the corresponding ones at Bob's. In order to emulate active switches between different analyzis settings at each user's location without being limited by driving speeds, we exploit instead a simple and elegant solution for passive implementation, referred to as Zbinden switches [18]. First, standard 50/50 beamsplitters connect the incoming photons to the measurement devices. Then, two quantum random number generators (QRNG) provide the inputs x for Alice and y for

BS

Loss

QRNG

Source

LHV strategy λ FIG. 2. A source emits supposedly entangled photons towards two measurement stations. In each station, a passive switch routes the individual incoming photons to one out of two pairs of detectors, corresponding to either settings 0 or 1. The boxes are given inputs x and y, which decide which pairs of detectors are switched on, and return outputs a and b depending on which detector fires. If the photon goes towards the inactive pair of detectors it is simply considered as lost and not registered. We compare this scenario to the scenario where a local hidden variable λ that can influence the losses and settings try to emulate our observed correlations.

Bob to randomly select one pair of detectors associated with one measurement. The other two detectors of each box remain inactive. Consequently, when, in any run of the experiment, a photon hits an inactive detector, it is merely disregarded. Hence, with perfect devices, the maximum attainable detection efficiency is 50%.

Our source produces partially entangled photon pairs close to the state

|Ψ = 1 √ 3 √ 5 -1 2 |00 + √ 5 + 1 2 |11 , (10)
and the 8 detectors are set up to perform measurements close to the ideal projective measurements

|A 0 (θ) = cos θ |0 + sin θ |1 , |A 1 (θ) =| A 0 (θ -π 4 ) , |B 0 (θ) = |A 0 (-θ) and |B 1 (θ) = |A 1 (-θ) , with θ = arccos 1 2 + 1 √ 5 .
Further experimental details are given in [15] as well as in the Supplemental Material [16]. Without any additional noise, the resulting correlations would violate inequality [START_REF] Colbeck | Quantum And Relativistic Protocols For Secure Multi-Party Computation[END_REF], ∀η min > 0 and ∀η max . Obviously, due to noise and imperfections even in the postselected case this is not fully realized. Our postselected correlations, meaning that we discard all the non-detection events, can be found in Table I. While we did not precisely determine the overall detection efficiency in our experiment, it is worth to note that it stands clearly below the required 2 3 efficiency that is necessary to close the detection loophole in the case of 2 parties with binary inputs and outputs [2,13]. However, assuming that the losses are the same for all input pairs, i.e., η xy = η x y , we can still conclude that we reveal quantum nonlocality for ηmin ηmax > 0.267. This follows from inequality [START_REF] Colbeck | Quantum And Relativistic Protocols For Secure Multi-Party Computation[END_REF]. There are two ways to illustrate this result. First, assume that a local hidden variable guides, on each side, the particle through the beam-splitter and influences the detection efficiency. Such a local hidden variable would need to have the power to reduce the overall detection efficiency down to below 26.7% in order to mimic nonlocal correlations. Next, consider an adversary partially controlling the detectors' efficiencies. Assuming standard 50/50 beam-splitters, we have that η max ≤ 0.5 and one gets η min = 0.134, i.e., such an adversary would have to lower the detector efficiencies corresponding to settings they do not like down to below 13.8% in order to have local correlations appear nonlocal. In our experiment, due to the beam-splitters, the losses in the fibres and the difficulties of perfectly aligning 8 detectors, we can safely consider that η max ≤ 0.1, leading to a required η min ≤ 0.027 in order to reproduce the correlations using a local hidden variable model. Additionally, the way our experiment was built and carried out clearly shows the link between limited detection efficiency and measurement dependence. At both Alice's and Bob's stations, the respective beam-splitter sends the photon to either the detectors corresponding to input 0 or to the ones corresponding to input 1. The inputs x and y determine which pair of detectors we consider as switched on each side. If the photon goes towards the detectors for input 1 while we input 0, then the detection is not registered and the photon is lost. The nondetection events thereby allow the photons to 'choose' the input they will reply to. The limited detection assumption then corresponds to limiting this choice, i.e., while the action of the beam-splitter can be biased, it cannot be fully biased. The link to measurement dependence is here obvious.

We can use theorem 1 to extend our analysis to the case where we consider our random number generators providing the inputs x and y to be biased. Let us assume that there is some measurement dependence as defined in (9), given by parameters ( , h), as well as some limited detection efficiency, given by parameters (η min , η max ). Using an MDL inequality provided in [14] and theorem 1, we then conclude that our experiment shows quantum nonlocality for

h η min η max 4 > 0.15529. ( 11
)
Conclusion -Losses and detection inefficiencies have been a long-lasting milestone in all dedicate experiments. This is why a loop-hole free Bell-test took so long to be successfully demonstrated. It is also one of the main weak points that an adversary may attack in any task whose security relies on quantum nonlocality. To help dealing with both of these issues from a theoretical point of view, we have introduced the additional assumption of limited detection efficiency (2). The assumption is that the inefficiencies in the setup are only partially exploited, an idea that we consider very intuitive in itself especially in the case of a fundamental Bell test, provided Nature is non-malicious. The case of an adversary is also interesting. If one assumes an adversary with full control over Alice's and/or Bob's devices, then one needs to resort to full Device-Independent Quantum Information Processing (DIQIP) [2,19]. However, semi-DIQIP is extremely appealing and timely as it allows one to lower the extremely high requirement of full DIQIP. For example, considering DIQIP as a way to check the proper functioning of our system, i.e., going back to the original idea of self-testing [20], it is very natural to assume that possible defects do not set η min = 0. In such a case, our approach provides much simpler self-tests. Even against an active adversary, there are cases in which it is reasonable to assume that they have only partial control on the detection efficiencies.

In addition, we introduced a generalized method to deal with detection inefficiencies. The method includes the usual techniques of postselection and assignment to an additional outcome as special cases and allows, on top of that, mixing the two. It is an open question whether or not for a given experiment one of this continuum of methods trumps the other ones or if they are all equivalent. We leave this question for future research.

We connected the ideas of limited detection locality and measurement dependent locality. We showed that results from studying measurement dependent local correlations can be applied to the case of limited detection locality. Moreover, it is possible to deal with detection efficiencies and lack of measurement independence at the same time, which we hope will be of use for future Bell experiments.

Finally, we carried out an experiment to bring these theoretical ideas to the real world. Our twin-photon based demonstration, having detection efficiencies too low to close the detection loophole [13], remains conclusive as long as we assume a limited detection efficiency such that ηmin ηmax > 0.267. It furthermore reveals very clearly the link between measurement dependence and limited detection. We believe that the methods depicted here will be of use to any frontier experiments or applications that do not manage to close the detection loophole completely.

Polytopal structure of Limited Detection Local correlations

Consider the case where N parties perform a nonlocality experiment. The input and outcome of the i-th party will be denoted by X i and A i respectively. We consider the case where non-detection events can occur, they will be denoted by A i taking the value ∅. We will denote by A i the outcome of party i after postselecting on having a detection. As discussed in the main text, we make the assumption of limited detection locality and show that these correlations form a polytope. The theorem stated here is more general than needed for the main text, where we only consider the case of 2 parties.

Definitions: Let {A i } N i=1 , {A i } N i=1 , {X i } N i=1 be sets of random variables with alphabets {1 • • • m i , ∅}, {1 • • • m i } and {1 • • • n i } respectively, m i , n i ∈ N .
In the following, the corresponding lower case letters will denote values in the respective alphabet. We will denote probability distributions over a random variable V by P V , the value of this distribution for a given value of V by P V (v). For ease of notation, we will often omit the random variable and just write P (v). We will denote conditional probability distributions over a random variable V conditioned on a random variable W by P V |W . In the case of continuous random variable we denote the probability density by ρ V . In the following we assume that all the probability distributions are well defined. The set of all probability distributions over V will be denoted by P V and of all conditional probability distributions over V conditioned on W by P V |W .

We define the following sets:

• The sets of 1-party distributions with limited detection:

LD i (η min , η max ) = P Ai|Xi ∈ P Ai|Xi : 1 -η min ≤ P Ai|Xi (∅|x) ≤ 1 -η max ∀x ∈ {0 • • • n i }
• The set of N -party limited detection local distributions:

LDL N ({η min,i } N i=1 , {η max,i } N i=1 ) = P A1...A N |X1...X N ∈P A1...A N |X1...X N : ∃Λ s.t. P (a 1 . . . a N |x 1 . . . x N ) = dλρ(λ) N i=1 P (a i |x i λ), P Ai|XiΛ=λ ∈ LD i (η min,i , η max,i )∀λ, ∀i
• The set of N -party postselected limited detection local distributions:

LDLPS N ({η min,i } N i=1 , {η max,i } N i=1 , {η x1...x N } x1...x N ) = P A 1 ...A N |X1...X N ∈ P A 1 ...A N |X1...X N : ∃Q A1...A N |X1...X N ∈ LDL({η min,i } N i=1 , {η max,i } N i=1 ), Q(a 1 = ∅ . . . a N = ∅|x 1 . . . x N ) = η x1...x N , P (a 1 . . . a N |x 1 . . . x N ) = Q(a 1 . . . a N |x 1 . . . x N ) η x1...x N .
We further define the following two sets, which we will prove to be the vertices of LD i (η min , η max ) and

LDL N ({η min,i } N i=1 , {η max,i } N i=1 ): V LDi (η min , η max ) = V Ai|Xi ∈P Ai|Xi : ∀x ∈ {1 . . . n i }∃!a x ∈ {1 . . . m i } and η x ∈ {η min , η max } s.t. V (a x |x) = η x , V (∅|x) = 1 -η x and otherwise V (a|x) = 0 V LDL N ({η min,i } N i=1 , {η max,i } N i=1 ) = V A1...A N |X1...X N ∈P A1...A N |X1...X N : ∃V i ∈ V LDi (η min,i , η max,i ) s.t. V (a 1 . . . a N |x 1 . . . x N ) = N i=1 V i (a i |x i )
With these definitions, we can now state the theorem. It refers to polytopes, which for the purposes of this work are simply seen as a convex structure with a finite set of vertices. Equivalently they can be defined by a finite set of inequalities.

Theorem:

For Proof: To ease notation we will omit writing N , {η min,i } N i=1 and {η max,i } N i=1 from now on. The first part of the theorem follows from the following two lemmas.

Lemma 1: LD i is a polytope with vertices given by V LDi .

Proof: Due to the normalisation of probability distributions, i.e., ai P (a i |x i ) = 1 ∀x i , we have that P (∅|x i ) = 1 -mi ai=1 P (a i |x i ) and we can therefore work in the lowerdimensional subspace given by a i ∈ {1 . . . m i }. In this subspace, we are then left with the polytope defined by the inequalities η min ≤ mi ai=1 P (a i |x i ) ≤ η max . This is the definition of a hypercube whose vertices are defined by the corresponding part of V LDi . The Lemma follows.

Lemma 2: Let Q and R be polytopes with vertices

V Q and V R respectively. Let S = S : ∃Λ s.t. S(u, v) = dλρ(λ)Q λ (u)R λ (v) with Q ∈ Q, R ∈ R . Let V S = V S : V S (u, v) = V Q (u)V R (v) with V Q ∈ V Q , V R ∈ V R .
Then S is a polytope whose vertices are a subset of V S .

Proof: By definition, S is convex and V S ∈ S. Let S ∈ S, then by definition we have:

S(u, v) = dλρ(λ)Q λ (u)R λ (v) = dλρ(λ) i q λ,i V i Q (u) j r λ,j V j R (v) = (ij) dλρ(λ)q λ,i r λ,j V i Q (u)V j R (v) = (ij) s ij V ij S (u, v)
In the first step we use the fact that any element of Q and R can be written as a convex combination of their vertices and we define q λ,i ≥ 0, i q λ,i = 1 and r λ,j ≥ 0, j r λ,j = 1. In the last step we defined s ij = dλρ(λ)q λ,i r λ,j , which fulfils s ij ≥ 0 and ij s ij = 1 and also used the definition of V ij S .

This proves the Lemma.

Using these two Lemmas in conjunction (and using Lemma 2 iteratively) proves that LDL is a polytope and that V L DL contains its vertices.

To finalize the proof of the theorem we need to show that LDLPS is a polytope as well. This can be seen directly since the set is obtained by slicing LDL with the hyperplanes defined by P (a 1 = ∅ . . . a N = ∅|x 1 . . . x N ) = η x1...x N . Cutting a polytope with hyperplanes results in another polytope. The final step is a simple rescaling of the entries (equivalent to rescaling the axes) and therefore the set remains again a polytope. This proves the theorem.

Limited Detection Locality and Measurement Dependent Locality

In this section we prove the link between limited detection local and measurement dependent local distributions. This can be proven more generally, here we only present the 2-party version. • Knowing less cannot result in knowing more, meaning that upper and lower bounds on P (µ|νσ) also hold for P (µ|ν): Assume P (µ|νσ) ≤ h, then

P (µ|ν) = σ P (σ)P (µ|νσ) ≤ h σ P (σ) = h
where we used that σ P (σ) = 1. The same holds for lower bounds ≤ P (µ|νσ). Due to this, condition 3 implies

η min ≤ P D A D B |Λ (11|λ) ≤ η max .
Using the implications above, we can show that the conditions imply bounds on P (xy|D A = 1, D B = 1, λ):

P (xy|D A = 1, D B = 1, λ) = ηmin≤•••≤ηmax P (D A = 1, D B = 1|xyλ) P (D A = 1, D B = 1|λ) ηmin≤•••≤ηmax P (xy|λ) ≤•••≤h ⇒ η min η max ≤ P (xy|D A = 1, D B = 1, λ) ≤ η max η min h.
We can now prove the theorem:

P (abxy|D A = 1, D B = 1) = dλρ(λ|D A = 1, D B = 1)P (xy|λ, D A = 1, D B = 1)P (ab|xyλ, D A = 1, D B = 1) = dλρ(λ|D A = 1, D B = 1) η min ηmax ≤•••≤ ηmax η min P (xy|λ, D A = 1, D B = 1) • P (a|xλ, D A = 1)P (b|yλ, D B = 1)
This is by definition an MDL-correlation:

P (ABXY |D A = 1, D B = 1) ∈ MDL( η min η max , η max η min h)

Experimental details

To produce the twin-photon partially polarisation entangled state, we employ the source depicted in the top-left hand side of FIG. 3. A pump laser at 404 nm is sent to two cascaded type-I BBO crystals to produce photon pairs at 808 nm in a coherent superposition of two orthogonal polarisations via the process of spontaneous parametric downconversion. To generate the twin-photon state of eq. ( 10), we set the desired weights of the coherent superposition and the phase between |V V and |HH by placing a half wave-plate and a tilted birefringent crystal in the path of the pump laser. The generation process introduces some distinguishability between the two components of the state which is erased by an adapted compensation crystal. Moreover, to remove the discernibility in terms of spatial modes (k-vectors), the paired photons are coupled, each, into a single mode fibre. Thanks to this configuration, we can produce the desired state with a fidelity of 99% (see also Ref. [15] for more details).

To perform the measurements, Alice and Bob have each two polarization analysers with fixed settings, defining two bases, {A 0 , A 1 } and {B 0 , B 1 }, respectively. These analysers are made of a half wave-plate and a Wollaston prism. Each prism of the setup has an extinction ratio of 6 OD. They are connected to the source via a 50/50 beam-splitter. To choose the measurement basis at both locations, two independent random bit sequences are utilized. This way, only the two randomly selected outputs associated with the given basis are considered, the other two being disregarded [18]. This configuration therefore requires the use of 8 single photon detectors, being all Silicon avalanche photodiodes. Alice employs 4 Excelitas SPCM-AQR-16-FC featuring ∼60% detection efficiency and dark count rates of ∼100 Hz, while Bob has 4 QuTools featuring ∼50% detection efficiency and dark count levels of ∼2000 Hz. Note that the detectors' configuration is chosen in such a way that it can satisfy, as much as possible, the assumption η x,y = η x ,y for all x, y, x , and y with an overall efficiency of about 5% (fibre coupling and propagation losses included). The detection arrival times of all the detection events are recorded by a time-to-digital converter (not shown) and the random choice of the basis is applied during the data post-processing thanks to two different quantum random number generators.

  FIG.1. Two boxes receive each a particle, emitted by a common source, each. They are given inputs x and y, which we depict here as the setting of an active switch, and return outputs a and b, respectively. There is the possibility for non-detection events, in which case the corresponding output variable takes the value ∅. Since these losses can be seen as happening inside the box, they can depend on the inputs x and y, respectively. We analyze the limited detection local case of this scenario, meaning that a local hidden variable λ not only fully describes the state of the particle, but can also influence whether or not a non-detection event occurs.

Definitions: 1 .

 1 We introduce the random variables D A and D B with alphabet {0, 1} such that D A = 0 if and only if A = ∅. We define the set of limited detection local distributions allowing for measurement dependence: MDLDL( , h, η min , η max ) = P AD A BD B XY :P (ad A bd B xy) = dλρ(λ)P (xy|λ)P (ad A bd B |xyλ),η min ≤ P D A D B |XY Λ (11|xyλ) ≤ η max , P (ad A bd B |xyλ) = P (ad A |xλ)P (bd B |yλ) ≤ P (xy|λ) ≤ h, dλρ(λ) = 1, ρ(λ) ≥ 0We also define the set of measurement dependent local correlations: MDL(h, ) = P ABXY :P (abxy) = dλρ(λ)P (xy|λ)P (ab|xyλ), P (ab|xyλ) = P (a|xλ)P (b|yλ)≤ P (xy|λ) ≤ h, dλρ(λ) = 1, ρ(λ) ≥ 0 Theorem: If P AD A BD B XY ∈ MDLDL( ,h, η min , η max ) then P ABXY |D A =1,D B =1 ∈ MDL( P (ad A bd B xy) = dλρ(λ)P (xy|λ)P (ad A bd B |xyλ) 2. P (ad A bd B |xyλ) = P (ad A |xλ)P (bd B |yλ) 3. η min ≤ P D A D B |XY Λ (11|xyλ) ≤ η max 4. ≤ P (xy|λ) ≤ h. Let us prove a few implications: • If (AD A |X) and (BD B |Y ) are local, meaning that they fulfil condition 2, then (D A |X) and (D B |Y ) are also local: P (d A d B |xyλ) = a,b P (ad A bd B |xyλ) = ab P (ad A |xλ)P (bd B |yλ) = P (d A |xλ)P (d B |yλ). • If (AD A |X) and (BD B |Y ) are local, then (A|D A X) and (B|D B Y ) are also local: P (ab|xd A yd B λ) = P (ad A bd B |xyλ) P (d A d B |xyλ) = P (ad A |xλ) P (d A |xλ) P (bd B |yλ) P (d B |yλ) = P (a|xd A λ)P (b|yd B λ).

FIG. 3 .

 3 FIG. 3. Experimental setup -SL: spherical lenses; HWP: half-wave plate; BS: birefringent crystal; CL: cylindrical lens; BBO: beta-barium-borate crystals; UVF: ultra-violet filter; CC: compensation crystal; L: lens; PC: polarization controller; BS: 50/50 beam-splitter; WP: Wollaston prism; D: single photon detector; QRNG: Quantum random number generator; Daq & P: Data acquisition & processing.

  fixed N , {η min,i } N i=1 and {η max,i } N i=1 , LDL N ({η min,i } N i=1 , {η max,i } N i=1 ) is a polytope whose vertices are a subset of V LDL N ({η min,i } N i=1 , {η max,i } N i=1 ). Furthermore, for fixed {η x1...x N } x1...x N ), LDLPS N ({η min,i } N i=1 , {η max,i } N i=1 , {η x1...x N } x1...x N ) is also a polytope.

Acknowledgments -We acknowledge Alex May who wrote down an early version of a proof similar to the one given in appendix 2 showing a connection between limited detection and measurement dependence, as well as Roger Colbeck, who mentioned the idea of limited detection in private discussions. We also thank Valerio Scarani for discussions and Laurent Labonté for his help in the data acquisition process. We further acknowledge financial support by the European project CHIST-ERA DIQIP, the Swiss NCCR-QSIT, the Fondation iXCore pour la Recherche, as well as the Fondation Simone & Cino Del Duca (Institut de France).