Pierre Six

Philippe Campagne-Ibarcq

Igor Dotsenko

Alain Sarlette

Benjamin Huard

Pierre Rouchon

Quantum state tomography with noninstantaneous measurements, imperfections, and decoherence

I. INTRODUCTION

Determining efficiently the state of a system whose preparation is imperfectly known is instrumental to quantum physics experiments. Contrary to classical physics, the determination of a quantum state ρ, its tomography, requires a large number N of independent measurements [START_REF] Paris | Quantum State Lecture Notes in Physics[END_REF][START_REF]Collège de France of Serge Haroche Cours 2009-2010: Synthèse et reconstructions d'états quantiques[END_REF]. The state of a quantum system is indeed a statistical quantity in essence, as it encodes the statistics of outcomes for any upcoming measurement. These measurements are usually modeled using a positiveoperator-valued measure (POVM) defined by non-negative self-adjoint operators π n such that n π n = I . The probability of measurement outcome n is then given by Tr[ρπ n ]. For N large enough, the reconstruction of ρ is based on the fact that Tr[ρπ n ] should be close to N n /N , with N n the number of outcomes n among the N independent measurements: n N n = N. Several reconstruction methods are available. Maximum entropy [START_REF] Buzek | Quantum Tomography from Incomplete Data via MaxEnt Principle[END_REF] and compressed sensing [START_REF] Gross | Quantum State Tomography via Compressed Sensing[END_REF] methods are well adapted to informationally incomplete sets of measurements. For informationally complete sets of measurements, maximum likelihood (MaxLike) reconstruction [START_REF] Lvovsky | Continuous-variable optical quantum-state tomography[END_REF] is usually used: it consists in taking as an estimate for ρ the value ρ ML that maximizes the likelihood function denoted by P(Y |ρ), the probability of the measurement data Y ≡ (N n ) n knowing ρ. For the POVM (π n ) n , the likelihood function is directly * pierre.rouchon@mines-paristech.fr given by

P(Y |ρ) = n (Tr[ρπ n ]) N n .
(

) 1
In such a usual setting, the measurement process is assumed to be instantaneous and free from imperfections and decoherence. In specific experimental situations, such as in [START_REF] Ourjoumtsev | Generation of optical /'Schrodinger cats/' from photon number states[END_REF], it is not very difficult to take into account some measurement imperfections. In general, the derivation of the likelihood function in the presence of imperfections and decoherence during the measurement has not been investigated. This is one of the precise goals of this paper. Since the seminal work of Belavkin, it is known how to take into account measurement imperfection and decoherence for quantum filtering [START_REF] Belavkin | Quantum stochastic calculus and quantum nonlinear filtering[END_REF]. We show here how to exploit the stochastic master equation governing ρ t , the conditional state of the quantum filter, to derive a general expression of the likelihood function: this expression, given by [START_REF] Lvovsky | Continuous-variable optical quantum-state tomography[END_REF], is a direct generalization of the above one where the π n are replaced by the adjoint states at the initial times of the N measurement sequences. These adjoint states obey a backward master equation appearing in quantum smoothing [START_REF] Tsang | Time-Symmetric Quantum Theory of Smoothing[END_REF][START_REF] Guevara | Quantum State Smoothing[END_REF], and they correspond to the effective operator E defining the past quantum state (ρ,E) introduced in [START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF].

When the support of the likelihood function is mainly concentrated around its maximum at ρ ML , it is known that ρ ML is a good approximation of the Bayesian mean estimate ρ BM defined by (see, e.g., [START_REF] Blume-Kohout | Optimal, reliable estimation of quantum states[END_REF])

ρ BM = D ρP(Y |ρ)P 0 (ρ)dρ D P(Y |ρ)P 0 (ρ)dρ , ( 2
)
where D is the convex set of density operators (here the underlying Hilbert space is of finite dimension) and P 0 (ρ) is some prior probability law of ρ (e.g., Gaussian unitary ensemble [START_REF] Mehta | Random Matrices[END_REF]). Such an approximation of ρ BM by ρ ML relies on the first terms of an asymptotic expansion of multidimensional Laplace integrals [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF] under some regularity conditions. We show here how such asymptotic expansion provides also a confidence interval for Tr(ρ ML A), where A is any Hermitian operator. Such a confidence interval is based on a similar approximation for the Bayesian variance of Tr(ρ ML A), denoted by σ 2 ML (A). We provide in [START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF] an explicit expression that depends only on the first and second-order derivatives of the log-likelihood function at its maximum ρ ML . When ρ ML has full rank, we recover the usual asymptotics of MaxLike estimators involving the Fisher information matrix and the Cramér-Rao bound (see, e.g., [START_REF] Cappé | Inference in Hidden Markov Models[END_REF]). When ρ ML is rankdeficient, the likelihood reaches its maximum on the boundary of D. In this case, such an explicit expression approximating the Bayesian variance is not usual and, as far as we know, seems to be new. From a practical viewpoint, our MaxLike estimator thus provides a statistically efficient reconstruction method-exploiting all measurements and information about the dynamics of the measurement process-along with a confidence bound about the estimate, provided the measurement model is correct. In principle, an uncertain parameter in the dynamics of the measurement model could also be MaxLike estimated by looking at the likelihood of the measurement data conditioned on the parameter value, although this is not covered in the present paper. Some robustness to model errors is illustrated in Sec. II D.

After developing the general theory, we report such quantum state reconstructions for two different sets of experimental data. The first set corresponds to the quantum nondemolition photon counting in a cavity using Rydberg atoms, and presented in [START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF]. The stochastic master equation is a discrete-time Markov chain whose state is the photon-number population. The second set corresponds to the fluorescence heterodyne measurements of a superconducting qubit presented in [START_REF] Campagne-Ibarcq | Observing Interferences between Past and Future Quantum States in Resonance Fluorescence[END_REF], a system described by a continuous-time stochastic master equation driven by two Wiener processes. In both cases, we compute the MaxLike estimates ρ ML of ρ, the initial state whose tomography we are supposed to make. For any time t between 0 and T , we can ignore the measurement outcomes between 0 and t and just retain the measurement outcomes between t and T for the tomography. Thus we can artificially investigate the result of such tomography on the quantum state at time t, namely the state that would result from decoherence between 0 and t. We also give, for a physically interesting observable A, the usual 95% confidence interval via the approximation Tr(ρ ML A) ± 2σ ML (A). This confidence interval just means that, if we perform another tomography with another similar dataset, the probability that the MaxLike estimation of Tr(ρA) remains between Tr(ρ ML A) -2σ ML (A) and (Trρ ML A) + 2σ ML (A) is greater than 95%. This probability has nothing to do with the quantum stochastic character of ρ. Here, ρ is considered, from a classical statistical point of view, as an unknown constant parameter of a probability law. More generally, any constant parameter appearing in the quantum filter governing the conditional state can be estimated in a similar way; see, e.g., [START_REF] Six | Parameter estimation from measurements along quantum trajectories[END_REF].

The following section is devoted to discrete-time systems. First, we show how to obtain, from the discrete-time formulation of the quantum filter, an explicit expression of the likelihood function; then, we give the expression of the confidence interval and use it on the first set of experimental data related to the detection of a photon creation quantum jump. In another section, we show how to apply the discretetime formulation to the continuous-time stochastic master equation driven by Wiener processes in order to obtain a numerical algorithm for computing the adjoint states and the likelihood function. Then, we use this numerical algorithm to estimate the initial state of a qubit relaxing toward its ground state and submitted to the heterodyne measurement of its fluorescence. In the Appendix, we give the main calculations yielding the asymptotic expression of the Bayesian variance for any observable A.

II. DISCRETE-TIME SETTING

A. Quantum filtering

We have at our disposal N realizations, starting from the same initial state ρ that we want to determine, and producing N measurement records (y (n) t ) t=0,...,T n , indexed by n ∈ {1, . . . ,N} and where the time t corresponds to an integer between 0 and T n , the duration of realization n. For each realization, the quantum filter provides the conditional state at t, denoted by ρ (n) t , conditioned on ρ (n) 0 = ρ and knowing the past measurements (y (n) 0 , . . . ,y (n) t-1 ). This filter, a discrete-time version of Belavkin's continuous-time filter, is defined by a family of completely positive maps K y,t indexed by the time t and the measurement outcome y (see, e.g., [START_REF] Dotsenko | Quantum feedback by discrete quantum non-demolition measurements: Towards on-demand generation of photon-number states[END_REF][START_REF] Wiseman | Quantum Measurement and Control[END_REF][START_REF] Somaraju | Design and stability of discrete-time quantum filters with measurement imperfections[END_REF]). Moreover, for each t, the completely positive map K t = y K y,t is trace-preserving. For each y and density operator ξ , Tr[K y,t (ξ )] is the probability to measure y at time t knowing that the quantum state at t is ξ . The quantum filter reads

ρ (n) t+1 = K y (n) t ,t ρ (n) t Tr K y (n) t ,t ρ (n) t , t = 0, . . . ,T n . (3)
A discrete-time quantum filter has this Markovian structure, with K y,t depending only on its physical settings. As the initial state ρ is unknown, one can only work with ρ (n) t generated by the above recurrence and starting from a guess ρ (n) 0 = ρ. To each measurement record (y (n) t ) t=0,...,T n we can associate a number P n (ρ) being the probability of getting this record, assuming the initial state was ρ

(n) 0 = ρ. Since Tr[K y (n) t ,t (ρ (n) t )
] is the probability of having detected y (n) t knowing ρ (n) t , a direct use of Bayes' law yields

P n (ρ) = T n t=0 Tr[K y (n) t ,t (ρ (n) t )] with ρ (n)
t generated thanks to [START_REF] Buzek | Quantum Tomography from Incomplete Data via MaxEnt Principle[END_REF]. Some elementary computations show that

P n (ρ) = Tr K y (n) Tn ,T n • • • • • K y (n) 0 ,0 (ρ
P n (ρ) = Tr ρ K * y (n) 0 ,0 • • • • • K * y (n) Tn ,T n (I ) ,
where I is the identity operator. Consider the normalized adjoint quantum filter with the adjoint state E t (see, e.g., [START_REF] Tsang | Time-Symmetric Quantum Theory of Smoothing[END_REF][START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF]), with final condition E (n) T n +1 = I and governed by the following backward recurrence:

E (n) t = K * y (n) t ,t E (n) t+1 Tr K * y (n) t ,t E (n) t+1 for t = T n , . . . ,0. ( 4
)
It defines a family of Hermitian non-negative operators (E (n) t ) of trace 1 and depending only on the measurement data Y . We have K *

y (n) 0 ,0 • • • • • K * y (n) Tn ,T n (I ) = c n E (n) 0 with c n depending only on (y (n) 0 , . . . ,y (n) T n ). Thus P n (ρ) = c n Tr[ρE (n) 0 ]

and we have

P(Y |ρ) = N n=1 c n Tr[ρE (n) ], (5)
where E (n) stands for E (n) 0 . This formula is a generalization of (1) where E (n) replaces π n .

C. Quantum state tomography

The maximum likelihood (MaxLike) estimate ρ ML of the hidden initial quantum state ρ underlying the measurement data Y is given by maximizing the likelihood function

P(Y |ρ). It is usual to consider the log-likelihood function f (ρ) = log [P(Y |ρ)]: f (ρ) = N n=1 log c n + N n=1 log(Tr[ρE (n) ]) = C + N n=1 log(Tr[ρE (n) ]), ( 6
)
where C is a constant independent of ρ. Thus,

ρ ML = argmax ρ∈D f (ρ), ( 7
)
where D is the set of density operators. Assume that the underlying Hilbert space is finite-dimensional. Then D is a closed convex set and f is a smooth concave function.

Thus this optimization problem can be efficiently solved numerically (see, e.g., [START_REF] Boyd | Convex Optimization[END_REF]). Moreover, ρ ML is characterized by the following necessary and sufficient conditions: there exists a non-negative scalar λ ML such that [ρ ML ,∇f ML ] = 0 and λ ML P ML ∇f ML λ ML I, [START_REF] Tsang | Time-Symmetric Quantum Theory of Smoothing[END_REF] where P ML is the orthogonal projector on the range of ρ ML , and ∇f ML is the gradient at ρ ML of the log-likelihood:

∇f ML = N n=1 E (n) Tr[ρ ML E (n) ] . ( 9
)
The necessary and sufficient condition ( 8) is just the translation of the standard optimality criterion for a convex optimization problem (see, e.g., [START_REF] Boyd | Convex Optimization[END_REF]): ρ ML maximizes the log-likelihood function over the convex set of density operators, if and only if, for all density operators ρ, Tr[(ρρ ML )∇f ML ] 0. When ρ ML has full rank, it belongs to the interior of D. Then P ML = I and ∇f ML is collinear to I . When ρ ML is rank-deficient, it lies on the boundary of D. Then P ML < I and ( 8) means that the gradient of the loglikelihood is pointing outward D and is orthogonal to the tangent space at ρ ML to the submanifold of density operators with the same rank as ρ ML .

When the likelihood function is concentrated around its maximum, ρ ML appears to be an approximation of the Bayesian mean estimate ρ BM whose definition has been recalled in [START_REF]Collège de France of Serge Haroche Cours 2009-2010: Synthèse et reconstructions d'états quantiques[END_REF]. It is proven in the Appendix that ρ BM ≈ ρ ML independent of the chosen prior distribution P 0 . Thus, for any Hermitian operator A, its Bayesian mean

A BM = D Tr[ρA] exp[f (ρ)]P 0 (ρ)dρ D exp[f (ρ)]P 0 (ρ)dρ can be approximated by A BM ≈ Tr[ρ ML A].
Similarly, we prove in the Appendix that its Bayesian variance

σ 2 BM (A) = D Tr 2 [(ρ -ρ ML )A] exp[f (ρ)]P 0 (ρ)dρ D exp[f (ρ)]P 0 (ρ)dρ
, which captures the mean uncertainty on the value of A BM , due to the fact that the hidden initial state ρ is unknown, can be approximated by

σ 2 BM (A) ≈ σ 2 ML (A) ≡ Tr[A R -1 (A )], (10)
where B = B -Tr[BP ML ] Tr[P ML ] P ML -(I -P ML )B(I -P ML ) is the orthogonal projector of any Hermitian operator B on the tangent space at ρ ML to the submanifold of Hermitian operators with zero trace and with ranks equal to the rank of ρ ML . Here above, the linear super-operator R reads for any Hermitian operator X,

R(X) = N n=1 Tr(XE (n) ) Tr 2 (ρ ML E (n) ) E (n) + 1 2 (λ ML I -∇f ML )Xρ + ML + 1 2 ρ + ML X(λ ML I -∇f ML ) (11)
with λ ML and P ML appearing in [START_REF] Tsang | Time-Symmetric Quantum Theory of Smoothing[END_REF] and ρ + ML the Moore-Penrose pseudoinverse of ρ ML . Notice that the superoperator R is symmetric and non-negative for the Frobenius product. Thus, σ 2 ML (A) is always non-negative, as it should be. When ρ ML has full rank, P ML = I and ∇f ML = λ ML I . Then R(X) corresponds to the orthogonal projection (for the Frobenius product) of -∇ 2 f ML (X) onto the subspace of Hermitian operators of zero trace. Here, ∇ 2 f ML corresponds to the Hessian of f at ρ ML :

∇ 2 f ML (X) = - N n=1 Tr(XE (n) ) Tr 2 (ρ ML E (n) ) E (n) .
We recover, up to this orthogonal projection, the standard Cramér-Rao bounds attached to MaxLike estimation: -∇ 2 f ML stands for the Fisher information matrix. The superoperator R defined by [START_REF] Blume-Kohout | Optimal, reliable estimation of quantum states[END_REF] is the prolongation of such a Fisherinformation matrix when ρ ML lies on the boundary of D.

Notice its dependence on boundary curvature due to the fact that (λ ML I -∇f ML )Xρ + ML does not vanish in general. Notice that the expressions of A BM and σ 2 BM (A) do not depend on the prior distribution of probability P 0 (ρ). This can be easily understood, as these expressions are the first terms of asymptotic expansions when f (ρ) grows large, i.e., when the amount of information brought by the measurement records makes the initial information P 0 (ρ) outdated. The computations underlying [START_REF] Blume-Kohout | Optimal, reliable estimation of quantum states[END_REF] are given in the Appendix: they rely on a specific application of asymptotic expansions for multidimensional Laplace integrals given in Chap. 8 of [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF].

D. Experimental validation for QND photon counting

We apply in the following the MaxLike reconstruction to the state ρ of the light field stored in a cavity based on the experiment reported in [START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF] and, more precisely, experimental data associated with Fig. 4(b) therein. The field of a very highquality superconducting cavity (with a frequency of 51 GHz and a photon lifetime of T c = 65 ms) is initially in a thermal state (with a temperature of 0.8 K and a mean number of thermal photons of n b = 0.06). At time t = 0 -, a single photon is injected into the field. Due to experimental imperfections, in reality 1.26 photons are injected on average. The QND photon number measurement consists of a long sequence of atomic probes (samples of individual atoms prepared in a highly excited circular Rydberg state) crossing the cavity mode one by one and separated by 86 μs. The measurement starts at t = -172 ms and has a duration of 344 ms, which is large compared to the photon lifetime. The main sources of measurement imperfections are random atomic occupation of samples, nonconstant atom-photon interaction from sample to sample, nonideal atom state detection, etc. The decoherence is mainly due to the limited cavity lifetime T c leading to photon losses. For more details on the considered experiment, please refer to [START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF] and references therein.

Here, the quantum and adjoint states ρ t and E t are diagonal in the Fock basis and truncated to a maximum of seven photons: they are described by vectors of dimension 8. The partial Kraus maps K y,t reduced to 8 × 8 real matrices, and the computations of the E (n) rely on the transpose of these real matrices. We do not detail here the precise expressions of the different K y,t : they can be deduced from [START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF]. For any t between -100 ms and +150 ms, we compute the MaxLike estimation ρ ML (t) of ρ(t), the state at time t, based on the measurement outcomes between t and +172 ms. In a tomographic spirit, this MaxLike reconstruction takes into account the precise model of the "measurement" of state ρ(t): QND interaction with the atomic probes while the cavity is decohering, between time t and our last information at +172 ms. However, it does not assume anything about the target state prior to measurement, i.e., [START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF] for a photon creation quantum jump induced at t = 0. The latter is not taken into account in the measurement model. Top: solid red line corresponds to the MaxLike mean photon number Tr[ρ ML (t)a † a] of the tomography target ρ(t), and based on measurements between t and +172 ms. The two dashed blue lines correspond to Tr(ρ ML a † a) ± 2σ ML (a † a), a 95% confidence interval. The black line corresponds to the mean photon number a † a t = Tr[ρ(t)a † a] given by [START_REF] Mehta | Random Matrices[END_REF]. Bottom: 2σ ML (a † a) based on [START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF]. See the text for more explanations. it neglects anything that happened before t. Moreover, the addition of a photon at t = 0 -is not taken into account in the model, which means that for all t < 0 there is in fact a mismatch in the model of the measurement process lasting from t to +172 ms; this allows us to get an idea about the robustness of ρ ML to model errors. For any time t, the rank of ρ ML (t) is strictly less than 8 (between 2 and 5), and we have checked that it satisfies the characterization [START_REF] Tsang | Time-Symmetric Quantum Theory of Smoothing[END_REF].

The results for N = 1390 measurement trajectories are illustrated in Fig. 1. To evaluate the precision of such a MaxLike estimation, we have computed the variance given in [START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF] for the photon-number operator a † a. The resulting 95% confidence intervals show that, for all t between -100 and +150 ms, the mean photon numbers are evaluated with estimated uncertainties between ±0.015 and ±0.065. We also compare the MaxLike reconstruction for t between -100 and +150 ms with the mean photon number a † a t = Tr[ρ(t)a † a] given by

a † a t = n b for t < 0, n b + ( a † a 0 -n b )e -t/T c for t 0, ( 12
)
where a † a 0 = 1.26; T c = 65 ms and n b = 0.06 are derived from [START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF]. We observe that a † a t remains almost inside the 95% confidence tube except for t between -50 ms and 0. This is a very positive result, and incidentally it is much better than the "backward reconstruction" based on future measurements performed in [START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF] [Fig. 4(b), blue curve], which features a significant offset. Let us briefly comment on the MaxLike estimation at each t. At time 0 ms, the MaxLike photon number reaches its maximum, 1.237 ± 0.045. It corresponds to the programed injection of [START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF]. Our MaxLike estimation is a tomography of that state, i.e., it never includes information about what we have injected into the system before t = 0, and hence this is an independent confirmation of the programed injection. For t > 0, we expect the mean photon number to decay due to the finite photon lifetime. The MaxLike estimate confirms such behavior-again, just by making the tomography of ρ(t) using measurements obtained after t, i.e., without any information on how ρ(t) might have been constructed. For t between -50 ms and 0, the observed mismatch is expected, since the photon jump at t = 0 is not taken into account in the tomography model for the MaxLike reconstruction. Yet for t < -50 ms we get a MaxLike mean photon number of 0.042 ± 0.015, close to the thermal photon number 0.06 observed in [START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF]. This is a notable result, illustrating that the MaxLike estimate is robust to the erroneous tomography model, which neglects the photon injection at t = 0. In other words, our MaxLike estimate appropriately gives more credit to measurements obtained right after ρ(t) was prepared than to measurements further in the future, although it takes these measurements into account with a (supposedly) appropriate weight. Hence, when t becomes negative, one simply has to collect enough measurements between t and 0 to recover a correct MaxLike estimation of ρ(t). Due to the moderate amount N of measurement trajectories here, having sufficiently many measurements means requiring that t < -50 ms.

III. ADAPTATION TO CONTINUOUS TIME

Most open quantum systems are modeled in continuous time. This section shows how to recover and exploit the discrete-time setting when dt, the sampling time, is much smaller than the characteristic time scales for systems modeled with stochastic master differential equations driven by Wiener processes.

A. Diffusive quantum filtering and adjoint state

The stochastic master equations admit here the following form (see, e.g., [START_REF] Belavkin | Quantum stochastic calculus and quantum nonlinear filtering[END_REF][START_REF] Barchielli | Quantum Trajectories and Measurements in Continuous Time: The Diffusive Case[END_REF]):

dρ t = - i [H,ρ t ]+ ν L ν ρ t L † ν - 1 2 (L † ν L ν ρ t +ρ t L † ν L ν ) dt + ν √ η ν (L ν ρ t + ρ t L † ν -Tr[(L ν + L † ν )ρ t ]ρ t )dW ν,t (13)
with dW ν,t = dy ν,t -√ η ν Tr[(L ν + L † ν ) ρ t ] dt given by the measurement y ν,t , attached to the operator L ν , with efficiency η ν between 0 and 1. Here, dW ν,t are independent scalar Wiener processes and H is the Hamiltonian that could depend on time t via, e.g., some time-varying coherent drives.

Following [START_REF] Six | Parameter estimation from measurements along quantum trajectories[END_REF][START_REF] Rouchon | Efficient quantum filtering for quantum feedback control[END_REF][START_REF] Ph | Quantum backaction and feedback in superconducting circuits[END_REF], such quantum filters admit the following infinitesimal discrete-time formulations based on Itō rules:

ρ t+dt = K dy t (ρ t ) Tr K dy t (ρ t )
,

where the complete positive maps K dy t depend on dy t = (dy ν,t ) according to

K dy t (ρ t ) = M dy t ρ t M † dy t + ν (1 -η ν )L ν ρ t L † ν dt
with

M dy t = I + - i H - 1 2 ν L † ν L ν dt + ν √ η ν dy ν,t L ν .
The probability of outcome dy t = (dy ν,t ) is given then by the following distribution based on Gaussian laws of variance dt:

P dy t ∈ ν [ξ ν ,ξ ν + dξ ν ] ρ t = Tr[K ξ (ρ t )] ν e -ξ 2 ν /2dt dξ ν √ 2πdt . ( 14
)
Take a sampling time dt much smaller than the time constant involved in the Hamiltonian H and in the decoherence operator L ν . Then we can exploit the above formulations similar to discrete-time quantum filtering and compute the normalized adjoint states E (n) t . They are associated with the measurement data (y (n) t ) 0 t T n corresponding to the quantum trajectory number n, via the following discrete-time formulation:

E t = K * dy (n) t (E t+dt ) Tr K * dy (n) t (E t+dt ) with E T n +dt = I/Tr(I )
with T n = N n dt for N n a large integer and where the adjoint K * dy t of K dy t reads

K * dy t (ρ t ) = M † dy t ρ t M dy t + ν (1 -η ν )L † ν ρ t L ν dt.
After having obtained, for the N quantum trajectories, the value at t = 0 of the adjoint states (E (n) ) n=1,...,N , we can directly use the MaxLike quantum-state tomography developed in the preceding section.

B. Quantum state tomography for a qubit

For a two-level system, the MaxLike estimation developed in the previous sections admits a simpler formulation with the Bloch sphere variables that can be used for both ρ and E:

ρ = I + xσ x + yσ y + zσ z 2 , E = I + e x σ x + e y σ y + e z σ z 2 ,
where (x,y,z) and (e x ,e y ,e z ) correspond to the coordinate of vectors, with σ x , σ y , and σ z the three Pauli matrices. Here the convex set D corresponds to the unit ball x 2 + y 2 + z 2 1 and the Frobenius product between operators to the Euclidian product between vectors in three-dimensional Euclidian space.

Then the gradient of the log-likelihood function ( 9) becomes the vector

∇f ML = n 1 1 + x ML e (n) x + y ML e (n) y + z ML e (n) z ⎛ ⎜ ⎝ e (n) x e (n) y e (n) z ⎞ ⎟ ⎠.
The characterization of ρ ML given in ( 8) becomes as follows: if

x 2 ML + y 2 ML + z 2 ML < 1, then ∇f ML = 0; if x 2 ML + y 2 ML + z 2 ML = 1, then ∇f ML = λ ML x ML y ML z ML
with λ ML 0. The superoperator R defined in ( 11) becomes a 3 × 3 symmetric non-negative matrix. It reads as follows:

(i) When x 2 ML + y 2 ML + z 2 ML < 1, we have R = n e (n) x e (n) x e (n) x e (n) y e (n) x e (n) z e (n) y e (n) x e (n) y e (n) y e (n) y e (n) z e (n) z e (n) x e (n) z e (n) y e (n) z e (n) z 1 + x ML e (n) x + y ML e (n) y + z ML e (n) z 2 .
It is usually of rank 3 and can be inverted on any operator of the form A = aσ x +bσ y +cσ z 2 associated with the vector

a b c
to get a variance estimation via [START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF] where the trace is replaced by the Euclidean scalar product.

(ii) When x 2 ML + y 2 ML + z 2 ML = 1 and λ ML > 0 large enough, we have R = n e (n) x e (n) x e (n) x e (n)

y e (n) x e (n) z e (n) y e (n) x e (n) y e (n) y e (n) y e (n) z e (n) z e (n) x e (n) z e (n) y e (n) z e (n) z 1 + x ML e (n) x + y ML e (n) y + z ML e (n) z 2 +λ ML ⎛ ⎜ ⎝ 1 -x ML x ML -x ML y ML -x ML z ML -y ML x ML 1 -y ML y ML -y ML z ML -z ML x ML -z ML y ML 1 -z ML z ML ⎞ ⎟ ⎠
where e (n) ξ = e (n) ξs (n) ξ ML for ξ = x,y,z and s (n) = e (n) x x ML + e (n) y y ML + e (n) z z ML . Its rank is less than or equal to 2. The inverse of R appearing in [START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF] corresponds here to the Moore-Penrose pseudoinverse. It is evaluated on the vector associated with A :

A = (a -sx ML )σ x + (b -sy ML )σ y + (c -sz ML )σ z 2
with s = ax ML + by ML + cz ML .

C. Experimental validation for a qubit with fluorescence heterodyne measurements

MaxLike quantum state tomography is conducted on a superconducting qubit whose fluorescence field is measured using a heterodyne detector [START_REF] Campagne-Ibarcq | Observing quantum state diffusion by heterodyne detection of fluorescence[END_REF]. For the detailed physics of this experiment, see [START_REF] Campagne-Ibarcq | Observing Interferences between Past and Future Quantum States in Resonance Fluorescence[END_REF][START_REF] Ph | Quantum backaction and feedback in superconducting circuits[END_REF]. The measurement model is described by a stochastic master equation of the form [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF] with H = 0 and ν = 1,2,3:

L 1 = 1 2T 1 σ x -iσ y 2 , L 2 = iL 1 , L 3 = 1 2T φ σ z
with η 1 = η 2 = 0.24 the efficiency of the heterodyne measurement and with η 3 = 0 corresponding to an unmonitored dephasing channel. The measurement and dephasing time constants are T 1 = 4.15 μs and T φ = 35 μs.

We have at our disposal N = 4 × 10 4 quantum trajectories with the same length T n = T = 9.2 μs with a sampling time dt = 200 ns ≈ 1 20 T 1 . Each trajectory is supposed to start at time t = 0 from the same initial state ρ 0 close to (|g + |e )/ √ 2. In a first test, probably closest to experimental needs, the goal is to check this fact by performing a MaxLike tomographic estimation of ρ 0 based on records of the continuous fluorescence signals between 0 and final time T . For the trajectory number n, the measurement record corresponds to 2 × 47 real values corresponding to t t-dt dy (n) 1 and t t-dt dy (n) 2 for t = dt,2dt, . . . ,47dt. From the measurements between 0 and T , we get an estimation of the quantum state ρ 0 with a 95% confidence interval using the above formula for σ 2 ML (A) and A = σ x , σ y , and σ z :

Tr(ρ 0 σ x ) = 0.99 ± 0.06, Tr(ρ 0 σ y ) = -0.03 ± 0.07, Tr(ρ 0 σ z ) = -0.10 ± 0.19. This estimated value of the initial state is consistent with a gate error of a few percent in the preparation of (|g + |e )/ √ 2 starting from the thermal state with less than 1% excitation.

To further validate how the MaxLike tomography can reconstruct different states of the qubit, we next perform, as for the previous experiment, the tomography of the state ρ t obtained at various times t, using N records of the continuous fluorescence signals between time t and final time T . Figure 2 shows, for t between 0 and 5 μs, the resulting estimates x ML (t) and y ML (t) (red solid lines) with their 95% confidence intervals (blue dashed lines). The black circle marks correspond to the average over a much larger set of N * = 3 × 10 6 trajectories of the normalized signals, 2T

= √ η ν Tr[(L ν + L † ν )ρ t ]
dt, these black circles are meant to provide reliable estimations of Tr(ρ t σ x ) and Tr(ρ t σ y ) at times t = 0,dt,2dt, . . . ,25dt, where ρ t starts at ρ 0 and follows the Lindblad master equation corresponding to unread measurements. Thus the message is that our MaxLike method appears to obtain consistent reconstructions, as its confidence interval covers the very precise statistics (black circles) of a more standard model.

Figure 3 corresponds, for the same set of N measurement trajectories, to the MaxLike reconstruction of z. Contrary to x and y, this Bloch coordinate cannot be recovered directly by The red solid lines correspond to x ML and y ML , while the blue dashed lines correspond to the confidence intervals x ML ± 2σ ML (σ x ) and y ML ± 2σ ML (σ y ) obtained from [START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF]. The black circles are generally admitted to represent reliable estimates of Tr(ρ t σ x ) and Tr(ρ t σ y ); see the text for more explanations.

an ensemble average of the measurement signal, so there are no black circles for validation.

In Fig. 4 we use the same set of N = 4 × 10 4 measurement records to compute the ensemble average of the normalized heterodyne signals (black circles). With this smaller dataset, the black circles do not provide a statistically accurate estimation of Tr(ρ t σ x ) and Tr(ρ t σ y ) anymore. On the contrary, the noise attached to such an ensemble average is large compared to MaxLike estimation of x and y. This shows that, as expected, our MaxLike estimation by exploiting all the data between t and T has superior statistical power compared to an estimation of ρ t from measurement outputs obtained at time t only.

IV. CONCLUDING REMARKS

We have proposed a method that takes into account imperfection and decoherence for quantum state tomography based on noninstantaneous measurements. This method is well adapted to MaxLike estimation since it provides directly the likelihood function. The latter is written in terms of sort of backpropagated POVM operators that depend on the measurement trajectory, and which characterize the information the trajectory of outputs carries on about the initial state to be estimated. We have given an approximation of the confidence interval to complete the MaxLike state estimate. In essence, given many copies of an initial quantum state, we can run it through any well-characterized device giving a sequence of outputs for each copy, and from this we can estimate efficiently what the initial state was, with an associated confidence interval. The estimator appears to rely mostly on measurement outputs obtained just after the state was prepared, much like a standard POVM; however, it can compensate for "missing statistical power" by relying on further evolution of the trajectory. This seems to guarantee robustness to uncertainties in the model of the dynamic measurement apparatus, as illustrated in Sec. II D.

The proposed method is directly applicable to reconstruct the state ρ 0 from a standard projective measurement performed on ρ t with t > 0. If on the measurement run n the final projective measurement at T n yields the known state ρ n , then the backward computations (4) just start with E (n)

T n = ρ n instead of E (n) T n = I/Tr(I ). If measurements are obtained as the state evolves from ρ 0 to ρ t , one just applies our standard filter with this different initial condition. If no measurements are obtained between 0 and t, e.g., the quantum state just undergoes decoherence, then one can obtain the corresponding filter by modeling the decoherence as hypothetical unread measurements in the environment, and replacing the corresponding K * y t ,t associated to read measurements by the unital linear map K * t = y K * y,t . In this case, the reconstructed MaxLike estimation will appropriately take into account that the POVM was performed not on ρ 0 , but on a state ρ t that has evolved from ρ 0 according to some known dynamics.

It appears possible to extend the proposed method to measurement protocols in which a meter is coupled to the system of interest, and, after a small time, a (projective) measurement on the meter only is done. For example, it may be useful for the Wigner tomography of a quantum oscillator, such as those realized, e.g., in [START_REF] Leghtas | Confining the state of light to a quantum manifold by engineered two-photon loss[END_REF], to take into account higher-order Hamiltonian distortions and some decoherence effects during the joint evolution of system and meter.

ACKNOWLEDGMENTS

This work was supported partially by the EMERGENCES grant QUMOTEL of Ville de Paris, by the IDEX program ANR-10-IDEX-0001-02 PSL * , and by the ANR, Projet Blanc EMAQS ANR-2011-BS01-017-01.

APPENDIX: ASYMPTOTIC EXPANSIONS UNDERLYING (10)

To formalize the fact that the likelihood function ρ → e f (ρ) is concentrated around its maximum, we set f (ρ) = f (ρ), > 0 and large with f (ρ) a normalized log-likelihood with bounded variations: for any density operators ρ 1 and ρ 2 , |f (ρ 1 )f (ρ 2 )| 1. Throughout this appendix, we assume that f (ρ) is maximal for a unique density matrix ρ ML and that its Hessian matrix ∇ 2 f ML is negative definite. The dimension of the underlying finite-dimensional Hilbert space is denoted here by the integer n.

For any scalar function g of ρ, we consider thus the asymptotic expansion versus of the following multidimensional integral of Laplace type:

I g ( ) = D g(ρ)e f (ρ) dρ.
The domain of integration D is a compact convex subset of the set of n × n Hermitian matrices of trace 1. Here, dρ stands for the standard Euclidian volume element on D, derived from the Frobenius product between n × n Hermitian matrices. We denote by D n-1 the set of density matrices of rank less than or equal to n -1. It corresponds to the boundary of D. Here, we denote by ∇f and ∇ 2 f the gradient and Hessian of f that is considered to be a scalar function on the set of Hermitian matrices of unit-trace.

Assume that ρ has full rank, i.e., f reaches its maximum in the interior of D and ∇f ML = 0. Then we can use Eq. 8.3.52 of Ref. [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF] to get the following equivalent for I g ( ) using dim D = n 2 -1: If g(ρ ML ) = 0, then I g ( ) = O( 1 ) and we have to use the next term in the asymptotic expansion. It appears that, when g and its gradient vanish at ρ ML , Eqs. 8.3.50 and 8.3.53 of Ref. [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF] yield an explicit expression of this term with respect to ∇ 2 g ML , the Hessian of g at ρ ML . For g(ρ) = h(ρ)Tr 2 [(ρρ ML )A], where h is any scalar function of ρ and A is any Hermitian matrix of zero-trace, we have ∇ 2 g ML (B) = 2h(ρ ML )Tr(AB)A for any zero-trace Hermitian matrix B. For such a special form of g, Eqs. 8.3.50 and 8.3.53 of [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF] give

I g ( ) = 2π n 2 -1 2 e f (ρ ML ) | det(∇ 2 f ML )| g(ρ ML ) + O 1 . ( A1
I g ( ) = - h(ρ ML ) + O 1 2 • • • ⎛ ⎝ 2π n 2 -1
2 e f (ρ ML ) Tr A∇ 2 f -1

ML (A) | det(∇ 2 f ML )| ⎞ ⎠ . (A3)
Thus with h = P 0 , we get D Tr 2 [A(ρρ ML )]e f (ρ) P 0 (ρ)dρ D e f (ρ) P 0 (ρ)dρ

= -

Tr A∇ 2 f -1

ML (A) + O 1 2 . ( A4
)
Notice that ∇ 2 f corresponds to the Hessian of the restriction of f to D. Some usual calculations show that this Hessian coincides with -R, where R is defined in [START_REF] Blume-Kohout | Optimal, reliable estimation of quantum states[END_REF] with ∇f MLλ ML I = 0 and P ML = I . This explains approximation [START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF] since A is of zero trace here and thus coincides with A because ρ ML has full rank. When ρ ML is rank-deficient, then f reaches its maximum on the boundary of D and the computations are more complicated. We only provide here the main steps for ρ ML of rank n -1. Moreover, we assume that ∇f ML = 0. The other cases of rank between 1 and n -2 can be conducted in a similar way and will be detailed in a forthcoming publication. According to Eqs. 8.3.10 and 8.3.11 of Ref. [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF], the first term of the asymptotic expansion of I g ( ) coincides with the first term of the asymptotic expansion of a boundary integral localized on V, an open small neighborhood of ρ ML on D n-1 :

J g ( ) = 1 V g(ρ)
Tr[∇f (ρ)N (ρ)]

Tr[(∇f (ρ)) 2 ] e f (ρ) d , where the Hermitian operator ∇f is the gradient of f , the Hermitian operator N (ρ) corresponds to the unitary normal to D n-1 at ρ, and where d is the volume element on V considered to be a Riemannian submanifold of the Euclidian space of Hermitian matrices equipped with the Frobenius product. With g 1 (ρ) = g(ρ) Tr[∇f (ρ)N(ρ)] Tr[(∇f (ρ)) 2 ] , we have to evaluate the integral V g 1 (ρ)e f (ρ) d . Once again, we exploit Eq. 8.3.52 of Ref. [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF] for this integral since f restricted to V reaches its maximum in the interior of V with an invertible Hessian. We get

J g ( ) = 1 K ML e f (ρ ML ) n 2 -2 2 g 1 (ρ ML ) + O 1 ,
where K ML is a positive constant, independent of g 1 , and it can be expressed via the Hessian of f restricted to the Riemannian submanifold V. After some simple computations, we recover the asymptotic expansion (A2).

For g(ρ) = h(ρ)Tr 2 [(ρρ ML )A] and the corresponding g 1 (ρ), we exploit Eqs. 8.3.50 and 8.3.53 of Ref. [START_REF] Bleistein | Asymptotic Expansions of Integrals[END_REF]. We get the following analog to (A3): Here, the Hermitian matrix A is given by the Hessian at ρ ML of the restriction of ρ → Tr 2 [(ρρ ML )A] to the Riemannian submanifold V. Thus A is equal to the orthogonal projection onto the tangent space at ρ ML to the submanifold V. Similarly, ∇ 2 f ML corresponds to the Hessian at ρ ML of f restricted to the submanifold V. With the two previous asymptotic expansions for the boundary integral J g ( ), we recover (A4) with A instead of A and ∇ 2 f ML instead of ∇ 2 f ML . Some additional calculations show that A corresponds to A , and ∇ 2 f ML corresponds to the opposite of R defined in [START_REF] Blume-Kohout | Optimal, reliable estimation of quantum states[END_REF].

J g ( ) = - h 1 (ρ ML ) + O 1 2 • • • ⎛ ⎝ K ML e f (ρ ML ) Tr A ∇ 2 f -1 ML ( A)

FIG. 1 .

 1 FIG.1. MaxLike quantum state tomography based on the QND photon measurements reported in Fig.4(b) of[START_REF] Rybarczyk | Forward-backward analysis of the photon-number evolution in a cavity[END_REF] for a photon creation quantum jump induced at t = 0. The latter is not taken into account in the measurement model. Top: solid red line corresponds to the MaxLike mean photon number Tr[ρ ML (t)a † a] of the tomography target ρ(t), and based on measurements between t and +172 ms. The two dashed blue lines correspond to Tr(ρ ML a † a) ± 2σ ML (a † a), a 95% confidence interval. The black line corresponds to the mean photon number a † a t = Tr[ρ(t)a † a] given by[START_REF] Mehta | Random Matrices[END_REF]. Bottom: 2σ ML (a † a) based on[START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF]. See the text for more explanations.

FIG. 2 .

 2 FIG. 2. Comparison of MaxLike reconstruction for x (left plot: solid and dashed curves) and y (right plot: solid and dashed curves) exploiting the measurement data set associated with N = 4 × 10 4 experimental trajectories, with an ensemble average of the normalized fluorescence signals (black circle) exploiting a much larger data set associated with N * = 3 × 10 6 experimental trajectories.The red solid lines correspond to x ML and y ML , while the blue dashed lines correspond to the confidence intervals x ML ± 2σ ML (σ x ) and y ML ± 2σ ML (σ y ) obtained from[START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF]. The black circles are generally admitted to represent reliable estimates of Tr(ρ t σ x ) and Tr(ρ t σ y ); see the text for more explanations.

FIG. 3 .

 3 FIG.3. MaxLike reconstruction for z exploiting a measurement data set with N = 4 × 10 4 experimental trajectories; the red solid line corresponds to z ML and the blue dashed line to the confidence interval z ML ± 2σ ML (σ z ) obtained from[START_REF] Gammelmark | Past Quantum States of a Monitored System[END_REF]; see the text for more explanations.

FIG. 4 .

 4 FIG.[START_REF] Gross | Quantum State Tomography via Compressed Sensing[END_REF]. Similar to Fig.2, but the ensemble average of the normalized fluorescence signals (black circles) uses the same dataset as the MaxLike reconstruction. The MaxLike reconstruction appears to be significantly less noisy, illustrating the efficiency of our method; see the text for more explanations.

  ) Thus, I g ( ) = I 1 ( )[g(ρ ML ) + O( 1 )]. With g(ρ) = Tr(ρA)P 0 (ρ) for any operator A, we get, assuming P 0 (ρ ML ) > 0, D Tr(Aρ)e f (ρ) P 0 (ρ)dρ D e f (ρ) P 0 (ρ)dρ = Tr(Aρ ML ) + O 1 . (A2)

n 2 2 ⎞⎠

 2 with h 1 (ρ) = h(ρ) Tr[∇f (ρ)N(ρ)]Tr[(∇f (ρ)) 2 ] .

Adjoint-state derivation of the likelihood function

  ) .

	knowing the initial state ρ, reads
	N
	P(Y |ρ) =	P n (ρ).
	n=1
	The adjoint map K * y,t of K y,t is defined by Tr[AK y,t (B)] ≡ Tr[K * y,t (A)B] for all Hermitian operators A and B. Thus
		Since the N measurement records are independent realizations
		from the same initial state, the probability P(Y |ρ) of the
		measurement data,
		Y = y (n)

t |n ∈ {1, . . . ,N},t ∈ {0, . . . ,T n } , B.


Marie Ferru
email: marie.ferru@univ-poitiers.fr

Marie-Pierre Bès

Michel Grossetti

Measuring social proximity in the geography of innovation: application for the French science-industry collaborations study

Keywords: O33, O31, O12 interpersonal ties, social proximity, science-industry partnerships, geography of innovation, mixed method analysis

Considering the role of interpersonal ties in the formation of innovation partnerships, the concept of social proximity has been increasingly used these recent years for the analysis of the geography of innovation. Nevertheless, the existing works reveal problems of measurement that prevent determining the precise weigh of social proximity in the partnerships build-up and deducing its effect on the geography of innovation. Our purpose is to fill in this gap by defining precisely social proximity as a modality of partners networking.

On this basis, we propose an original method for empirically tracking the weigh and effects of social proximity, and we apply it to more than 200 cases of French science-industry collaborations. The combination of qualitative and quantitative approaches to construct and analyze relational data makes it possible to reaffirm both the significant and stable role of interpersonal ties and its significant but variable effect on the geography of partnerships.

Introduction

An abundant literature about the geography of innovation has been developed since the nineties as scholars observed the spatial concentration of innovation activities. The very first authors drew attention to the fact that firms locate close to universities in order to absorb knowledge considering that tacit knowledge was difficult to transmit from a distance. For some years now, while they recognized the relevance to study partners networking, authors have been criticizing this argument (see notably [START_REF] Breschi | Localised knowledge spillovers vs. innovative milieux: Knowledge "tacitness" reconsidered[END_REF]) and have considered the main challenge is the study of the dynamics of network formation [START_REF] Autant-Bernard | Spatial knowledge diffusion through collaborative networks[END_REF] ; [START_REF] Boschma | Some Notes on Institutions in Evolutionary Economic Geography[END_REF] ; [START_REF] Ferru | Formation process and geography of science-industry partnerships : the case of the University of Poitiers[END_REF]). In this context, they recognized the structuring role of interpersonal ties of the actors of innovation (ie. partners such as researchers or industrial project leaders) giving rise to the concept of social proximity.

Beyond the various denominations, a relative consensus has emerged around the idea that social proximity matters in the partners networking process and therefore impacts the geography of collaborative innovation. Regarding its increasing use (based on the seminal typology of proximity developed by [START_REF] Boschma | Proximity and innovation: a critical assessment[END_REF] and the development of the social networks analysis), it has become an umbrella and ambiguous concept; in addition, the data used to assess social proximity suffer from the lack of relational data and thus appear objectionable in some cases (see [START_REF] Bernela | Collaboration networks within a French cluster: do partners really interact with each other?[END_REF] notably). In this regard, to enrich the analysis of spatial dynamics of collaborative innovation through social proximity, the challenge is twofold: we need a clear and stabilized definition of social proximity and a robust methodology to construct data that gives a precise measure.

We propose to move in this direction through this paper by providing first a redefinition of social proximity based on the social embededdness thesis. We therefore reveal the importance of interpersonal ties in the networking of innovative partners and distinguish it from other modalities of networking (sometimes integrated in social proximity). Based on this definition, we thereafter build up an original methodology to collect data about social proximity (through the qualitative "analysis of relational chain") and measure its effects (using the "quantified narrations method"). We applied this method to a set of 244 cases of collaboration between academic laboratories and companies in France. Our empirical work seeks to measure the role of social proximity of and to test "its spatial impact". In other words, it searches to answer the following questions: did partners use interpersonal ties to initially connect each other? How important is that modality of networking in comparison to others? Is this result independent from the partners' characteristics? Does the use of interpersonal ties impact the spatial dimension of partnerships? Are social and spatial proximities correlated?

We first give an overview of the studies underlying the role of social proximity in the geography of innovation (section 2). We then offer a redefinition of the concept and we show the need to develop a robust methodology to measure it (Section 3). We present the empirical work conducted on the basis of this methodology and the building of a relational database (Section 4), and we do a summary of the results from the quantitative treatment of the collected data (Section 5). The final part gives the conclusions from this work (Section 6).

Social proximity and the geography of innovation: insights and limits of the existing works

To open the black box of spatial externalities, authors increasingly pay attention to the structuration of innovation partnerships: they therefore highlight the importance of interpersonal relationships of star scientists (Hagedoorn and Schakenrad, 1994;[START_REF] Shan | Interfirm Cooperation and Startup Innovation in the Biotechnology Industry[END_REF][START_REF] Powell | Interorganizational collaboration and the locus of innovation: networks of learning in biotechnology[END_REF][START_REF] Zucker | Intellectual human capital and the birth of U.S. biotechnology enterprises[END_REF] and reveal the innovative networks and their structures. [START_REF] Breschi | Tracing the links between science and technology: An exploratory analysis of scientists' and inventors' networks[END_REF] or [START_REF] Lissoni | Academic inventors as brokers[END_REF] for instance focus on the important role of author-inventors in networking between researchers and innovators. For many researchers the spatial agglomerations of innovation activities (already demonstrated particularly by [START_REF] Acs | Innovation in Large and Small Firms[END_REF][START_REF] Jaffe | Real effects of academic research[END_REF]Audretsch and Feldman, 1994 and many others) can be explained by this social networks and the fact that for the most part these are local. More generally, existing works show that social relationships are formed more easily in the neighborhood: "the greater the distance, the less contact and support" (Mok et al.,2007, p.434). Some empirical studies [START_REF] Wellman | Are personal communities local? A Dumptarian Reconsideration[END_REF] on a sample of residents of Toronto, [START_REF] Fischer | To dwell among friends[END_REF] on the population of San Francisco and [START_REF] Grossetti | Are French networks different?[END_REF] on Toulouse) confirm that personal networks include a large share of local relationships. This is also the case in studies directly inspired by the new economic sociology, like the already cited study by [START_REF] Powell | Competitive cooperation In biotechnology : learning through networks ?[END_REF], which explicitly refer to it, or like those of [START_REF] Saxenian | Regional advantage: culture and competition In Silicon Valley and Route 128[END_REF], or [START_REF] Ferrary | The role of venture capital firms in Silicon Valley's complex innovation network[END_REF] on Silicon Valley.

Based on this scholars, authors increasingly used the concept of social proximity to highlight the role of interpersonal ties on the geography of innovation [START_REF] Boschma | Proximity and innovation: a critical assessment[END_REF][START_REF] Maggioni | Space versus networks in the geography of innovation: A European analysis[END_REF][START_REF] Agrawal | How do spatial and social proximity influence knowledge flows? Evidence from patent data[END_REF][START_REF] Massard | Proximity and innovation through an « accessibility to knowledge » lens[END_REF][START_REF] Huber | On the role and interrelationship of spatial, social and cognitive proximity: personal knowledge relationships of R&D workers in the Cambridge information technology cluster[END_REF]Caniels et al., 2014;[START_REF] Hansen | Substitution or overlap? The relations between geographical and nonspatial proximity dimensions in collaborative innovation projects[END_REF]Steinmo, 2015, etc.) and are studying the overlapping between social and spatial proximities in innovation partnerships. Besides other forms of proximity 1 , social proximity could be defined in terms of socially embedded relations between agents at the micro-level based on friendship, kinship and past experience, etc. [START_REF] Boschma | Proximity and innovation: a critical assessment[END_REF]. The concept is sometimes denoted as personal proximity 2 [START_REF] Schamp | Dimensions of proximity in knowledge-based networks: the cases of investment banking and auto-mobile design[END_REF] or as relational proximity [START_REF] Coenen | Nodes, networks and proximities: on the knowledge dynamics of the Medicon Valley biotech cluster[END_REF]) to express the same idea. These authors have enhanced the empirical approach through social network analysis to measure the significance of this social proximity. Most of them use geolocation databases of individuals and their inter-relations, from which they can identify collaboration networks and formalize network analysis. Despite recent technical refinements, these works face a data problem. Either they are based on relational data derived from "output data" such as co-patent or co-publication 3 and the social proximity (ie. the existence of interpersonal ties) between partners are identified through their results; either, they are based on consortium data (european projects, clusters projects, etc.) and the partners networks are studied through the questionable hypothesis of complete graph representation (see [START_REF] Breschi | Unveiling the texture of a European research area: emergenceof oligarchic networks under EU framework programmes[END_REF][START_REF] Autant-Bernard | Spatial knowledge diffusion through collaborative networks[END_REF][START_REF] Roediger-Schluga | R&D collaboration networks in the European Framework Programmes : data processing, network construction and selected results[END_REF][START_REF] Balland | Proximity and the Evolution of Collaboration Networks : Evidence from Research and Development Projects within the Global Navigation Satellite System (GNSS) Industry[END_REF]Vonortas, 2013[START_REF] Grandclement | Vers une géographie des pôles de compétitivité : réseaux, construction politique et organisation territoriale[END_REF]Levy and Talbot, 2014). According to this hypothesis every dyad of partners involved in a common innovation project interacts in a homogeneous way. Authors transform bipartite (or 2-mode) network projections -which link actors to the projects in which they are both involved -into unipartite (or 1-mode) projections linking together pairs of actors involved in the same project. The work of [START_REF] Bernela | Collaboration networks within a French cluster: do partners really interact with each other?[END_REF] gives empirical evidence of the abuse of the hypothesis since 8,5 % of ties between project participants are characterized by the absence 2 On the contrary, by using personal proximity, based on the homophily principle, some authors which to insist on property that is endogenous to the individual (see [START_REF] Werker | The Role of Personal Proximity in Collaborations: The Case of Dutch Nanotechnology[END_REF][START_REF] Broekel | Knowledge networks in the Dutch aviation industry: the proximity paradox[END_REF]etc.). 3 "A growing number of studies use patent information to apply social networks analysis (…). Some authors link inventors directly by assuming relations between inventors who jointly worked on patents" [START_REF] Graf | Public Research in Regional Networks of Innovators: A Comparative Study of Four East German Regions[END_REF]Henning, 2009, p. 1353, underlined by the authors). of real interaction. Alongside these works based on the hypothesis of complete graph representation, authors measure social proximity (and the existence of interpersonal ties) thanks to the observation of former participations in a common project. The interaction between partners is still assumed: partners can be integrated into a project without necessarily having met all the partners and / or without having really worked together (as it is common observed for large projects such as European one since they require a large number of participants from various domains). Empirical works searching to assess social proximity face therefore a data problem due to the lack of relevant data. Aware of these limits, some authors recently developed new methodology to perform the social proximity measurement (see notably Steinmo and Rassmussen, 2016).

Redefining and measuring social proximity

Social proximity as the embeddedness of actors in interpersonal ties

Regarding the acceptance of the social proximity concept in the literature, we also adopt by simplicity this denomination but it appears necessary to give precisions. We first consider the concept of to be rooted in the socio-economic literature and particularly in the embeddedness thesis developed by Granovetter in a famous article published in 1985. Granovetter defended the contention that economic activities depend on interpersonal relationships in which the actors are involved and called this dependence embeddedness. This proposition has several implications. On the one hand, economic activity is dependent on more general social structures that are not social groups or categories but networks [START_REF] Wellman | Introduction: studying social structures[END_REF]. Second, the relevant level of economic action is not that of companies or organizations in general but that of the individual actors and their relationships. Thus, relying on a study by [START_REF] Eccles | The quasi firm in the construction industry[END_REF], Granovetter shows that relations among companies (prime manufacturers and subcontractors in the construction field) are underlain by interpersonal relationships, thereby sketching a "community." The embeddedness theory as advanced by Granovetter was also the subject of a certain number of criticisms (for example, [START_REF] Portes | Social capital : its origins and applications in modern sociology[END_REF] or [START_REF] Baret | Management et réseaux sociaux Jeux d'ombres et de lumières sur les organisations[END_REF]).

In particular, the concept of embeddedness does not really explain the logic that makes it possible to share without going through interpersonal relations. In his study of the labor market, Granovetter distinguished different ways of getting a job: "personal contacts" (56% of cases), formal means (placement agencies, etc.) (19%), "direct approaches" (spontaneous applications) (19%) and miscellaneous other means (6%). Granovetter was especially interested in personal contacts, leaving out the 44% of cases that did not involve social relationships. This premise of the primacy of social networks can be seen as a form of relational reductionism, ignoring the reality of organizations [START_REF] Grossetti | Where do social relations come from? A study of personal networks in the Toulouse area of France[END_REF].

To study the spatial dynamic of collaborative innovation, we consider here the role of social proximity as a modality of networking with partners and not as a result of innovation activities.

In this perspective, social proximity is used to analyze the process of partnerships build-up and is therefore a mean to reintegrate the necessary dynamic perspective of the innovative collaboration; we also restrict social proximity to the embeddedness in interpersonal ties leading to distinguish it accurately from the other modalities of partners networking. When the actors involved in an interaction are not connected by interpersonal relationships, they indeed rely on other modalities of partners networking. These resources may include material systems (directories, Internet sites, newspapers, etc.) as well as organizations or people whose role is to put people in contact (transfer centers, promotion centers, innovation agencies) (notably). We consider these coordination resources to be anything that allows an exchange without going through social-relationship chains, all systems and institutions have made it possible to put public researchers and their industrial counterparts in touch with one another.

We therefore distinguish, beside interpersonal ties and social proximity, two subcategories of coordination resources mobilized in the formation of collaboration:

1) the market resources to which most firms and scientists have access (advertisements, conferences, reputation, media, professional-training markets and contracts) as already noticed by [START_REF] Carayol | Objectives, agreements and matching In science-industry collaborations: Reassembling the pieces of the puzzle[END_REF], [START_REF] Mansfield | Academic research and industrial innovation[END_REF], Bozeman andWittmer (2001) Levy et al. (2009), etc.

2) the specific organizational resources of the partners or institutions dedicated to putting laboratories and enterprises in touch with one another (innovation agencies, professional interface institutions, theme days, etc.) as already underlined by [START_REF] Ponomariov | Industry Involvement in University Patents: Before and After the Invention[END_REF], [START_REF] Eom | Determinants of industry-academy linkages and their impact on firm performance: The case of Korea as a latecomer in knowledge industrialization[END_REF] notably.

To coordinate with one another, the actors therefore have the opportunity between turning to social relations, calling on market resources, or making use of organizational resources see table 1). This distinction between interpersonal relations and coordination resources is in part similar to the classic opposition in neo-institutionalist economics between network and coordination by market or organization [START_REF] Powell | Neither market nor hierarchy: Network forms of organization[END_REF]. Of course, actors may combine all three modalities, but most often we can identify the main form of networking by focusing on the first contact between partners, or by using appropriate methods of narration (cf. infra).

Table 1: Modality of partners networking

Networking procedure Subcategories

Interpersonal ties / social proximity

Professional relations (research colleagues, business colleagues)

Nonprofessional relations (family, childhood, associations, friendships)

Relations associated with education (students, teachers)

Market resources

Contracts market (contracts with clients, subcontractors, contract-givers)

Traineeship market4

Research market (tenders)

Reputation

Seminars, conferences, trade shows, fairs (large-scale)

Media (press, Internet, publications, etc.)

Organisational resources

Projects (competitiveness clusters)

Public and semi-public structures (transfer centres)

Closed meetings and invitation-only theme days Professional bodies (clubs, technology associations, expert commissions)

Relations associated with the organisation of the firm or laboratory

Methodology to measure the weigh and spatial effect of social proximity

As stated previously, since relational data are unavailable or not accurate enough to assess the role of social proximity in the formation of collaborations, it is crucial to have a sufficiently sound method for obtaining them and revealing interpersonal ties and social proximity. In this perspective, our proposition is twofold: we propose to collect relational data directly from the people involved in the collaborations and to simultaneously give systematic results and test their correlation between social proximity (ie. interpersonal ties) and spatial proximity (ie.

local collaborations). In this perspective, we use a mixed method that associates qualitative approaches allowing to collect relevant and precise data and quantitative approaches permitting to realize statistical and econometric treatments. More precisely, the collect of data is based on the relational chains method in which it is not a matter of analyzing static structures but rather of recourse to interpersonal relations in order to access resources5 . Only relations actually mobilized in concrete actions are taken into account. The method was used in the seminal work of [START_REF] Granovetter | Getting a job[END_REF], dealing with obtaining a job. However, it fell into oblivion face development approaches so-called comprehensive and personal networks. This method has been particularly developed in France during the last ten years. Since the initial work of Grossetti and Bes (2001) developed for the analysis of innovation, the method has been refined on the one hand (see Grossetti, 2011 notably) and expanded to others contexts and research topics (see for instance [START_REF] Grossetti | Dynamiques des réseaux interpersonnels et des organisations dans les créations d'entreprises[END_REF], [START_REF] Berrou | Dynamique des réseaux sociaux et trajectoires d'entreprises informelles à Bobo-Dioulasso (Burkina Faso)[END_REF]Gondard-Delcroix (2011), Ferru (2010)).

Applied to the case of innovative collaborations, this method consists of reconstructing the build-up process of collaborations by focusing on the initial modalities of partners' networking. We did this based on in-depth interviews 6 with academic researchers and company heads. The analytical unit is therefore neither the laboratory nor the enterprise but the collaboration between them. A collaboration implies the existence of a contract between the parties, but it may encompass several successive contracts when they involve the same partners. For each collaboration, we try to go back as far as possible by placing it in the career of the researchers or company heads. Hence the history of a collaboration starts well before it led to the signing of a contract. We are especially careful to hear about the genesis of the collaboration, the objective being to identify the modality that initiated the networking of the partners, or in other words to determine how the partners met at the time of their first collaboration. During interviews, we use specific questions that will yield precise information on social relations. When the interviewee cites a person's name, we ask whether this is someone he or she knew beforehand, in what context this person was met, and for how long the person has been known. After the interviews, the researcher prepares a detailed account of the collaboration and, when necessary to clarify information, he submits this account to the interviewees. (cf. box 1).

Box 1: A collaboration story

Story 022: Collaboration between an automotive manufacturer (located in Ile-de-France) and an engineering science laboratory (L47)

The partnership (R1) between the two partners is not new (they had collaborated once before).

The partners renewed a former partnership for the new research project. The automotive manufacturer wanted to conduct thermo-aerodynamic research on automotive brakes. One of the firm's scientific directors went to a conference where a researcher explained his work about the item of interest to the firm. "The scientific director was impressed by the results," explained the researcher. Hence, the manufacturer went to the researcher to discuss and find out if his laboratory might be interested in working with them on this subject. Hence, the contact was made possible by a conference, which is considered a market resource.

This account forms the basis for coding: we grouped together the situations we encountered and coded them according to the categories of networking procedures proposed earlier. In the first category -mobilization of interpersonal ties -contact results from the existence of a prior relational chain linking the people deciding on the collaboration. In the second category -use of operational coordination resources -contact is established under the aegis of an outside organization that causes, willingly or not, interactions among members of organizations who will subsequently be led to collaborate. Finally, in the third categorymobilization of market coordination resources -contact results from the initiative of one of the partners who relies on the available information resources (directories, databases, Internet sites) or their meeting at a collective event (conference, trade show), which can be defined as a market coordination resource. Above the modalities of partners networking, data related to the location, scientific specialization of partners and data about the spatial dimension of partnerships are also coded and integrated in the database in order to realize the statistical treatments: 1) cross tabulations and tests of independence to measure the weigh of interpersonal ties and spatial proximity and to check its independence from the partners characteristics; 2) econometric model (logit) to test the correlation between social proximity between partners and the geography of innovative partnerships. Non coded parts of interviews are used as a discourse analysis and will be mobilized for a better understanding of the formation process of partnerships and to interpret the quantitative results.

Operationalization of the method: data collected and treatments implemented

Building relational data relating to nearly 250 cases of partners networking allows us to measure and test the correlation between social proximity and the spatial dimension of collaborations through different statistical and econometric treatments.

Study corpus and sample

Applying the methodological framework presented earlier, we collected qualitative data about the initial networking of partners and analyzed them (through coding and discourse analysis) in order to realize quantitative treatments, interpreted thanks to verbatim. The following chart sums up the operationalization of the method in our case. With regard to the CVR database, we note that on the laboratory side the engineer and chemical sciences researchers are especially active in terms of industrial research, accounting for more than 70% of the contracts. On the manufacturer side, they are for the most part establishments belonging to the aeronautic and automotive construction sectors or specializing in R&D and engineering. Hence, we find as recurring partners such companies as Snecma, Airbus, Renault, PSA, and major research centers such as Anjou Recherche, Centre de R&D de Veolia. As these are contracts signed by CNRS units, we also note the extreme concentration of the contracts in favor of a few laboratories (50% of the contracts are signed by just 9% of the laboratories) and a few firms (50% of the contracts involve only 3% of companies). The most-represented scientific departments of the CNRS are the chemical sciences (29%), life sciences (20%) and engineering sciences (17%).

We built our sample based on these two databases; more specifically, we selected researchers who held scientific responsibility for contracts and asked them to tell us about several collaboration stories. Each story collected from a researcher was complemented with interviews of the manufacturer partners or other participants (another researcher, doctoral student funded as part of the collaboration, outside body that was involved, etc.). The investigators conducted 65 initial interviews with the researchers, which we complemented with 129 interviews, (83 with manufacturer partners, 46 with other researchers or participants). We were thus able to reconstruct 244 collaboration histories.

While our sample is representative of both of the mobilized databases, it is not representative of all science-industry collaborations in France. It is however sufficiently varied in composition so that we can study the correlations between embeddedness in social networks and the characteristics of the laboratories and companies7 . As we shall see that the share of collaborations initiated by chains of interpersonal relations is not very closely linked to the other characteristics tested, the corpus analyzed provides a good estimate of the magnitude of this proportion.

For our sample as a whole, an essential share of the manufacturers involved in the collaborations studied (61%) corresponds to large-scale industrial groups. Furthermore, based on previous studies [START_REF] Ferru | Formation process and geography of science-industry partnerships : the case of the University of Poitiers[END_REF][START_REF] Bès | Pares y redes en los contratos de investigación entre laboratorios de CNRS y sus colaboradores[END_REF], we know that scienceindustry collaborations are characterized by repetition of the contracts, that while there is some volatility in the contracts, which are often fleeting, the laboratories have a certain loyalty to their manufacturer partners, especially with large companies (EDF, ELF, RHONE POULENC, SNECMA, ARIBUS, etc.).

With regard to the geography of the collaborations studied, 22% are conducted "locally" (i.e., the research laboratory and the firm are located in the same region), nearly three-fourths of the collaborations are conducted with a French partner outside the region and located primarily in Ile-de-France, this region being involved in nearly 40% of all the collaborations studied and in more than one French collaboration out of two.

Statistical and econometric treatments of qualitative data

After collecting these qualitative data and ensuring their consistency for both corpuses, we have a set of original variables, presented in the following table. a N=1 refers to the number of data having the value 1.

b N=0 refers to the number of data having the value 0.

In order to complete the qualitative analysis of the descriptive statistics, econometric treatments were done on 235 data (9 networks remaining undetermined as we see in Table 2).

After measuring the weight of social proximity (ie. the use of interpersonal ties) relating to recourse to other modality of partners networking, independence tests are done between the various networking procedures (interpersonal relations, market resources, organizational resources) and various characteristics of the partners (laboratory's scientific domain and size of the city) and of the partnership (geography of the collaboration). By testing these different relationships of dependence, we can verify whether the characteristics of the actors involved in the collaborations influence recourse to social proximity and whether this latter is significantly associated with local collaborations.

Finally, we create a logistic regression model so as to clarify the links of dependency that prove The associated probability is therefore written as:

We test two versions of this model to measure the influence of network modalities on the local dimension of collaborations at a level more or less disaggregated from interpersonal ties.

Results

Weigh of social proximity: the crucial and stable role of interpersonal ties

The data collected reveal the importance of social proximity in the formation of scienceindustry collaborations since 43% of the partnerships having been made possible by an interpersonal tie. The interpersonal relations mobilized to establish collaborations may be of different kinds. The largest share (47% of cases) concerns strictly professional relations among former colleagues or people having worked together on projects. For example, in one of our cases, an engineer with an interprofessional body who became the director of a university laboratory was able to negotiate contracts for his team with his former colleagues, enabling his team to grow quickly. Another frequent (39%) situation is the use of interpersonal relations deriving from the educational system (professor -former doctoral candidate or former students from the same graduating class). The importance of relations associated with education, and especially with former doctoral students, is well known [START_REF] Bozeman | Editor's introduction: building and deploying scientific and technical human capital[END_REF]. Thus, in our interviews, one researcher states that "They are a key means 8 ", while another adds that "They facilitate contact with the manufacturer, because they are familiar with the laboratory's know-how". In one of our histories, for example, an engineer starts work after his training in a company that has regular collaborations with one of the laboratories at his school. After a few years, he changes jobs. Asked by his new employer to 8 Verbatim have been translated by the authors.

set up a central R&D department, he immediately calls on the director of his school's laboratory and negotiates funding with it for a doctoral student: "I've known the laboratory director for thirty years […], it saves time […], it made it possible to get started more easily".

One researcher explains that one of his collaborations with a foreign firm would not have been possible without this relation associated with education: before this former doctoral student was recruited into the company, the laboratory was already interested in the firm's knowhow and wanted to collaborate with it; it had tried to contact it via the Internet to propose collaborative projects, but the firm had never followed up. [START_REF] Fleming | The evolution of inventor networks In the Silicon Valley and Boston regions[END_REF], [START_REF] Todling | Do different types of innovation rely on specific kinds of knowledge interactions?[END_REF] and [START_REF] Giuliani | Why do researchers collaborate with industry? An analysis of the wine sector In Chile, South Africa and Italy[END_REF] have already noted the importance of the recruitment of former doctoral students into companies for the establishment of scienceindustry collaborations. Finally, there are family or friendship relations with no tie to professional activity, which account for just 14% of cases of social proximity. In one of our histories, a collaboration between a laboratory and a large firm was established at the initiative of a doctoral student whose father had worked at the firm in question. It was the doctoral student's father who facilitated his son's and the laboratory's contact with the head of the department that subsequently entered into the collaboration.

The role of social proximity in partners networking is not, however, exclusive: coordination resources, whether market or organizational, seem essential and complementary to the interpersonal ties while allowing the networking of the partners in 57% of the collaborations studied. Market coordination resources are important (24% of cases). Conferences are an example of this. In one collaboration story, a firm in the automotive sector wanted to conduct thermo-aerodynamic research on braking systems. One of the firm's scientific directors then went to a conference on this topic and there met various university researchers presenting their results. The scientific director was interested in the studies presented by one of the researchers from a Poitiers university laboratory. The manufacturer then went to talk to the researcher to find out whether his laboratory might be interested in research work in cooperation with their firm. But many collaborations get established simply because one of the two partners has received information about the other through the media, specialized or otherwise.

Contact between the worlds of science and industry can also be established through a person that is specifically responsible for this kind of work (33% of collaborations were initiated by an organizational resource), like transfer centers or competitive clusters. In one case, the founder of a small firm submitted a problem on controlling the strength of materials to an academic who heads up a regional center promoting technology transfer. The academic put him in contact with a researcher he knew in the field of the physics of materials. The researcher led them to one of his colleagues, a specialist in non-destructive control, with whom the manufacturer established a collaboration. The role of these transfer centers, however, is less significant than might be expected. Scientific or industrial bodies are a more common organizational resource in partners networking. In one of the cases studied, an academic and an engineer working in an interprofessional group are members of the same topical commission in a governmental structure. As a sideline to the commission's activity, they decide to establish a collaboration between their respective teams. Another example involves an engineer with an electrical manufacturing company; put in charge of establishing relations with engineering schools and laboratories, he joined the Arc Électrique club of Électricité de

France. At a club meeting, he met a researcher with whom he began a collaboration that encompassed three different subjects and continued for some twenty years.

By cross-tabulating the three network modalities with the various characteristics of the partners, we find that the weight of social proximity remains relatively stable: whatever the scientific field to which the researchers belong, the collaborations were initiated through interpersonal relations in 40 to 50% of the cases. Likewise, whether the researcher is located in a large or small urban center, the weight of social proximity remains the same. The independence tests allow more rigorous testing of the links between the modalities of networking and the various characteristics of the partners involved in the collaborations. These results show the especially stable nature of social proximity in the formation process of partnerships. The geographic or discipline-related characteristics of the partners have no significant impact on the way in which science-industry collaborations begin: the density of the location does not favor recourse to interpersonal ties, nor does belonging to a specific scientific field encourage the use of organizational resources. the researchers' scientific specialization has no impact on mobilization of a particular modality of partners networking.

In addition, the likelihood of recourse to interpersonal ties does not increase significantly with the density of the actors' city.

Correlation between social proximity and spatial dimension of partnerships

by cross-tabulating the variables associated with the nature of the networking with those having to do with the geography of the collaborations, we find that the interpersonal ties are more frequently used when the collaboration is local. Conversely, recourse to coordination resources leads in a great majority of cases to the establishment of a national partnership. The khi² tests confirm part of these results (cf. Table 4): the nature of the networking influences the geography of the collaborations. If we break down the networking by interpersonal relations even further, a simple statistical cross-tabulation seems to show that nonprofessional relations allow the establishment of collaborations at a more restricted spatial scale (cf. Table 5). By definition, the khi² tests do not indicate the direction of the relationship of dependence between these two variables. We know however that interpersonal ties and coordination resources, whether market or organizational, exist before collaborations: they made possible the partners' networking and the formation of the collaboration. This leads us to think that the nature of the networking influences the geography of the partnerships, and to clarify this relationship using a logit-type econometric model in which the explained variable corresponds to the local (vs. non-local) dimension of the collaborations and to the initial network modality.

The partners' discipline-related and geographic characteristics are integrated into the model as control variables.

The tested model provides several complementary results (sum up in the following table). The first version of the logit reaffirms the positive influence of social proximity on the local dimension of collaborations: the use of interpersonal relations multiplies the likelihood of establishing a partnership in the region by a factor of 2.5. Organizational resources seem to have no significant influence on the geography of collaborations. The second version of the model, with a finer breakdown of network modalities, makes it possible to further clarify the link between spatial proximity and social proximity. Whereas non-professional relations positively and significantly influence intra-regional partnerships -by multiplying the likelihood of establishing collaborations at this spatial scale by a factor of 14 -professional relations do not significantly influence the geography of collaborations. By cross-tabulating the variable associated with the size of the laboratory's location, we note that interprofessional relations are even negatively correlated to the local dimension, which can be explained by the small size of the local labor market for the small cities studied (cf. Poitiers). Relations associated with education (reference variable in our model) generate local collaborations, whether in large centers or medium-sized centers, but to a lesser extent than non-professional relations. As these are geographic and scientific characteristics integrated into the model as control variables, we find that location in an average-sized urban center (Poitiers or Clermont-Ferrand) is a significant disadvantage in terms of local collaborations, as their density does not make it possible to find a potential partner locally. However, this control variable has a far more limited effect than network modalities, its odd ratio being around 2. This being the laboratories' scientific field, it plays only a very negligible role: the ES variable, significant at the 10% threshold in the first version of the model, is not significant in the second.

The results of this model complement and qualify the literature relating to the geography of social relationships. We show that it is not possible to systematically associate interpersonal ties with the local dimension of partnerships. Our work underscores the need to break down interpersonal ties according to their precise nature; non professional ties seem to favor the establishment of partnerships that are more geographically circumscribed than relations associated with education or professional activity. By intuition, this result may differ according to the territorial economic context (size, institutions, specialization, etc.) in particular job market; this result calls for additional cases to test this hypothesis.

Conclusion and discussion

Whereas the role of social proximity in the geography of innovative partnerships appears obvious in many empirical studies, there are very few studies assessing precisely its importance in relation to other modality of partners networking. Our article provides the first data to fill in this gap, proposing a redefinition of the concept -distinguishing social proximity (ie. the use of interpersonal ties) from other coordination resources more classic in economics, like market and organizations -and a sound method for empirically tracking interpersonal relations. Application of this method allows the construction of relational data and contributes a new measurement of the role of social proximity in science-industry collaborations. We also checked various determinants of this social proximity and verified its correlation with the spatial proximity of partnerships.

A qualitative and quantitative analysis of these data reveals the greater weight of social proximity (interpersonal ties making possible the establishment of 43% of collaborations) and the complementary role of market and organizational resources. The role of social proximity varies little as a function of the discipline-related and geographic characteristics of the partners. Finally, our work confirms the existence of a link between social proximity and spatial proximity, while clarifying it: only non-professional relations significantly favor local collaborations.

Although the literature on science-industry collaborations focuses primarily on performance indicators, our work shows the need to explore the means and methods of analyzing the dynamics of these partnerships by considering the different time frames (genesis, negotiation, start of research theses or projects, exchange of data and results, end and continuation) and their differentiated issues.

Our results should of course be refined using larger and more varied corpuses, in terms both of disciplines and national contexts.

Figure 1 :

 1 Figure 1 : Operationalisation of the method

  significant. This enables us to verify the effects of social proximity deriving from the use of interpersonal relations by testing whether this network modality has a real correlation with the local dimension of collaborations. More specifically, we will test a binary logit. In the model tested, the variable explained corresponds to the local dimension of the collaboration, noted as , and the explanatory variables to the various potentially determining factors: the nature of the network modalities and several characteristics of the partners. More specifically, we write the latent variable as and the associated binary variable as which takes the value 1 if the collaboration is established at the local level and the value 0 otherwise.With is the matrix of explanatory variables and the vector of the associated parameters.

Table 2 : Data used Variable name Variable meaning Procedure name Procedure meaning a N=1 b N=0

 2

			𝐼𝑇	Information technology	80	164
	𝑆𝐶𝑑𝑜𝑚𝑎𝑖𝑛 𝑘	Laboratory's scientific domain	𝐸𝑆	Engineering sciences Biological sciences, chemical	126	118
			𝑜𝑡ℎ𝑒𝑟𝑠	sciences, humanities and social	38	206
				sciences, physical sciences
	𝑐𝑖𝑡𝑦_𝑠𝑖𝑧𝑒 𝑘	Size laboratory's city of the	𝑙𝑎𝑟𝑔𝑒 𝑐𝑒𝑛𝑡𝑒𝑟 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑐𝑒𝑛𝑡𝑒𝑟	Toulouse, Bordeaux, Grenoble 122 Poitiers, Clermont-Ferrand 122	122 122
	𝑙𝑜𝑐 𝑘	Spatial dimension of collaborations	𝑛𝑜𝑛_𝑙𝑜𝑐𝑎𝑙 𝑙𝑜𝑐𝑎𝑙	Non-local collaborations Local collaborations	185 59	59 185
			𝑖𝑛𝑡𝑒𝑟𝑝𝑒𝑟𝑠𝑜𝑛𝑎𝑙 𝑡𝑖𝑒 Social relations	101	143
	𝑚𝑜𝑑_𝑐𝑜𝑛𝑡𝑎𝑐𝑡 𝑘	modality of the networking	𝑚𝑎𝑟𝑘𝑒𝑡 𝑜𝑟𝑔𝑎 𝑟𝑒𝑠	Market resource Organisational resource	57 77	187 167
			𝑢𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒𝑑 undetermined	9	235

Table 3 : Results of khi² tests Market resource

 3

	Organisatio nal resource	Interpers onal ties	Total	Results of khi²

Table 4 : Tests of independence: networking modality and spatial dimension of collaborations

 4

			Organisatio nal resource	Market resource	interperson al ties	Total	Results of khi²
		Local	16	9	32	57	X²=5.76
	𝑙𝑜𝑐 𝑘	Non-local	61	48	69	178	p=0.06
		Total	77	57	101	235	Dependence
	NB:X² corresponds to the sum of the differences between the theoretical values and the observed
	values, and p corresponds to the likelihood of an associated error.

Table 5 : Type of interpersonal ties and local vs non-local collaborations.

 5

	Interpersonal ties	local	Non-local
	Education-linked relation	12	27
	Non-professional relation	11	3
	Professional relation	9	38

Table 6 : Result of the binary logit model Explained variable: 𝑙𝑜𝑐 𝑘

 6

					1 st version	2 nd version
			C		-1.84***	-1.46***
					(-3.57)	(-3.47)
	Tested	𝑚𝑜𝑑_𝑐𝑜𝑛𝑡𝑎𝑐𝑡 𝑘	Market resource	Ref	-
	variables		Organisational	0.18	-
			resource		(0.37)
			Interpersonal ties	0.91**	-
					(2.10)
			Education-linked	-	Ref
			relation
			Non-professional	-	2.64***
			relation			(3.78)
			Professional		-	-0.14
			relation			(-0.32)
	Control	𝑐𝑖𝑡𝑦_𝑠𝑖𝑧𝑒 𝑘	Average	urban	-0.77**	-0.77*
	variables		centre		(-1.87)	(-1,79)
			Large Urban Centre Ref	Ref
		𝑆𝐶𝑑𝑜𝑚𝑎𝑖𝑛 𝑘	ES		0.72*	0.64
					(1.67)	(1.50)
			IT		Ref	Ref
			Other		0,87	0.73
					(1.55)	(1.23)
						Wald Chi² test:
						non prof rel.=prof rel
						12.91*** a
		N			235	101
		Log-likelihood			-121.67	-115.40

Notes:*P<0,1, **P<0,05, ***P<0,00 .t-statistics in brackets a Chi² statistic for the hypothesis that the difference of the marginal effects of non professional relation and professional relations is zero

Following the seminal typology of proximity developped by[START_REF] Boschma | Proximity and innovation: a critical assessment[END_REF], besides social proximity four dimensions of proximity are considered :cognitive, organizational, institutional and geographical. By cognitive proximity, it is meant that people sharing the same knowledge base and expertise may learn from each other.Organizational proximity is defined as the extent to which relations are shared in an organizational arrangement, either within or between organization. Institutional proximity is associated with the institutional framework at the macro-level. Geographical proximity refers to the spatial or physical distance between economic actors, both in its absolute and relative meaning.

Some collaborations originate from a student traineeship, obtained in response to an offer put out by the firm to educational institutions.

In guides to analysing social networks[START_REF] Degenne | Introducing Social Networks[END_REF] for example), it is customary to make a distinction between approaches by personal networks, in which relational neighbourhoods are studied around selected actors without making any hypothesis on the relations they might have among them (and without being concerned about these relations), and approaches by complete networks, in which one starts with the existence of resources common to certain actors, and hence likely links among them, in order to select the latter. Analyses of personal networks or complete networks are not suitable for understanding the role of interpersonal ties in the networking of partners.

Questionnaires tend to underestimate the mobilization of personal relationships( Chauvac , 2011); Similarly, a self-administered questionnaire makes it more random entry relational channels , especially when they go beyond a single intermediary and appear difficult to implement given the variation and complexity stories and contexts.

Concerning the sample size, the saturation of the categories is increasingly the reference for estimating the end of the data collection and the sample size[START_REF] Pourtois | et instrumentation en sciences humaines[END_REF].