V Vidal

M Ripepe

T Divoux

D Legrand

J.-C Géminard

F Melo

Dynamics of soap bubble bursting and its implications to volcano acoustics

In order to assess the physical mechanisms at stake when giant gas bubbles burst at the top of a magma conduit, laboratory experiments have been performed. An overpressurized gas cavity is initially closed by a thin liquid film, which suddenly bursts. The acoustic signal produced by the bursting is investigated. The key result is that the amplitude and energy of the acoustic signal strongly depend on the film rupture time. As the rupture time is uncontrolled in the experiments and in the field, the measurement of the acoustic excess pressure in the atmosphere, alone, cannot provide any information on the overpressure inside the bubble before explosion. This could explain the low energy partitioning between infrasound, seismic and explosive dynamics often observed on volcanoes.

Introduction

Volcanic explosions generate both seismic and acoustic waves propagating in the ground and in the atmo-sphere, respectively. Monitoring the acoustic emissions thus represents, together with the seismic signals monitoring, an attractive tool to investigate the source of volcanic explosions. In particular, the simultaneous recording of the seismic and acoustic signals might provide clues to constrain the source process (e.g. [START_REF] Vergniolle | Strombolian explosions 1. A large bubble breaking at the surface of a lava column as a source of sound[END_REF]]). However, the link between the seismic and acoustic waves and the explosive source dynamics is still poorly understood. Some authors claim that seismic and acoustic waves are generated by an unique shallow process (< 500 m depth) [START_REF] Kobayashi | Small inflation sources producing seismic and infrasonic pulses during the 2000 eruptions of Miyake-jima, Japan[END_REF], Johnson (2007)]. Others propose that the acoustic waves are produced by the bursting of meter-sized gas bubbles, while the seismic waves result from the pressure variations, in the magma column, associated with the rise of the gas bubbles toward the surface [START_REF] Ripepe | Time constraints for modeling source dynamics of volcanic explosions at Stromboli[END_REF], [START_REF] Chouet | Source mechanisms of explosions at Stromboli Volcano, Italy, determined from momenttensor inversions of very-long-period data[END_REF], [START_REF] James | Pressure changes associated with the ascent and bursting of gas slugs in liquid-filled vertical and inclined conduits[END_REF]]. Nonetheless, almost all studies assert that acoustic waves are generated either by the bursting of the gas bubbles [START_REF] Ripepe | Infrasonic waves and volcanic tremor at Stromboli[END_REF], Johnson (2003)] or by the oscillation of the magma membrane covering the gas slug just before the burst-ing [START_REF] Vergniolle | Strombolian explosions 1. A large bubble breaking at the surface of a lava column as a source of sound[END_REF]].

The acoustic wave characteristics, in the infrasonic range, are commonly related to the properties of the bursting bubble, such as its volume and overpressure before bursting [START_REF] Vergniolle | Strombolian explosions 1. A large bubble breaking at the surface of a lava column as a source of sound[END_REF]]. However, most of these analysis are theoretical and numerical (e.g. [START_REF] Vergniolle | Strombolian explosions 1. A large bubble breaking at the surface of a lava column as a source of sound[END_REF]]). Only a few laboratory experiments were dedicated to characterizing the acoustics of bubble bursting in conditions that are relevant to volcanology [START_REF] James | Pressure changes associated with the ascent and bursting of gas slugs in liquid-filled vertical and inclined conduits[END_REF], [START_REF] James | Degassing at low magma-viscosity volcanoes: Quantifying the transition between passive bubble-burst and Strombolian eruption[END_REF]].

Here we investigate experimentally the bursting, in static conditions, of a 'slug' whose parameters (geometry and overpressure) are accurately controlled. The characteristics of the acoustic signal emitted at bursting (frequency, energy) are compared with the initial bubble geometry (volume) and overpressure. This experiment focuses on the physical mechanisms at stake when the overpressurized cavity suddenly opens. Because the dynamics of bubble bursting on volcanoes is much more complex, we will not compare directly our experiment with the field situation. However, the physical processes we describe here are likely to be involved when a large gas bubble explodes at the top of a volcano vent (Figure 1a). We therefore comment the results in regard to potential implications for large bubbles bursting on volcanoes. We discuss further (section 5) the limits of this application.

Experimental results

Our experimental setup consists of a cylindrical cavity drilled in a plexiglas slab (Figure 1b). Following [START_REF] Vidal | Acoustic signal associated with the bursting of a soap film which initially closes an overpressurized cavity. Experiment and theory[END_REF]], we close the cavity by stretching a thin soap film. Air is injected inside and, due to the increase in the inner pressure, the thin soap film deforms and bulges out. Injection is stopped when a chosen overpressure ∆P is reached. The system then remains in mechanical equilibrium, while the soap film drains the liquid aside [START_REF] Mysels | Soap Films, Studies of their Thinning and a Bibliography[END_REF]] and eventually bursts. This controlled experiment makes possible to easily vary the length and volume of the cavity, and the gas overpressure before the film bursting. Different tube lengths L (from 2 to 23 cm) and diameters d (6, 8 or 10 mm) have been used (aspect ratio α = L/d ranging from 2 to 23) in order to quantify the role of the conduit geometry.

The rupture of the film results in a sudden drop of the inner overpressure, which excites resonant modes inside the cavity. The inner standing waves are damped due to dissipation along the walls and radiation out of the open end of the cavity. A microphone inside the tube (Figure 1b, bottom) records the pressure variation at the cavity bottom (P int ) while a microphone outside (Figure 1b, top) monitors the radiated acoustic waves (P ext ). As expected for a resonating tube, both signals, inside and outside, exhibit the same spectral content, with a fundamental frequency associated with the wavelength in air λ 0 ∼ 4L and odd harmonics [START_REF] Kinsler | Fundamentals of Acoustics[END_REF]]. Note that the location of the inner microphone is pertinent, as the amplitude of the pressure variation associated with all the harmonics is maximum at the cavity bottom.

Partitioning of the acoustic pressure

When the soap film breaks at the top of the cavity, the overpressure recorded by the bottom microphone 2 (Figure 1b) drops from +∆P to -P int . Before the bursting, the overpressure ∆P inside the cavity is constant. We thus expect, in the absence of significant energy loss, to measure P int = ∆P . We found that this is true only for long tubes (α = 23, large gas volume) whereas for short tubes (α = 2, small gas volume) a large scatter of the pressure drop is observed, with P int ≤ ∆P (Figure 2). This scatter can be explained by taking into account the film rupture dynamics, and in particular its typical rupture time τ burst [START_REF] Vidal | Acoustic signal associated with the bursting of a soap film which initially closes an overpressurized cavity. Experiment and theory[END_REF], [START_REF] Divoux | Acoustic emission associated with the bursting of a gas bubble at the free surface of a non-Newtonian fluid[END_REF]]. It is approximated, from the experimental data, as the time necessary for the overpressure at the cavity bottom to drop from +∆P to -P int . The characteristic film rupture time is compared to the propagation time τ prop of the acoustic wave inside the tube, defined as

∆P / ∆P max 0 0.2 0.8 0.6 0.4 1 0 0.5 1 P int / ∆P τ burst < τ prop τ burst > τ prop
τ prop = 2L/c (1)
where c is the sound velocity (c = 340 m/s). The experiment indicates that the initial relative amplitude P int /∆P is a strongly decreasing function of the ratio τ burst /τ prop (Figure 3a). In other words, for a given geometry, the larger the characteristic rupture time is, the smaller is the amount of energy transferred to the resonant modes [START_REF] Vidal | Acoustic signal associated with the bursting of a soap film which initially closes an overpressurized cavity. Experiment and theory[END_REF]].

For τ burst /τ prop > 1, we observe a drastic drop in the amplitude P int of the signal inside the cavity. Long cavities (large aspect ratio, e.g. α = 23) are not sensitive to the film rupture time, as they always fulfill the condition τ burst < τ prop . In this case, the acoustic amplitude inside the tube well approaches the initial overpressure, and P int /∆P ≃ 1 (Figures 2 and3a).

To the contrary, for short cavities, the system is sensitive to the film rupture time, which is on the order of τ prop , and a large scatter is observed. The uncontrolled bubble rupture time thus accounts for the scatter of the acoustic energy measured outside the cavity (P ext , see section 4).

On volcanoes, we do not have access to the acoustic waves inside the slug, and we measure the acoustic signal propagating outside. When a slug bursts, part of the energy is radiated as infrasonic waves in the atmosphere, while part remains trapped in the volcanic body and propagates as seismic waves. We consider here that the overpressure recorded by the internal microphone (P int , Figure 1b) represents the amplitude (and/or the energy) of the seismic signal generated by the explosive process, while the pressure outside the cavity (P ext ) is comparable to the amplitude of the acoustic waves recorded in the atmosphere as infrasonic waves.

Therefore, by comparing the amplitude ratio between the internal and external pressure variations (Figure 3b, inset), we assume to observe a process similar to the amplitude (or energy) partitioning between seismic and acoustic waves associated with the same explosive dynamics. Our experiment indicates that the inside and outside pressure partitioning (P ext /P int ) changes as function of the tube length hence, of the bubble volume. Longer tubes have acoustic waves with long propagation time and with large damping effects due to viscous dissipation. As a consequence, the amplitude of the acoustic signal outside the cavity will be much smaller than for short tubes (Figure 3b, inset).

In other words, long tubes (e.g. L = 23 cm) are more efficient in terms of acoustic wave energy trapped inside the cavity (Figures 2 and3a), but are not efficient to transmit acoustic energy outside inset). In contrast, short tubes (e.g. L = 2 cm) are more efficient in radiating acoustic energy outside (Figure 3b, inset), but are also more sensitive to the rupture time (τ burst /τ prop > 1, Figure 3a). If the energy loss for long tubes is mainly due to the non-efficient radiation process (small P ext /P int ), the energy loss for short tubes (higher P ext /P int ) is mainly due to the film rupture time (τ burst /τ prop > 1). In both cases, energy is dissipated.

Energy balance

In order to quantify the total energy balance in the system, we estimate the acoustic energy E a from the signal measured outside:

E a = 2πr 2 ρc ∞ t=0 P 2 ext (t) dt (2)
where r is the distance between the cavity aperture and the microphone, ρ the gas density, and c the sound velocity in air. The potential energy stored inside the 'slug' before bursting can be written as:

E p = 1 2 V ∆P 2 ρc 2 (3)
where V is the volume of the gas slug. Figure 4 displays the acoustic energy measured outside, as a function of the initial slug overpressure, before bursting. The three different experimental conditions represent the field situation, where the conduit radius is constant in time, but the slug length can vary from one explosion to another [START_REF] Vergniolle | Strombolian explosions 1. A large bubble breaking at the surface of a lava column as a source of sound[END_REF]]. We report in Figure 4 the maximum total acoustic energy estimated from a series of 10 measurements, performed in similar conditions (same ∆P ). Each point then represents the acoustic energy obtained when the rupture time is the smallest. Its effect is therefore considered negligible (τ burst /τ prop < 1).

For small initial overpressure ∆P , the acoustic energy E a behaves as E a ∼ ∆P 2 . When ∆P increases, the bubble deforms and the soap film curvature increases. Consequently, when the film bursts, the pressure front entering the cavity is spherical, and thus no longer matches the planar geometry of the resonant modes. As a consequence, the efficiency of the energy transfer decreases and E a drops down when ∆P is increased (Figure 4). Finally, we point out that this simple bubble bursting cannot build up gas overpressure above the maximum threshold. For a bubble bursting in static condition, the threshold pressure ∆P max is given by ∆P max = 8σ/d (semispherical film), where σ represents the film surface tension.

In summary, two mechanisms limit the energy tranfer to the acoustic waves. First, the characteristic rupture time of the film breaking (section 3); Second, the film curvature before bursting. Indeed, as the film curvature increases, the pressure of the acoustic wave front generated at bursting departs from a planar wavefield. It is therefore less efficient in exciting the cavity resonant modes. This explains why, in the experiment, the fraction of energy, E a /E p , measured outside is small (∼15%) [START_REF] Vidal | Acoustic signal associated with the bursting of a soap film which initially closes an overpressurized cavity. Experiment and theory[END_REF]], and points out the importance of the geometry and dynamics of the film rupture in controlling the amount of energy released in the atmosphere as acoustic waves.

On volcanoes, the velocity of the film rupture and the film geometry itself are much more complex than in our experiments performed in static conditions, and depend largely on the dynamics of the bubble rising and on viscosity, temperature and volatile content of the magma film layer above the bubble.

Discussion and Conclusion

This simple experiment provides an insight into the physical mechanisms involved in the bursting of a slug of well-controlled geometry and overpressure, in static conditions. Even in a fully controlled laboratory experiment, the amplitude and energy of the pressure wave propagating into the atmosphere after bursting cannot be predicted from the initial slug overpressure -and vice versa. We demonstrated that two processes are responsible for this unpredictability: (1) the rupture time of the bubble film, which cannot be controlled in the experiments; and (2) the energy loss due to the film curvature at bursting, which excites more or less efficiently the cavity.

When the rupture time τ burst is larger than the characteristic propagation time τ prop inside the cavity, the acoustic signal amplitude (and, thus, the energy) drops. The energy fraction (E a /E p ) transferred into the acoustic signal radiated outside decreases drastically when the rupture time τ burst increases.

A quantitative comparison with the much more complex field situation is out of the scope of this paper. Indeed, when a slug bubble rises and bursts in a volcanic conduit, viscous and inertial forces -two processes not investigated in this work -play an important role. On the one hand, by limiting the bubble expansion when rising, these forces are thought to be responsible for the large overpressure stored inside the slug before bursting [START_REF] James | Modelling the rapid nearsurface expansion of gas slugs in low-viscosity magmas[END_REF]]. On the other hand, viscous effects may strongly affect the dynamics of the film aperture [START_REF] Debrégeas | Viscous bursting of suspended films[END_REF]]. Experiments investigating bubbles bursting in either a Newtonian [START_REF] James | Modelling the rapid nearsurface expansion of gas slugs in low-viscosity magmas[END_REF]] or non-Newtonian [START_REF] Divoux | Acoustic emission associated with the bursting of a gas bubble at the free surface of a non-Newtonian fluid[END_REF]] fluids pointed out the importance of the rising velocity and, more generally, of the bursting dynamics, on the acoustic wave amplitude.

However, even if in static conditions, the physical mechanisms described here are likely to be at stake in the field. Different rupture dynamics and film thicknesses largely affect the rupture time, and the acoustic signal emitted at bursting. In particular, we point out that any interpretation of the measured acoustic amplitude, or energy, in terms of gas overpressure in the bubble before bursting requires a good knowledge of the physics controlling the opening of the bubble at bursting. This suggests that on volcanoes also, the amplitude of the acoustic waves generated by the bubble bursting strongly depends on the thickness and on the rupture velocity of the bubble cap at bursting. Both features are uncontrolled in the field, and could explain the low correlation observed at Stromboli volcano between the amplitude of the acoustic wave and the vigor of the explosive event both in terms of mass of ejected fragments [START_REF] Marchetti | Tracing the differences between Vulcanian and Strombolian explosions using infrasonic and thermal radiation energy[END_REF] All data represent the maximum pressure recorded outside over a series of ten measurements. We assume that this value corresponds to the signal recorded when the rupture time is negligible (τ burst /τprop < 1). [Symbol,α]: [•,2.5]; [ ,5]; [⋄,6.9].

Figure 1 :

 1 Figure 1: (a) Sketch of a slug exploding at the top of a volcanic conduit. b) Experimental setup of a soap film bursting at the surface of a cavity of well controlled geometry (length L, diameter d, volume V ) and initial overpressure (∆P ).

Figure 2 :

 2 Figure 2: Normalized amplitude Pint/∆P of the acoustic signal at bursting, inside the cavity, as a function of the initial normalized overpressure ∆P/∆Pmax. Pint/∆P < 1 (gray region) indicates a slow dynamics of the film opening. [Symbol,α]: [•,2]; [•,8]; [△,23].

Figure 3 :

 3 Figure3: Normalized amplitude of the acoustic signal at bursting, inside (a) and outside (b) the cavity, as a function of the ratio between the bursting time τ burst , and the propagation time τprop. Pext is recorded at r = 5 cm from the cavity aperture. The light gray region indicates a slow rupture dynamics of the film (τ burst /τprop > 1, same as Figure2). Insets: (a) Semi-log plot of Pint/∆P as a function of τ burst /τprop. The efficiency of the bubble bursting to transmit pressure waves in the air drastically drops for slow rupture dynamics (τ burst /τprop > 1). (b) Amplitude ratio Pext/Pint. The most efficient energy partitioning occurs for the shorter tube (dark gray region).[Symbol,α]: [•,2];[•,8];[△,23].

Figure 4 :

 4 Figure 4: Acoustic energy Ea as a function of the initial overpressure ∆P (normalized to ∆Pmax) for different lengths (same diameter d = 8 mm).All data represent the maximum pressure recorded outside over a series of ten measurements. We assume that this value corresponds to the signal recorded when the rupture time is negligible

  ] and gas volume[START_REF] Mcgonigle | Spectroscopic capture of 1 Hz volcanic SO 2 fluxes and integration with volcano geophysical data[END_REF]].

			1.0			1.0
						0.8	τ burst > τ prop
	P int / ∆P	0.4 0.6 0.8		P int / ∆P	0.1 0.0 0.2 0.4 0.6 τ burst < τ prop 2 τ burst / τ prop 1 4 6 2 4 6	10	2
			0.2
			0.0
			0	2	4		τ burst / τ prop 6	8	10	12
			4
			2
	P ext / ∆P	4 6 0.01
			2
	0.001
			8
			0.1		1	τ burst / τ prop 10	100

8

∆P / ∆P max

Acknowledgments We thanks M. Ichihara for the fruitful discussions on a previous version of this manuscript. S. Lane is acknowledged for his useful com-ments, which improved the manuscript. D.L. was supported by FONDECYT Project #1061253. V.V., D.L., J.C.G. and F. M. were supported by CNRS/CONICYT Project #18640. V.V. and J.C.G. thank the Centre National de la Recherche Scientifique (CNRS, France) for supporting the research of their members in foreign laboratories.


Alain Thomas
email: thomas@cmi.univ-mrs.fr

CAN AN INFINITE LEFT-PRODUCT OF NONNEGATIVE MATRICES BE EXPRESSED IN TERMS OF INFINITE LEFT-PRODUCTS OF STOCHASTIC ONES?

Keywords: 2000 Mathematics Subject Classification: 15A48 LCP sets, RCP sets, products of nonnegative matrices, products of stochastic matrices

Can an infinite left-product of nonnegative matrices be expressed in terms of infinite left-products of stochastic ones?

Introduction

The problem of the convergence of M 1 . . . M n or M n . . . M 1 for all the sequences (M n ) with terms in a finite set of complex matrices, is studied for instance in [START_REF] Daubechies | Sets of matrices all infinite products of which converge[END_REF], [START_REF] Daubechies | Corrigendum/addendum to: Sets of matrices all infinite products of which converge[END_REF] and [START_REF] Elsner | Norm conditions for convergence of infinite products[END_REF]. The same problem in case of stochastic matrices is also classical, see for instance [9, chapter 4].

On the other side there exist much results about the distribution of the product matrix M 1 . . . M n where the M i are taken in a set of stochastic matrices endowed with some probability measure. In [START_REF] Mukherjea | On the distribution of the limit of products of i. i. d. 2 × 2 random stochastic matrices[END_REF] and [START_REF] Mukherjea | On the continuous singularity of the limit distribution of products of i. i. d. d × d stochastic matrices[END_REF], Mukherjea, Nakassis and Ratti give conditions for the limit distribution of M 1 . . . M n to be discrete or continuous singular; this contains for instance the case of the Erdős measure [START_REF] Erdős | On a family of symmetric Bernoulli convolutions[END_REF]. It is well known that in much cases the normalized product M 1 . . . M n , if the M i are taken in a finite set of nonnegative matrices endowed with some positive probability P , converges P -almost everywhere to a rank one matrix; but this general result should be more consistent if one can to specify the P -negligeable set of divergence.

Notice that the Erdős measure is studied more in detail in [START_REF] Olivier | On the Gibbs properties of Bernoulli convolutions related to β-numeration in multinacci bases[END_REF] by an other method, using a finite set of matrices and the asymptotic properties of the columns in the products of matrices taken in this set.

In the present paper we first consider ( §1) the left-products and the right-products of 2 × 2 matrices (resp. 2 × 2 stochastic matrices). For the left-products of stochastic matrices, the hypothesis that the matrices belong to a finite set is not necessary. We recover the known results by giving several formulations of the necessary and sufficient conditions of convergence. In §2 we associate to any sequence (M n ) of nonnegative d × d matrices, some sequences of stochastic ones, let

S (i) n for 1 ≤ i ≤ t. The convergence of M n . . . M 1 is equivalent to the one of the S (i) n . . . S (i)
1 and some additional condition. 1.2. Case of stochastic matrices. The case of the matrices

S n = 1 v 1 0 0 1 v 2 M n v 1 0 0 v 2 from
S n = x n 1 -x n y n 1 -y n , x n , y n ∈ [0, 1],
is also trivial because one can compute the left-product

(1)

Q n := S n . . . S 1 = t n 1 -t n s n 1 -s n where    s n := n i=1 y i det Q i-1 t n := s n + det Q n Q 0 := I.
To find the conditions for the sequence of matrices (Q n ) to converge one can use the relation ( 2)

s n = s n 0 -1 + s n 0 ,n det Q n 0 -1 , where s n 0 ,n := n i=n 0 y i n 0 ≤j<i det S j .
s n 0 ,n belongs to [0, 1] because it is one of the entries of the stochastic matrix S n . . . S n 0 . Hence, in case det Q n has limit 0 the relation (2) implies that (s n ) is Cauchy; so (s n ), (t n ) and (Q n ) converge. Suppose now that det Q n do not have limit 0. Since det

Q n = n i=1 (x i -y i ) with x i -y i ∈ [-1, 1]
, the non-increasing sequence (| det Q n |) has a positive limit δ hence |x ny n | has limit 1; (x n , y n ) cannot have other limit points than (0, 1) and (1, 0).

In case (0, 1) is one of its limit points, y n | det Q n-1 | do not tend to 0 hence the series n y n det Q n-1 diverges and (Q n ) also do. In case (1, 0) is the unique limit point of (x n , y n ), x ny n is positive from a rank n 0 .

Since the series n log |x ny n | converges to log δ, the inequalities n≥n 0 log(x ny n ) ≤ n≥n 0 log(1y n ) ≤ -n≥n 0 y n prove that the series n y n converges. Since det Q n has limit δ or -δ according to the sign of det Q n 0 -1 , the sequences (s n ), (t n ) and (Q n ) converge.

Consider now the right-product P n := S 1 . . . S n and suppose that the S n belong to a finite set. As noticed in [START_REF] Elsner | Norm conditions for convergence of infinite products[END_REF] it is clear that the nonnull rows of the limit matrix P , if this matrix exists, are nonnegative left-eigenvectors -for the eigenvalue 1 -of each matrix S such that S n = S for infinitely many n, because the equality P n = P n-1 S implies P = P S. So P n can converge only if the S n , for n greater or equal to some integer n 0 , have a common nonnegative left-eigenvector for the eigenvalue 1.

We suppose there exists such a left-eigenvector, let L, and we search the condition for (P n ) to converge. Notice that the S n for n ≥ n 0 commute: since any 2 × 2 stochastic matrix S has left-eigenvector 1 -1 for the eigenvalue det S, both vectors L and 1 -1 are orthogonal to the columns of S n S n ′ -S n ′ S n and consequently this matrix is null. So we fall again in the case of the left-products.

In case S n 0 . . . S n diverges, nevertheless the sequence of the row-vectors LS n 0 . . . S n converges (it is constant). Let L ′ be some row-eigenvector not colinear to L; considering the invertible matrix M whose rows are L and L ′ , MS n 0 . . . S n obviously diverges hence L ′ S n 0 . . . S n also do. Consequently S 1 . . . S n diverges if and only if at least one of the rows of S 1 . . . S n 0 -1 is not colinear to L. We have proved the following Proposition 1.2. Let (S n ) be a sequence of 2 × 2 stochastic matrices, namely

S n = x n 1 -x n y n 1 -y n .
(i) The left-product Q n = S n . . . S 1 converges if and only if n k=1 (x ky k ) has limit 0 or (x n , y n ) has limit (1, 0) when n → +∞. (iii) If it converges, lim n→∞ Q n = s + q 1sq s 1s where s := n y n det Q n-1 and

q := lim n→∞ (det Q n ).
(iv) Suppose now that the S n belong to a finite set. Then the right-product P n = S 1 . . . S n converges if and only if there exists n 0 such that the matrices S n for n ≥ n 0 have a common left-eigenvector for the eigenvalue 1 and either lim n→∞ n k=n 0 (x ky k ) = 0, or lim n→∞ (x n , y n ) = (1, 0), or the rows of S 1 . . . S n 0 -1 are colinear to the left-eigenvector.

(v) Suppose the S n , n ≥ 1, have a common left-eigenvector with respect to the eigenvalue 1.

Then the S n commute and P n = Q n .

Left products of d × d nonnegative matrices

Let us first give one example in order to illustrate the proposition that follows: we consider the products Q n = M n . . . M 1 , where the M i belong to the set of nonnegative matrices of the form

M =            a b 3 -3a -2b e 4 -4a -3b -5e 1 0 c d 2 -3c -2d f 3 -4c -3d -5f 0 1 0 0 1 0 0 1 0 0 0 0 g 5 -5g 0 1 0 0 0 h 1 -5h 1 0 0 0 0 0 0 x 1 -x 0 0 0 0 0 y 1 2 -y           
where a, b, c, d, e, f, g, h, x, y are some reals such that M has exactly twenty four nonnull entries. Since the eigenspace associated to the eigenvalue 1 is generated by

         3 2 1 0 0 0 0          and          4 3 0 5 1 0 0         
, we can associate to each matrix M two submatrices with positive eigenvectors: the submatrix M {1,2,3} of the entries of M with row and column indexes in {1, 2, 3} and the submatrix M {1,2,4,5} of the entries of M with row and column indexes in {1, 2, 4, 5}.

Then we associate two stochastic matrices S = ∆ -1 M {1,2,3} ∆ and

S ′ = ∆ ′ -1 M {1,2,3} ∆ ′ , where ∆ =   3 0 0 0 2 0 0 0 1   and ∆ ′ =     4 0 0 0 0 3 0 0 0 0 5 0 0 0 0 1     ; namely S =   a 2b/3 1 -a -2b/3 3c/2 d 1 -3c/2 -d 0 0 1   and S ′ =     a 3b/4 5e/4 1 -a -3b/4 -5e/4 4c/3 d 5f /3 1 -4c/3 -d -5f /3 0 0 g 1 -g 0 0 5h 1 -5h     .
Now we obtain 21 of the 49 entries of

Q n = M n . . . M
= ∆ ′ S ′ n . . . S ′ 1 ∆ ′ -1 .
The products S n . . . S 1 and S ′ n . . . S ′ 1 converge to rank 1 matrices: use for instance [START_REF] Lorenz | Convergence of products of stochastic matrices with positive diagonal and the opinion dynamics background[END_REF], or use the previous section and the formula for the products of triangular-by-blocks matrices that is,

1 i=n A i B i 0 D i = 1 i=n A i n i=1 A n . . . A i+1 B i D i-1 . . . D 1 0 1 i=n D i . The limit of Q n is a rank 2 matrix of the form          0 0 3 4α 4β 4γ + 6 4δ + 6 0 0 2 3α 3β 3γ + 4 3δ + 4 0 0 1 0 0 2 2 0 0 0 5α 5β 5γ 5δ 0 0 0 α β γ δ 0 0 0 0 0 0 0 0 0 0 0 0 0 0          .
The following proposition generalizes what we have seen on the example. For any d × d matrix M and for any K, K ′ ⊂ {1, . . . , d} we denote by M K the submatrix of the entries of M whose row index and column index belong to K and by M K,K ′ the submatrix of the entries of M whose row index belongs to K and column index belongs to K ′ . By commodity we use the same notation for the row-matrices L (resp. the column-matrices V ): L K (resp. V K ) is the row-matrice (resp. column-matrice) of the entries of L (resp. V ) whose index belong to K.

Proposition 2.1. Let (M n ) be a sequence of nonnegative d × d matrices that belong to a given finite set. The left-product Q n,n 0 := M n . . . M n 0 converges to a nonnull limit when n → ∞ -for each positive integer n 0 -if and only if (M n ) satisfies both conditions:

(i) there exist some subsets of {1, . . . , d}, let K 1 , . . . , K t with complementaries K c 1 , . . . , K c t , and some diagonal matrices with positive diagonals, let ∆ 1 , . . . , ∆ t , such that the S (i) (ii) setting K = ∪ i K i , lim n→∞ M K c n . . . M K c n 0 is the null matrix for any n 0 and the series

n = ∆ i -1 M K i n ∆ i
∞ i=n 0 M K n . . . M K i+1 M K,K c i M K c i-1 . . . M K c n 0 converges.
Proof. If the conditions (i) and (ii) are satisfied, the entries of M n . . . M n 0 with column index in K converges either to 0 or to the entries of the matrices lim n→∞ ∆ i S

  any rank n 0 converges; or the M n have from a certain rank 1 0 for common positive right-eigenvector with respect to the eigenvalue 1, the sum ∞ n=n 0 b n d n-1 . . . d n 0 is finite and d n . . . d n 0 converges, for any n 0 , when n → ∞; or the M n have from a certain rank 0 1 for common positive right-eigenvector with respect to the eigenvalue 1, the sum ∞ n=n 0 c n a n-1 . . . a n 0 is finite and a n . . . a n 0 converges, for any n 0 , when n → ∞.

(

  ii) It diverges only in the case where n (1 -|x ny n |) converges and S n do not tend to 1

  are stochastic, the M K c i ,K i n are null, and lim n→∞ S (i) n . . . S (i) n 0 exists for any i and n 0 ;

  n 0 ∆ i -1 , i = 1, .. . , t. Consequently M n . . . M n 0 converges, by using the formula of product of

1 .

 1 Products of 2 × 2 matrices 1.1. Convergent left-products. Suppose that the left-product M n . . . M 1 of some nonnegative matrices M n = a n b n c n d n converges to a nonnull limit, let Q, and that the set {M ; ∃n, M = M n } is finite. It is clear that from a rank n 0 , the M n belong to the set

	{M ; there exists infinitely many n such that M = M n }
	and consequently, for any M in this set, MQ = M. In other words the nonnull columns of Q
	are eigenvectors -for the eigenvalue 1 -of any M n , n ≥ n 0 . If for instance this eigenvector is v 1 0 , then M n = 1 b n 0 d n 1 n i=n 0 b i d i-1 . . . d n 0 and M n . . . M n 0 = 0 d n . . . d n 0
	Proposition 1.1. M n . . . M n 0 converges to a nonnull limit, for any fixed n 0 and when
	n → ∞, if and only if
	either the M n have from a certain rank, a common right-eigenvector	v 1 v 2	for the eigen-
	value 1, with v 1 v 2 > 0, and the left-product of the stochastic matrices

  1 in function of two products of stochastic matrices, and the other entries in the first five columns of Q n are null: indeed Q n has for submatrices M

	{1,2,3} n	. . . M 1 {1,2,3}	= ∆S n . . . S 1 ∆ -1 and M n {1,2,4,5}	{1,2,4,5} 1 . . . M

triangular-by-blocs matrices:

Conversely suppose that Q n,n 0 = M n . . . M n 0 converges to a nonnull limit for any n 0 ∈ N.

Since the M n belong to a finite set, we can choose n 0 large enough such that all the matrices M that are equal to M n for at least one n ≥ n 0 , are also equal to M n for infinitely many n. Then the nonnull columns of the limit matrix Q are right-eigenvectors of all the M n , n ≥ n 0 , for the eigenvalue 1, because the equality

, and denote by V 1 , . . . , V t the linearly independent columns of Q. We denote also by K i the set of the indexes of the nonnull entries in V i , and by ∆ i the diagonal matrix whose diagonal entries are the nonnull entries

On the other side, denoting by K the union of the K i , the product

The first converges to 0: by the definitions of K and the vectors V i , the rows of Q whose indexes belong to K c are null. Since

converges to the null matrix.

Remark 2.2. The two conditions of (ii) are satisfied -assuming that the ones of (i) areif all the submatrices M K c n have spectral radius less than 1, or if their eigenvalues greater or equal to 1 disappear in the product M K c n+h . . . M K c n for some fixed h and for any n. Let (M n ) be a sequence of complex-valued matrices such that R n converges. Since the limit matrix R has norm 1, it has some nonnull columns; let us prove that they are righteigenvectors of each matrix M that occurs infinitely many times in the sequence (M n ).

The nonnegative real λ n := MR n-1 is bounded by M and satisfy MR n-1 = λ n R n for any n such that M n = M, so it has at least one limit point λ that satisfy MR = λR, and the columns of R are right-eigenvectors of M for the eigenvalue λ.

Suppose that λ = 0 -perhaps it is not possible that λ = 0. The convergence of R n can hold without the convergence of 1 λ M n . . . 1 λ M 1 , see for instance the case where all the M n are equal to 1 1 0 1 . But conversely the convergence of this last product to a nonnull matrix, for some λ = 0, implies obviously the one of R n .


Taoufik Amri
email: amri.taoufik@gmail.com

Quantum Behavior of Measurement Apparatus

Keywords: numbers: 03.65.Ta, 42.50.Dv, 03.65.Wj, 03.67.-a

We precise for the first time the quantum behavior of a measurement apparatus in the framework of the usual interpretation of quantum physics. We show how such a behavior can also be studied by the retrodiction of pre-measurement states corresponding to its responses. We translate in terms of these states some interesting properties of the behavior of an apparatus, such as the projectivity, the fidelity, the non-Gaussian character, or the non-classicality of measurements performed by this one.

We also propose an experimental procedure allowing the tomography of these pre-measurement states for optical detectors. We illustrate the relevance of these new notions for measurements, by evaluating them for two detectors widely used in quantum optics: the avalanche photodiode and the homodyne detection.

I. INTRODUCTION

The measurement apparatus plays an important role not only in quantum physics experiments, by providing information about the measured system, but also in the foundations of quantum theory by leading to the famous measurement problem [1,[START_REF] Wheeler | Quantum Theory and Measurement[END_REF]. This one is in part linked to our ability to prepare the measured system in a particular state, by using information available after its interaction with an apparatus. This state, conditioned on the measurement result, is given in the simplest case by Von Neumann's projection rule [1]. In the most general case, this is a non-linear transformation of the initial state that we will specify in the following.

However, such a conditioning needs a precise knowledge about the performed measurements. It is surprising that the first experimental characterization of measurements has been realized only very recently, in quantum optics [4], while the quantum tomography of states (QST) and processes (QPT) [START_REF] Brune | [END_REF] are now well-established in this same field. Indeed, the quantum detector tomography (QDT) is the reconstruction of positive operator valued measures (POVMs) [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF], describing any measurement device. We probe the behavior of its responses with a set of known states, in order to reconstruct the POVM elements giving the probabilities which are the closest to those measured. This was proposed for instance in Ref. [START_REF] Fiurasek | [END_REF] with a maximum-likelihood estimation (MaxLik), which is a reconstruction method widely used for QSTs.

Despite attempts, the interpretation of QDT results is not yet exploited, and to our knowledge, it has never been treated in the framework of the usual interpretation of quantum physics, which mainly deals with states in its formalism. The aim of this paper is to show that the retrodictive approach of quantum physics [7,8], in addition to the usual predictive one, provides unexpected insights into the quantum behavior of a measurement apparatus. It brings a meaning to the non-classicality of a measurement and allows us to introduce other interesting properties for a measurement, such as its projectivity, its fidelity with another measurement, and its non-Gaussian character. Moreover, we evaluate some of these properties for two detectors widely used in quantum optics: the avalanche photodiode and the homodyne detection.

II. STATES AND PROPOSITIONS

As a preliminary step, it is necessary to focus on some important notions from the usual interpretation of quantum physics, which is predictive about measurement results and retrodictive about state preparations. These tasks need conditional probabilities linked by Bayes' theorem, and in the most general case, the expression of probabilities on the Hilbert space H is given by the recent generalization (2003) [9] of Gleason's theorem [10].

This theorem is only based on very general requirements about the probabilities and the mathematical structure of the Hilbert space, which is the starting point of any quantum description. It states that any system can be described by a density operator. More precisely, when we propose a property P n about a system corresponding to a precise value for a given observable, the proposition about the state of this system is represented by the projector on the eigenstates corresponding to this value. In the most general case, such a proposition can also be represented by a hermitian and positive operator Pn in the Hilbert space.

The probability Pr (n) of checking this property on the system should satisfy the following conditions:

1. 0 ≤ Pr (n) ≤ 1 for any proposition P n .

2.

n Pr (n) = 1 for any exhaustive set of propostions such that n Pn = 1. According this theorem for a system needing predictions (i.e. with a Hilbert space of dimension D ≥ 2), this probability is given by Pr (n) = Tr{ρ Pn } in which ρ is a hermitian, positive, and normalized operator, allowing us to make predictions about any properties of the system. This is the reason why we call this operator the state of the system, and the probabilities are in fact conditioned on this state of knowledge. When we reduce this operator in its eigenbasis, we retrieve a statistical mixture usually used in teaching for introducing the density matrix. Thus, each time that we can write the probabilities concerning any exhaustive set of propositions about the system in this previous form, we can then determine the state of the system thanks to this theorem.

Pr (n

III. PREPARATIONS AND MEASUREMENTS

A. The Game

In quantum physics, any situation is based on preparations and measurements. In such a game, the preparation of the system in a state ρm can be associated to a classical information that we call the choice 'm'. The measurement, corresponding to the POVM element Πn , gives another classical information which is simply the result 'n'. However, we can only make predictions about these choices 'm' and these results 'n'. We have then two approaches in quantum physics, that we will examine in details in the following. Each approach needs a quantum state and propositions, allowing predictions about the measurement results or retrodictions about the state preparations, as pictured on Fig. 1.

B. Predictive approach

We usually prepare the system in a given quantum state based on a choice 'm', and we make propositions about the results of any subsequent measurement which will be performed on this system. The conditional probability of obtaining a certain result 'n', after that the system was prepared in the state ρm , is then given by:

Pr (n|m) = Tr{ρ m Πn }. (1)
This result constitutes the Born's rule, in which the proposition operator Πn is nothing else than the POVM element corresponding to the result 'n'. In the framework of states and propositions, the POVM elements should be understood as propositions about the state of the system interacting with the measuring device. These operators also describe the behavior of responses of this measuring device, and the corresponding propositions are simply labeled by the responses of the apparatus performing their tests.

Moreover, any proposition can be reduced to more simple propositions corresponding to projectors on a larger Hilbert space. This is the Neumark's extension [11] which can often be interpreted as a noise influence on ideal measurements. Such a modeling is in fact common in quantum optics. An imperfect optical measurement can be viewed as an ideal measurement onto the signal field, which has interacted with an appropriate noise field.

C. Retrodictive approach

Contrary to the previous one, this approach is less usual and leads to difficulties linked to an inappropriate use of its tools (see for example [START_REF] Aharonov | Resolution of a "Retrodiction Paradox[END_REF]). We start with an example in order to "demystify" such a approach.

We consider a perfect photon counter able to discern the number of absorbed photons.

When such a device displays n counts, the proposition about the measured system simply corresponds to the projector Pn = |n n|. The pre-measurement state, which is sufficient to describe this result, is then this photon number state Pn . This one should allow us to predict the states in which the measured system was prepared, before its interaction with the apparatus giving the result on which we base such a retrodiction.

The link between the proposition checked by the apparatus and the pre-measurement state is obvious in this ideal case. We will generalize it by using the generalization of Gleason's theorem [II]. Indeed, we can already write an expression for the retrodictive probability. The probability of preparing the measured system in a given state ρm , when we have the result 'n', can be written as

Pr (m|n) = Tr{ρ [n] retr Θm }. (2)

In this expression, ρ[n]

retr is the pre-measurement state retrodicted from the result 'n' labeling the POVM element Πn . Θm is a hermitian and positive operator, corresponding to a proposition about the state of the measured system just after the process preparing the state ρm .

In order to have an exhaustive set of propositions about the preparations of the system, these operators should constitute a resolution of the Hilbert space H: m Θm = 1.

(

) 3
As it was previously noticed by S. Barnett et al. [8], the expressions of the retrodicted state and proposition operators can be derived from Born's rule with Bayes' theorem. However, it is interesting to give them in the light of the generalization of Gleason's theorem, which justifies the expression of retrodictive probabilities and brings interesting insights into these tools.

Retrodiction requires no additionnal assumption other than the projection rule which could be used in the preparation of states. Indeed, Bayes' theorem gives for the retrodictive probability:

Pr (m|n) = Pr (n|m) Pr(m)/Pr(n). (4)
The marginal probability Pr (n) of having the result 'n' is obtained by summing the joint probability on all 'candidate' states:

Pr (n) = m Pr (n|m) Pr(m) = Tr{ρ [?] Πn }, (5)
in which we introduce the state ρ[?] = m P m ρm . In a conditional preparation, such a state corresponds to the state of the measured system after an 'unread' measurement [START_REF] Haroche | Exploring the quantum : Atoms, Cavities, and Photons[END_REF]: a mixture of states conditioned on each result 'm' and weighted by their respective success probabilities P m = Pr(m). This mixture could also be obtained with preparations based on random choices 'm', as we will see it in our scheme for the tomography of pre-measurement states.

Then, for writing the retrodictive probabilities (4) in their most general form (2), this statistical mixture should be such that ρ[?] = 1/D in a Hilbert space H of finite dimension D. The pre-measurement state simply corresponds to the normalized POVM element: Finally, we propagate these retrodicted states backward in time contrary to prepared states, as pictured on Fig. 1. Indeed, when we take the Heisenberg picture for the POVM elements in the predictive approach, we retrieve the pre-measurement state (6) in the Schrödinger picture with the time-evolution operator Û (-t) = Û † (t). The complementarity between the predictive and retrodictive approaches is illustrated on Fig. 2.

ρ[n] retr = Πn Tr{ Πn } , (6)

IV. HOW TO STUDY AN APPARATUS ?

The POVM elements should be understood as propositions about the measured system.

These propositions are actually labeled by the responses of the measuring device, and this one appears as a physical implementation of tests checking these propositions. However, the POVM elements only allow us to study the behavior of an appartus, by giving the predictive probabilities (1) of having a certain response. Thus, how could we go beyond these simple conditional probabilities in order to study the measurement apparatus ?

A. Measurement effects on a system

The POVM elements are not sufficient for studying such effects in the general case.

Indeed, the effects of a generalized measurement on a state are given by the following transformation (see for example [START_REF] Hornberger | Introduction to decoherence theory[END_REF]):

ρ → ρ[n] cond = µ Mn,µ ρ M † n,µ P n , (8)
where Mn,µ are the measurement operators, also called Kraus operators [START_REF] Kraus | States, Effects and Operations[END_REF], describing completely the measurement process corresponding to the result 'n'. The probability P n of having this result is then given by the POVM element:

Πn = µ M † n,µ Mn,µ . (9)
The normalization of expression (8) by the probability P n is the manifestation of the socalled projection rule, leading to a non-linear transformation of the initial state contrary to an unread measurement. When we perform the measurement on only one part B of a bipartite entangled resource ρAB , the state of the other part A -conditioned on the result of this measurement -depends on the POVM element corresponding to the expected result 'n':

ρ[n] A,cond = 1 P n Tr B {ρ AB 1A ⊗ Πn }. (10)
In quantum optics, such protocols have been recently implemented for preparing more "exotic" non-classical states from gaussian resources, such as Fock states [16] or Schrödinger's cat states [17].

Thus, in the predictive approach [III B], the study of an apparatus is restricted to nondestructive measurements. Indeed, it needs a kind of QPT for determining the measurement operators Mn,µ . Such a approach also depends on the initial state of the measured system, and cannot lead to a study of the apparatus as an implementation of tests checking propositions. Furthermore, different measurement processes { Mn,µ } µ can check one same proposition, since the decomposition of a given POVM element ( 9) is not unique. In other words, there is no general rule to design a measuring device with a given POVM, underlying the importance of an experimental determination of POVM elements describing an apparatus [4]. Therefore, the predictive approach is more adapted for studying the measurement effects on a system, instead to really translate its behavior in terms of states.

B. 'State' translation of measurements

The retrodictive approach [III C] seems to be more close to such a goal. Indeed, from a

given result, we make predictions about the preparations of the system before its interaction with the apparatus checking the property corresponding to this result. The main tool of this approach is the retrodicted pre-measurement state, that we propagate backward in time for predicting these preparations as depicted on Fig. 1. We can therefore determine in which kind of states the system was prepared for leading to such a result. Actually, the pre-measurement states of an apparatus can directly be reconstructed with the same tools as a QST. This experimental procedure is pictured on Fig. 3 for optical detectors.

In such a experiment, we can probe the behavior of an optical detector with a statistical mixture of coherent states ρ[?] = m p m |α m α m |, obtained by amplitude and phase modulations randomly performed on each light pulse. As a first step, the QST of this mixture could be necessary and leads to its Cholesky decomposition, ρ[?] = σ † σ. We must have det{ρ [?] } = 0 for generating the proposition operators:

Λm = σ-1 † p m |α m α m | σ-1 . (11)
With these operators, we realize an usual QST of the state ρn giving the conditional probabilities Pr (m|n) = Tr{ρ n Λm }, measured directly in the experiment. For this purpose, we replace the POVM elements describing the measurements in a QST method by these proposition operators (11). We determine then the state ρn giving the closest probabilities Pr (m|n) to those measured. The pre-measurement state, retrodiced from the response 'n' of the apparatus, is obtained from this state by:

ρ[n] retr = σ-1 ρn (σ -1 ) † Tr{σ -1 ρn (σ -1 ) † } . (12)
Finally, it's interesting to note that the retrodictive probabilities (4) can also be obtained from data taken in QDT experiments [4], in which we directly measure the predictive probabilities (1) for each response 'n' of an apparatus. Indeed, if the preparation rate is the same for all the probe states ρm , the probability of preparing the state ρm is simply given by Pr (m) = 1/M, where M is the number of probe states. These retrodicted probabilities are then given by:

Pr (m|n) = Pr (n|m) M m ′ =1 Pr (n|m ′ ) . ( 13
)
We can therefore realize the QST of pre-measurement states, retrodicted from each response of the device, with the procedure previously depicted.

V. BEHAVIOR OF A MEASUREMENT APPARATUS

We translate some interesting aspects of such a behavior in terms of pre-measurement states. This translation motivates the implementation of our experimental procedure for the tomography of these states, or the interpretation of results from realized experiments [4]. We also illustrate the relevance of these new properties for measurements on two wellknown detectors from quantum optics: the Avalanche Photodiode (APD) and the Homodyne Detection (HD).

A. Non-classicality of a measurement

There are different signatures for the non-classicality of states such as the negativity in particular quasi-probability distributions or the contextuality of hidden variable models trying to describe the measurement results [19,20]. These two notions are in fact equivalent, as it was recently shown in Ref. [21].

The non-classicality of a measurement corresponds to the non-classicality of its premeasurement state, for which such a notion is well-established with different signatures.

We illustrate the relevance of such a correspondance for optical detectors in the conditional preparation of non-classical states of light. In such experiments [16,17], we generally reconstruct from experimental data the Wigner representation of the conditioned state (10) given by:

W [n] cond (x, p) = N dx ′ dp ′ W AB (x, p; x ′ , p ′ ) W [n] retr (x ′ , p ′ ) , (14)
where W AB and W In pratice, we use for such a task an Avalanche Photodiode (APD) which is a singlephoton detector with only two responses, called 'off' and 'on'. The response 'off' corresponds to the imperfect detection of zero photon (see for example [START_REF] Barnett | Imperfect photodetection as projection onto mixed states[END_REF]):

Πoff (η, ν) = e -ν ∞ n=0 (1 -η) n |n n|, (15)
in which η and ν are respectively the detection efficiency and the mean number of dark counts. On the other side, the Wigner representation [18] of the POVM element Πon = 1 -Πoff is given by:

W on (x, p) = 1 2π - e -ν π ∞ m=0 (η -1) m e -(x 2 +p 2 ) L m 2 x 2 + p 2 . ( 16
)
where L m (x) = e x ∂ m x (x m e -x ) /m! are the Laguerre polynomials. When we read the result 'on', the non-classicality of the measurement performed by the APD can be measured by the negativity of the Wigner representation of its pre-measurement state. To avoid some mathematical difficulties, we choose for the negativity N on (η, ν) = W on (0, 0), since we have Tr{ Πon } > 0. The evolution of this signature of non-classicality, under the noise influence, is pictured on Fig. 4. The black line corresponds to a negativity N on (η, ν) = W on (0, 0) = 0.

Thus, this signature stays negative for a mean number of dark counts such that ν < ln (1η/2). We can easily check this for a given APD, by simply measuring its detection efficiency η and its dark noise ν. Moreover, we can compare such a signature of nonclassicality for different "on/off" photon detectors, by simply plotting their points (η, ν) on Fig. 4.

B. Projectivity

An ideal measurement checks a simple proposition corresponding to a projector Πn = |ψ n ψ n | in the Hilbert space. However, in more realistic situations, a measuring device is characterized by POVM elements which are not at all projectors. We have actually an evaluation of the projectivity of a measurement with the purity π n of its pre-measurement state:

π n = Tr ρ[n] retr 2 , ( 17
)
such that 1/D < π n ≤ 1.

When the pre-measurement state is a pure quantum state with π n = 1, the measurement performed by the apparatus is projective for the response 'n', but it may be non-ideal.

Indeed, the POVM element corresponding to such a projectivity (π n = 1) is given by

Πn = η n |ψ n ψ n | (18)
where η n = Tr{ Πn } can be viewed as the detection efficiency of the state |ψ n , by using the predictive approach in which the predictive probability Pr (n|ψ n ) = η n . We clearly see on this property the complementarity between the predictive and retrodictive approaches in quantum physics.

For an APD, the projectivity is only relevant for the measurement corresponding to the response 'off' which is characterized in the ideal case (η = 1) by the vacuum state |0 . Indeed, the pre-measurement state retrodicted from the result 'off' is

ρ[off] retr (η, ν) = η ∞ n=0 (1 -η) n |n n|, (19)
and the projectivity of this measurement is given by:

π off (η, ν) = η 2 -η . (20)
Contrary to the POVM element [START_REF] Kraus | States, Effects and Operations[END_REF], we can see that this state and its properties do not depend on the dark noise ν. We can easily understand this since the dark counts correspond to results 'on'. Thus, when we have the result 'off', the properties of the performed measurement only depends on the detection efficiency η. If we have a quasi-ideal measurement η ≃ 1, we are sure of the pre-measurement state even for a dark noise arbitrarely large, which may be interesting for conditional protocols based on such measurements.

C. Fidelity with a projective measurement

We define this fidelity as the overlap between the pre-measurement state ρ[n] retr retrodicted from a certain result 'n' and a target state |ψ tar , in which we would like checking the system before its interaction with the apparatus. Such a fidelity [START_REF] Jozsa | Fidelity for Mixed Quantum States[END_REF] can be written as

F n (ψ tar ) = ψ tar |ρ [n] retr |ψ tar . (21)
With the retrodictive approach, we have an interesting interpretation for this overlap. This is nothing else than the retrodictive probability (2) of preparing the system in the target state |ψ tar , before the measurement process giving the result 'n':

F n (ψ tar ) = Pr (ψ tar |n) = Tr{ρ [n] retr Θtar }. (22)
The proposition operator (7) about the state of the system, just after its preparation, is

Θtar = |ψ tar ψ tar |. (23)
We note that the marginal probability P tar = Pr (ψ tar ) of preparing the system in this target state is 1/D in this particular case. It ensures a maximum mixture of pure states constituting a basis of the Hilbert space. This basis is composed of the target state |ψ tar and (D -1) other pure states. Thus, for having an exhaustive description of the system, we need at least one other proposition:

Θ tar = 1 -Θtar . ( 24
)
It corresponds to the preparation of the system in the state ρ tar = Θ tar /Tr{ Θ tar } with the probability P tar = (D -1) /D. When the measurement giving the result 'n' is sufficiently faithful F n (ψ tar ) ≃ 1, the most probable state in which the system was prepared before its interaction with the apparatus is this target state |ψ tar .

We now illustrate this notion for the measurements performed by an APD. For the measurement corresponding to the result 'off', the fidelity with an ideal photon number-resolved detection (PNRD) of n photons is given by,

F off (n, η) = Pr (n|off) = η (1 -η) n . ( 25
)
The evolution of this fidelity with the detection efficiency η is plotted for different photon numbers n on Fig. 5 (a). As we can expect it for the response 'off' of an efficient APD, the only PNRD which is the most faithful to the measurement performed by the APD corresponds to the photon number n = 0. In terms of retrodictive probabilities [START_REF] Barnett | Imperfect photodetection as projection onto mixed states[END_REF], this means that the most probable state in which the system was prepared, before the measurement giving the result 'off', is the vacuum state. For a low efficiency η → 0, all the photon-number states become equally probable. We can understand this since the measurement is not at all projective (π off ≃ 0) for such efficiencies. For the measurement corresponding to the result 'on', the situation is more delicate since the POVM element corresponding to this result leads to divergences in the retrodiction of the pre-measurement state. Indeed, we can only determine an asymptotic equivalent for this pre-measurement state:

ρ[on] retr (η, ν) ∼ D→+∞ 1 D D n=0 1 -e -ν (1 -η) n |n n|, ( 26
)
where D is in fact the dimension of the Hilbert space. The fidelity between this measurement and a PNRD of n photons is given by

F on (n, η, ν) = Pr (n|on) ∼ D→+∞ 1 D Pr (on|n) , (27)
in which we recognize the predictive probability (1) of having the result 'on' when the light was prepared in the photon-number state |n :

Pr (on|n) = 1 -e -ν (1 -η) n . (28)
Thus, the behavior of the fidelity F on (n, η, ν) is the same as that of the predictive probability [START_REF] Banaszek | Operational theory of homodyne detection[END_REF]. Its evolution with the photon number n and the detection efficiency η is pictured on Fig. 5 (b). For a given efficiency, the large photon-number states are the most probable preparations leading to the result 'on', with or without dark noise. We can also see the effect of this one on Fig. 5 (b), which is mainly a non-zero probability for a totally inefficient detector (η = 0). This probability is the same for all the photon-number states. We understand this since the APD has actually no interaction with the signal field.

D. Non-Gaussian character of a measurement

The non-Gaussian character of a measurement is essential in many quantum information protocols such as entanglement purification [25]. This property can be measured by the non-Gaussian character of its pre-measurement state. We can compute for instance its "non-Gaussianty" (non-G) [26] defined by the relative Von Neumann entropy between this pre-measurement state and its reference gaussian state, which is the gaussian state with the same covariance matrix. The non-G is therefore equal to zero only for gaussian states.

Then, there is an interesting result based on Hudson-Piquet's theorem [24]. It states that any pure state, characterized by a non-negative Wigner representation [18], is a gaussian state with a gaussian distribution for its Wigner representation. We have here an important link between the Gaussian character of a measurement and its projectivity : When a measurement is projective with a non-negative Wigner representation for its pre-measurement state, then this measurement is a Gaussian measurement.

In order to illustrate such a Gaussian character, we determine the pre-measurement states retrodicted from results of a measuring device widely used in quantum optics : the Homodyne Detection (HD). Indeed, the inefficient HD of the value x I for the quadrature observable xφ [START_REF]e -iφ â + e iφ â † / √ 2, where â[END_REF] is characterized by the following POVM element [START_REF] Banaszek | Operational theory of homodyne detection[END_REF]:

ΠHD (x I ; φ) = 1 π (1 -η) exp - x I - √ ηx φ 2 1 -η , (29)
in which η is the detection efficiency linked to quantum efficiencies of photodiodes used in such a device. The Wigner representation of this POVM element is then given by:

W HD (x φ , p φ ) = 1 8π 3 η σ 2 x (η) exp - x φ -x I / √ η 2 2 σ 2 x (η) , (30)
with the variance σ 2 x (η) = 1-η 2η . The Gaussian character of this measurement is obvious as pictured on Fig. 6, in which we can distinguish the evolution of the variance σ 2

x (η) for different values of the detection efficiency η. We can therefore say that an HD performs non-classical measurements in the same sense as squeezed states are non-classical states. Indeed, the squeezing of the pre-measurement state (31) is ensured for a detection efficiency such that η > 50 %.

In the continuous-wave (cw) regime, the detection efficiency η can currently reach values around 98% [29], corresponding to a really impressive squeezing level s dB ≃ -17 dB. At our knowledge, one of best levels recorded in the cw regime [30] is -10 dB, corresponding to an efficiency around 90%. Obviously, the pre-measurement state retrodicted from such a detection has a lower purity also corresponding to the projectivity of the HD.

VI. CONCLUSION

In this paper, we have shown that the retrodiction of pre-measurement states, in addition to the usual predictive approach, leads to a complete study of the measurement apparatus as an implementation of tests checking propositions about the system interacting with it. These propositions simply correspond to POVM elements describing the behavior of its responses in the predictive approach. When a certain proposition is checked, the pre-measurement state retrodicted from this one highlights the behavior of the apparatus in terms of states, which is the main language of quantum physics. We have also proposed an experimental procedure, essentially based on the retrodictive approach, allowing the direct tomography of pre-measurement states for making such a translation. We hope that this experimental procedure will be realized for probing each response of a measurement device. These results, or those from the predictive version [4], could then be interpreted in the framework of our

1 or n 2

 2 or ...) = Pr (n 1 ) + Pr (n 2 ) + ... for any non-exhaustive set of propositions such that Pn 1 + Pn 2 + ... ≤ 1.

Figure 1 :

 1 Figure 1: (color online) The game of preparations and measurements in quantum physics: the prediction and the retrodiction need quantum states and propositions that we respectively note ρm , Πn and ρ[n] retr , Θm . The time-evolution operator Û (t) allows us to propagate forward or backward in time these states between the preparation and the measurement, in order to make these predictions.

Figure 2 :

 2 Figure 2: (color online) Complementarity between the predictive and retrodictive approaches of quantum physics -the successful test of a proposition in one of these approaches gives the state of the other one, with which we can make predictions about measurement results or preparation choices.

Figure 3 :

 3 Figure 3: (color online) Scheme for the quantum tomography of pre-measurement states for optical detectors: Each light pulse is randomly prepared in a coherent state |α m , with a probability P m , by an amplitude modulation (AM) and a phase modulation (PM).

  are respectively the Wigner representations for the resource ρAB and the pre-measurement state ρ[n] B,retr retrodicted from the expected result 'n'. N is a positive normalizing constant of W [n] cond . When the resource has a non-negative representation as gaussian states, the necessary condition for preparing a non-classical state ρ[n] cond is to perform a non-classical measurement, in the sense of a non-positive Wigner representation W [n] retr .

Figure 4 :

 4 Figure 4: (color online) Evolution of the non-classicality of a measurement performed by an APD, displaying the result 'on', with its detection efficiency η and the mean number of dark counts ν.

Figure 5 :

 5 Figure 5: (color online) Fidelities between the measurements performed by an APD and PNRDs : (a) Evolution of the fidelity, between the measurement corresponding to the result 'off' and projective measurements of n photons, with the quantum efficiency η. (b) Evolution of the predictive probability Pr (on|n) with the quantum efficiency η and the photon-number n, for two mean numbers of dark counts ν.

Figure 6 : 1 π 1 + 2 s (η) - p 2 φ 1 /

 611221 Figure 6: (color online) (a) Wigner representation of the POVM element ΠHD (x I ; φ) for x I = 1 and η = 0.75 in the phase-space (x φ , p φ ). (b) Evolution of this Wigner representation with the detection efficiency η in the phase-space (x,p).


Tim Mazzarol
email: mazzarol@gsm.uwa.edu.au

Sophie Reboud
email: sreboud@escdijon.com

Customers as predictors of rent returns to innovation in small firms -an exploratory study

Keywords: Small Business, Entrepreneurship, Innovation, Risk Assessment

Small to medium enterprises (SME) are frequently associated with high levels of innovation. However, it is difficult for such firms to fully assess the merits of a particular innovation. This study examined the risk-return profile of future investment in innovation by SME with respect to anticipated 'rent' or financial returns. A survey of highly innovative SME was undertaken that examined management perceptions of the key strategic influences on rent returns. Regression analysis suggests that the firm's assessment of the rent returns from their innovation may be influenced by the value it is likely to deliver to the customer, the customer's expected use of the innovation to generate new sales and the ease of integrating the new innovation into existing technologies. The findings have implications for how entrepreneurs from early stage ventures are assisted.

Innovation in Small Firms

Innovation is recognised as a key driver for economic and social development and a critical element in the future success of industry (Senge, Carstedt, & Porter, 2001). As a process within organisations, innovation is about new product or process leading to the enhancement of value for customers and shareholders [START_REF] Drucker | The Discipline of Innovation[END_REF]. Innovation requires change, both to the customer and the supplier, as well as to the firm that drives the initiative. Such change can be incremental in nature or highly radical and disruptive [START_REF] Tushman | Organizing for Innovation[END_REF]. Due to the potentially disruptive nature of innovation it is an activity containing inherent risk, with uncertainty in future technical, commercial and financial returns to the initial investment required [START_REF] Gatignon | The Impact of Risk and Competition on Choice of Innovations[END_REF][START_REF] Dziura | Innovations: Sources and Strategies[END_REF]. An important consideration in the decision to invest in future innovations is therefore the trade off between risk and return.

For small to medium enterprises (SME) (e.g. those with less than 200 employees), the ability to assess risk and evaluate potential returns is frequently difficult. Such firms are frequently viewed as having an important role to play in the generation of new innovations [START_REF] Freel | Do Small Innovating Firms Outperform Non-Innovators?[END_REF][START_REF] Hansen | Environmental Innovations in Small and Medium Sized Enterprises[END_REF][START_REF] Mogee | Foreign Patenting Behaviour of Small and Large Firms[END_REF]. For example, it has been estimated that around 67 percent of all inventions and 95 percent of radical inventions patented in the United States since 1945 were attributed to small entrepreneurial firms [START_REF] Ncoe | Embracing Innovation: Entrepreneurship and American Economic Growth[END_REF]. The role of small firms in Europe (European Commission, 2003) and Asia (APEC, 2003) has also been noted.

Despite the importance of SME to enhancing innovation, it remains difficult for the entrepreneurs associated with such businesses to fully assess the relative merits of their innovation in relation to risk-return trade offs. Successful diffusion of an innovation frequently involves consideration of a range of variables associated with market forces, including the relative power of competitors, suppliers and customers and regulatory agencies. For most small business operators, the evaluation of these many factors remains complex and difficult. Thus an evaluation tool designed to assist owner-managers of small firms assess the merits of their innovations is likely to be of significant value.

The Concept of RENT

While financial models (e.g. Net Present Value) can offer a guide to the potential attractiveness of innovations, these measures are limited in their capacity to fully assess the non-financial variables likely to impact on the investment, such as the competitive reactions of the environment.¹ Moreover the innovator might be convinced of the value of its innovation but unable to analyse it or even prove its reality. There can be three main reasons to explain this.

First an innovator often places more emphasis on the anticipated absolute value of the innovation without considering the potential difficulties associated with its launch [START_REF] Martin | The Nature of Innovation Market Failure and the Design of Public Support for Private Innovation[END_REF]. This evaluation frequently can be overly optimistic either because the anticipated interest is overestimated or because it does not take into account the erosion of the rent due to the bargaining power of such environmental actors as suppliers and customers; and due to the effects of competition.

Second the innovator may have had an unstructured way of reasoning, thus they may act on an intuition rather than on a rational reasoning [START_REF] Mockler | Prescription for Disaster: Failure to Balance Structured and Unstructured Thinking[END_REF]. This is a high probability among small firms where the owner-manager is relatively isolated and lacks the support of an evaluation team with the skills to conduct appropriate feasibility analysis. Third, the innovation frequently involves a complex, non-linear process in which feedback over progress is difficult to anticipate. [START_REF] Santi | Modèle de valorisation et de protection intellectuelle des innovations des PEI[END_REF] have suggested a risk assessment framework for SME to assess the strategic risk-return of a future investment in innovation. This measures the anticipated "RENT" or financial return anticipated from a particular investment in an innovation [START_REF] Miles | Modelling corporate entrepreneurship as rentseeking competition[END_REF].

In order to help the innovator not only to correctly assess the initial anticipated value of their innovation but also the erosion effects they will face when launched on the market, the process of analysis follows several steps [START_REF] Santi | Modèle de valorisation et de protection intellectuelle des innovations des PEI[END_REF][START_REF] Reboud | Intérêt d'un outil d'évaluation du risqué lié à l'innovation pour les PME[END_REF]: i) analysis of anticipated rent (the "potential rent"), related to the kind of innovation and the number and size of the potential using markets; ii) analysis of the characteristic of the environment of the potential using market, and the related erosion effects; iii) analysis of the competitive strengths the innovation will have to face, leaving only a "residual rent"; and iv) analysis of the competitive situation of the SME and its ability to launch the innovation, able to catch an "appropriable rent".

Innovation is in economic theory a potential source of competitive advantage [START_REF] Nemeth | Managing Innovation -When Less is More[END_REF]. The rent associated to this competitive advantage is the condition for its success and further development. An innovator is thus going to develop a new resource or competence or going to create a new combination of existing resources and competencies [START_REF] Schumpeter | The Theory of Economic Development[END_REF]. A firm with a competitive advantage should be able to generate a rate of profit higher than the mean rate of profit in its industry and even more to keep this rate higher during a given period. In many cases of innovation the durability of the advantage can be protected thanks to the patent system.² The innovator is then the only legal user of the innovation for a twenty-year period securing a legal monopoly and creating a sustainable competitive advantage. This emulates the concept of Ricardian rent.³

A Model of Risk Assessment

While financial models (e.g. Net Present Value) can offer a guide to the potential attractiveness of innovations, these measures are limited in their capacity to fully assess the non-financial variables likely to impact on the investment, such as the competitive reactions of the environment. Moreover the innovator might be convinced of the value of its innovation but unable to analyse it or even prove its reality. There can be three main reasons to explain this.

Determination of the potential rent-return from a future innovation is a function of three variables (Santi, et.al 2003 andReboud &[START_REF] Reboud | Intérêt d'un outil d'évaluation du risqué lié à l'innovation pour les PME[END_REF]): i) Volume -as measured by volume of sales over one year; ii) Rate of Margin -as measured by profits generated from the innovation; and iii) Length -or duration or life cycle of the innovation. This amount will be the combination of three components, which will vary independently one from the other. This can be shown as follows: RENT = VOLUME X RATE X LENGTH.

Figure 1 illustrates the relationship between these variables in relation to the diffusion life-cycle of an innovation. Table 1 also shows each of these three elements and their potential indicators. (Santi, et.al. 2003) Source: [START_REF] Santi | Modèle de valorisation et de protection intellectuelle des innovations des PEI[END_REF]Reboud &[START_REF] Reboud | Intérêt d'un outil d'évaluation du risqué lié à l'innovation pour les PME[END_REF] As the combination of these three variable components, a rent can be characterised by its extreme profile: large/small volume; high/low rate of margin and short/long life cycle. With two possibilities for each variable a total of six combinations can be identified. These are illustrated in Figure 2, where the options are shown with their various trade-offs of volume, rate and length within the rent equation. Each configuration involves different levels of volume, rate and length thereby determining the anticipated rent to be derived from the innovation. As shown in Figure 2 these configurations are labelled: "Shrimp", "Champion", "Gadget", "Joker", "Lark Mirror" (Flash in the Pan), or "Oasis". These may be further described in the following terms.

1. Shrimp -a configuration offering low rent potential due to its modest levels of volume, rate and length, as such it is unlikely to be of much interest;

2. Champion -a configuration with high potential rent;

3. Gadget -a configuration offering low volume and length but high rate, leading to little interest overall. Such a configuration would not justify significant investment:

4. Joker -configuration with high volume and length but low rate making it little better than the "Gadget" despite its apparent unattractiveness;

5. Lark Mirror or Flash in the Pan -a configuration with good volume but poor length and may experience both high or low rate, making it challenging for the investor that may need to outlay substantial capital to secure the return over the short life cycle; and 6. Oasis -a configuration that offers good length but low volume and high or low rate.

Oasis

Source: [START_REF] Santi | Modèle de valorisation et de protection intellectuelle des innovations des PEI[END_REF]Reboud &[START_REF] Reboud | Intérêt d'un outil d'évaluation du risqué lié à l'innovation pour les PME[END_REF] While the "Champion" configuration appears the most desirable, Santi et.al (2003) have suggested that the "Oasis" configuration is more suitable for an SME because the small overall volume of sales may be more readily exploited by the small firm. Such an innovation opportunity is essentially that found in a niche market. However, the capacity of the SME to secure a desirable rent return from its innovation is likely to depend on its resources, the nature of the innovation and the characteristics of the market environment into which it is seeking to diffuse the innovation. Key forces likely to influence the market environment include the power of customers and their capacity and willingness to adopt the new innovation, the power of competitors, and the capacity for new market entrants and substitution threats to erode the competitive advantage of the innovation [START_REF] Porter | Competitive Strategy: Techniques for Analyzing Industries and Competitors[END_REF].

A Model of Risk Assessment

To evaluate the efficacy of the RENT configuration model outlined in Figure 2, a survey instrument was prepared based upon the original framework used by Santi, et.al. (2003). This used the questions and followed the steps of the original model, asking the respondent to evaluate items using Likert-type scales rating his/her responses. The survey was piloted with owner-managers of SME known for their high level of innovation investment in new products.

The questionnaire was subsequently distributed to SME identified as highly innovative. These firms were registered with the Western Australian State Government Department of Industry and Resources (DOIR), who facilitated the data collection process. A total of 550 firms were surveyed, drawing from the database of innovator companies held by the Western Australian Government. Mailed questionnaires with accompanying covering recruitment letters were sent to all these firms and a total of 57 usable responses were returned (10% response rate).

The questionnaire was targeted at persons within the firms who could report on behalf of the entire organisation. Within the final sample 42 percent were owner-managers, 23 percent were executive managers and principal shareholders, 25 percent were executive manager shareholders and 10 percent were executive managers without shareholdings. Thus the majority of corespondents were both executive managers and owners. All firms within the sample had less than 100 employees, with the majority (75%) being enterprises with less than 20 employees. The final distribution of firms by size was: i) 35 percent -micro-enterprises (e.g. less than 5 employees); ii) 40.5 percent -small enterprises (e.g. 5-20 employees); and iii) 14.5 percentmedium-sized enterprises (e.g. 21-200 employees). Annual turnover ranged from less than AUD$0.5 million to over AUD$20 million, however 83 percent had less than AUD$5 million, with the average around AUD$1 million. The majority (76%) of respondent firms indicated that they had prepared a formal business plan within the previous three years. Eighty-percent of these firms indicated that they had engaged in some form of export activity, and 58 percent claimed to have some formal process for innovation or new product development.

All respondents were asked if they had plans to introduce a new innovation within the next three years. The majority (96.5%) indicated that they had plans for some new innovation of which 54 percent felt their innovation would create a new market, while 55 percent felt their innovation was compatible with existing products and processes. Sixty-four percent of the firms had generated their innovation alone.

This demographic profile suggests a particularly high level of innovation intensity among the respondent firms with above average investment in R&D and export activity when compared to other SME in Australia and Europe (ABS, 2001;[START_REF] Evangelista | Measuring Innovation in European Industry[END_REF]. The sample drawn for this study, although small in size, was representative of high innovator SME and therefore a good example of the type of firms most likely to experience problems of making rent-return trade off assessments when seeking to exploit a new innovation.

Exploratory Regression Analysis

To evaluate the relationship between the six configurations of RENT and various predictors within the SME a regression analysis was employed. However, first a dependent variable had to be created to measure RENT. In order to estimate this model the dependent variable had to be constructed from three measures of VOLUME, RATE, and LENGTH. The first of these was estimated using an item relating to the potential annual sales estimated to be secure on a worldwide basis as measured in dollars. The second component was measured using an item that estimated the potential rate of profit the SME felt the innovation was likely to yield. Finally, length was measured using an item that estimated the ease with which an innovation could be copied. This allowed the development of a dependent variable that provided an estimate of low to high levels of potential RENT while also creating a proxy measure for the six RENT configurations as outlined in Figure 2.

A total of 28 independent variables were used in the model that measured the respondent"s assessment of the customer"s likely perception of the innovation and how readily the customer might adopt the innovation. This included customer perceptions of risk, compatibility of the innovation with existing technologies, price, and value, ease of understanding, and pre and post testing ability. These independent variables also examined power of customers and suppliers.

The analysis used a stepwise methodology that produced a model after three steps as shown in Table 2. It can be seen that this model had an adjusted R-square of 0.432 suggesting that the model explains around 43 percent of the variation in the data. This model suggests that the predictors of RENT were: i) the level of value likely to be offered to those who adopt the innovation; ii) the relative importance of potential sales turnover to lead customers; and iii) the compatibility of the innovation to the existing technologies operated by customers.

Discussion of the Findings

The regression model outlined above suggests that the respondent"s assessment of the RENT returns from their innovation may be influenced more by their anticipation of the value it is likely to deliver to the customer, the customer"s expected use of the innovation to generate new sales and the ease of integrating the new innovation into the customer"s existing technologies. Such findings are consistent with those of [START_REF] Rogers | Diffusion of Innovations[END_REF] who highlighted the importance of the relative advantage of the innovation to the customer or adopter, and the compatibility of the innovation to existing processes.

The most important influence on the decision making of small firms is likely to be the customer. Small firms are likely to focus on their leading customers as a priority due to their ability to provide immediate sales revenues for the innovation. [START_REF] Sundbo | The Strategic Management of Innovation: A Sociological and Economic Theory[END_REF] has also shown the importance of close interactions between the firm and its lead customers as a means of enhancing innovation.

While still tentative, these findings indicate that the small innovator firm is likely to gauge the merits of their proposed innovation largely in terms of how it will be received by its leading customers. This is not surprising and supports other findings of the way in which small, high technology firms interact within their industry [START_REF] Mazzarol | Strategic Networking Among High Technology Firms: Evidence from the Western Australian ICT Sector[END_REF].

Research into the innovativeness of small firms has placed emphasis on the role of the entrepreneur, the market and the firm itself [START_REF] De Jong | Determinants of the Innovative Ability of SMEs: Literature Review[END_REF]. Such firms will need to have internal cultures in which there is a close relationship between the entrepreneurial leadership and the employees, in which quality and the willingness to strive for continuous improvement manifest [START_REF] Mazzarol | Innovativeness in Small Firms: An Exploratory Study of the Perspectives of Growth Oriented Owner-Managers[END_REF].

The role of the customer as a driver of innovation within small firms might be explained in terms of the prospecting behaviour of the entrepreneur [START_REF] Kickul | Prospecting for Strategic Advantage: The proactive entrepreneurial personality and small firm innovation[END_REF]. The findings from this present study support this and point to the need for the small innovator firm to have entrepreneurial management that is closely in touch with leading customers and able to adapt its innovations to suit their needs and deliver value in terms of new sales growth, but without having to make significant changes to their existing technology base.

The most important of the predictors to the determination of RENT appears to be the potential value the innovation offers to the customer, as reflected in the beta for this item. Interesting the beta for the item relating to the compatibility of the innovation with the customer"s existing technologies was negative. This may reflect the lower rental from an innovation that is more incremental than radical within the industry.

However, while the focus on the willingness of a customer to adopt an innovation can be understood in the context of a small entrepreneurial firm seeking quick returns to its investment in a new product, there should be a note of caution. Entrepreneurs that become too focused on the immediate needs of their customers may ignore the more disruptive innovations that may require longer time for diffusion into markets, and that may require time for customers to adopt them. If a small firm with a highly disruptive technology cannot find ready acceptance from a potential customer, it is likely to struggle with its financial sustainability. In such cases -examples may be found in the biotechnology sector -the entrepreneur may seek venture financing to assist in bridging the gap between the technology development and the market adoption.

Much attention has been given to the role of venture capital financing in the development of small high technology firms. However, the findings from this study point to the role of the customer in providing cash flow to the smaller firm through sales of products that can be adopted quickly by the market. Fifty-one percent of the sample drawn for this study reported having formal joint venture agreements with leading customers, but only 31 percent reported having sought to raise venture capital requiring equity sharing in the previous three years. Further, only 6 percent reported having developed their innovation in conjunction with a third party research centre from either a university or publicly funded research institution or wider industry network. The majority of firms (64%) reported having developed their innovation alone, with 21 percent reporting having collaborated with their leading customer in the generation of the innovation.

These findings suggest that many small firms may be developing their innovation alone and with minimal input from either the formal venture capital sector, or the publicly funded research community. Their capacity to develop new innovations depends on establishing a relationship with leading customers and using this to assist in the development and diffusion process. For the entrepreneur in the small innovator firm, this pattern of behaviour may result in a form of entrapment where they are caught in a relationship of customer dependency. Innovations that are not readily accepted by their customers will be at risk of not being developed, and if their customer base is too narrow they will be restricted in the ultimate RENT return that they can expect as volume levels may be low.

Conclusions

This regression model is exploratory in nature and does not seek to provide definitive findings. Further research is required with a larger sample and some adjustment to the questionnaire before any strong conclusions can be shown. Tentatively this analysis highlights the importance of the leading customer for small innovator firms in relation to RENT returns particularly the ability of the innovation to offer value to customers and customers who place high value on sales turnover. However, enhanced RENT may accrue where the innovation is more radical than incremental in nature. In the case of a disruptive technology that might ultimately return greater RENT, the small firm that is overly focused on its customer"s willingness to adopt the innovation may chose not to proceed. This suggests that public policy seeking to assist the small firms sector to adopt more innovations should focus upon the customer-firm interface, with attention given to working with larger leading firms that build relationships with the SME sector to allow new products to be developed as part of a wider market diffusion process. Such a system might involve the participation of larger firms that serve as a focal point for a local production network. These firms would work with their local SME supply network, with the support of venture capital firms and publicly funded research agencies, to co-develop innovations that would be mutually beneficial to all parties. Public policy support for such a scheme would seek to work via the larger firm as leading customer and co-financier, rather than directing funding entirely at the SME that frequently seeks to then on sell to the customer.

As the findings from this study indicate, the key predictors for returns to investment in innovation as perceived by the management teams of small firms are the acceptance of the new product by customers. Small firms seeking to invest in innovation are likely to be guided more by the market pull of customer acceptance of their new product than consideration of the intrinsic merits of the technology itself. The need for relatively quick market diffusion of the innovation can be explained in terms of the need within the small firm to secure sales revenue to ensure sufficient cash flow to fund the growth of the business. Unlike their larger counterparts the small firm is usually deficient in working capital and lacks the ability to withstand long periods before investments reach break-even. For this reason the small firm is likely to focus on the needs of the customer and the customer"s ability to readily adopt the innovation. It is noteworthy that the majority of firms in this sample were engaged in the development of their innovation alone, with few seeking to develop technologies in concert with university researchers or other publicly funded research centres. This is consistent with other findings from similar studies [START_REF] Mazzarol | Strategic Networking Among High Technology Firms: Evidence from the Western Australian ICT Sector[END_REF] and may suggest that innovation in many small firms is more likely to be incremental than radical in nature. Future research will focus on further development of the model and its evaluation across a wider range of SME with the prospect of multi-country studies designed to measure the possible impact of national innovation systems.

Notes

Figure 1 :

 1 Figure 1: Components of RENT

Figure 2 :

 2 Figure 2: Typology of configurations of rent of an innovation

Table 1 : measure of potential rent

 1

	Components of RENT		Indicators
	VOLUME: (potential annual sales) -	Size of industry or market sector
		-	Geographic diffusion potential
		-	Size of end-user markets
		-	Limits due to prior patents
	RATE: (potential profit margin)	-	Type of innovation process involved
		-	Type of innovation involved
		-	Level of prior protection for intellectual property
	LENGTH: (potential life-cycle)	-	Technological basis of innovation
		-	Innovation intensity of the user centre
		-	Legal and technical protection of intellectual
		-	property

Table 2 : Model Summary

 2

	Model	R	R Square	Adjusted		Std Error of
				R Square		the Estimate
	3	.685	.469	.432		.57296
	Predictors: (Constant), Q22 value offer to customer, Q51 importance of sales turnover to
	customers, Q27 customer compatibility of technology.
	Dependent Variable: RENT
	Coefficients
	Model			B Std Error	Beta	t Sig.
	(Constant)			1.091	.555	1.964 .056
	Q22 value offer to customer		.341	.079	.487	4.345 .000
	Q51 importance of sales turnover to customers	.184	.058	.358	3.201 .003
	Q27 customer compatibility of technology	-.160	.054 -.329 -2.946 .005
	Dependent Variable: RENT

Tim Mazzarol is Director of the Centre for Entrepreneurial Management and Innovation (CEMI)