A Ouahbi

♭ A Iggidr
email: iggidr@loria.fr

M El Bagdouri
email: elbagdouri@ucam.ac.ma

Project ‡ Conge

Stabilization of an Exploited Fish Population *

Keywords: Continuous system, Feedback control, Population dynamics, Stabilization

.

We give a formula for the fishing effort as a feedback control that allows to stabilize the system.

Introduction

The use of dynamic population models has become common in the study of exploited fish populations. These models are used to provide qualitative and quantitative descriptions of events in various fisheries, to give predictions of the future evolution of a given population and they are also useful for making policy decisions about fisheries.

In this paper we consider the following stage-structured model that has been built in [START_REF] Touzeau | On the stock-recruitment relationships in fish population models[END_REF]. It includes (n + 1) stages represented by their abundance X i (t), stage 0 being the pre-recuits stage. Each stage is characterized by its fecundity, mortality and predation rates (for more details, see [START_REF] Touzeau | On the stock-recruitment relationships in fish population models[END_REF]):

                
Ẋ0 (t) = -αX 0 (t) -m 0 X 0 (t) + n i=1 f i l i X i (t) -n i=0 p i X i (t)X 0 (t), Ẋ1 (t) = αX 0 (t) -αX 1 (t) -m 1 X 1 (t), . . . Ẋn (t) = αX n-1 (t) -αX n (t) -m n X n (t), [START_REF] Jurdjevic | Controllability and stability[END_REF] where • m i : linear mortality rate.

• α: linear aging coefficient.

• p 0 : juvenile competition parameter.

• p i : predation rate of class i on class 0.

• f i : fecundity rate of class i.

• l i : reproduction efficiency of class i.

As suggested in [START_REF] Touzeau | On the stock-recruitment relationships in fish population models[END_REF], the mortality coefficient can be written as a sum of the natural mortality rate M i and the fishing mortality coefficient F i . Hence one can write:

m i = M i + q i E
where q i is the catchability of stage i and E is the fishing effort that can be seen as a control term. The goal of this paper is to show how to control this system by acting on the fishing effort E. To this end, we rewrite it as follows:

                 Ẋ0 (t) = -α 0 X 0 (t) + n i=1 f i l i X i (t) -n i=0 p i X i (t)X 0 (t) Ẋ1 (t) = αX 0 (t) -(α 1 + q 1 E(t))X 1 (t) . . . Ẋn (t) = αX n (t) -(α n + q n E(t))X n (t) (2)
where • M i is the natural mortality rate.

• α i = α + M i • q i is the relative catchability coefficient of class i.
• E is the fishing effort.

Our aim is to construct a feedback control E = u(X) to ensure the survival of the population. More precisely, we shall compute u(X) in such a way that the closed loop system [START_REF] Khalil | Nonlinear systems[END_REF] with this feedback has a nontrivial equilibrium state which is globally asymptotically stable.

In the following we shall use these notations:

IR n+1 + = {X = (X 0 , X 1 , . . . , X n ) ∈ IR n+1 | X i ≥ 0 , i = 0 . . . n + 1}. Ω = Int(IR n+1 + ) is the interior of IR n+1 + , that is Ω = {X ∈ IR n+1 | X i > 0 , i = 0 . . . n + 1}. X(t, X 0 ) is the solution of (2) with initial state X 0 , that is X(0, X 0 ) = X 0 .
2 Main result

Stability with a constant fishing effort

Consider the system with a constant fishing effort Ē:

     Ẋ0 (t) = -α 0 X 0 + n i=1 f i l i X i (t) -n i=1 p i X i (t)X 0 (t) -p 0 X 2 0 (t) Ẋi (t) = αX i-1 (t) -(α i + q i Ē)X i (t) for i = 1, . . . , n. (3)
The origin is an equilibrium point which corresponds to an extinct population and is therefore not very interesting. Under the following nonlinearity and survival conditions ( [START_REF] Touzeau | On the stock-recruitment relationships in fish population models[END_REF]):

(H1)

n i=0 p i = 0 (H2) n i=1 f i l i π i > α 0
where

π i = α i i j=1 (α i + q i Ē) system (3) has an other equilibrium X *      X * 0 = n i=1 f i l i π i -α 0 p 0 + n i=1 p i π i X * i = π i X * 0 . (4)
In ([6], pp 65-74), it has been shown that the positive orthant Ω = {X ∈ IR n+1 | X i > 0 , i = 0 . . . n+1} is invariant and the following stability results

for system (1) has been derived. It has been proved that if assumptions H1, H2 are satisfied and if moreover

(H3) f n l n = 0 (H4) X * 0 < µ = min i=1,...,n f i l i p i for f i l i p i = 0
then:

(i) the origin is unstable.

(ii) there exists an invariant domain D = n i=1 [a i , b i ], with 0 < a i < b i , such that the nontrivial equilibrium X * belongs to D, X * is asymptotically stable and its attraction domain contains D.

The proof of (i) uses the properties of positives matrices in order to show that the linearized system around the origin has a positive eigenvalue associated to a positive eigenvector. The proof of (ii) uses the properties of cooperative systems [START_REF] Smith | Systems of ordinary differential equations which generate an order preserving flow. A survey result[END_REF]. The numbers a i can be chosen as small as one needs but the numbers b i are bounded by some functions of the parameters f i , l i and π i (see [6] for more details).

Here we shall prove that the attraction domain of X * is the whole positive orthant Ω. Our proof uses the Lyapunov theory. The advantage of this method is that we get a global result and the Lyapunov function allows us to construct a variable fishing effort as a function of the state of the system and that stabilizes the population at its nontrivial equilibrium.

Proposition 2.1 Let Ē be any positive constant fishing effort. If assumptions (H1), (H2), (H3) and (H4) are satisfied then the state X * is a globally asymptotically stable equilibrium state for system (3) defined on Ω.

Proof: Let us consider the following Lyapunov function

V (X) = 1 2 (X 0 -X * 0 ) 2 + n i=1 n j=i k j π j α i + q i Ē X i -X * i π i 2
where

k i = f i l i -p i X * 0 i = 1, . . . , n
V is a positive definite and proper function of X • V (X) > 0 for X = X * , and V (X * ) = 0

• lim X →+∞ V (X) = +∞
The derivative of V along the solutions of system (3) is

V (X) = -α 0 X 0 + n i=1 f i l i X i - n i=1 p i X i X 0 -p 0 X 2 0 (X 0 -X * 0 ) + n i=1 n j=i k j π j α i + q i Ē αX i-1 -(α i + q i Ē)X i π i X i -X * i π i Since αX * i-1 -(α i + q i Ē)X * i = 0 for all i = 1, . . . , n (5)
and

-α 0 X * 0 + n i=1 f i l i X * i - n i=1 p i X * i X * 0 -p 0 X * 0 2 = 0, (6)
we get

V (X) = n i=0 k i (X i -X * i ) - n i=0 p i (X i -X * i )(X 0 -X * 0 ) (X 0 -X * 0 ) + n i=1 n j=i k j π j α i + q i Ē α(X i-1 -X * i-1 ) π i -(α i + q i Ē)( X i -X * i π i )
where

k 0 = -(α 0 + p 0 (X 0 + X * 0 ) + n i=1 p i X * i ). Thus V (X) = k 0 - n i=1 p i (X i -X * i ) (X 0 -X * 0 ) 2 + n i=1 k i (X i -X * i )(X 0 -X * 0 ) + n i=1 n j=i k j π j α α i + q i Ē X i-1 -X * i-1 π i - X i -X * i π i X i -X * i π i
Taking into account that α

(α i + q i Ē)π i = 1 π i-1
, and remarking that

k 0 - n i=1 p i (X i -X * i ) ≤ -α 0 -p 0 X * 0 , it follows V (X) ≤ -(α 0 + p 0 X * 0 )(X 0 -X * 0 ) 2 + n i=1 k i (X i -X * i )(X 0 -X * 0 ) + n i=1   ( n j=i k j π j ) ( X i-1 -X * i-1 π i-1 )( X i -X * i π i ) -( X i -X * i π i ) 2  
The inequality ab ≤ 1 2 (a 2 + b 2 ) that holds for any pair of numbers a and b allows to write the inequalities (π 0 = 1 by convention)

V (X) ≤ -(α 0 + p 0 X * 0 )(X 0 -X * 0 ) 2 + n i=1 k i (X i -X * i )(X 0 -X * 0 ) + n i=1   n j=i k j π j 1 2 ( X i-1 -X * i-1 π i-1 ) 2 - 1 2 ( X i -X * i π i ) 2   ≤ (-α 0 -p 0 X * 0 + 1 2 n i=1 k i π i )(X 0 -X * 0 ) 2 + n i=1 k i (X i -X * i )(X 0 -X * 0 ) + 1 2 n-1 i=1 n j=i+1 k j π j X i -X * i π i 2 - 1 2 n i=1 n j=i k j π j X i -X * i π i 2 ≤ -α 0 -p 0 X * 0 + 1 2 n i=1 k i π i (X 0 -X * 0 ) 2 + n i=1 k i π i X i -X * i π i (X 0 -X * 0 ) - n i=1 k i π i X i -X * i π i 2
¿From the expression (4) we check that

-α 0 -p 0 X * 0 + 1 2 n i=1 k i π i = - 1 2 n i=1 k i π i . Thus V (X)≤ - 1 2 n i=1 k i π i (X 0 -X * 0 ) 2 + 2( X i -X * i π i )(X 0 -X * 0 ) -( X i -X * i π i ) 2 ≤ - 1 2 n i=1 k i π i (X 0 -X * 0 ) -( X i -X * i π i ) 2 ≤ 0
The equilibrium X * is then Lyapunov stable for system [START_REF] Lasalle | Stability by Liapunov's direct method with applications[END_REF]. Note that all solutions of the system are bounded because V is proper and its derivative is non positive. It remains to show the attraction of the equilibrium. Let

W = {X ∈ IR n+1 + | V (X) = 0}
According to LaSalle theorem [START_REF] Lasalle | Stability by Liapunov's direct method with applications[END_REF], all the solutions starting in Ω tend to L the largest invariant set contained in W . Let X ∈ L, we have

V (X) = 0 ⇔ X 0 -X * 0 = X 1 -X * 1 π 1 = ... = X n -X * n π n (7)
Substituting ( 5) in (7) and by using the equalities

π i = α α i + q i Ē π i-1 , it follows that αX i-1 -(α i + q i Ē)X i = 0 for all i = 1, ..., n.
Which implies that for all i = 1, ..., n, Ẋi = 0 and

X i = π i X 0 . ( 8
)
Since L is invariant, we get Ẋ0 = 0, so

Ẋ0 = -α 0 X 0 + n i=1 f i l i π i ( X i π i ) - n i=1 p i π i ( X i π i )X 0 -p 0 X 2 0 = 0.
Exploiting equality (8), we deduce that

-α 0 + n i=1 f i l i π i - n i=1 p i π i X 0 -p 0 X 0 X 0 = 0.
This implies that X 0 = 0 or X 0 = X * 0 and so, by (8) we deduce that L ⊂ {0, X * }.

But the origin can not be an omega-limit point: suppose there exists a state X 0 ∈ Ω such that the solution X(t, X 0 ), issued from X 0 , tends to the origin as t tends to +∞, then by continuity of V , V (X(t, X 0 )) -→ V (0) as t -→ +∞. Since V is nonincreasing along the solutions of the system, we have

V (X 0 ) ≥ V (X(t, X 0 )) ≥ V (0), ∀t ≥ 0. ( 9
)
On the other hand, we have

V (0) -V (X) = 1 2 X * 2 0 + n i=1 n j=i k j π j α i + q i Ē X * i π i 2 - 1 2 (X 0 -X * 0 ) 2 + n i=1 n j=i k j π j α i + q i Ē X i -X * i π i 2 = 1 2 X * 2 0 + n i=1 γ i X * 2 i -(X 0 -X * 0 ) 2 - n i=1 γ i (X i -X * i ) 2 = 1 2 X 0 (2X * 0 -X 0 ) + n i=1 γ i X i (2X * i -X i ) .
where

γ i = n j=i k j π j (α i + q i Ē)π 2 i . Hence, V (0) > V (X) for all X ∈ U = {X ∈ IR n+1 + : 0 < X i < 2X * i , i = 0, . . . , n}.
This is a contradiction to (9). So, we conclude that all the solutions tend to the nontrivial equilibrium X * .

Remark 2.1 A sufficient condition to get X * 0 < µ is Ē ≥ α n i=i 0 (f i l i -p i µ) q i 0 (p 0 µ + α 0 -α α 1 i 0 -1 i=1 (f i l i -p i µ))
.

with i 0 = min{i ∈ {1, ...n} / q i = 0} In fact for Ē ≥ α n i=i 0 (f i l i -p i µ) q i 0 (p 0 µ + α 0 -α α 1 i 0 -1 i=1 (f i l i -p i µ))
. we have

(p 0 µ + α 0 ) ≥ α q i 0 Ē n i=i 0 (f i l i -p i µ) + α α 1 i 0 -1 i=1 (f i l i -p i µ)
Taking into account that π i ≤ α q i 0 Ē for all i = i 0 , ...n, and

π i ≤ α α 1 for all i = 1, ...i 0 -1, we get (p 0 µ + α 0 ) ≥ n i=1 (f i l i -p i µ)π i hence µ ≥ n i=1 f i l i π i -α 0 p 0 + n i=1 p i π i = X * 0 2.

Stabilization with a variable fishing effort

Now we suppose that the fishing effort is time varying and we propose to construct it as a state feedback control that allows to stabilize the system around its nontrivial equilibrium. Thus we want to find a smooth function u(x) in such a way that the state X * is a globally asymptotically stable equilibrium point for the closed-loop system (2) with E(t) = Ē + u(X(t)):

                   Ẋ0 = -α 0 X 0 + n i=1 f i l i X i -n i=0 p i X i X 0 Ẋ1 = αX 0 -(α 1 + q 1 Ē)X 1 -q 1 uX 1 . . . Ẋn = αX n -(α n + q n Ē)X n -q n uX n (10)
In a condensed form this system can be written:

Ẋ = F (X) + uG(X), (11)
where

F (X) =          -α 0 X 0 + n i=1 f i l i X i -n i=0 p i X i X 0 αX 0 -(α 1 + q 1 Ē)X 1 . . . αX n -(α n + q n Ē)X n          , G(X) =          0 -q 1 X 1 . . . -q n X n          .
To stabilize this system we shall use and adapt the Jurdjevic-Quinn [START_REF] Jurdjevic | Controllability and stability[END_REF] stabilization procedure:

A candidate stabilizer is u = Φ(X) = -G(X), ∇V (X) where ., . is a scalar product on IR n+1 and

∇V (X) =          ∂V ∂X 0 ∂V ∂X 1 . . . ∂V ∂Xn          .
Using the Lyapunov function introduced above,we get:

Φ(X) = n i=1 γ i q i X i (X i -X * i ). ( 12
)
Where

γ i = n j=i k j π j (α i + q i Ē)π 2 i .
The derivative of V along the trajectories of the closed loop system is:

V = F (X), ∇V (X) + u(X) G(X), ∇V (X) = F (X), ∇V (X) -Φ(X) 2 ≤ - 1 2 n i=1 k i π i (X 0 -X * 0 ) -( X i -X * i π i ) 2 -Φ(X) 2 ≤ 0
As in section 2.1, LaSalle Invariance Principle allows to conclude that X * 0 is globally asymptotically stable.

However, the function Φ is unbounded and takes positive as well as negative values, hence E = Ē + u(X) can take negative values which has no sense (E is a fishing effort). Thus, instead of using the feedback given by the formula (12), we use the following bounded function:

u(X) = 2Φ(X) 1 + Φ 2 (X) Ē . (13)
We then have |u(X)| ≤ Ē which ensures that 0 ≤ E(X) ≤ 2 Ē for all X ∈ Ω.

3 Remarks and simulations

1. The results of [START_REF] Touzeau | On the stock-recruitment relationships in fish population models[END_REF]6] ensures the attraction of the equilibrium only for

α i + q i Ē 1.3 1 1 1 1
Table1: The parameter values of a harvested fish population.

It is clear that this parameter values satisfy hypotheses (H1), (H2), 2. Our proof and the one of [START_REF] Touzeau | On the stock-recruitment relationships in fish population models[END_REF]6]

α i + q i Ē 1.3 1 1 1 1
For these parameters X * 0 = 15.2495, µ = 9.375. We have X * 0 > µ. However the simulations show the attraction of the equilibrium X * .

initial states that belong to D = n i=0 [

 i=0 a i , b i ]. Our result proves that the attraction domain is the whole positive orthant. Here we give a simulation corresponding to a harvested fish population characterized by the parameter values given in table1, which are checked on from [

(

  H3) and (H4). For these parameters we have b 0 = 50, b 1 = 40, b 2 = 32, b 3 = 17.06, b 4 = 10.9. For the initial state X init = (100, 80, 15, 20, 9) which is outside the domain D, we obtain the following curves that confirm our results.

  use the condition X * 0 < µ. But we think that this condition is not necessary as it is suggested by the following

	example:
	Stage i:	0	1	2	3	4
	p i	0.2	0	0.1	0.1	0.8
	f i		0	0.5	0.5	0.5
	l i		0	10	20	15
	m i	0.5 0.2	0.2	0.2	0.2
	q i	0	0.1 0.25 0.25 0.5
	M i	0.5 0.2	0.2	0.1 0.05
	α			0.8
	α i	1.3 0.98 0.95 0.95 0.9
	Ē			0.2


Marcin Markiewicz

Anna Przysiezna

Stephen Brierley

Tomasz Paterek

Tomasz Paterek Genuinely

Anna Przysi¦»na

Genuinely multi-point temporal quantum correlations

Due to a restrictive scenario of sequential projective measurements, multi-point temporal quantum correlations were thought to factorize into two-point ones. We show that sequential generalized measurements lead to genuinely multi-point temporal correlations, which enable us to translate typical spatial genuinely multipartite nonlocal phenomena, like Greenberger-Horne-Zeilinger paradox, into the temporal scenario.

Introduction. Temporal quantum correlations have

been widely studied since the seminal paper of Leggett and Garg [1]. They proposed Bell-like inequalities involving two-point correlations between measurement outcomes of a single observable measured at dierent instances in time on an evolving system. The Leggett-Garg inequalities hold whenever the measurement results are assumed to be well-dened before the actual measurement takes place and are not altered by the act of measurement. A related scenario was proposed by Brukner et al. [2] in which there are two time instances and one of two dierent observables is measured at each time. Under these assumptions a CHSH-like temporal inequality holds. This approach was further generalized to several temporal Bell-type inequalities involving many instances of time [3,4].

The common feature of all these inequalities is that they involve only two-point temporal correlations. It was believed that multi-point temporal quantum correlations always factor into at most two-point cases, even though a proof exists only in the case of projective measurements on qubits [2]. To the best of our knowledge, there are no constructive examples of genuinely multi-point temporal quantum correlations in a scenario of sequential measurements on a single system. A non-constructive example is given by Budroni et al. [4], who show that three-point temporal quantum correlations that arise from sequential projective measurements can violate a Bell inequality proposed in [5]. When the three parties are spatially separated such a violation can only occur using superquantum correlations what shows that sequential projective measurements can simulate super-quantum spatial correlations. However no concrete physical implementation is known. Another approach to demonstrate genuinely multi-point temporal quantum eects has been proposed by ukowski [6], however in this scenario a sequence of unitary operations rather than measurements is considered.

Here we consider a single system that is measured from time to time in order to establish probabilities for various sequences of outcomes.

We show in a constructive way that it is indeed possible to obtain genuinely multi-point temporal quantum correlations by sequential POVM measurements on a single qubit, which overcomes the no-go result of [2]. Our approach is based on the fact that every pure quantum state of nite dimension can be expressed in the form of a Matrix Product State (MPS) and therefore can be generated in a sequential manner [7]. We utilize the sequential procedure to construct a series of POVM measurements on a single qubit giving Temporal quantum correlations basic concepts and denitions. The most general temporal correlation function for a sequence of N generalized quantum measurements is dened as follows:

E(m 1 , . . . , m N ) = i1,...,i N i 1 . . . i N P (i 1 , . . . , i N |m 1 , . . . , m N ), (1)
where

P (i 1 , . . . , i N |m 1 , . . . , m N ) (2)
is the probability to observe a particular sequence of outcomes {i 1 , . . . , i N } conditioned on the settings {m 1 , . . . , m N }.

Let us now discuss the case of sequential projective measurements. Consider a single qubit prepared in an initial state described by a Bloch vector s. The qubit is measured at time instances t 1 , . . . , t N with the corresponding dichotomic observables parameterised by Bloch vectors m 1 , . . . , m N , i.e. the measurement outcomes i 1 , . . . , i N are all equal to ±1. Sequential projective measurements on a single qubit form a Markov chain [2,6], so that the probability of a sequence of outcomes is given by

P (i 1 , . . . , i N | m 1 , . . . , m N ) = P (i 1 | m 1 )P (i 2 |i 1 , m 1 , m 2 ) × P (i 3 |i 2 , m 2 , m 3 ) . . . P (i N |i N -1 , m N -1 , m N ), (3)
where P (i j+1 |i j , m j , m j+1 ) denotes the probability of measuring outcome i j+1 on a qubit which collapsed in the previous measurement to the state i j m j , i.e. P (i j+1 |i j , m j ) = 1 2 (1 + i j+1 i j m j+1 • m j ) with • being the inner product. Straightforward calculation of the temporal correlations reveals that for odd N we have:

E( m 1 , . . . , m N ) = ( m 1 • s)( m 2 • m 3 ) . . . ( m N -1 • m N ), (4)
whereas for N even:

E( m 1 , . . . , m N ) = ( m 1 • m 2 )( m 3 • m 4 ) . . . ( m N -1 • m N ). (5)
The temporal correlations between outcomes of multiple projective measurements always factor into correlations between at most two measurements. In this sense a qubit never gives rise to multi-point correlations in time.

This property of projective measurements is related to the fact that they can be treated as a preparation of an initial state of the system, which fully describes all future probabilistic predictions [2]. As we will see, in the case of generalized measurements this intuition is no longer valid.

For generalized measurements, the probability

P (i 1 , . . . , i N |m 1 , . . . , m N ) no longer possesses the
Markov property, it can only be expressed in the general form

P (i 1 , . . . , i N |m 1 , . . . , m N ) = P (i 1 |m 1 )P (i 2 |i 1 , m 1 , m 2 ) × . . . P (i N |i 1 , . . . , i N -1 , m 1 , . . . , m N ), (6)
where m k is some parametrization of measurement settings of an arbitrary POVM, dened by measurement operators M k . The conditional probabilities in the above formula are given by

P (i k |i 1 , . . . , i k-1 , m 1 , . . . , m k ) = Tr ρ k M † k M k , (7)
where the post-measurement state is dened as a recursive formula

ρ k = M k-1 ρ k-1 M † k-1 Tr M k-1 ρ k-1 M † k-1 . (8)
The correlations of a sequence of POVM measurements depend directly on the measurement operators M k , corresponding to dierent possible implementations of given

POVM elements E k = M † k M k . The
|Ψ = d i1...in=1 A [1] i1 ...A [N ] i N |i 1 ...i N . (9)
The rst and the last matrices are vectors and each matrix A n in has a maximum dimension D × D. D is so called bond dimension and is an important parameter characterising MPS state. Namely, D is the largest rank of the reduced density matrix with respect to every cut. It was

shown in [15] that the rank of the reduced density matrix is a measure of entanglement and therefore the parameter D contains information about the entanglement structure of the state.

For suciently large bond dimension any state of a Hilbert space can be represented as an MPS [15]. However in practice the most commonly used is the class of MPS with low bond dimension as it describes well low energy states of 1D systems with local interactions. In the context of quantum information theory we can also nd number of states playing an important role in information processing that are MPS with small bond dimension. For example multi-qubit GHZ states [16] and W states [17] have bond dimension D = 2.

To sequentially generate an MPS state of bond dimension D we start from a chain of N initially uncorrelated d-dimensional particles and implement one of two methods. The rst is due to Schön et al. [7] who use an ancilla system with dimension D and sequentially couple it to the d-dimensional particles performing unitary operations on each party and the ancillary system. In the last step the ancilla decouples from the system enabling us to generate a pure entangled state. The second method [18] is the one we focus on in this paper. In this method we apply unitary operations on m neighbouring particles in a sequential manner and as a result we obtain a state with bond dimension D = d m-1 . Given MPS form of a state with N matrices A i k k we can nd (N -1) unitary matrices U k by use of singular value decomposition as it is explained in [7].

Not only pure states can be generated this way we can easily generalize the method for a class of mixed states. Group of the particles can be traced out in an eect leaving us with a multipartite mixed state with the bond dimension D ρ ≤ D ψ , where D ψ describes an original state. The more mixed state we want to obtain the higher dimensional must be the cut bond. So to create n-partite mixed state with bond dimension D that has density matrix of the rank D we need to sequentially create a purication of this state that contains n-qubits and one particle of dimension D , i.e. n + log D qubits.

In particular it is instructive to consider the case of generating pure qubit states with D = 2 using two-qubit unitary gates. The scheme is depicted in Fig. 1.

Sequential POVM measurements. As described, for any MPS state with bond dimension D = 2 we can determine its sequential generation scheme. Now we can utilize this scheme to nd a sequence of measurements performed on a single qubit such that the correlations between these measurements are exactly the same as correlations measured on the MPS state. Such transition from the spatial to the temporal correlations is depicted in Fig. 1. As illustrated in Fig. 1a, at the kth step of the preparation procedure one of the particles prepared in the previous step is being entangled by the gate U k with the particle that has not been used up to now. This particle is next subject to the gate U k+1 , whereas the other particle is left untouched during the rest of the procedure. This important characteristic of the sequential preparation allows one to perform a projective measurement on the rst particle right after the gate U k is applied without disturbing the latter process of preparing the state (see Fig. 1b).

The entire protocol can equivalently be seen as a quantum channel with a single particle input and output that realises a two outcome POVM measurement (Fig. 1c).

U k |ψ ⊗ |0 = i=±1 M (k) i |ψ ⊗ V |i , ( 10
)
where V is a unitary rotation from the standard basis to the measurement basis of the k-th observer.

All the above discussion can be summarized in the following proposition being a consequence of equivalence of quantum circuits depicted in Fig. 1.

Proposition 1. For any N -partite MPS state ψ with bond dimension 2, the arbitrary spatial correlation function for local projective measurements can be simulated by a sequence of N -1 consecutive two-outcome POVM measurements followed by a projective measurement in the nal step.

The above construction can be experimentally implemented with only two qubits (Fig. 1d). Such qubit recycling has already been implemented in an experimental realization of the Shor's algorithm [START_REF] Martin-Lopez | [END_REF].

Temporal GHZ paradox. The previous literature [2,6] suggests, that in the scenario in which a single particle is measured sequentially N times, there is no possibility for genuinely N -partite quantum correlations to arise. However, Proposition 1 shows that this is not the case when we allow generalized quantum measurements at some steps. Especially interesting is the case of Npartite correlations of a GHZ-state

|GHZ = 1 √ 2 (|0 . . . 0 + |1 . . . 1 ) (11)
which give rise to the so called GHZ paradox. The tripartite version is commonly known [16], whilst the N -partite version [21] is equivalent to the modulo-4 sum problem in communication complexity theory [22]. The general form of a correlation function of a GHZ state is given by

E(θ 1 , φ 1 , . . . , θ N , φ N ) = sin(θ 1 ) • . . . • sin(θ N ) × cos (φ 1 + . . . + φ N ) (12)
where (θ k , φ k ) denote a Bloch-sphere parametrization of a projective measurement on a k-th qubit. Since to produce sequentially an N -partite GHZ state one uses a CNOT operation as a unitary entangling gate U k in eq. ( 10), it is easy to calculate the POVM measurement operators for this case:

M (k) -1 = cos(θ k /2) 0 0 e -iφ k sin(θ k /2) M (k) 1 = e iφ k sin(θ k /2) 0 0 -cos(θ k /2) (13)
They give rise to the following POVM elements:

E (k) -1 = cos(θ k /2) 2 0 0 sin(θ k /2) 2 E (k) 1 = sin(θ k /2) 2 0 0 cos(θ k /2) 2 (14)
The above implementation of a POVM allows us to obtain the general correlation function for local measurements on a GHZ state by performing sequential POVM measurements on a single qubit followed by a projective one at the last step. Thus we can in fact implement the GHZ paradox in time.

Note that in the above construction, the POVM ele-

(I)(t(I)+1)) 2 E(I) ≤ 2 n-1 (15)
where I = (i 1 = {0, 1}, . . . , i n = {0, 1}) denotes the set of measurement settings for each observer, E(I) denotes the correlation function for the measurements and t(I) is the number of occurrence of index 1 in the sequence I. GHZ state [23].

We can treat these inequalities as temporal Bell inequalities in the scenario proposed by Brukner et al. [2].

A single system is measured sequentially at n time instances, and at each stage two possible settings are chosen. A violation of the inequalities given in Eq. (15) indicates that the temporal correlation function is not bi-separable. Due to properties (4) and ( 5) every correlation function arising from a sequence of projective measurements on a qubit is factorisable, therefore a sequence of such measurements cannot violate the above inequalities. However, using our construction, we can violate it up to √ 2•2 n-1 using a sequence of POVM measurements given by ( 13) and ( 14), which reconstructs the correlation function (12) of the GHZ state.

We leave it an open question whether there exist a sequence of POVM measurements on qubits which vio- isting experiments would allow our construction to be implemented in the laboratory, likely giving a chance of observing the temporal GHZ paradox for higher number of measurements than the number of particles for which this paradox has been observed in space. In analogy to the experiment presented in [START_REF] Martin-Lopez | [END_REF] our construction can be used to implement quantum algorithms which require genuine n-partite correlations in a temporal scenario.

  rise to temporal correlations identical to the spatial correlations of the corresponding MPS state. In this way we obtain a temporal version of the Greenberger-Horne-Zeilinger paradox and temporal Bell inequalities violation of which detects genuinely multi-point temporal correlations.

FIG. 1 .

 1 FIG. 1. Sequential generation of a multipartite MPS state with bond dimension two, and its temporal counterpart. a) Twoparticle unitary gates U k (grey rectangles) are sequentially performed on qubits (red circles). After such generation qubits are in the entangled state Ψ (big red shape). In the end projective measurements (blue shapes) can be done on qubits. Dashed lines indicate how unitary gates are related to MPS matrices; b) Measurements can be shifted in time; c) the circuit can be rearranged into a sequence of quantum channels performing POVM measurements; d) We can reduce the required resources to two qubits. After each measurement, one of the qubits is reset and recycled into the remaining protocol.

  The corresponding measurement operators M (k) i can be determined from the relation[START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]

ments ( 14 )

 14 do not depend on the phase φ, which enters the non-classical part of the correlation function of the GHZ state, whereas the measurement operators (13) directly depend on this parameter. This shows that the information about the phase is somehow encoded in the way the state collapses at each stage of the sequence of measurements and not in the statistics of individual measurements. This illustrates that knowledge about POVM elements alone is not sucient to determine temporal correlations. Multipartite temporal Bell inequalities. The possibility of generating genuinely multi-point temporal correlations enables us to consider multipoint temporal Bell inequalities, a violation of which detects truly n-fold temporal quantum correlations. Let us consider n-partite Bell inequalities with two settings per observer proposed by Seevinck and Svetlichny [23]:

  t

  These inequalities are fullled by any correlations arising from conditional probabilities of outcomes which are biseparable. Using quantum mechanical resources one can obtain the maximal violation √ 2 • 2 n-1 with an n-partite

  late temporal Svetlichny inequalities up to the algebraic bound. Conclusions. We have shown the possibility of obtaining genuinely multi-point temporal quantum correlations in a scenario of sequential generalized quantum measurements on a single particle. In this way we recover in a temporal experiment an arbitrary correlation function of any multipartite quantum state that has a MPS representation with bond dimension 2. Our result sheds a new light on a relation between spatial and temporal quantum correlations, showing that the similarities between these scenarios are much deeper than was presented in previous works. Moreover, a simple modication of ex-

Acknowledgements. We would like to thank Marek ukowski and Marcin Pawªowski for helpful discusions. MM and AP are supported by the International PhD Project Physics of future quantum-based information technologies grant MPD/2009-3/4 from Foundation for Polish Science and by the University of Gda«sk grant 538-5400-B169-13. This work is supported by the National Research Foundation, the Ministry of Education of Singapore, start-up grant of the Nanyang Technological University, and NCN Grant No. 2012/05/E/ST2/02352.