Dominicus Paulus Borgers

Romain Postoyan
email: romain.postoyan@univ-lorraine.fr

Adolfo Anta
email: aantam@gmail.com

Paulo Tabuada
email: tabuada@ee.ucla.edu

Dragan Nešić

Maurice Heemels
email: m.heemels@tue.nl

D P Borgers
email: d.p.borgers@tue.nl

W P M H Heemels

Periodic event-triggered control of nonlinear systems using overapproximation techniques

Keywords: Event-triggered control, Digital implementation, Polytopic overapproximation, Nonlinear systems

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

Introduction

In digital control applications, the control task consists of sampling the outputs of the plant and computing and implementing new actuator signals. This procedure is typically executed in a time-triggered fashion, which may lead to a waste of communication and energy resources, as the execution of the control task is done irrespective of whether there actually is a need for a control update or not. To mitigate the unnecessary waste of resources, various event-triggered control (ETC) strategies have been proposed in the recent literature, see, e.g., [START_REF] Cassandras | The event-driven paradigm for control, communication and optimization[END_REF][START_REF] Dolk | Outputbased and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF][START_REF] Girard | Dynamic triggering mechanisms for eventtriggered control[END_REF][START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF][START_REF] Henningsson | Sporadic event-based control of first-order linear stochastic systems[END_REF][START_REF] Lunze | A state-feedback approach to event-based control[END_REF][START_REF] Miskowicz | Send-on-delta concept: An event-based data reporting strategy[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF][START_REF] Tallapragada | Decentralized eventtriggering for control of nonlinear systems[END_REF]. In ETC, the control task is executed after the occurrence of an event, generated by some well-designed triggering condition, rather than after a fixed period of time, as in conventional periodic sampled-data control. In this way, ETC is capable of significantly reducing the number of control task executions, while retaining a satisfactory closed-loop performance.

A main implementation issue of ETC (for which we will use the term continuous event-triggered control (CETC) from here on) is that the event function has to be monitored continuously, which is difficult to realize on digital platforms. A solution to this problem is periodic event-triggered control (PETC), in which the event function is only checked periodically at fixed equidistant time instances, thereby enabling (easier) implementation on a digital platform. Note that PETC differs from standard periodic sampled-data control, as in PETC the event times are only a (specific) subset of the sampling times and can be aperiodic. Of course, event-triggered control schemes for discrete-time systems (e.g., [START_REF] Cogill | Event-based control using quadratic approximate value functions[END_REF][START_REF] Eqtami | Event-triggered control for discrete-time systems[END_REF][START_REF] Heemels | Modelbased periodic event-triggered control for linear systems[END_REF][START_REF] Li | Weakly coupled event triggered output feedback system in wireless networked control systems[END_REF][START_REF] Molin | On the optimality of certainty equivalence for event-triggered control systems[END_REF][START_REF] Yook | Trading computation for bandwidth: reducing communication in distributed control systems using state estimators[END_REF]) can also be interpreted as PETC schemes, but these do not take into account the inter-sample behavior. In the past few years, various PETC strategies have been proposed, see, e.g., [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF][START_REF] Heemels | L 2 -gain analysis for a class of hybrid systems with applications to reset and event-triggered control: A lifting approach[END_REF][START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF][START_REF] Henningsson | Sporadic event-based control of first-order linear stochastic systems[END_REF][START_REF] Postoyan | Periodic event-triggered control for nonlinear systems[END_REF]. However, to the best of the authors' knowledge, there are hardly any design methods for PETC for nonlinear continuous-time systems. Two exceptions are the works in [START_REF] Sanfelice | Lyapunov analysis of sampleand-hold hybrid feedbacks[END_REF][START_REF] Wang | Stabilization of nonlinear systems using statefeedback periodic event-triggered controllers[END_REF]. In [START_REF] Sanfelice | Lyapunov analysis of sampleand-hold hybrid feedbacks[END_REF], the sample-and-hold implementation of general hybrid controllers for nonlinear systems is analyzed, which covers the PETC implementation of a nonlinear CETC system as a subcase. These results ensure that, under general conditions, if a compact set is uniformly globally asymptotically stable (UGAS) for the original CETC system, then this property is semiglobally and practically preserved for the emulated PETC system by taking the sampling period sufficiently small. In the recent work [START_REF] Wang | Stabilization of nonlinear systems using statefeedback periodic event-triggered controllers[END_REF], an approach has been proposed for the design of PETC state-feedback controllers to stabilize nonlinear systems, which ensures uniform global asymptotic properties and provides an explicit bound on the sampling period.

In this paper, we present a method to transform a general nonlinear CETC system into a PETC system which preserves the control performance guarantees of the given CETC system. Our method consists of two steps. First, we upper bound the evolution of the event function of the given CETC system between two successive sampling instants by a linear time-invariant (LTI) system. Based on this LTI system, we can formulate a redesigned event function for the PETC implementation which would involve checking an infinite number of conditions at every sample time. To overcome this issue, we use convex techniques to overapproximate the evolution of the LTI system over a sampling period, and end up with a redesigned event function which is implementable in practice.

In contrast to [START_REF] Sanfelice | Lyapunov analysis of sampleand-hold hybrid feedbacks[END_REF], our method provides an explicit bound on the sampling period, it fully preserves the control performance guarantees of the given CETC system, and is not limited to stability of a compact set a priori. Compared to [START_REF] Wang | Stabilization of nonlinear systems using statefeedback periodic event-triggered controllers[END_REF], we do not focus on stabilization and we can cope with a larger class of triggering conditions. Preliminary results have been presented in [START_REF] Postoyan | Periodic event-triggered control for nonlinear systems[END_REF], in which we were only able to approximately preserve the control performance guarantees of the given CETC system. In addition, the new results presented here are based on less stringent conditions compared to [START_REF] Postoyan | Periodic event-triggered control for nonlinear systems[END_REF] (see Remark 3 below for more details).

Nomenclature. Let R = (-∞, ∞), R 0 = [0, ∞), N = {1, 2, . . .} and N 0 = {0, 1, 2, . . .}.
Given N ∈ N, we denote the set {1, 2, . . . , N } by N . For a vector x ∈ R n , we denote by x :=

√

x x its 2-norm, and for a matrix A ∈ R n×m , we denote by A := λ max (A A) its induced 2-norm. For a signal w : R 0 → R n , we denote the right limit at time t ∈ R 0 by w(t + ) = lim s↓t w(s), when it exists. The solution z of a time-invariant dynamical system at time t ∈ R 0 starting with the initial condition z(0) = z 0 will be denoted by z(t, z 0 ) or simply by z(t) when the initial state is clear from the context. The notation x stands for the largest integer smaller than or equal to x ∈ R.

Problem statement

We consider a nonlinear plant of the form

ẋ(t) = f (x(t), u(t)), (1)
where x(t) ∈ R nx is the state and u(t) ∈ R nu is the control input at time t ∈ R 0 . We assume that we have designed a continuous event-triggered state-feedback controller for plant (1), given by

x(t) = x(t k ), for t ∈ (t k , t k+1 ] (2a) u(t) = k(x(t)) (2b) t 0 = 0 t k+1 = inf{t > t k | Γ (x(t), x(t), χ(t)) > 0}, (2c)
where the function k(x) defines the feedback law, x is the state information available to the controller, and χ ∈ R nχ is used to capture other relevant variables such as timers, counters, or possibly even the state of an auxiliary dynamical system [START_REF] Dolk | Outputbased and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF][START_REF] Girard | Dynamic triggering mechanisms for eventtriggered control[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF]. The event function Γ is designed such that some desired control performance (e.g., asymptotic stability with a certain decay rate) is achieved as long as it remains non-positive along the system's trajectories.

Writing the triggering law as in (2c) allows us to consider various event-triggers considered previously in the literature, which we illustrate by the following two examples.

In [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], the condition Γ(x, x) = γ( x-x )-σα( x ) 0 (for specific functions γ, α and σ ∈ (0, 1)) ensures that a Lyapunov function V has a guaranteed decay rate (1 -σ)α( x ) along the solutions to system (2) (which guarantees global asymptotic stability of the system).

In [START_REF] Dolk | Outputbased and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF], we have that χ = (τ, κ, η) (where τ is a timer, κ a counter, and η the state of an auxiliary dynamical system), and that the condition Γ(x, x, χ) = -η 0 ensures that the system is UGAS with a guaranteed decay rate. Another example is provided in Section 4.

Let z = (x, x, χ) ∈ R nz with n z = 2n x + n χ . We model the closed-loop system (1)-(2) (and possibly auxiliary dynamics for χ) as an impulsive system like in [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF], which gives

ż = g(z), for t ∈ (t k , t k+1 ] (3a) z(t + k ) = b(z(t k )) (3b) t 0 = 0 t k+1 = inf{t > t k | Γ (z(t)) > 0}, (3c)
for k ∈ N 0 , and appropriate g : R nz → R nz and b : R nz → R nz . In case n χ = 0, we have that

g(z) = f (x, k(x)) 0 and b(z) = x x .
For the definition of the functions g and b in case n χ = 0 we refer to Section 4 for an example.

Solutions to (3) are interpreted as follows. In between the event times t k , k ∈ N, determined by (3c), the system evolves according to the differential equation (3a), where z(t + k ) given by the update (3b) denotes the starting point for the solution to (3a) in the interval (t k , t k+1 ], k ∈ N. Hence, the solutions we consider are left-continuous signals. Note that t 0 = 0, and hence, we start with an update according to (3b).

Remark 1 The analysis presented in this paper is based on system (3). Therefore, our design applies to any CETC configuration that can be written in the format of (3), including the case where the control input u in (1) is generated by a dynamic controller. The states of the controller would then be incorporated in the vector x and we would obtain a model of the form (3). Similarly, the case in which the controller is not implemented using zero-order-hold functions can be considered as long as the problem can be modeled by [START_REF] Cassandras | The event-driven paradigm for control, communication and optimization[END_REF]. For instance, when using the model-based technique of [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], x would be equal to x s in [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], which is the model-based estimate of x.

In order to transform the CETC system (3) into a PETC system, we require the following three assumptions.

Assumption 1 There exists a nonempty set Ω ⊆ R nz such that for all z 0 ∈ Ω, all corresponding solutions z(t, z 0 ) to (3) are defined for all time t ∈ R 0 and satisfy z(t, z 0 ) ∈ Ω for all t ∈ R 0 .

Assumption 1 holds for most CETC designs proposed in literature, including [START_REF] Abdelrahim | Stabilization of nonlinear systems event-triggered output feedback controllers[END_REF][START_REF] Dolk | Outputbased and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF][START_REF] Girard | Dynamic triggering mechanisms for eventtriggered control[END_REF][START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF][START_REF] Henningsson | Sporadic event-based control of first-order linear stochastic systems[END_REF][START_REF] Li | Weakly coupled event triggered output feedback system in wireless networked control systems[END_REF][START_REF] Lunze | A state-feedback approach to event-based control[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF][START_REF] Tallapragada | Decentralized eventtriggering for control of nonlinear systems[END_REF]. It implies that the set Ω is forward invariant for the closedloop CETC system (3) (i.e., that all solutions starting in Ω remain in Ω for all t ∈ R 0 ), and rules out finite escape times and Zeno-behaviour (an infinite number of events in a finite time interval) that would prevent global existence of all solutions starting in Ω. Note that we do not exclude Zeno behavior or finite escape times for solutions starting outside Ω.

Because the system starts with an update according to (3b) (as t 0 = 0), Assumption 1 also implies that the function b satisfies b(Ω) ⊆ Ω. Hence, solutions do not leave the set Ω, even if we allow the generation of events when Γ(z(t)) < 0, which will typically be the case in the PETC implementation that we envision.

Assumption 2 There exists T > 0 such that the jumps induced by Γ on the CETC system (3) are spaced by at least T units of time, i.e., for all z 0 ∈ Ω and all solutions

z(•, b(z 0 )) to (3) it holds that inf{t > 0 | Γ(z(t, b(z 0 ))) > 0} T. (4)
Assumption 2 implies that there exists a uniform positive minimum inter-event time T for the CETC system in the set Ω. Sufficient conditions for this assumption to hold are given in [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF], and most available event-triggering schemes in the literature either provide a positive lower bound on T (e.g., [START_REF] Girard | Dynamic triggering mechanisms for eventtriggered control[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]), or enforce a positive lower bound by design (e.g., [START_REF] Abdelrahim | Stabilization of nonlinear systems event-triggered output feedback controllers[END_REF][START_REF] Dolk | Outputbased and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF][START_REF] Tallapragada | Decentralized eventtriggering for control of nonlinear systems[END_REF]).

Assumption 3 There exists p ∈ N such that Γ is ptimes continuously differentiable on Ω, g is (p -1)-times continuously differentiable on Ω, and there exist real numbers c, ς j , j ∈ {0, 1, . . . , p -1}, satisfying

L p g Γ(z) p-1 j=0 ς j L j g Γ(z) + c, (5)
for all z ∈ Ω, where L j g Γ is the j-th Lie derivative of Γ along the flow dynamics ż = g(z), with L 0 g Γ = Γ, (L g Γ)(z) = ∂Γ ∂z g(z) and

L j g Γ = L g (L j-1 g Γ) for j ∈ N.
Assumption 3 is a condition on the evolution of Γ along the solution to (3a), i.e., along the solutions to the system between two successive updates. It plays a crucial role in our design as it allows us to upper-bound the evolution of Γ by the solution to a linear system, as explained later in Section 3.2. A similar assumption is made in the context of self-triggered control in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF]. In case Ω is compact, inequality (5) can always be satisfied when g and Γ are (p -1)-times and p-times continuously differentiable, respectively, as it suffices to take c = max z∈Ω L p g Γ(z) and ς j = 0 for j ∈ {0, 1, . . . , p -1} to ensure [START_REF] Dolk | Outputbased and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF]. However, this particular choice may be conservative and tighter estimates of L p g Γ(z) can be obtained by using the other terms L j g Γ(z) in the right-hand side of (5). The parameter p may be increased to further reduce the conservatism of the upper bounds on L p g Γ(z) in ( 5) at the price of a higher computational complexity.

In this paper, we describe a method for designing PETC strategies for nonlinear system (1), given that a CETC scheme (2) has already been designed. We do this by redesigning the event function Γ to a new event function Γ, while we keep using the pre-designed feedback law. The envisioned closed-loop PETC system is described by

ż = g(z), for t ∈ ( tk , tk+1 ] (6a) z( t+ k ) = b(z( tk )) (6b) t0 = 0 tk+1 = min{t > tk | Γ (z(t)) 0, t = nh, n ∈ N}, (6c)
where h > 0 is the sampling period at which the triggering condition Γ is evaluated. Our aim is to provide tools for the design of h and Γ to guarantee that Γ remains non-positive along all solutions to (6) starting in Ω, such that the stability and performance guarantees of the original CETC system (3) are preserved. To do so, we will use the following design requirement on h and Γ.

Design Requirement 1 For all z 0 ∈ Ω such that Γ(z 0 ) < 0 it holds that Γ(z(t, z 0 )) < 0 for all t ∈ [0, h] for all solutions z(•, z 0 ) to ż = g(z).

Remark 2 In general it is not possible to satisfy Design Requirement 1 by choosing Γ = Γ, see for example [START_REF] Postoyan | Periodic event-triggered control for nonlinear systems[END_REF].

We will also demonstrate this in Section 4.

Remark 3 Based on Assumptions 1, 2, and 3, we present in the next section our method to transform the CETC system (3) into a PETC system [START_REF] Eqtami | Event-triggered control for discrete-time systems[END_REF]. Compared to the conference version of this work [START_REF] Postoyan | Periodic event-triggered control for nonlinear systems[END_REF], we provide the following improvements:

(1) We guarantee preservation of the control performance of the original CETC system (which we were not able to in [START_REF] Postoyan | Periodic event-triggered control for nonlinear systems[END_REF]), see Theorem 8. (2) We do not require that the set Ω is compact, see Assumption 1. As a result, we can now also apply our PETC design to CETC systems which have additional variables in the event-generator that are not forward invariant with regard to any compact set, such as timers or counters (see, e.g., [START_REF] Abdelrahim | Stabilization of nonlinear systems event-triggered output feedback controllers[END_REF][START_REF] Dolk | Outputbased and decentralized dynamic event-triggered control with guaranteed Lp-gain performance and Zeno-freeness[END_REF][START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF][START_REF] Tallapragada | Decentralized eventtriggering for control of nonlinear systems[END_REF]). (3) We do not require that all solutions z(t, z(t s j )) to ż = g(z) lie inside Ω for all t ∈ [0, h]. Consequently, we do not require [21, Assumption 3], which basically states that Ω is a forward invariant set for the system ż = g(z) (without resets), which would mean that the plant already satisfies the desired control performance in open-loop.

These observations reveal a significant relaxation with respect to [START_REF] Postoyan | Periodic event-triggered control for nonlinear systems[END_REF].

Note that items (1) and (2) also support our claims with respect to [START_REF] Sanfelice | Lyapunov analysis of sampleand-hold hybrid feedbacks[END_REF].

3 Main results

Sampling period selection

Under the PETC strategy, the input can be updated only when the triggering condition is evaluated, that is, every h units of time. Hence, an event should be triggered at a sampling time t = nh, n ∈ N, before Γ becomes positive, and thus it is necessary that the sampling interval h is less than the minimum inter-event time T of the CETC system (which exists in view of Assumption 2).

In this way, after a jump, we know that Γ will remain non-positive at least until the next sampling instant, and that Design Requirement 1 may be satisfied. Therefore, we select h such that

0 < h T (7)
with T as in (4).

Remark 4

We do not need to verify the triggering condition for the next T h sampling instants following a control input update due to Assumption 2, which allows to reduce the computational load of the event-trigger.

As we aim at guaranteeing that Γ remains non-positive along the solutions in Ω of the PETC system, we would like to verify at each sample time t s j = jh, j ∈ N 0 , whether the condition Γ(z(t)) > 0 may be satisfied for t ∈ [t s j , t s j+1 ] (as then we will have to trigger an event at t s j ). To do this, we would need to analyze the evolution of the triggering function Γ along the solutions to ż = g(z), which is difficult to do when g(z) is a nonlinear function.

We first propose to upper bound the evolution of Γ on flows by a linear system.

Analysis of the evolution of Γ(z) on flows

To upper bound the evolution of Γ by a linear system, we resort to similar techniques as in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF], which rely on Assumption 3. This assumption allows to bound the evolution of Γ by a linear differential equation as stated in the lemma below, which is a variation of [2, Lemma V.2].

The proof directly follows from the comparison lemma ([14, Lemma 3.4]).

Lemma 5 Consider any solution z(•, z 0 ) to ż = g(z) with initial state z 0 ∈ Ω, and define

t * := inf{τ ∈ R 0 | z(τ, z 0 ) / ∈ Ω}. If Assumption 3 holds, then Γ(z(t, z 0 )) y 1 (t, µ p (z 0 )) for all t ∈ [0, t * ],
where y 1 (t, µ p (z 0 )) is the first component of the solution to the linear differential equation

     ẏj = y j+1 , j ∈ {1, 2, . . . , p -1} ẏp = p-1 j=0 ς j y j+1 + y p+1 ẏp+1 = 0 (8)
with initial condition y(0) = µ p (z 0 ) given by

µ p (z 0 ) = (Γ(z 0 ), L g Γ(z 0 ), . . . , L p-1 g Γ(z 0 ), c). (9)
As formalized in the next theorem, for each initial state z 0 ∈ Ω, we can now check if Γ remains non-positive for the complete sampling interval [0, h], by checking the solution of the linear system (8) with initial condition [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF].

Theorem 6 Suppose Assumptions 1 and 3 hold and that z 0 ∈ Ω, and consider the solution y(•, µ p (z 0 )) to (8) with initial condition [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF]. If y 1 (t, µ p (z 0 )) < 0 for all t ∈ [0, h], then for all solutions z(•, z 0 ) to ż = g(z) with initial state z 0 , it holds that Γ(z(t, z 0 )) < 0 for all t ∈ [0, h].

PROOF. First, note that Assumption 1 is an invariance property of system (3), while here we consider the system ż = g(z), which is system (3a) (without the discrete dynamics (3b) and (3c)). These are therefore two different dynamical systems, and the solutions to ż = g(z), may leave Ω, while those of (3) will not because of the well-designed discrete dynamics (jumps) in (3b) and (3c).

Consider any solution z(•, z 0 ) to ż = g(z) with initial state z 0 ∈ Ω, and suppose that y 1 (t, µ p (z 0 )) < 0 for all t ∈ [0, h].

Case 1: z(t, z 0 ) ∈ Ω for all t ∈ [0, h].
From Lemma 5, it directly follows that Γ(z(t, z 0 )) y 1 (t, µ p (z 0 )) < 0 for all t ∈ [0, h]. 0}. For all t ∈ [0, t 1 ), it holds that Γ (z(t, z 0 )) < 0, and thus in this interval the solutions z(t, z 0 ) to ż = g(z) coincide with the solutions to the CETC system (3) with initial condition z(0) = z 0 . As Ω is forward invariant for CETC system (3), it follows that z(t, z 0 ) ∈ Ω for all t ∈ [0, t 1 ), which leads to the observation that t 1 t * h. Furthermore, Γ(z(t 1 , z 0 )) = 0 as Γ is continuous in view of Assumption 3. Finally, since according to Lemma 5, y 1 (t, µ p (z 0 )) Γ(z(t, z 0 )) for all t ∈ [0, t * ) we will have that y 1 (t 1 , µ p (z 0 )) 0, which is in contradiction with the assumptions of the theorem. As a result, Case 2 cannot occur.

2

Remark 7

The self-triggering formulas in [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] tend to provide accurate estimates of the time instants when Γ becomes positive provided the bound (5) is tight, which may be difficult to achieve in practice. In the proposed PETC approach, the evolution of Γ is investigated over shorter horizons than in STC, namely at time-intervals of length h. Hence, the bound in (5) does not necessarily need to be accurate to provide satisfactory results, as we expect these estimates to be tighter whenever times are shorter based on previous experience [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF].

Design of Γ

The analytic expression of y 1 (t, µ p (z 0 )) is given by

y 1 (t, µ p (z 0 )) = C p e Apt µ p (z 0 ) (10)
with

A p =       0 1 • • • 0 0 . . . . . . . . . 0 0 • • • 1 0 ς 0 ς 1 • • • ς p-1 1 0 0 • • • 0 0       and C p =     1 0 . . . 0     .
At each sampling instant t s j , j ∈ N 0 , the current state z(t s j ) is measured and a transmission should occur if y 1 (t, µ p (z(t s j ))) 0 for some t ∈ [0, h], as then Γ(z(t, z(t s j ))) may be positive for some t ∈ [0, h]. Otherwise, according to Theorem 6 a transmission is not necessary to satisfy Design Requirement 1. However, verifying whether y 1 (t, µ p (z(t s j ))) < 0 for all t ∈ [0, h], involves an infinite number of conditions and is computationally infeasible. In [START_REF] Postoyan | Periodic event-triggered control for nonlinear systems[END_REF], this problem was (approximately) solved by evaluating y 1 (t, µ p (z(t s j ))) only at a finite number of points in the interval [0, h]. The price paid for this solution is that the non-positivity of Γ(z(t)) along the solutions of ( 6) can no longer be ensured, but only approximated.

The solution we propose here satisfies Design Requirement 1, and thus guarantees non-positivity of Γ(z(t)) for all t ∈ R 0 , thereby preserving the stability and performance guarantees of the CETC system. Our method uses convex techniques to overapproximate the set {y 1 (t, µ p (z(t s j ))) | t ∈ [0, h]} in a computationally tractable manner.

To find a convex overapproximation of (10), we first define

Φ = C p e Apt | t ∈ [0, h] . (11)
By using overapproximation techniques as described in [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF], the set Φ of matrices can be embedded as

Φ ⊆ N i=1 α i (F i + G i ∆H i ) α ∈ A, ∆ ∈ ∆ , (12)
in which

F i ∈ R 1×p , G i ∈ R 1×q , H i ∈ R q×p
are suitably constructed matrices, N is the number of vertices in the polytopic overapproximation (which can be freely selected), ∆ is a specific set of structured matrices in R q×q with norm bound ∆ 1, α := (α 1 , . . . α N ), and

A = α ∈ R N 0 N i=1 α i = 1 . (13)
In [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF], overapproximations as in [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF] with ∆ absent (G i = 0 and H i = 0) are also given.

As y 1 (t, µ p (z 0 )) ∈ Φµ p (z 0 ) for t ∈ [0, h] and ( 12), it follows that y 1 (t,

µ p (z 0 )) < 0 for all t ∈ [0, h] if max N i=1 α i (F i + G i ∆H i )µ p (z 0 ) α ∈ A, ∆ ∈ ∆ < 0,
which holds when

F i µ p (z 0 ) + G i H i µ p (z 0 ) < 0 for all i ∈ N , (14)
where we used that ∆ 1. Consequently, we can choose Γ in (6c) as

Γ(z) := max i∈ N {F i µ p (z) + G i H i µ p (z) } (15)
to obtain the guarantee Γ(z(t, z 0 )) 0 for all t ∈ R >0 , z 0 ∈ Ω, where z is the corresponding solution to (6), as formalized in the next theorem, whose proof directly follows from Theorem 6 and the developments above.

Theorem 8 Suppose Assumptions 1, 2, and 3 hold and consider the PETC system (6). If h satisfies (7) and Γ is defined by [START_REF] Li | Weakly coupled event triggered output feedback system in wireless networked control systems[END_REF], then the set Ω is forward invariant for the system (6), and for any solution z to (6) with initial state z 0 ∈ Ω it holds that Γ(z(t)) 0 for all t ∈ R >0 .

PROOF. Define z P (t, z 0 ) as the solution to the PETC system (6), and z C (t, z 0 ) as the solution to the CETC system (3), both starting in z 0 ∈ Ω. Hence, z P (t 0 , z 0 ) = z C (t 0 , z 0 ) = z 0 . At t = 0, both (3) and ( 6) start with an update. Hence, z P (t + 0 , z 0 ) = z C (t + 0 , z 0 ) = b(z 0 ). Next, due to Assumption 2 and h satisfying [START_REF] Girard | Dynamic triggering mechanisms for eventtriggered control[END_REF], it follows that z P (t, z 0 ) = z C (t, z 0 ) for all t ∈ [0, h]. At t = h, the PETC system (6) may or may not generate an event. In case an event is generated by the PETC system (6), we have that t1 = h < T , and hence that t1 < t 1 . In case an event is not generated by the PETC system (6) at t = h, it still follows that t1 t 1 , as Γ defined by (15) satisfies Design Requirement 1. The above proves that z P (t, z 0 ) = z C (t, z 0 ) for all t ∈ [0, t1 ], and hence that Γ(z P (t, z 0 )) 0 for all t ∈ (0, t1 ] as the solution to (3) satisfies Γ(z C (t, z 0 )) 0 for all t ∈ R >0 .

As b(Ω) ⊆ Ω, the same reasoning can be used to show that Γ(z P (t, z 0 )) 0 for all t ∈ ( t1 , t2 ], and by induction, that Γ(z P (t, z 0 )) 0 for each interval ( tk , tk+1 ], k ∈ N.

The proof is completed by observing that lim k→∞ t k = ∞.

2

Note that the redesigned event function Γ may become positive in between sampling instants along solutions to [START_REF] Eqtami | Event-triggered control for discrete-time systems[END_REF]. However, when Γ(z(t s j )) < 0 at sample time t s j , j ∈ N 0 , our design of Γ given by [START_REF] Li | Weakly coupled event triggered output feedback system in wireless networked control systems[END_REF] ensures that Γ(z(t)) < 0 will be satisfied for all t ∈ [t s j , t s j+1 ] = [t s j , t s j + h]. Alternatively, when Γ(z(t s j )) 0, then our periodic eventgenerator (6c) triggers an event, after which Assumption 2 ensures that Γ(z(t)) 0 will be satisfied for all t ∈ [t s j , t s j + T ]. Hence, by selecting h < T , our proposed periodic event-generator ensures that Γ(z(t)) 0 for all t ∈ R >0 , and thus preserves the control performance guarantees of the original CETC system (3). Remark 9 When the numerical complexity of calculating µ p (z) is O(φ(p)), for some function φ, then evaluating the event function [START_REF] Li | Weakly coupled event triggered output feedback system in wireless networked control systems[END_REF] has numerical complexity O(N qp + φ(p)) (or O(N p + φ(p)) when ∆ is absent). Thus, from a computational point of view, it makes sense to choose p and N small. On the other hand, choosing p and N large usually increases the accuracy of the overapproximation of the evolution of Γ along the solutions to ż = g(z) in view of [START_REF] Anta | Exploiting isochrony in self-triggered control[END_REF] and [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF], which leads to larger inter-event times. As such, there is a trade-off between computation and communication, as is also apparent from the example in Section 4 below.

Remark 10

The plant (1) can be extended to include process disturbances w (i.e., ẋ(t) = f (x(t), u(t), w(t))) as long as the continuous event-triggering law (2c) also satisfies Assumptions 1 and 2. In order to satisfy Assumption 3 we then either require that w does not show up in L j g Γ(z) in (5) for j ∈ {1, 2, . . . , p} (which might be the case when Γ(z) does not depend on the complete plant state x), or we need to know (estimates of ) bounds on w and of its time-derivatives which show up in (5).

Illustrative example

Consider the following example inspired by [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], in which the plant is given by

ẋ(t) = x 2 (t) -x 3 (t) + u(t), (16)
and the controller by

u(t) = -2x(t), (17)
where x is the sampled version of x as in (2a). We design a continuous event-triggering condition as in [START_REF] Abdelrahim | Stabilization of nonlinear systems event-triggered output feedback controllers[END_REF]. The idea is to wait a fixed amount of time T > 0 before checking a state-dependent criterion of the same form as in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. We introduce an extra variable χ ∈ R 0 to keep track of the time between two events. Hence, χ evolves according to χ(t) = 1 for t ∈ [t k , t k+1 ) and χ(t + k ) = 0. The function Γ in (2) is given in this case by Γ (x, x, χ) = min χ -T, γ 2 |x -x| 2 -δ(x) , [START_REF] Molin | On the optimality of certainty equivalence for event-triggered control systems[END_REF] with parameter γ ∈ R >0 and δ a positive definite function to be designed. Hence, the CETC system (3) is given by z

= [x, x, χ] , g(z) = [x 2 -x 3 -2x, 0, 1] , b(z) = [x, x, 0
] , and Γ as in [START_REF] Molin | On the optimality of certainty equivalence for event-triggered control systems[END_REF]. Note that this CETC system satisfies Assumption 2 by design for any set Ω.

To construct Γ, we have verified that the conditions required in [START_REF] Abdelrahim | Stabilization of nonlinear systems event-triggered output feedback controllers[END_REF] hold, by following similar lines as in [19, Example 1]. We have taken the Lyapunov function R of the emulated PETC implementation with h = and Γ = Γ. Clearly, this PETC implementation does not ensure that Γ(z(t)) 0 for all t ∈ R >0 , and thus does not preserve the control performance guarantees of the CETC system.

(z) = V (x) + max{0, λζ(χ)W 2 (x -x)}, with V (x) = σ 2 α 2 x 2 + β 4 x 4 , W (x -x) = |x -x|, ζ : R 0 → R the solution to ζ = -2Lζ -λ(ζ 2 + 1), ζ(0) = θ -1 , (19)
To overcome this issue, we construct Γ as in [START_REF] Li | Weakly coupled event triggered output feedback system in wireless networked control systems[END_REF]. For this purpose we need Γ to be p-times continuously differentiable, for some p ∈ N, which is currently not the case in view of [START_REF] Molin | On the optimality of certainty equivalence for event-triggered control systems[END_REF]. Still, we can apply the procedure of Section 3.2 for Γ 1 = χ -T and Γ 2 = γ 2 |x -x| 2 -δ(x) separately. The case of Γ 1 is trivial as we know that it is violated after T units of time. We proceed by constructing Γ2 for Γ 2 . We have verified numerically using SOSTOOLS [START_REF] Papachristodoulou | SOSTOOLS: Sum of squares optimization toolbox for MATLAB[END_REF] that for Γ 2 , Assumption 3 holds in the forward invariant set

Ω := {z | R(z) 0.15}. ( 20
)
The obtained values of c and ς i , i ∈ {0, 1, . . . , p -1}, for different values of p are reported in Table 1. Finally, we design the PETC strategy by following the procedure in Section 3 and by using the gridding and norm bounding (GNB) overapproximation technique as described in [12, Section III], with N equally distanced grid points.

A simulation of this redesigned PETC implementation with p = 3 and N = 2 is shown in Figure 4 (again with z(0) = z 0 and h = 0.005) . Clearly, the redesigned PETC implementation does ensure that Γ(z(t)) 0 for all t ∈ R >0 , and the control performance of the CETC system is preserved.

In Table 2, the average inter-event time τ avg for the first 10 events of the proposed PETC implementation is shown for different parameter settings. The average inter-event time for the first 10 events of the original CETC system is τ avg = 0.0757. As expected, choosing

Table 2 Average inter-event time τavg for the first 10 events of the PETC implementation with Γ as in [START_REF] Li | Weakly coupled event triggered output feedback system in wireless networked control systems[END_REF]. h smaller leads to a better approximation of the CETC system, and thus the inter-event times of the PETC implementation approach those of the CETC implementation. Similarly, increasing N leads to larger inter-event times when p = 2, and increasing p to p = 3 also leads to a large improvement (especially for large values of h). However, increasing p or N beyond p = 3, N = 2, does not further improve the results.

Conclusions

We have presented a systematic method for designing periodic event-triggered controllers for nonlinear systems, given that a continuous event-triggered controller is already available. Using convex overapproximation techniques, we were able to preserve the control performance guarantees of the given CETC system.

Case 2 :

 2 z(t, z 0 ) ∈ Ω for some t ∈ [0, h]. Define t * := inf{τ ∈ [0, h] | z(τ, z 0 ) / ∈ Ω} and t 1 := inf{τ > 0 | Γ (z(τ, z 0 )))

  and parameters σ = 1, α = β = 3, L = 2, ρ = 0.1, η = 2, λ = σ α 2 + β 2 + σ -2 ρ + η, and θ = 0.9, and we have obtained that the choice T = 0.010, δ(x) = δx 2 with δ = 0.5, and γ = σ α 2 + β 2 + σ -2 ρ, guarantees uniform global asymptotic stability of the set A := {z ∈ R nz | x = 0} for the CETC system. The evolution of Γ along the solution to the CETC system with initial condition z(0) = z 0 = [0.3, 0.3, 0] is shown in Figure 4. This figure also includes a simulation

Fig. 1 .

 1 Fig. 1. Evolution of Γ(z(t)) along the solution to the CETC system, the PETC implementation with Γ = Γ, and the PETC implementation with Γ as in (15).

Table 1

 1 Parameters of Assumption 3 for Γ2 in the invariant set[START_REF] Papachristodoulou | SOSTOOLS: Sum of squares optimization toolbox for MATLAB[END_REF].

		c	ς0	ς1	ς2	ς3
	p = 2 359.7 -26.46	3.085
	p = 3	0	222.8	-36.37	10.1
	p = 4	0	4384	-645.9 145.3 -17.36

This work is supported by the Innovational Research Incentives Scheme under the VICI grant "Wireless control systems: A new frontier in automation" (No. 11382) awarded by NWO (The Netherlands Organization for Scientific Research) and STW (Dutch Science Foundation), by the ANR under the grant COMPACS (ANR-13-BS03-0004-02), and by the Australian Research Council under the Discovery Projects Scheme.


Pablo Arrighi

Christophe Patricot

The Conal representation of Quantum States and Non Trace-Preserving Quantum Operations

Keywords: numbers: 03.65, 03.67, 03.67.d Keywords: Cones, Bloch sphere, Generalized measurements, Information gain versus disturbance

We represent generalized density matrices of a d-complex dimensional quantum system as a subcone of a real pointed cone of revolution in R d 2 , or indeed a Minkowskian cone in E 1,d 2 -1 . Generalized pure states correspond to certain future-directed light-like vectors of E 1,d 2 -1 . This extension of the Generalized Bloch Sphere enables us to cater for non-trace-preserving quantum operations, and in particluar to view the per-outcome effects of generalized measurements. We show that these consist of the product of an orthogonal transform about the axis of the cone of revolution and a positive real linear transform. We give detailed formulae for the one qubit case and express the post-measurement states in terms of the initial state vectors and measurement vectors. We apply these results in order to find the information gain versus disturbance tradeoff in the case of two equiprobable pure states. Thus we recover Fuchs and Peres' formula in an elegant manner.

I. INTRODUCTION

The space of pure states of finite d-dimensional Quantum Mechanics CP d , set of rays in the complex Hilbert space C d , is, as most complex spaces, not easy to visualize. Physical motions, let alone unitary time evolutions, have no clear geometric interpretation. However, the set of hermitian operators on C d , Herm d (C), is a (d 2 -1)-dimensional real vector space, and as such is certainly easier to represent geometrically. States, or more generally density matrices, form of course a subset of Herm d (C). The Generalized Bloch Sphere Representation ([1]- [START_REF] Schlienz | Description of entanglement[END_REF]) is a famous application of this fact which has proved to be popular and elucidating: a given density matrix can be represented as a real vector inside a (hyper) sphere. It turns out that this representation defined for density matrices, or unit trace positive elements of Herm d (C), is only good at handling unitary or trace preserving quantum operations on density matrices: the former induce rotations of the Bloch vector, the latter affine transformations [START_REF] Zanardi | A Note on Quantum Cloning in d dimensions[END_REF]. Individual outcomes of generalized measurements, for example, are not directly representable. Considering the insight the Bloch Sphere representation gave to unitary and trace-preserving operations, it seems interesting, for the mere sake of geometry at first, but mainly to give a useful picture to tackle Quantum Information problems, to extend it to cater for non-tracepreserving quantum operations. This is further motivated by the fact that the space of (semi-definite or def-inite) positive hermitian operators, thereafter denoted Herm + d (C), is a closed convex cone, and that all admissible quantum operations should be a subset of the transformations of this cone.

In spite of being so central in Quantum Information Theory, the tradeoff between how much Shannon Information one may gain about a quantum system versus how much Disturbance the observation must necessarily cause to the system, remains extremely difficult to quantify. Quantum cryptographists tend to circumvene the problem: most of their proofs are a witty blend of the particular symmetries of the protocol in question, together with a convoluted machinery. A few attempts have been made to solve the tradeoff [START_REF] Banaszek | Information gain versus disturbance for a single qubit[END_REF] [START_REF] Barnum | Information-disturbance tradeoff in quantum measurement on the uniform ensemble and on the mutually unbiased bases[END_REF], but only one [START_REF] Fuchs | Information Gain vs. State Disturbance in Quantum Theory[END_REF] deals with discrete ensembles -namely the case of two equiprobable pure states, and this is already something. Unfortunately the approach involves lengthy algebra and a number of assumptions. One should be able to find a method which gives a glimpse of intuition about the geometry of optimal measurements, and for this purpose, we think that our approach is useful.

In section II we consider general quantum systems of d complex dimensions. We give a representation of the set of positive hermitian matrices Herm + d (C) as a subcone of a real Minkowskian cone in R d 2 , and analyse geometrical properties of generalized measurements in this setting. We find that our approach is particularly useful to represent per-outcome post-measurement states, and that pure states correspond to certain light-like vectors of the cone. Unitary operators on the complex system become real orthogonal transforms, while positive operators become real positive transforms. Section III should be of special interest for quantum information theorists: we treat the d = 2 one qubit case in full detail. We find further geometrical relations between measurement vec-tors, state vectors and post-measurement state vectors and give explicit formulae. In section IV we apply our results to a typical Information gain versus Disturbance tradeoff scenario in which Alice gives Eve two equiprobable pure states. Thus we recover Fuchs and Peres' formula in an elegant and geometrical manner.

II. CONAL REPRESENTATION OF d-DIMENSIONAL QUANTUM SYSTEMS

The state of such a system is described by a d × d density matrix. We shall express hermitian matrices as real linear combinations of Hilbert-Schmidt-orthogonal hermitian matrices, and then restrict this representation to elements of Herm + d (C), or generalized density matrices. Herm + d (C) turns out to be "isomorphic" to a convex subcone of a cone of revolution in R d 2 , or indeed a Minkowskian future cone in E 1,d 2 -1 . We then analyse the effects of quantum operations on density matrices in this representation.

A. Hermitian matrices

Let {τ i }, i ∈ {1, . . . , d 2 -1}, be a Hilbert-Schmidt orthogonal basis (as in (1)) of d × d traceless hermitian matrices, and let τ 0 be the identity matrix I. Throughout this article latin indices will run from 1 to d 2 -1, greek indices from 0 to d 2 -1, and repeated indices are summed unless specified. We take the τ µ 's to satisfy by definition:

∀ µ, ν Tr(τ µ τ ν ) = dδ µν (1)
with δ the Kronecker delta. {τ µ } µ is a basis of Herm d (C), and any hermitian matrix A ∈ Herm d (C) decomposes on this basis as

A = 1 d Tr(A)I + Tr(Aτ i )τ i = 1 d Tr(Aτ µ )τ µ (2)
Letting A = (A µ ) ∈ R d 2 with A µ = Tr(Aτ µ ) be the component vector of A in this particular basis, we have

∀ A, B ∈ Herm d (C), AB = 1 d 2 A µ B ν τ µ τ ν hence TrAB = 1 d A.B ≡ 1 d A µ B µ (3)
We shall call A the vector in R d 2 , -→ A = (A i ) the restricted vector in R d 2 -1 , and φ the coordinate map:

φ : Herm d (C) → R d 2 A → A Equation (3) says that φ is an isometric isomorphism of (Herm d (C), Tr( )) onto (R d 2 , (1/d)( . )).
Therefore any linear operator L on Herm d (C) defines via φ and φ -1 an operator on

R d 2 , M (L) = φ • L • φ -1
. This definition yields the following "morphism" property :

Lemma 1 If L 1 , L 2 are linear operators on Herm d (C), then M (L i ) = φ • L i • φ -1 for i = 1, 2 are endomorphisms of R d 2 and satisfy M (L 1 • L 2 ) = M (L 1 )M (L 2 ) (4)
In particular, any complex d × d matrix A defines via Ad A : ρ → AρA † a linear operator on Herm d (C) which corresponds to a real endomorphism M (Ad A ) : ρ → M (Ad A )ρ ; and

Ad AB = Ad A • Ad B implies M (Ad AB ) = M (Ad A )M (Ad B ).
As a direct consequence of this and the previous definitions , calling GL n (K) the group of invertible n × n matrices on the field K, we get :

Lemma 2 For any subgroup G of GL d (C), the following mapping Proof: Let ρ = (1/d)(Tr(ρ)I + ρ i τ i ) a hermitian matrix. Using (3) and the fact that Ad U is trace-preserving for U unitary:

ψ : G → ψ(G) ⊂ GL d 2 (R) A → M (Ad A ) = φ -1 • Ad A • φ (5) is a group homomorphism. ψ(G) is a subgroup of GL d 2 (R).
U ρU † .U ρU † = (Trρ) 2 + ψ(U )ρ i ψ(U )ρ i = dTr(U ρU † U ρU † ) = dTrρ 2 = ρ.ρ = (Trρ) 2 + ρ i ρ i
In addition to preserving the first component ρ 0 = Trρ, ψ(U ) preserves the R d 2 -1 scalar product ρ i .ρ i . For all U ∈ SU (d), there exist t ∈ R and B ∈ su(d) such that U = U (t) = exp(tB). Since det ψ(U (0)) = 1 and t → det ψ(U (t)) is continuous and has values in {±1}, det ψ(U ) = 1. Thus ψ(U ) is a special rotation about the I -axis of R d 2 . By Lemma 2, ψ(SU (d)) is a subgroup of the special orthogonal group SO(d 2 -1) ⊂ SO(d 2 ). As for all θ ∈ R, Ad U = Ad e iθ U , ψ(U (d)) = ψ(SU (d)). Since the { √ -1τ i } span the Lie algebra su(d), ψ(SU (d)) is the Adjoint group of SU (d). For d = 2, we get the whole of SO(3), but this is not the case for d > 2, as is easily seen looking at the dimensions:

dim SU (d) = d 2 -1 dim SO(d 2 -1) = 1 2 (d 2 -1)(d 2 -2)
and dim SU (d) < dim SO(d 2 -1) for d > 2. ✷ All the results of this section remain true of course when we just consider Herm + d (C). From now on, for any A complex d × d matrix we shall denote ψ(A) = M (Ad A ) the real endomorphism of R d 2 .

B. Generalized density matrices

In [START_REF] Zanardi | A Note on Quantum Cloning in d dimensions[END_REF], Zanardi showed using a restriction of a mapping analogous to φ : A → A that d × d density matrices lie in a convex subset of a ball S ⊂ R d 2 -1 . We shall extend this to a convex cone by considering generalized density matrices, by which we mean elements of Herm + d (C). This seems more natural in the sense that we like to think of the space of states of most physical theories and indeed Quantum Mechanics as a space invariant under positive linear combinations and not just convex combinations. In addition, this bigger space allows a per-outcome representation of generalized measurements. We define generalized pure states to be generalized density matrices which yield pure states after rescaling them to unit trace. Note that these are not the "states of partial purity" of the complex d-dimensional system, which are singular density matrices. In other words, generalized pure states are not the elements of the boundary of Herm + d (C) in the sense of characteristic functions of cones (see [START_REF] Gauntlett | BPS States of D = 4 N = 1 Supersymmetry[END_REF] for example).

Proposition 1 The cone of positive hermitian matrices Herm

+ d (C) is isomorphic to a convex subcone C of the following cone of revolution in R d 2 : Γ = {(λ µ ) ∈ R d 2 / d 2 -1 i=1 λ 2 i ≤ (d -1)λ 2 0 , λ 0 ≥ 0} (6)
The set of generalized pure states verifies C = C ∩ ∂Γ, where ∂Γ stands for the boundary of Γ.

Proof : We begin as in [START_REF] Zanardi | A Note on Quantum Cloning in d dimensions[END_REF]. Let P denote the space of (not generalized) pure states in Herm d (C). In addition to being positive, ρ ∈ P satisfies Tr(ρ 2 ) = Tr(ρ) = 1, so we have

Tr(ρ 2 ) = 1 d ρ.ρ = 1 d (Trρ) 2 + ρ i ρ i = 1 d 1 + ρ i ρ i = 1, hence ρ i ρ i = d -1 (7)
The restricted vector (ρ i ) is on a (d 2 -2)-sphere of ra-

dius √ d -1, ∂S d 2 -2
, where S is the corresponding ball. In R d 2 , ρ pure sits in the intersection of the cylinder [START_REF] Gauntlett | BPS States of D = 4 N = 1 Supersymmetry[END_REF] and the ρ 0 = Tr(ρ) = 1 hyperplane, in other words on ∂S d 2 -2 "centered" at (1, 0, . . . , 0). Any density matrix can be expressed as a positive (convex) linear combination of pure states, and any positive (convex) linear combination of pure states defines a density matrix. Calling D the set of (not generalized) density matrices, D ⊂ Hull(P) and Hull(P) ⊂ D. Since D is closed, D = Hull(P), a well-known result. As φ : Herm d (C) → R d 2 is linear and bi-continuous, φ(D) = φ Hull(P) = φ(Hull(P)) = Hull(φ(P)). This set is a closed convex subset of S "centered" at (1, 0, . . . , 0):

φ(P) ⊂ ∂S d 2 -2 ⇒ Hull(φ(P)) ⊂ S
Calling S + ≡ Hull(φ(P)) the image set of density matrices as a subset of R d 2 -1 , we get:

ρ ∈ D ⇔ Tr(ρ) = ρ 0 = 1 and (ρ i ) ∈ S + Now a non-zero ρ ∈ Herm d (C) is positive if and only if (1/Trρ)ρ is positive, that is if and only if (1/Tr(ρ))ρ i ∈ S + . In R d 2 , recalling that Tr(ρ) ≡ ρ 0 , this reads ρ ∈ Herm + d (C) ⇔ ρ ∈ {(λ 0 , (λ i )) ∈ R d 2 /(λ i ) ∈ λ 0 S + } (8) This clearly defines a cone C in R d 2 .
As S + ⊂ S, C is a subcone of the cone of revolution Γ given by [START_REF] Fuchs | Information Gain vs. State Disturbance in Quantum Theory[END_REF]. φ being an isomorphism, C is convex and isomorphic to Herm + d (C). As pure states correspond to some points on the sphere ∂S d 2 -2 , generalized pure states lie in the boundary of Γ. Calling C the set of vectors of C corresponding to generalized pure states, we have C ⊂ C ∩ ∂Γ. Moreover C ⊃ C ∩ ∂Γ follows from the fact that any rescaled positive matrix ρ such that Tr(ρ 2 ) = Trρ = 1 is a pure state. Remember that C is not the boundary of C, but a cone over φ(P), the image set of pure states. ✷ As we shall see in detail in section III in d = 2 dimensions, generalized pure states correspond to futuredirected light-like vectors of Minkowski space of signature [START_REF] Hioe | N-level Coherence Vector and Higher Conservation Laws in Quantum Optics and Quantum Mechanics[END_REF][START_REF] Schlienz | Description of entanglement[END_REF]. We have shown that this remains true to a certain extent in d-complex dimensions, Γ being the future light-cone of Minkowski space E 1,d 2 -1 with metric η µν = Diag(d -1, -1, . . . , -1). Thus the appearance of a Minkowski product is to be expected.

As a consequence of Lemma 3, unitary transforms, since they leave Herm + d (C) invariant, yield rotations which leave C (globally) invariant. This fact deserves to be analysed in detail to understand the geometry of C. As Unitary transforms act transitively on pure states, S + is the closed convex hull of a homogeneous subspace φ(P) of ∂S d 2 -2 . For the moment however, we shall consider the geometric representation of general quantum operations in C.

C. Generalized measurements

We call a generalized measurement [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF] a finite set {M m } m of complex d × d matrices which satisfy :

m M † m M m = I. The set of {E m } m = {M † m M m } m de- fines a Positive Operator Valued Measure (POVM), as E m ∈ Herm + d (C)
′ m = (1/Tr(E m ρ))(M m ρM † m ). We shall call ρ m = M m ρM † m ∈ Herm + d (C) the unrescaled post-measurement state.
Recall that any complex matrix can be polar-decomposed into a product of a unitary matrix and a positive matrix. For all m, there exists

U m ∈ U (d) and A m ∈ Herm + d (C) such that M m = U m A m . As E m = M † m M m = A m A m , A m = √ E m , the positive square root of E m . Using this polar decomposition, ρ ′ m is represented in the cone C by ρ ′ m ≡ φ(ρ ′ m ) = 1 Tr(E m ρ) φ(U m E m ρ E m U † m ) = 1 Tr( √ E m ρ √ E m ) ψ(U m )( E m ρ E m )
Thus when outcome m occurs, the post-measurement state ρ ′ m of {M m } m is the same as that of { √ E m } m up to a rotation ψ(U m ), and similarly for the unrescaled states. As a consequence we shall consider the geometrical effects of generalized measurements { √ E m } m where E m and √ E m are in Herm + d (C) and m E m = I, bearing in mind that the most general measurements just involve rotations on the post-measurement state vectors. For example, in section IV, Eve is free to perform unitary transforms on her post-measurement states, and can decide this according to the outcome m. One might wonder here why not just rescale all the post-measurement states and only consider the density matrices ρ ′ m . The reason for not doing so is that the unrescaled states encode extra information: their "height" in the cone, the first component ρ m 0 = Tr(E m ρ), is simply the probability of their outcomes. Under a given generalized measurement, post-measurement vectors with identical first components are equiprobable. Thus the sections of C of constant λ 0 have a clear physical interpretation. We shall need the following simple properties:

Lemma 4 For A ∈ Herm d (C) and B, C ∈ Herm + d (C), Tr(BC) ≥ 0 Tr(BABA) ≥ 0 Proof: Let B = √ B √ B, then Tr(BC) = Tr( √ BC √ B) ≥ 0 since √ BC √ B ∈ Herm + d (C). Then polar decompose A into A = U |A|, with U unitary and |A| ∈ Herm + d (C). As A ∈ Herm d (C), A = |A| U † = A † , and
Tr(BABA) = Tr(BU |A| B |A| U † ) = Tr(U † BU |A| B |A|)
p(m) = 1 d E m .ρ (9)
Proof : By using (2) successively, we have

ρ m µ = Tr( E m ρ E m τ µ ) = 1 d Tr( E m τ ν E m τ µ )ρ ν (10)
≡ M m µν ρ ν Clearly M m µν is real symmetric by cyclicity of the trace and the fact that √ E m τ ν √ E m and τ µ are hermitian. (Actually ψ(A) is real for any complex d × d matrix A, and real symmetric for any A hermitian

). Let v = (v µ ) ∈ R d 2 .
Using [START_REF] Levitin | Information and Distinguishability of Ensembles of Identical Quantum States[END_REF] we get

v T ψ( E m )v = v µ M m µν v ν = 1 d v µ Tr( E m τ ν E m τ µ )v ν = 1 d Tr( E m (v ν τ ν ) E m (v µ τ µ )) ≥ 0 This follows from Lemma 4 since v µ τ µ ∈ Herm d (C).
Hence M m µν is a positive real (symmetric) matrix. The properties on purity simply follow from general facts on quantum operations on density matrices which remain true for generalized density matrices: For |u >< u| and |v >< v| generalized pure states, for any A complex d × d matrix and any generalized density matrix ρ, A|u >< u|A † = |A u >< A u| and |v >< v|ρ|v >< v| =< v|ρ|v > |v >< v| [START_REF] Levitin | Optimal quantum measurements for two pure and mixed states, Quantum Communications and Measurement[END_REF] are generalized pure states. Relation (9) follows from (2) and Tr(

√ E m ρ √ E m ) = Tr(E m ρ). ✷
The following properties will help to give a geometrical intuition of the action of the ψ(

√ E m )'s. For √ E m = |v >< v| pure, < v|ρ|v >= Tr( √ E m ρ) = (1/d) √ E m .
ρ. Thus using [START_REF] Levitin | Optimal quantum measurements for two pure and mixed states, Quantum Communications and Measurement[END_REF]:

ψ( E m )ρ = 1 d ( E m .ρ) E m So ψ( √ E m
) is as was expected a non-normalized projection. For any

√ E m ∈ Herm + d (C) , the d 2 eigen-vectors v σ of ψ( √ E m ) with eigen-values λ σ correspond to d 2 hermitian matrices M σ ≡ φ -1 (v σ ) which satisfy √ E m M σ √ E m = λ σ M σ (no summation). As a conse- quence, if ρ ∈ C is
ρ m = 1 d 3 √ ε µ ρ ν √ ε σ τ µ τ ν τ σ
Expanding this using τ 0 = I and grouping the products of the τ i 's in hermitian terms, we easily derive:

ρ m = 1 d 3 √ ε 0 ρ 0 √ ε 0 I + (2 √ ε 0 ρ 0 √ ε i + √ ε 0 ρ i √ ε 0 )τ i + ( 1 2 √ ε i ρ 0 √ ε j + √ ε 0 ρ i √ ε j )(τ i τ j + τ j τ i ) + 1 2 √ ε i ρ j √ ε k (τ i τ j τ k + τ k τ j τ i ) (12)
To push the general d-dimensional analysis further, we need a particular choice of τ i 's whose anti-commutation relations are convenient. This is subject to current work. We now treat in full detail the d = 2 (one qubit) case and apply our geometric approach to a challenging quantum information theoretical problem.

III. THE QUBIT CASE PUSHED FURTHER

Applied to qubit states the representation yields two of the most familiar objects in fundamental physics: the 2 × 2 density matrices yield a Minkowskian future-lightcone in E 1,3 whose vertical sections are nothing but Bloch spheres. The correspondence between light-like vectors and fully determined spins is puzzling, but it requires a little more than one qubit to be investigated further. Meanwhile in this simple case we are able to give explicit coordinates for states posterior to non trace-preserving quantum operations. These formulae remain simple provided Minkowskian products are introduced alongside the Euclideans. They constitute a sufficient armoury to deal, using only four-vectors, with the most general evolutions to happen on a qubit.

A. The Cone and the Bloch Sphere suitable Hilbert-Schmidt orthogonal basis for 2 × 2 traceless hermitian matrices is given by the set of Pauli matrices:

τ 1 = X = 0 1 1 0 τ 2 = Y = 0 -i i 1 τ 3 = Z = 1 0 0 -1
Together with the identity

τ 0 = I = 1 0 0 1
one may express any 2 × 2 hermitian matrix as a sum A = 1 2 A µ τ µ with the A µ 's real. The positivity conditions for those matrices turns out simple.

Lemma 5

The cone of positive hermitian matrices Herm + 2 (C) is isomorphic to the following cone of revolution in R 4 :

Γ = {(λ µ ) ∈ R 4 / λ 2 0 - 3 i=1 λ 2 i ≥ 0, λ 0 ≥ 0}
Generalized pure states lie on the boundary of Γ.

Proof : The eigenvalues of A are given by

λ ± = 1 2 (A 0 ± A i A i ).
A is positive if and only if λ + λ -≥ 0 and λ + + λ -≥ 0. This is equivalent to:

η µν A µ A ν ≥ 0 ∧ A 0 ≥ 0 ( 13
)
with η µν = Diag(1, -1, -1, -1). The purity condition is an obvious consequence of Proposition 1. ✷ Thus the generalized (not necessarily normalized) density matrices of a qubit cover the whole Minkowskian future-light-cone in E 1,3 . Taking a vertical cross-section of the cone is equivalent to fixing the trace A 0 of the density matrix, which might be thought of physically as the overall probability of occurrence for the state. By doing so we are left with only the spin degrees of freedom along X, Y, Z, therefore each vertical cross-section is a Bloch sphere with radius a = A 0 .

The ability to represent states with different traces is convenient when dealing with quantum ensembles {(p x , ρ x )} x . When we seek to represent non tracepreserving quantum operations the feature becomes absolutely crucial.

B. The Post-measurement State

As we have seen in subsection II C, the most general quantum operation can be described as {M m } m = {U m √ E m } m with U m unitary and √ E m positive (the only exta feature Kraus operators allow is the possibility to ignore one's knwoledge of some measurement outcomes, but in our setting this is easily catered for by adding up the undistinguished non-normalized postmeasurement states). While the action of U m is well understood in terms of four-vectors (as a mere rotation in the Bloch Sphere, see Lemma 3), the authors of this paper are not aware of a solid geometrical framework for representing the effects of √ E m -other than the one presented here. In Lemma 6, if A ≡ √ E m while ρ is the initial state, then AρA stands for the (not renormalized) 'post-measurement' state when outcome m has occurred (up to a unitary evolution U m ). Lemma 6 Let A and ρ be two matrices in Herm + 2 (C). Then:

AρA = 1 8 [-ρ 0 (η µµ ′ A µ A µ ′ ) + 2A 0 (A.ρ)]τ 0 + 1 8 [ρ 1 (η µµ ′ A µ A µ ′ ) + 2A 1 (A.ρ)]τ 1 + 1 8 [ρ 2 (η µµ ′ A µ A µ ′ ) + 2A 2 (A.ρ)]τ 2 + 1 8 [ρ 3 (η µµ ′ A µ A µ ′ ) + 2A 3 (A.ρ)]τ 3 = 1 8 [η νν ′ ρ ν (η µµ ′ A µ A µ ′ ) + 2A ν ′ (A.ρ)]τ ν ′ (14) Proof : Consider A = [ α β γ δ ] ρ = [ a x y z ]
We have:

AρA = 1 8 [a(α 2 + β 2 + γ 2 + δ 2 ) + 2α(βx + γy + δz)]τ 0 + 1 8 [x(α 2 + β 2 -γ 2 -δ 2 ) + 2β(αa + γy + δz)]τ 1 + 1 8 [y(α 2 -β 2 + γ 2 -δ 2 ) + 2γ(αa + βx + δz)]τ 2 + 1 8 [z(α 2 -β 2 -γ 2 + δ 2 ) + 2δ(αa + βx + γy)]τ 3 (15)
This formula can be be obtained either by brute force calculation using the Pauli multiplication relations, or by exploiting the fact that Pauli matrices form a Clifford Algebra I.e. {τ i , τ j } = 2δ ij τ 0 together with equation [START_REF] Townsend | The Jordan formulation of quantum mechanics: A review[END_REF]. Regrouping the terms gives formula (14). ✷ Corollary 1 Let A and ρ be two matrices in Herm + 2 (C). AρA can be expressed as a linear combination of ρ, A and the Identity:

AρA = 1 2 4 (A.ρ) A + 1 2 5 (η µµ ′ A µ A µ ′ )(ρ -ρ 0 τ 0 )
This last corollary provides much geometrical insight on non trace-preserving quantum operations. We find that the effect of √ E m is not that difficult to visualize: the resulting state is a weighted sum of √ E m , the initial state and the identity, with real coefficients.

It is a somewhat strange fact that the structure equation (15) does not become apparent until one brings the Minkowskian product to the rescue. The spurious appearance of special relativistic products in quantum mechanics bears some explanation in this setting however, since the Minkowski metric is intrinsically related to the characteristic function of pointed cones of revolution.

Finally it is important to notice that the results expressed in these two last subsections are invariant under any orthogonal change of basis {τ i } i . This is because rotations about the vertical axis leave the Minkowskian product invariant. The Pauli matrices have been helpful in computing those results, but from now and in the rest of the paper we may consider ourselves in the more general setting of section II.

C. Square and Square Root

In our quest towards representing non trace-preserving quantum operations in the cone we have managed to obtain the probability of occurrence p(m) in terms of E m (Proposition 2). In the previous subsection we have also worked out the evolved state ρ m , but unfortunately this was done in terms of √ E m . In order to deal fully with these operations in the Cone formalism we need to understand ways of switching back and forth from E m to √ E m . The next Lemma is a direct consequence of equation (14) when ρ = I.

Lemma 7

The square of a matrix A in Herm + 2 (C) is given by:

A 2 = A 0 A - 1 4 (η µν A µ A ν )τ 0
Inversely the square root operation obeys:

√ A = 1 r (A + 1 2 η µν A µ A ν τ 0 ) with: r = A 0 + η µν A µ A ν Note that A is proportional to √ A if
η µν √ A µ √ A ν = 2 η µν A µ A ν A 2 .ρ = A 0 (A.ρ) - 1 2 ρ 0 (η µν A µ A ν ) √ A.ρ = 1 r (A.ρ + ρ 0 η µν A µ A ν ) with: r = A 0 + η µν A µ A ν
On the whole taking the square root of E m is not so easy.

It would be much more convenient if we could make all calculations in terms of E m , with the added advantage condition:

m E m = 2τ 0 (16)
is easily visualized. Results in the following subsection are most useful for this purpose.

D. Inner Products Through Quantum Operations

Consider two states ρ 0 , ρ 1 . Suppose they undergo a quantum operation {M m } m = {U m √ E m } m and outcome m occurs. Rather than seeking the coordinates of the rescaled post-measurement states ρ 0 m ′ and ρ 1 m ′ , we may be interested in their positions relative to one another. Note this subsection reuses a number of notational conveniences introduced in section II.

Lemma 9 Let ρ 0 ,ρ 1 be two initial states in Herm + 2 (C) and √ E m a measurement element in Herm + 2 (C). The inner products of the post-measurement states satisfy:

ρ 0 m .ρ 1 m = (17) 1 4 [2(E m .ρ 0 )(E m .ρ 1 ) -(η µµ ′ E m µ E m µ ′ )(η νν ′ ρ 0 ν ρ 1 ν ′ )] ρ 0 m ′ .ρ 1 m ′ = 2 - (η µµ ′ E m µ E m µ ′ )(η νν ′ ρ 0 ν ρ 1 ν ′ ) (E m .ρ 0 )(E m .ρ 1 ) -→ ρ 0 m . -→ ρ 1 m = 1 4 [(E m .ρ 0 )(E m .ρ 1 ) -(η µµ ′ E m µ E m µ ′ )(η νν ′ ρ 0 ν ρ 1 ν ′ )] --→ ρ 0 m ′ . --→ ρ 1 m ′ = 1 - (η µµ ′ E m µ E m µ ′ )(η νν ′ ρ 0 ν ρ 1 ν ′ ) (E m .ρ 0 )(E m .ρ 1 ) (18)
Proof: By using (3) we have:

ρ 0 m .ρ 1 m = E m ρ 0 E m . E m ρ 1 E m = 2T r( E m ρ 0 E m E m ρ 1 E m ) = 2T r(E m ρ 0 E m ρ 1 ) = E m ρ 0 E m .ρ 1
From there we readily obtain equation ( 17) by applying equation ( 14) once. ✷ By letting ρ 0 = ρ 1 = ρ in the above Lemma we get:

||ρ m || 2 = 1 4 [2(E m .ρ) 2 -(η µµ ′ E m µ E m µ ′ )(η νν ′ ρ ν ρ ν ′ )] ||ρ m ′ || 2 = 2 - (η µµ ′ E m µ E m µ ′ )(η νν ′ ρ ν ρ ν ′ ) (E m .ρ) 2 || -→ ρ m || 2 = 1 4 [(E m .ρ) 2 -(η µµ ′ E m µ E m µ ′ )(η νν ′ ρ ν ρ ν ′ )] || --→ ρ m ′ || 2 = 1 - (η µµ ′ E m µ E m µ ′ )(η νν ′ ρ ν ρ ν ′ ) (E m .ρ) 2 (19)
Equation ( 19) clearly exhibits the general property we stated in Proposition 2: that is if the initial state is generalized pure

(η µµ ′ ρ µ ρ µ ′ = 0) or the measurement is generalized pure ( η µµ ′ E m µ E m µ ′ = 0) then we have || --→ ρ m ′ || = 1 (pure)
, which implies that ρ m is generalized pure.

The above lemma enables us to determine all the relative positions (angles and norms) of quantum states using relatively compact formulae which do not involve √ E m . It is only when the coordinates of each post-measurement state are required that one needs to take the impractical square root of E m . But remember we are allowed an arbitrary rotation U m in order to complete the quantum operation. This means we have full freedom to fix the absolute coordinates at will (so long as the relative positions are respected).

Most Quantum Information Theoretical problems seek to evaluate the limits of quantum operations, e.g. quantum cloning [START_REF] Cerf | Asymmetric quantum cloning machines in any dimensions[END_REF], distinguishability [START_REF] Levitin | Information and Distinguishability of Ensembles of Identical Quantum States[END_REF], Infomation Gain versus Disturbance tradeoff [START_REF] Banaszek | Information gain versus disturbance for a single qubit[END_REF]. In these situations the precise individual coordinates of the states after Ad √ Em tend not to matter; usually they will need to be rotated anyhow into a position which optimizes the fidelity measure in question. What counts is the relative position of the post-measurement states. Therefore these problems can be treated comfortably in our framework. Section IV provides a good example of such an application.

There are, however, some rare situations where we would like to see quantum operations act step by step, yielding precise coordinates -instead of just fixing the coordinates of the final state as we would do in order to avoid taking the square root of E m . This is the case for instance in quantum complexity, where one needs an appreciation of how many basic computational operations it takes to accomplish some calculation. Yet in this type of problems it turns out that the basic operations can be taken to be unitary operators, with measurements only performed at the end (principle of delayed measurement [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]). Therefore these scenarios may still be analyzed comfortably within our conal representation: the basic unitary operators will just be a set of chosen real orthogonal rotations, and the final measurement statistics will be evaluated straight from E m .

IV. APPLICATION: INFORMATION GAIN VERSUS DISTURBANCE TRADEOFF

The following idealized scenario captures a key situation for any quantum cryptographic protocol: Alice owns a random variable X = {( 1 2 , 0), ( 1 2 , 1)}. According to the outcome x she prepares either |ψ 0 or |ψ 1 , i.e. she runs |x |0 U → |x |ψ x . Eve knows U and the distribution X, but not the particular outcome Alice has drawn. Later Eve gains access to |ψ x and may use of this opportunity to try and learn about x. How much she learns is quantified using Information theoretical notions. Even though Alice has had to expose |ψ x , still she really wanted to keep x secret. But now she gets a chance of checking upon Eve's honesty -by asking her to return |ψ x . Suppose Eve's measurement and further manipulations have modified |ψ x into ρ x . Alice then measures {|ψ x ψ x |, Id -|ψ x ψ x |} and has a probability 1 -ψ x |ρ x |ψ x of detecting the felony.

The point is that most quantum cryptographic protocols rely upon the fact that Eve cannot eavesdrop a state without causing it an irreversible, detectable damage. In spite of their central role, Information Gain versus Disturbance tradeoffs upon discrete ensemble states remain largely unknown, due to the mathematical difficulties they raise. In 1995 Fuchs and Peres accomplished the mathematical feat of obtaining an analytic formula for the above case of two non-orthogonal states. But the method they used relies upon a number of "plausible" assumptions -and does not provide a geometrical intuition of what the family of optimal measurements looks like.

The Cone, by enabling a per outcome geometrical representation of generalized measurements, permits us to overcome some of these shortcomings and greatly facilitate the derivation of Fuchs and Peres' formula. We hope this illustrates the power of the geometrical framework developed in this paper.

A. Information Contribution, Disturbance Contribution

Suppose the {|ψ x } x=0,1 states Alice prepares verify the following basic relations:

v x = φ(|ψ x ψ x |) v 0 .v 1 = d = 1 -c 2
By choosing a suitable basis in the Bloch Sphere and since the {|ψ x } x are pure we may fix:

v 0 = [ 1 c d 0 ]
(20)

v 1 = [ 1 -c d 0 ] (21)
The most general thing Eve can ever do is to attack the states with a measurement {M m } m . This procedure equivalent to first measuring { √ E m } m , and then, conditional to m, applying the unitary transformation U m , with E m and U m defined as in subsection II C. It is rather interesting to observe that the second step has no other use but to "repair" the post-measurement states as much as is possible. The first step on the other hand may partially destroy the initial states so as to collect the Information Eve seeks. This is the step we now study in order to quantify her Information Gain.

Let Y be the random variable arising from the measurement outcomes, i.e. Y {(p(m), m)} m . Eve's Information Gain is given by:

I = H(X : Y ) = H(Y ) -H(Y |X) = m p(m) log(p(m)) -
ε m = [ α β γ δ ] = φ(E m )
By making use of the relations (20),( 21) and ( 9) one can express I m geometrically in terms of scalar products in the cone:

I m = -(p m + q m ) log(p m + q m ) + p m log(2p m ) + q m log(2q m ) with p m = α + βc + γd 4 = ε m .v 0 4 ≡ p(0, m) (22)
q m = α -βc + γd 4 = ε m .v 1 4 ≡ p(1, m) (23)
Notice that if ε m is orthogonal to v 1 (resp. v 0 ) then I m = p m (resp. q m ). Such a measurement element may be said to be "all or nothing": it brings a whole bit of information when it occurs, but does so only with probability p m (resp. q m ). Taken individually these measurement elements seem ideal: they fully identify |ψ x and thus they let you reconstruct the initial state perfectly, with no disturbance at all. The downside is that failure to occur comes at a high price. In order to verify the condition (16) the other measurement elements generally become rather inefficient with respect to the tradeoff. The family of the optimizing {M m } m is not constructed in such simple ways.

Next we seek an expression of the Disturbance Contribution brought by each measurement element. For this purpose we must first assume outcome m has occurred. Eve knows it, and now she will try to maximize her chances of fooling Alice by applying a carefully tailored unitary evolution U m . First we will give D m as a function of U m , next proceed to the maximization which determines U m . Remember that upper indices x distinguish initial states, while lower indices m specify the measurement outcome.

p(fool|m) = x p(x|m)Tr(|ψ x ψ x |U m ρ x m ′ U m ) ≡ x x .r x m 2 = 1 + x p(x|m) -→ v x . -→ r x m 2 where r x m = [ 1 -→ r x m ] ≡ φ(U m ρ x m ′ U m ) = φ(U m √ E m |ψ x ψ x | √ E m U m ) p(m|x)
Negating back to the Disturbance we obtain:

D = m D m with D m = p(¬fool, m) = p(m) -x p(x, m) -→ v x . -→ r x m 2 = p(m) -x p(x, m) -→ v x -→ r x m cos ( -→ v x , -→ r x m ) 2
In our scenario the {|ψ x } x are pure. Thus by Lemma 2 or equation (19) we have

-→ v x -→ r x m = 1. Now let us deal with cos ( -→ v x , -→ r x m
) by making the following definitions:

θ = ( -→ v 0 , -→ v 1 ) θ m = ( -→ r 0 m , -→ r 1 m ) ∆ m = θ -θ m ω m = ( -→ r 0 m + -→ r 1 m , -→ v 0 + -→ v 1 )
ω m is the angle between the bisector of ( -→ r 0 m , -→ r 1 m ) and that of ( -→ v 0 , -→ v 1 ). Given that we want to minimize D m in terms of U m we can safely assume

-→ r 0 m , -→ r 1 m , - → v 0 , -→ v 1
to be coplanar. Thus D m may now be rewritten in terms of those angles as well as p m and q m :

D m = p m + q m -p m cos(∆ m -ω m ) -q m cos(∆ m + ω m ) 2
In this equation the values of p m , q m and ∆ m are fully determined by ε m , as described in ( 22),( 23), (17). ω m on the other hand solely depends on U m : it can be chosen at will by rotation in the Bloch Sphere. We now show how Eve must tune ω m so as to minimize D m .

∂D m ∂ω m = 0 ⇒ p m sin(∆ m -ω m ) -q m sin(∆ m + ω m ) = 0
The minimum occurs at:

ω m = arcsin p m -q m p 2 m + q 2
m + 2p m q m cos(2∆ m ) which yields, after simplification:

D m = p m + q m -p 2 m + q 2 m + 2p m q m cos(2∆ m ) 2 B. The Tradeoff
How many elements should Eve's measurement contain? Levitin has proved that there exists a two-element measurement {M m } m=0,1 which maximizes Eve's Information Gain [START_REF] Levitin | Optimal quantum measurements for two pure and mixed states, Quantum Communications and Measurement[END_REF]. While this was never formally shown to be the case for the measurements which optimize the Information Gain versus Disturbance Tradeoff, there is strong numerical evidence in support of this assumption [START_REF] Fuchs | Information Gain vs. State Disturbance in Quantum Theory[END_REF]. Suppose this is the case and let ε m µ denote the µ th coordinate of ε m . Using the constraint equation ( 16) we have: δε 0 µ = -δε 1 µ and thus:

∀f ∂f ∂ε 0 µ = - ∂f ∂ε 1 µ (24)
Optimizing the Tradeoff implies finding a stationary point for the Disturbance while keeping the Information Gain fixed. We need to find ε 0 such that

µ ∂D 0 ∂ε 0 µ δε 0 µ = µ ∂D 1 ∂ε 1 µ δε 0 µ (25) subject to µ ∂I 0 ∂ε 0 µ δε 0 µ = µ ∂I 1 ∂ε 1 µ δε 0 µ (26)
Guided by the geometrical picture of the scenario one may consider the following attack:

ε 0 = [ 1 β 0 0 ] ε 1 = [ 1 -β 0 0 ]
The fact that this is indeed a solution follows from its obvious symmetries:

For µ = 2 ∂D 0 ∂ε 0 µ = ∂D 1 ∂ε 1 µ and for µ = 2 ∂D 0 ∂ε 0 µ = - ∂D 1 ∂ε 1 µ (27) For µ = 2 ∂I 0 ∂ε 0 µ = ∂I 1 ∂ε 1 µ and for µ = 2 ∂I 0 ∂ε 0 µ = - ∂I 1 ∂ε 1 µ (28)
Substituting (28) in the constant Information constraint (26), we get δε 0 2 = 0. Using this fact together with equation (27) it becomes clear that condition (25) is fulfilled. Thus ε 0 is a stationary point. We may now proceed to compute the values of the Disturbance and the Information Gain under this family of optimal attacks. First by making a few additional observations: √ E m } m , where the polar decomposition was applied. In order to represent M m 's per-outcome effect upon the real vectors we defined ψ : A → φ • Ad A • φ -1 and showed that ψ(U m ) is a real orthogonal transform while ψ( √ E m ) turns out to be a real positive matrix. Thus the geometrical effect of a generalized measurement can be viewed in terms of real transformations only. Such a nice correspondence suggests quantum mechanics could be expressed elegantly over the real numbers in this manner, quite differently from its formulation in terms of real Jordan algebras [START_REF] Townsend | The Jordan formulation of quantum mechanics: A review[END_REF]. However we first need to gain more geometrical intuition about the set of real vectors φ(Herm + d (C)), and the sets of allowed orthogonal and positive transforms. For now we know that φ(Herm + d (C)) is a subcone of the future-light-cone Γ = {(λ µ ) ∈ R d 2 / d 2 -1 i=1 λ 2 i ≤ (d -1)λ 2 0 , λ 0 ≥ 0}. One of the advantages of defining φ upon Herm + d (C) instead of the restricted set of density matices is that E m = M m M † m can be visualized. In order to characterize its effects we derived rather compact and powerful formulae for the qubit case, such as the one giving the scalar product of the post-measurement states:

p 0 = q 1 = p p 1 = q 0 = q D 0 = D 1 =
1 4 [2(E m .ρ 0 )(E m .ρ 1 ) -(η µµ ′ E m µ E m µ ′ )(η νν ′ ρ 0 ν ρ 1 ν ′ )]
looking at such expressions it becomes apparent that Minkowskian products have a crucial role to play in our framework, and even more so as we showed that pure quantum states correspond to light-like vectors (i.e. they sit on the boundary of Γ), even in dimensions greater than 2. It seems interesting to notice that BPS states of supersymmetric theories can also be thought of as lying on the boundary of a cone of positive operators [START_REF] Gauntlett | BPS States of D = 4 N = 1 Supersymmetry[END_REF], and that their stability is related to that fact. Somehow our setting seems to single out generalized pure states in a more natural way than merely characterizing them as unit rank elements of the boundary of Herm + d (C). The reminiscence of special relativity must be investigated further; this will be a subject for future work. Pauli matrices together with special relativistic considerations have already brought some fruitful results to quantum information theory. This is the case for instance in [START_REF] Gisin | Quantum cloning without signaling[END_REF], where some limits of quantum cloning are derived by using the no-signalling condition. Armed with the present representation one should be able tackle more of these difficult quantum information theoretical problems. Already in this paper we recovered Fuchs and Peres' information gain versus disturbance formula simply and geometrically. In the future we should be able to extend our analysis to the case of two non-equiprobable states. Some highly symmetric n-states scenarios may well cease to be out of reach.

  Note that since ψ(I) = ψ(-I) = I, ψ is not necesseraly injective. Moreover ψ is certainly not linear. An interesting subgroup is the Special Unitary groupSU (d) = {U ∈ GL d (C) / U U † = I, det U = 1}. We call SO(n) = {O ∈ GL n (R) / OO T = I, det O = 1} the special orthogonal group in n-dimensions. Lemma 3 Special Unitary transformations on Herm d (C), Ad U : ρ → U ρU † with U ∈ SU (d), induce rotations of R d 2 about the I-axis. In fact the linear transforms ψ(U ) : ρ → ψ(U )ρ are special orthogonal and ψ(SU (d)) is a subgroup of SO(d 2 -1). It is a proper subgroup when d ≥ 3. Moreover, ψ(U (d)) = ψ(SU (d)).

  and m E m = I. Given a quantum state or density matrix ρ ∈ D, the generalized measurement {M m } m on ρ yields outcome m with probability p(m) = Tr(E m ρ), and if outcome m occurs, the post measurement state is ρ

  The procedure we use to find the Disturbance is to first measure with { √ E m } m and then maximise on Unitary transforms acting upon post-measurement states. Using the conal representation, both sets of vectors {E m } m and { √ E m } m are in C, and m E m = (d, 0, . . . , 0). This enables us to represent elements of a measurement inside C, and visualize the action of a particular non-trace-preserving operation √ E m on a given density matrix ρ, in other words find ρ m in terms of E m or √ E m . D. Quantum operations represented in C

  such an eigenvector, then the rescaled density matrix ρ is such that ρ = ρ ′ m , i.e. ρ is unchanged if outcome m occurs. We now give the general expressions for ρ m in terms of ρ ≡ (1/d)ρ µ τ µ and √ E m ≡ (1/d) √ ε ν τ ν , where we drop the index m and do not underline the components of the vectors ρ and √ E m for convenience. By definition:

  , m) log(p(m|x)) ≡ m I m with I m = p(m) log(p(m))x p(x, m) log(p(m|x)) I m must be understood as the Information Contribution brought by the measurement element:

  δε 0 µ are subject to the additional constraint: µ ∂I ∂ε 0 µ δε 0 µ = 0 Using equation (24) and D = D 0 + D 1 and I = I 0 + I 1 this gives:

D/ 2 I 0 = I 1 = I/ 2 D = 1 2 -p 2 + 2 -c 4 )(β 2 - 2 + 2 1 -β 2 ) 1 +

 2012222422121 q 2 + 2pq + cos(2∆ m ) I = 1 + 2p log(2p) + 2q log(2q) and second by plugging in the relations (18), (20)-(23), we reproduce the exact content of Fuchs and Peres' formulae: βc) log(1 + βc) + (1 -βc) log(1 -βc)) V. CONCLUSION In this paper we considered a linear embedding taking d × d positive hermitian matrices into vectors of d 2 real entries, φ : ρ → ρ = (Tr(ρτ µ )) µ . It is a well-known fact that the most general evolution a density matrix ρ may undergo is a generalised measurement {M m } m = {U m

  This is non-negative by the previous result since U † BU, |A| B |A| ∈ Herm + d (C). ✷ Unitary transforms induce rotations in C, and generalized measurements have the following geometric properties:

	Proposition 2 The linear transforms ψ( √ E m ) : ρ → ρ m associated to a generalized measurement { √ E m } m correspond to real symmetric matrices which are positive.
	They individually map C into itself. In addition, for any
	generalized pure state θ, ψ(θ) maps C into C.
	The probability of outcome m for a quantum system in
	state ρ is given by

  and only if A is generalized pure or A ∝ I. But when we seek to express a function of E m in terms of √ E m (or the reverse) the next formulae become convenient. Lemma 8 Let A and ρ be two matrices in Herm + 2 (C).

	The following relations hold: