A Novel Family of Geometric Planar Graphs for

Wireless Ad Hoc Networks Xu Li † ‡ , Nathalie Mitton † , Isabelle Simplot-Ryl † , and David Simplot-Ryl † † INRIA/CNRS/Univ. Lille 1, France, {firstname.lastname}@inria.fr ‡ ECE, University of Waterloo, Canada

Abstract-We propose a radically new family of geometric graphs, i.e., Hypocomb, Reduced Hypocomb and Local Hypocomb. The first two are extracted from a complete graph; the last is extracted from a Unit Disk Graph (UDG). We analytically study their properties including connectivity, planarity and degree bound. All these graphs are connected (provided the original graph is connected) planar. Hypocomb has unbounded degree while Reduced Hypocomb and Local Hypocomb have maximum degree 6 and 8, respectively. To our knowledge, Local Hypocomb is the first strictly-localized, degree-bounded planar graph computed using merely 1-hop neighbor position information. We present a construction algorithm for these graphs and analyze its time complexity. Hypocomb family graphs are promising for wireless ad hoc networking. We report our numerical results on their average degree and their impact on FACE routing [START_REF] Bose | Routing with guaranteed delivery in ad hoc wireless networks[END_REF]. We discuss their potential applications and some open problems.

I. INTRODUCTION

A planar graph is a sparse graph where edges intersect only at their end vertices. It has been widely adopted in different domains to solve various problems, e.g., circuit layout design on computer chips, image segmentation in computer vision, facility layout design in operations research, just to mention a few. In these applications, the position of all vertices is known, and edges can be added between any two vertices. Planarization is equivalent to an edge removal process on a complete graph with connectivity preservation. In some other cases, edge addition is subject to distance constraint, bringing about the problem of planarization on a Unit Disk Graph (UDG). In the sequel, we always assume connected UDG, and two intersecting (or crossover) edges imply that the two edges intersect, but not at their end vertices.

Define the unit circle C γ (a) of a vertex a as the circle of radius equal to a unit distance γ and centered at a. The unit disk D γ (a) of a is the area enclosed by C γ (a). In UDG, there is an edge between two vertices a and b and they are said 'adjacent to' or 'neighboring' each other if and only if b ∈ D γ (a) (equivalently, a ∈ D γ (b)). We denote by V N BR (a) the closed neighborhood (neighbor set) of a (including a) and by V N BR (a, b) the closed common neighborhood of a and b.

Wireless ad hoc networks (e.g., sensor networks) where nodes have the same maximum transmission range γ (unit distance) are commonly modeled as UDG. In such networks, each node is static and assumed to know its own geographic position by attached GPS device or some other means. Two nodes are neighbors (i.e., have an edge in between) if and only if they are within each other's transmission range (i.e., unit disk). Periodic 'hello' message is a basic ad hoc networking technique for neighborhood discovery [START_REF] Ingelrest | A Turnover based Adaptive HELLO Protocol for Mobile Ad Hoc and Sensor Networks[END_REF]. By this technique, each node is able to gather the location information of all neighboring nodes. In the past decade, several well-known position-based ad hoc routing protocols [START_REF] Frey | On Delivery Guarantees of Face and Combined Greedy-Face Routing Algorithms in Ad Hoc and Sensor Networks[END_REF] were proposed. They all rely on planar network topology for guaranteeing packet delivery. In general, UDG is not planar. A planar subgraph has to be extracted through a planarization procedure.

In wireless networks, nodes share the communication media and have limited channel capacity. The main communication cost is therefore message transmissions. To minimize the control overhead on the network, graph planarization ought to be carried out in a distributed fashion without resorting to any global knowledge and with a minimal total number of message transmissions per wireless node. Ideally, it involves no message transmission in addition to the built-in 'hello' message. Packets have constant size at MAC layer. Transmission of a long message requires message fragmentation and leads to increased number of transmissions. Long message consumes more transmission power than short message, are more likely to cause error and should be avoided. Thus as an additional requirement, no modification should be made to the default 'hello' message (normally containing constant-sized information such as sender position) during planarization. In summary, graph planarization in wireless ad hoc networks is expected to be a strictly localized procedure, where each node makes consistent planarization decision independently using 1-hop neighborhood information only.

There are a few strictly localized planar graphs such as GG, RNG [START_REF] Jaromczyk | Relative neighborhood graphs and their relatives[END_REF] and PDel [START_REF] Li | Partial Delaunay Triangulation and Degree Limited Localized Bluetooth Scatternet Formation[END_REF] and a few non-strictly localized planar graphs such as LMST [START_REF] Li | Applications of k-Local MST for Topology Control and Broadcasting in Wireless Ad Hoc Networks[END_REF] and LDel [START_REF] Gao | Geometric spanner for routing in mobile networks[END_REF], [START_REF] Li | Localized Delaunay Triangulation with Application in Ad Hoc Wireless Networks[END_REF]. The degree ∆(G) of a graph G is the maximum node degree in the graph. It is often desirable that ∆(G) is small and bounded above by a constant. In wireless communications, a small node degree reduces the contention and interference and helps to mitigate the hidden and exposed terminal problems at MAC layer. In bluetooth scatternets, each node is required to have maximum degree 7. All the above local planar graphs but LMST have unbounded degree in nature, while LMST construction is not strictly localized (requires 2-hop information). Li et al. [START_REF] Li | Approximate MST for UDG Locally[END_REF] modified RNG construction such that the degree is limited to a small constant. However, the modification requires each vertex to be associated a unique identifier (ID), which does not necessarily exist in, e.g., sensor networks. Li et al. [START_REF] Li | Partial Delaunay Triangulation and Degree Limited Localized Bluetooth Scatternet Formation[END_REF] showed that degree can be limited to a constant with connectivity and planarity preservation by applying Yao structure [START_REF] Yao | On constructing minimum spanning trees in k-dimensional spaces and related problems[END_REF].

In this paper, we propose a radically new family of geometric planar graphs, completely different from any known graph, and focus on their theoretical properties. We first introduce Hypocomb (Hypotenuse-comb), which is the 'dual' (an abused use of term duality) of a truncated mesh [START_REF] Li | Localized Distance-Sensitive Service Discovery in Wireless Sensor and Actor Networks[END_REF] referred to as Besh (Blocked-mesh). Given a set of vertices in the Euclidean plane, Besh is constructed by drawing rays synchronously from each vertex in four directions and allowing distancebased blocking when they meet each other. Hypocomb is obtained by linking vertices that have a ray-blocking relation in Besh. We prove that Hypocomb is connected planar with unbounded degree. Then we propose to reduce its degree to 6 by applying constrained edge creation rule, without jeopardizing its connectivity and planarity: link two vertices if and only if they have a mutual ray-blocking relation. The resultant Hypocomb is called Reduced Hypocomb. After that, we present Local Hypocomb on the basis of UDG. It is constructed in a strictly localized manner, by removing any UDG edge that does not belong to the Reduced Hypocomb of the closed common neighborhood of its end vertices. We prove that Local Hypocomb remains connected planar and has slightly larger degree 8. Local Hypocomb is the first strictlylocalized, degree-bounded planar graph computable using 1hop neighbor position information only. It may serve as alternative graph in geographic routing for providing delivery guarantee in wireless ad hoc networks. We present, along with complexity analysis, a construction algorithm for Hypocomb family graphs. Through simulation we study their average degree and their impact on the well-know FACE routing protocol [START_REF] Bose | Routing with guaranteed delivery in ad hoc wireless networks[END_REF], in comparison with widely-adopted Delaunay triangulation and Gabriel Graph. Simulation results imply that Local Hypocomb is superior to Gabriel Graph. We indicate that Hypocomb and Reduced Hypocomb may be built in a localized way among actor nodes in emerging wireless sensor and actor networks and provide a generic solution to the challenging actor-actor coordination problem.

The rest of this paper is organized as follows. We review existing local planar graphs in Sec. II. We propose and analyze Hypocomb family graphs in Sec. III -V, along with numeric results being reported in Sec. VI. We conclude the paper by describing some of their potential applications and open problems for future research in Sec. VII.

II. RELATED WORK

There is only a few localized planar graphs in the literature. Given a vertex set V in the Euclidean plane, in the following we will briefly introduce how to construct these graphs. The containment relations among these graphs are given below.

MST ⊆

LMST RNG' ⊆ RNG ⊆ GG ⊆ PDel ⊆ Del LDel A Minimum Spanning Tree (MST) is a subgraph connecting all the vertices with weighted edges that lead to minimum total weight. If edges are weighted by the Euclidean distance of their end vertices (as in our context here), it is called Euclidean MST, and it has degree bounded above by 6 [START_REF] Monma | Transitions in geometric minimum spanning trees[END_REF]. In general, V may have many MST unless each edge has a unique weight. MST can not be computed locally, i.e., each node can not determine which edges are in MST by purely using the information of the nodes within some constant hops [START_REF] Li | Approximate MST for UDG Locally[END_REF]. MST is not spanner, i.e., having no constant spanning ratio. The spanning ratio of a graph is the maximum ratio of the Euclidean length of the shortest path connecting two arbitrary vertices and their direct Euclidean distance.

A Local MST (LMST) [START_REF] Li | Design and Analysis of a MST-Based Topology Control Algorithm[END_REF] is a connected subgraph of UDG, constructed locally using 2-hop neighborhood information as follows: at each vertex u, compute the MST of the sub-graph of V N BR (u); add incident edge uw to LMST if and only if the edge is in both M ST (V N BR (u)) and M ST (V N BR (w)). LMST contains MST as subgraph and has the same degree bound 6. In [START_REF] Li | Applications of k-Local MST for Topology Control and Broadcasting in Wireless Ad Hoc Networks[END_REF], it is proved that LMST is also planar, and the notion is extended to k-Local MST (LMST k ) with k-hop neighborhood information being used. LMST is not spanner.

Gabriel Graph (GG) is built by connecting any two vertices u and w if and only if the closed disk disk(u, w) with uw as diameter contains no other vertex from V , while Relative Neighborhood Graph (RNG) is built by connecting u and w if and only if the interior of their lune lune(u, w) (i.e., the intersection of the two circles of radius |uw| centered at u and w) contains no other vertex. Both GG and RNG are connected planar if the original graph is UDG. Each of them can be constructed strictly locally by each vertex checking the construction condition for its neighbors only. GG and RNG are so-called proximity graphs [START_REF] Jaromczyk | Relative neighborhood graphs and their relatives[END_REF]. Neither of them has constant bounded spanning ratio or bounded degree. A study of their spanning ratio was presented in [START_REF] Bose | On the spanning ratio of gabriel graphs and beta-skeletons[END_REF].

Assuming each vertex is associated with a unique ID, a modified RNG, called RNG', was proposed in [START_REF] Li | Approximate MST for UDG Locally[END_REF]. RNG' contains all edges uw such that the interior of lune(u, w) contains no vertex, and (2) there is no vertex v on the boundary of lune(u, w) such that ID(v) < ID(w) and |vw| < |uw|, and (3) there is no vertex v on the boundary of lune(u, w) such that ID(v) < ID(u) and |vu| < |uw|, and (4) there is no vertex v on the boundary of lune(u, w) such that ID(v) < ID(u), ID(v) < ID(w), and |vu| = |uw|. RNG' is a subgraph of RNG. It is proved that RNG' has maximum degree 6 and contains MST as subgraph.

A Delaunay triangulation (Del) is built by connecting any two vertices u, w ∈ V if and only if the circumcircle of the triangle defined by u, w and any other vertex v ∈ V is empty. Given V , there may be more than one Delaunay triangulation, but only if V contains four or more co-circular vertices. Del has constant spanning ratio [START_REF] Keil | Classes of graphs which approximate the complete euclidean graph[END_REF]. Del can not be constructed locally, because it may contain arbitrary long edges.

A connected planar was proposed for UDG on the basis of Del and under the assumption of no four co-circular vertices in [START_REF] Li | Partial Delaunay Triangulation and Degree Limited Localized Bluetooth Scatternet Formation[END_REF]. The graph is a subset of Del and thus named Partial Delaunay triangulation (PDel). It contains only a few more edges than GG. To construct PDel, each node u for each w ∈ V N BR (u) checks the following conditions: (1) disk(u, w) is empty (i.e., uw belongs to GG); (2) disk(u, w) contains vertices only on one side of uw, with x being one of those vertices that maximizes ∠uxw in triangle ∆uxw such that [START_REF] Gao | Geometric spanner for routing in mobile networks[END_REF] and [START_REF] Li | Localized Delaunay Triangulation with Application in Ad Hoc Wireless Networks[END_REF]. As PDel, it contains GG as subgraph; unlike PDel, it has good spanning ratio. The planarization process works as follows:

∀u ∈ V , compute Del(V N BR (u)); ∀w ∈ V N BR (u), uw is added to LDel if uw ∈ Del(V N BR (u)) and ∄v ∈ V N BR (u) such that u, w ∈ V N BR (v) and uw / ∈ Del(V N BR (v))
. Construction of LDel obviously requires 2-hop neighborhood information. LDel has unbounded degree. In [START_REF] Wang | Localized Construction of Bounded Degree Planar Spanner for Wireless Networks[END_REF], the degree of LDel is limited to 19 + 2π/α, where 0 < α ≤ π/3, by applying Yao [START_REF] Yao | On constructing minimum spanning trees in k-dimensional spaces and related problems[END_REF] structure, without scarifying its spanning property. Note that Yao graph itself does not guarantee planarity.

III. HYPOCOMB

Given as creating points a vertex set V in the Euclidean plane, we show how to build a novel connected planar graph, named Hypocomb, by adding edges between them. This is equivalent to removing edges from a complete graph of V . For ease of understanding, we divide our graph planarization process into two steps and present them separately.

For all a, b ∈ V and a = b, they are said collinear if they have the same X or Y coordinate. Define north (south) as the positive (resp., negative) direction of the Y axis, and east (west) as the positive (resp., negative) direction of the X axis. T = {north, west, south, east}. For each dir ∈ T , dir is the opposite direction, and d ir the set of perpendicular directions. For example, if dir = north, then dir = south and d ir = {west, east}. The border of V is the smallest rectangle containing V and parallel to the two axes.

A. The first step: Besh

At the first step, we build an auxiliary structure, referred to as Besh [START_REF] Li | Localized Distance-Sensitive Service Discovery in Wireless Sensor and Actor Networks[END_REF]. We synchronously grow from all v ∈ V four rays

R north v , R west v , R south v and R east v with mutual angle of π 2
, respectively in the north, west, south and east directions. The growth of these rays is limited by the border of V . If there was no further constraint, we would obtain a mesh. However we indeed apply a blocking rule [START_REF] Li | Localized Distance-Sensitive Service Discovery in Wireless Sensor and Actor Networks[END_REF] to control ray growth.

Definition When this happens, we say 'b blocks a at u. In the first two cases (orthogonal blocking), it is expressed as b u a (or,

1 (Blocking rule): ∀a, b ∈ V , a = b and ∀dir, dir ′ ∈ T , dir = dir ′ , if R dir a and R dir ′ b meet at point u, R dir
R dir ′ b u R dir a ); in the last case (collinear blocking), it is expressed as a u b (or, R dir a u R dir ′ b
). Use of the blocking rules causes some rays to stop growing early, before hitting the border of V , and yields a truncated mesh, which is our so-called Besh (standing for blocked mesh). The Besh, denoted by BS(V ), is defined by a vertex set and an edge set. The former contains the creating points V and added Besh points, where the blocking rule is engaged; the later contains the edges between the vertices. In BS(V ), each cell is a rectangle. The creating points (i.e., vertices in V ) whose rays define the perimeter of a cell is called the defining points of the cell. Each cell obviously has at least two, and at most four, defining points. For a Besh cell, with respect to a given corner vertex (which is either a defining point or a Besh point), the diagonal defining points are the defining points that are not collinear with the vertex.

Figure 1(a) shows a Besh structure created using 8 points a, b, . . . , h, whose border is marked by a thick rectangle. The solid small dots are Besh points; the thin dashed lines are Besh edges. Besh cell bswt is defined by a, b and h. For this cell, the diagonal defining points with respect to b is a and h, and that with respect to s is a.

′ ∈ d ir, R dir a and R dir ′ a define a quadrant Q a (dir, dir ′ ).
As such, a has four different quadrants.

Lemma 1: ∀a, b ∈ V, a = b, dir ∈ T, dir ′ ∈ d ir, b ∈ Q a (dir, dir ′ ) and |bb ′ | ≤ |ab ′ | with b ′ being the projection of b on R dir a , if ∃c ∈ V , c ∈ Q a (dir, dir ′ ) and c = b such that c blocks R dir ′ b at u, then |cc ′ | ≤ |ac ′ | where c ′ is the projection of c on R dir a . Proof:
We prove this lemma by case study with illustrations being given in Fig. 1(b). According to the way that c blocks R dir ′ b , we have three cases to consider.

1) R dir c u R dir ′ b : This is the case of c = c 1 , c ′ = c ′ 1 and u = u 1 . We know |bb ′ | ≤ |ab ′ | and |cu| ≤ |bu|. Then |cc ′ | = |bb ′ | -|bu| ≤ |ab ′ | -|cu| = |ac ′ |. 2) R dir c u R dir ′ b : This is the case of c = c 2 , c ′ = c ′ 2 and u = u 2 . We have |cc ′ | = |bb ′ | -|bu| ≤ |bb ′ | ≤ |ab ′ | ≤ |ab ′ | + |b ′ c ′ | = |ac ′ |. 3) R dir ′ c u R dir ′ b : This is the case of c = c 3 , c ′ = c ′ 3 = b ′ and u = u 3 . |cc ′ | < |bc ′ | < |ac ′ |. Note that c ′ is within distance |bb ′ | from b ′ .
Lemma 1 tells us an important property of the blocking rule: if a node b blocks a orthogonally at u in the case that a and b are the only vertices in V , then a must be blocked by a vertex c (possibly identical to b) orthogonally at u ′ within distance |bu| from u when V contains also other vertices. On the basis of this result, we develop a computer algorithm named Blocking-Detection to support Besh construction. Given a ∈ V and dir ∈ T , this algorithm returns the set of vertices (at most 2 by the prioritized blocking policy) that block R dir a and the associated Besh point (a single point). If no vertex blocks R dir a , it returns an empty set. The pseudo codes are given in Algorithm 1. Functions First() and Second() return respectively the first and the second element of an input pair. Function arg() returns the argument of an input function.

Examine Algorithm 1. In Line is orthogonally blocked by a vertex and be blocked by that same vertex. Hence, the result from Lines 4-13 is a coarse upper bound of the length of R dir a in Besh, stored in variable dist. It is infinity in the case that c does not exist (without considering the constraint from the border of V ).

The upper bound dist is derived under the assumption that R dir a is not blocked by anybody else. In the latter half of the algorithm, we remove this assumption. Lines 15 and 16

S1 := {(b, p)|b ∈ V such that R dir ′ b p R dir c in the case of V = {b, c}} 8
:

S2 := {(b, p)|b ∈ V such that R dir ′′ b p R dir c in the case ofV = {b, c}} 9
:

S ′ 1 := {m|m ∈ S1 and |Second(m)c| ≤ dist such that ∄t ∈ S1, First(t) blocks R dir ′ First(m) in the case of V = {First(m), First(t)}}
10:

S ′ 2 := {m|m ∈ S2 and |Second(m)c| ≤ dist such that ∄t ∈ S2, First(t) blocks R dir ′′ First(m) in the case of V = {First(m), First(t)}} 11: if S ′ 1 ∪ S ′ 2 = ∅ then 12: dist := |Second(arg(min m∈S ′ 1 ∪S ′ 2 |Second(m)c|))a|
13:

end if 14: end if 15: W1 := {(b, p)|b ∈ V such that R dir ′ b p R dir a in the case of V = {a, b}} 16: W2 := {(b, p)|b ∈ V such that R dir ′′ b p R dir a in the case of V = {a, b}} 17: W ′ 1 := {m|m ∈ W1 and |Second(m)a| ≤ dist such that ∄t ∈ W1, First(t) blocks R dir ′ First(m) in the case of V = {First(m), First(t)}} 18: W ′ 2 := {m|m ∈ W2 and |Second(m)a| ≤ dist such that ∄t ∈ W2, First(t) blocks R dir ′′ First(m) in the case of V = {First(m), First(t)}} 19: if W ′ 1 ∪ W ′ 2 = ∅ then 20: dist ′ := min m∈W ′ 1 ∪W ′ 2
|Second(m)a| 21: ) time. The computation in these four lines is similar to that in Lines 7-10. In the case that the union of W ′ 1 and W ′ 2 is not empty, the associated blocking point nearest to a in these two sets is identified in O(|V |) time (Line 20). It is the true blocking point, i.e., Besh point. The rational is that R dir a can be blocked only at a single point, and after that no blocking is possible at any point further away from a. Thus the set of blocking vertices associated with this point are found and returned (Line 21). This final step takes another O(|V |) time. In the case that the union of W ′ 1 and W ′ 2 is empty, the return value is singleelement set {(c, d)} if c exists (Line 23), and ∅ otherwise (Line 25). The computation time is constant O(1). All the other lines in the algorithm take O(1) time.

ret := {m|m ∈ W ′ 1 ∪ W ′ 2 such that |Second(m)a| = dist ′ } 22: else if c = null then 23: ret := {(c,
In total O(|V | 2 ) is the complexity of Algorithm 1. More efficient algorithms may be developed, but beyond the scope of this paper. The correctness of Algorithm 1 simply follows from the above analysis. Then we may construct Besh within O(|V | 3 ) time, by running this algorithm for every vertex in V four times, each time for a different direction in T .

Although Besh is a transit product of our graph planarization process, it has its own importance in real life applications. In [START_REF] Li | Localized Distance-Sensitive Service Discovery in Wireless Sensor and Actor Networks[END_REF], we derived that Besh has good proximity property like Voronoi diagram through analytical study and simulation experiments, showed how to accomplish Besh in a localized way, without knowing V , and proposed a Besh-based localized distance-sensitive service discovery algorithm for wireless sensor and actor networks.

Before proceeding to the second drawing step, we would like to introduce a few important definitions and lemmas (whose proof can be found in [START_REF] Li | Graph Planarization in Wireless Ad hoc Networks[END_REF]) to be used in the sequel.

Definition 4 (Emptiness and Cleanness): A region is empty if and only if there are no vertices located in it; a region is clean (with respect to Besh) if and only if it does not contain any Besh edge. A clean region must be empty, while the converse is obviously not necessarily true.

Lemma 2: ∀a, b ∈ V, a = b, dir ∈ T and dir ′ ∈ d ir, if R dir a u R dir ′ b
then the region defined by triangle ∆aub (including its perimeter) is empty.

Lemma 3: ∀a, b ∈ V and a = b, if they have a mutual blocking relation, then ab is clean in BS(V ). Here ab is the region defined by the rectangle (including its perimeter) parallel to the X and Y axes and with ab being diagonal line.

Lemma 4: ∀a, b ∈ V and R dir a w R dir ′ b in BS(V ), if R dir b R dir ′ a
is not present in BS(V ) (namely, there is no mutual blocking between a and b), then 1) for the Besh cell BC w (dir, dir ′ ) cornered at w in Q w (dir, dir ′ ), there is exactly one diagonal definition point c with respect to w, and 2) c has a blocking relation with both a and b, and 3) max(mlen( ac), mlen( bc)) < mlen( ab), where mlen(•) is the length of the longest side of the box.

B. The second step: 'Dual' of Besh

Having obtained BS(V ), we start the second step. At this step, we create the 'dual' of BS(V ) by adding edges between the creating points (points in V ) that have a blocking relation. Here term 'dual' is from an abused use of duality. It is of importance to remember that inter-vertex blocking relation is subject to the prioritized blocking policy. Formally, we define Definition 5 (HC edge creation rule): ∀a, b ∈ V and a = b, create edge ab if and only if a and b have a blocking relation.

The dual of Besh BS(V ) is composed of the given vertex set V and the added edge set. We name it Hypocomb (standing for Hypotenuse-comb) and denote it by HC(V ). The name 'Hypocomb' owns its inspiration to the fact that each edge ab due to a u b is the hypotenuse of the right triangle ∆aub.

In Fig. 1 , for example. Ray extension occurs from a toward dir in a cascaded fashion until a vertex, called terminal node, whose ray growing in direction dir is not blocked (by any other vertex) is reached. Cascaded ray extension defines a directed acyclic graph DAG(a, dir), where nodes are the vertices involved and edges imply direct ray extension relation. Define DAG(a, dir, dir) = DAG(a, dir)∪ DAG(a, dir). It spans the space enclosed by the border of V . Because direct ray extension implies blocking relation, each edge in DAG(a, dir, dir) corresponds to an edge with the same end nodes in HC(V ). As such, this DAG is mapped to a subgraph of HC(V ), denoted by M DAG (a, dir, dir), which is connected due to the reachability from a to every other node in DAG(a, dir, dir). For all a ′ ∈ V , a ′ =a and dir ′ ∈ d ir, DAG(a ′ , dir ′ , dir ′ ) must have some node(s) in common with DAG(a, dir, dir). It is due to the spanning property and perpendicularity of the two DAGs. As a consequence, M DAG (a, dir, dir) and M DAG (a ′ , dir ′ , dir ′ ) are connected. By definition, HC(V ) = a∈V (M DAG (a, dir, dir) ∪ M DAG (a, dir ′ , dir ′ )). The connectivity of HC(V ) follows.

Theorem 2: HC(V ) is planar.

Proof: Assume for the sake of contradiction ab, cd ∈ HC(V ) and they intersect. Let u be a blocking point of a and b, and let v be a blocking point of c and d. Consider the two triangles ∆aub and ∆cvd. By Lemma 2, they are both empty, that is to say, a, b / ∈ ∆cvd and c, d / ∈ ∆aub. Then the two triangles must intersect, with their hypotenuses being across. In this case, one of the catheti of ∆aub, say au, intersects with one of the catheti, say cv, of ∆cvd. Let the crossover point be w. A blocking relation between a and c occurs at w. This renders either the blocking of a and b at u or that of c and d at v invalid. A contradiction is reached.

Theorem 3: ∆(HC(V )) ≤ |V | -1.
Proof: It is obvious that ∆(HC(V )) can not be larger than |V | -1 which is the degree of the complete graph of V . We just need to show that it is possible to have ∆(HC(V )) = |V | -1. Examine a particular vertex arrangement given in Fig. 1(c), where |au| = |cu|. Any vertex on the line segment bc will be blocked by a, and thus has an incidental edge with a in the corresponding Hypocomb. If all the other vertices in V are located on bc, vertex a will have degree exactly n -1. This completes the proof.

IV. REDUCED HYPOCOMB In previous section we presented a novel planar graph, Hypocomb, which is extracted from a complete graph and has unbounded degree. In this section we simplify Hypocomb, reducing the number of edges, by applying a constrained edge creation rule (see Definition 6) at the second drawing step. We refer to the resultant simplified Hypocomb as Reduced Hypocomb and denote it by RHC(V ).

Definition 6 (RHC edge creation rule): ∀a, b ∈ V and a = b, create edge ab iff a and b have a mutual blocking relation.

Corollary 1: RHC(V ) ⊆ HC(V ).

In Fig. 1(a), only solid thick lines belong to Reduced Hypocomb. Corollary 1 is derived immediately from the RHC edge creation rule. With Algorithm 1, Reduced Hypocomb construction is straightforward and has the same complexity O(|V | 3 ) as Hypocomb construction. In the following we show that Reduced Hypocomb not only remains connected planar but also possesses the desired bounded-degree property.

Theorem 4: RHC(V ) is connected. Proof: Since RHC(V ) is a subgraph of HC(V ), the construction of RHC(V ) can be viewed an edge removal process in HC(V ), where we remove non-RHC edges one by one. Consider an arbitrary non-RHC edge ab ∈ HC(V ). By definition, a and b have no mutual blocking relation. Without loss of generality, let the inclusion of ab in HC(V ) is due to R dir a w R dir ′ b with dir ∈ T and dir ′ ∈ d ir. By Lemma 4, we have ac, bc ∈ HC(V ) where c is the unique diagonal definition point of the Besh cell cornered at w and located in Q w (dir, dir ′ )) with respect to w. If we remove ab and only ab from HC(V ), a and b remain connected via c. We call such an edge removal action 'connectivity division' and call ac and bc the results of division of ab by c. Because it is possible that ac and bc are also removed, connectivity division would not preserve connectivity unless no division loop is induced.

Below we prove that no division loop occurs. Assume for the sake of contradiction that there are division loops. Take a smallest loop where each edge appears only once. We express this loop by

u 0 v 0 w1 → u 1 v 1 w2 → u 2 v 2 w3 → • • • wn → u 0 v 0 . Let u n v n = u 0 v 0 . For i = 1, • • • , n, u i-1 v i-1 wi → u i v i indicates that u i v i is a result of the connectivity division of u i-1 v i-1 by w i , where u i ∈ {u i-1 , v i-1 } and v i = w i . We know u i-1 and v i-1 have no mutual blocking relation and R diri-1 ui-1 wi R dir ′ i-1 vi-1 (or R dir ′ i-1 vi-1 wi R diri-1 vi-1 ) for dir i-1 ∈ T and dir ′ i-1 ∈ dir i-1 . Recall that mlen( u i-1 v i-1 ) is the maximum side length of u i-1 v i-1 .
In this case, applying lemma 4 along the division loop, we have mlen( u can not hold at the same time (even if w = t, by the prioritized blocking policy).

0 v 0 ) > mlen( u 1 v 1 ) > • • • > mlen( u n-1 v n-1 ) > mlen( u 0 v 0 ), which is impossible.
Theorem 6: ∆(RHC(V )) ≤ 6.

Proof: ∀a ∈ V , there are at most 4 clean ab in BS(V ) in the four quadrants of a (one in each quadrant), and at most 4 clean ab (which reduces to ab) along the X and the Y axis respectively in the four directions. Hence a has at most 8 incidental edges in RHC(V ), 4 quadrant edges and 4 axis edges. By Lemma 5, two axis edges must be either separated by more than one quadrant edge or adjacent to each other. This constraint then lowers the upper bound to 6. The scenario of degree 6 is that a has 4 quadrant edges and 2 collinear axis edges either along the X axis or the Y axis.

V. LOCAL HYPOCOMB Till now, we have successfully bounded the degree of Hypocomb above by a small constant 6, by applying constrained edge creation rule and yet without jeopardizing its connectivity and planarity properties. Hypocomb and Reduced Hypocomb are built with complete knowledge of V and with no constraint on edge length, i.e., extracted from a complete graph of V . In this section we investigate how to build Reduced Hypocomb on UDG with limited local knowledge. UDG has the following important property (proof is in [START_REF] Li | Graph Planarization in Wireless Ad hoc Networks[END_REF]).

Lemma 6: In UDG, if two edges intersect, then one end vertex of one edge neighbors the two end vertices of the other.

Specifically, ∀a ∈ V , when we draw incidental edges for it, we merely have the position information of vertices b located in the unit disk D γ (a) of a. In this case, we propose a local edge creation rule (see Definition 7) ). Below we show that Local Hypocomb surprisingly remains connected planar and has degree bounded above by 8 (just slightly larger than the degree bound 6 of Reduced Hypocomb).

Theorem 7: LHC(V ) is connected.

Proof: We view LHC(V ) construction as an edge removal process in U DG(V ). For every ab ∈ U DG(V ) and ab / ∈ LHC(V ) (i.e., a removed edge), by definition we have ab / ∈ RHC(V N BR (a, b)). This implies either ab / ∈ HC(V N BR (a, b)) or, ab ∈ HC(V N BR (a, b)) and there exists a unique c ∈ V N BR (a, b) such that it divides ab into ac, bc ∈ HC(V N BR (a, b)) (Lemma 4). In the former case, the removal of ab does not affect the connectivity between a of b since we know HC(V N BR (a, b)) is connected (Theorem 1). In the later case, a and b remain connected (through c) from the local view of a and b after removing ab. To prove the connectivity of LHC(V ), it is sufficient to prove that local connectivity division (edge removal) actions do not generate division loop in a global sense. The loop-free property can be proved similarly as in Theorem 4. The key is to explore the stability of mlen( ab) (i.e., it is the same in any vertex's local view) and the monotonically decreasing nature of mlen( ac) and mlen( bc) relative to mlen( ab).

Lemma 7: Any two crossover edges ab, cd ∈ U DG(V ) do not belong to LHC(V ) at the same time if ac, ad ∈ U DG(V ) and one of bc and bd appears in U DG(V ).

Proof: Assume for the sake of contradiction ab, cd ∈ LHC(V ). Without loss of generality, let bc ∈ U DG(V ). This implies that c is in the same situation with respect to a as a with respect to d. By the same argument, we conclude that there exits such a vertex f around w for c (like e for a).

By simple geometry, the four vertices a, b, c and d are all neighboring e if |su| ≤ |sv|, and f otherwise. Without loss of generality, we consider |su| ≤ |sv| since the other case is symmetric. In BS(V N BR (a, b)), R dir ′ e must be blocked by a vertex g 0 at a point x on segment ee ′′ . According to the blocking rule, g 0 must be located in a square area (shaded in the figure) with e as corner and with the diagonal line defined by two other corners lying on segment aw. Trivially, all points in the square are common neighbor of a, b, c and d. Since in BS(V N BR (c, d)), g 0 does not block R dir ′ e , there must be a vertex g 1 that blocks g 0 at a point y on segment g 0 x. This vertex 2(c). It is because, otherwise, |cd| > γ (given cd intersects ab) can not belong to U DG(V ). In this case and being with the constraint ab ∈ U DG(V ), a must be located in the arc segment area defined by c ′ and d ′ . And, it must be located outside cd so that cd ∈ RHC(V N BR (c, d)). This additional restriction limits the location of a to be within the arc segment defined by the intersection points p and q of cd and arc c ′ d ′ . In Fig. 2(c), cd is shown by a dotted rectangle. The tangent of C γ (b) at s has a π/6 angle with st. The angle of the tangent at p therefore has an angle less than π/6 with pq. We have ∠acq < ∠apq < π/6 < π/4. Recall a ∈ V N BR (c, d). In right triangle ∆ca ′ a, where a ′ is the projection of a on cq, . And obviously this vertex must be located in either of the two squares with aa ′ as common edge. These two squares are between p and q due to the fact that ∠apq < π/4 and ∠pqa < π/4. Thus the mutual blocking relation of c and d is broken. This finally contradicts cd ∈ RHC(V N BR (c, d)).

|aa ′ | < |ca ′ |. This implies R dir a a ′ R dir ′ c if no other vertex blocks R dir a before it reaches R dir ′ c . It is possible some vertex m ∈ V N BR (c,
Theorem 8: LHC(V ) is planar.

Proof: Any edge in LHC(V ) is also in U DG(V ). For any pair of crossover edges ab and cd in U DG(V ), without loss of generality, let ac, ad ∈ U DG(V ) by Lemma 6. Then regardless the containment relations of bc and bd in U DG(V ), ab and cd do not appear in LHC(V ) at the same time according to Lemma 7 and 8. Thus the theorem holds.

Lemma 9: ∀dir ∈ T, dir ′ ∈ d ir, ab, ac ∈ LHC(V ), ab, ac ∈ Q a (dir, dir ′ ) and ab = ac, ∄ad ∈ LHC(V ) and ad = ab, ac such that ad ∈ Q a (dir, dir ′ ).

Proof: We first derive |bc| > γ, where γ is unit distance. It is because, otherwise, b and c would be in each other's closed common neighborhood with a, and in this case, by Lemma 5 at most one of ab and ac would belong to LHC(V ). Then c must be located in the differential area of D γ (a) and the D γ (b) in Q(dir, dir ′ ). Let s and t respectively be the intersection point of R dir a and R dir ′ a with C γ (a), as shown in Fig. 2(d). For such a residence area of c to exist, b must be in one of the shaded areas, which are defined by C γ (s) and C γ (t). Symmetrically, c must be in the other shaded area. For the sake of contradiction, assume ∃ad ∈ LHC(V ), ad = ab, ac, with dir ∈ T , then ∄ab, ac ∈ LHC(V ) and ab = ac such that ∃dir ′ ∈ d ir, ab, ac ∈ Q a (dir, dir ′ ).

Proof: Assume for the sake of contradiction that such ab and ac exist. Observe Fig. 2 Proof: It follows immediately from Lemmas 9 and 10. Lemma 9 indicates that in LHC each node has at most 2 edges in each quadrant; Lemma 10 indicates that in LHC, if a node has an axis edge, then it has at most 1 edge in each of the 2 quadrants adjacent to that edge. Thus the scenario of degree 8 is that a vertex has 2 edges in each quadrant or that it has 4 axis edges and 4 quadrant edges, one in each quadrant.

VI. NUMERICAL RESULTS

We now study the average degree of Hypocomb family graphs and their impact on FACE routing [START_REF] Bose | Routing with guaranteed delivery in ad hoc wireless networks[END_REF], in comparison with Del and GG, through extensive simulation. We run simulation experiments using a custom C simulator to build these graphs over the same random node (i.e., vertices) distribution. To do so, we compute a virtual l × l grid and place n nodes at n randomly selected unique grid points. For GG and LHC computation, a UDG is generated with a properly selected unit distance to ensure connectivity. An example construction of these graphs when n = 20 and l = 10 can be found in Fig. 3. We run FACE over each graph for a randomly picked pair of source and destination. Indeed, FACE has to be run on a planar graph only, and it was supported by GG in [START_REF] Bose | Routing with guaranteed delivery in ad hoc wireless networks[END_REF]. Below we report our numerical results, which are obtained from 1000 simulation runs with l = 20 and n varying from 20 to 300.

Figure 4(b) verifies our theoretical findings about degree bound: HC has unbounded degree while the degree of RHC and LHC is bounded above by 6 and 8, respectively. Figure 4(a) shows the average degree (reflecting how sparse or dense a graph is topologically), which as expected slowly increases with the overall number n of nodes. For RHC and LHC, it never exceeds the corresponding degree bound. We observe that their curves become flat after a turning point of n = 200, 250 respectively. Del, HC and RHC are extracted from complete graph and therefore comparable to each other. Among them Del and RHC are respectively the densest and the sparsest. GG and LHC are both local graphs and thus competitors. LHC is a bit denser than GG before the turning point (n = 250) and is increasingly sparser afterwards as GG has no degree bound. Generally speaking, the higher the average degree (i.e., the denser the network), the smaller average face size, and therefore more likely to find direct paths (composed of relatively long links though). This expectation is confirmed by our simulation results plotted in Fig. 4(c) and 4(d). Notice that for a dense UDG (n > 250), although GG is denser than LHC, they lead to almost the same FACE routing performance.

It is well-known that GG contains short edges and FACE routing suffers from long routing paths in a sparse UDG when GG is used for planarization. Our simulation reveals that FACE will benefit from replacing GG with LHC. In addition, note that using the long edges provided by LHC may help in saving energy when used in a ETE [START_REF] Elhasfi | Energy-aware Georouting with Guaranteed Delivery in Wireless Sensor Networks with Obstacles[END_REF] fashion, i.e., when reaching the next hop by following an energy weighted shortest path.

VII. CONCLUSIONS

In this paper we proposed Hypocomb family graphs and proved their planarity, connectivity and degree bound. Proofs omitted due to space limit can be found in [START_REF] Li | Graph Planarization in Wireless Ad hoc Networks[END_REF]. This work opened a new line of research. Various follow-up works are possible. From theoretical point of view, it is an interesting topic to study the spanning ratio of Hypocomb family graphs, for example. In [START_REF] Li | Graph Planarization in Wireless Ad hoc Networks[END_REF], we showed through a counter example that these graphs may not contain MST as subgraph. It is therefore also interesting to study whether or not they are lowweight graphs. A structure is called low-weight if its total edge length is within a constant factor of the total edge length of the MST [START_REF] Li | Localized low weight graph and its applications in wireless ad hoc networks[END_REF]. Another research topic is to develop graph construction algorithms more efficient than Algorithm 1.

In emerging wireless sensor and actor networks, Hypocomb and Reduced Hypocomb can be constructed among actors in a localized way, as Besh [START_REF] Li | Localized Distance-Sensitive Service Discovery in Wireless Sensor and Actor Networks[END_REF], by using directional message transmission to simulate ray drawing from each actor. Each node where blocking happens informs the sender actor about the blocking so that the latter knows about who it is blocking and whom it is blocked by. The goal is to obtain an actor overlay network bearing a planar topology so that existing data communication protocols can be run directly on it to realize, for example, actor-to-actor broadcasting, any-casting and multi-casting, which are central to actor-actor and sensor-actor coordination. However, this construction method does not produce exactly these graphs due to generally non-straight-line message transmission and thus inaccurate blocking relation, unless the underlaying network has a grid topology. A future research direction is to study and improve the performance of this construction method and eventually develop new and better distributed/localized solutions. Comparative study of Local Hypocomb and other local planar graphs is also desired.

3 Fig. 1 .

 31 Fig. 1. Illustrations for Besh and Hypocomb

a

  will stop growing only in any of the following cases: 1) |au| > |bu|; 2) |au| = |bu|, dir ′ ∈ d ir and dir = north or south; 3) |au| = |bu| and dir ′ = dir.

  Examples of blocking case 2 are b u a, a t b and h w a. An example of blocking case 3 is b s c. Notice that |bs| = |cs| < |ds| = |hs|. By the blocking rule, d (similarly, h) can be blocked by all the other three vertices at s. To reduce this ambiguity, we define over the blocking rule the following important prioritized blocking policy, by which only the blocking from b and c is recognized at s in Fig. 1(a). Definition 2 (Prioritized blocking): ∀a, b, c ∈ V and a = b = c, when a u b and c u b are both possible for the same u, b is considered being blocked by c rather than a. Definition 3 (Quadrant): Given a point a, ∀dir ∈ T and ∀dir

Algorithm 1 4 :

 14 Blocking-Detection(V, a, dir) Require: a ∈ V and dir ∈ T 1: dist := ∞ 2: Let dir ′ and dir ′′ be the two elements in dir 3: c := arg(min b∈V,b =a,b∈R dir a |ab|) if c = null then 5: dist := 1 2 |ac| 6: d := mid point of a and c 7:

1 :

 1 (a), Hypocomb edges are drawn in thick links. By the HC edge creation rule and Algorithm 1, we can trivially build Hypocomb in O(|V | 3 ) time. Below we analyze the connectivity, planarity and degree bound of Hypocomb. Theorem HC(V ) is connected. Proof: For all a, b ∈ V and dir, dir ′ ∈ T , if R dir ′ we say R dir b is an extension of R dir a . A ray has at most 2 extensions. In Fig. 1(a), R south d is extended by both R south c and R south b

Theorem 5 :

 5 RHC(V ) is planar. Proof: It follows from Theorem 2 and Corollary 1. Lemma 5: ∀ab, ac ∈ RHC(V ), creation of ab is due to R dir a u R dir b and ac is due to R dir ′ a w R dir ′ c with dir ∈ T and dir ′ ∈ d ir, ∄ad ∈ RHC(V ) such that ad ∈ Q a (dir, dir ′ ). Proof: Assume for the sake of contradiction that such ad exists. By definition, a and d must have a mutual blocking relation. Without loss of generality, let R dir a u R dir ′ d and symmetrically R dir d w R dir ′ a for some u and w. However R dir d w R dir ′ a and R dir ′ a t R dir ′ c

  , which adds ab according to its inclusion in the Reduced Hypocomb graph of the closed common neighbor set of a and b. And obviously, the creation decision on edge ab is symmetric for a and b. Definition 7 (LHC edge creation rule): ∀a ∈ V , b ∈ V N BR (a) and a = b, create edge ab if and only if ab ∈ RHC(V N BR (a, b)). This local edge creation rule is dependent on 1-hop neighbors position information only. It gives the graph construction process strictly localized feature. The resultant Hypocomb variant is therefore called Local Hypocomb, and denoted by LHC(V ). We know |V N BR (a, b)| ≤ d(a) + 1, where d(a) is the degree of a in UDG. Using Algorithm 1 each node a is able to build RHC(V N BR (a, b)) for each neighbor b in O(d(a) 3 ) time, and the total cost (per node) of determining LHC edges is therefore O(d(a) 4 ). Because d(a) ≤ ∆(U DG(V )), an upper bound of the computation cost on each node for Local Hypocomb construction is O(∆(U DG(V )) 4

10 Fig. 2 .

 102 By definition 7, ab ∈ RHC(V N BR (a, b)). From Lemma 3, ab is clean in BS(V N BR (a, b)) and thus c / ∈ ab; likewise, cd is clean in BS(V N BR (c, d)) and a / ∈ cd. Under these (a) Lemma 7 -Case 3Illustrations for Local Hypocomb constraints, by varying the relative position of b and d to cd and ab we obtain the following cases: (1) b ∈ cd and d / ∈ ab; (2) d ∈ ab and b / ∈ cd; (3) b / ∈ cd and d / ∈ ab. Obviously, b ∈ cd and d ∈ ab can not hold at the same time; thus this case is not in our consideration. In case (1), |bd| ≤ |cd| because ∠cbd is not acute in triangle ∆cbd. In case (2), |bd| ≤ |ab| because ∠adb is not acute in ∆adb. In these two cases, d ∈ V N BR (a, b) and b ∈ V N BR (c, d), and obviously ab and cd do not appear in LHC(V ) at the same time, contradicting our assumption. Case (3) has two sub-cases. Below we derive a contradiction from both of them. Case 3.1 (Fig. 2(a)) : We first put ourselves under the condition: a would block d at u if V contained only a, d. We have |au| ≤ |du| by the blocking rule. Because a ∈ V N BR (c, d) and cd ∈ RHC(V N BR (c, d)), R dir a must be blocked by a vertex e 0 in BS(V N BR (c, d)) before reaching segment dp. If e 0 has a projection e ′ 0 on dp, then by Lemma 1 we have |e 0 e ′ 0 | ≤ |de ′ 0 | and are facing the same situation as with a, and therefore the same argument can be made for e 0 . By these means, we are presented a blocking chain in BS(V N BR (c, d)) that ends at a vertex e n (n ≥ 0) that has no projection on dp. Let e ′′ i be the projection of e i on R dir a . By the blocking rule and Lemma 1, we easily have |e i e ′′ i | ≤ |ae ′′ i | < |au| < |du| for i = 0, . . . , n. Since |du| > |au|, e n can not be around vertex d but point p. For ease of presentation, let e = e n and e ′′ = e ′′ n , as shown in the figure. Notice |cr| = |pu| ≤ |ee ′′ | ≤ |ae ′′ | < |au| < |ar|.

8 :

 8 g 1 is again located in the square area and thus neighboring a, b, c and d. The argument can be made iteratively, alternate between BS(V N BR (a, b) and BS(V N BR (c, d), giving us a set of vertices g 0 , . . . , g m all located in the square area and neighboring a, b, c and d. Hence the blocking relations among them appear in both BS(V N BR (a, b) and BS(V N BR (c, d), and ab and cd can not appear in LHC(V ) at the same time, contradicting our assumption. We now consider the opposite condition: d would block a if a and b were the only vertices in V . We have |au| ≥ |du| (≥ |ds|) and |bd| < |ds| + |bs| ≤ |au| + |bs| = |bt| ≤ |ab|, i.e., d ∈ V N BR (a, b). Then the only situation worth investigation includes the combination of the following conditions: b would block d at s if V contained only b, d (|ds| ≥ |bs| ≥ |bv|), c would block b at v if V contained only b, c (|bv| ≥ |cv| ≥ |cr|), and a would block c at r if V contained only a, c (|cr| ≥ |ar| ≥ |au|). It is because any other situation is equivalent to the previous one after vertex renaming and therefore leads to a similar contradiction. Under this circumstance, we have |au| = |ds| = |bv| = |cr| = |au| and |su| = |sv| = 0. Then either a and b or, c and d, do not have blocking relation according to the prioritized blocking policy, a contradiction. Case 3.2 (Fig. 2(b)) : As in case (3.1), we first investigate under the condition that a would block d at u if V contained only a, d. Then with respect to a, we may conclude a similar set of vertices e 0 , . . . , e n ∈ V N BR (c, d) in order to enable cd ∈ RHC(V N BR (c, d)). Among them, e n is around p. Let e be the one closest to segment bt and on the same side as p and e ′′ the projection of e on R dir a . In right triangle ∆ee ′′ a, |ae| 2 = |ae ′′ | 2 + |ee ′′ | 2 < |au| 2 + |du| 2 = |ad| 2 . Let j be the intersection point of ae and bt and k the intersection point of ee ′′ and bt. Trivially, |ek| ≤ |kj|. In right triangle ∆ekb, |be| 2 = |ek| 2 + |kb| 2 ≤ |kj| 2 + |kb| 2 ≤ (|tj| + |kj| + |kb|) 2 + |at| 2 = |ab| 2 . Hence e ∈ V N BR(a, b). By the same technique we may derive a similar contradiction as in case (3.1). Likewise, we can derive a contradiction under the condition that c would block b at v if V contained only b, c. We only remain to consider the combination of the opposites of the two conditions, where |au| ≥ |du| and |cv| ≥ |bv|. Observe |bt| = |au| + |sv| + |bv|, |dp| = |cv| + |at| + |du|, |ds| = |at| + |du| and |cp| = |sv|. Let δ 1 = |bd| 2 -|ab| 2 and δ 2 = |bd| 2 -|cd| 2 . In right triangle ∆cpd, |cd| 2 = |dp| 2 + |cp| 2 = (|cv| + |at| + |du|) 2 + |sv| 2 . In right triangle ∆bsd, |bd| 2 = |bs| 2 + |ds| 2 = (|bv| + |sv|) 2 + (|at| + |du|) 2 . In right triangle ∆atb, |ab| 2 = |at| 2 +|bt| 2 = |at| 2 +(|au|+|sv|+|bv|) 2 . Then δ 1 = |du| 2 -|au| 2 + 2|du||at| -2|au|(|sv| + |bv|) and δ 2 = |bv| 2 -|cv| 2 +2|bv||sv|-2|cv|(|at|+|du|). Recall |du| ≤ |au| and |bv| ≤ |cv|. If |at| ≤ |sv| + |bv|, then δ 1 ≤ 0 (i.e., |bd| ≤ |ab|); otherwise, |sv| < |at|-|bv| < |at|+|du| and thus δ 2 ≤ 0 (i.e., |bd| ≤ |cd|). This implies bd ∈ U DG(V ). Thus current situation is equivalent to the first situation examined (after switching the name of a and d and other vertex remaining), and we may derive a contradiction similarly. Lemma Any two crossover edges ab, cd ∈ U DG(V ) do not belong to LHC(V ) at the same time if ac, ad ∈ U DG(V ) and bc, bd / ∈ U DG(V ). Proof: Clearly, cd must intersect the unit circle C γ (b) of b as, otherwise, ab / ∈ U DG(V ). Let c ′ and d ′ be intersection points of cd and C γ (b). |cd| > |c ′ d ′ |. Then c ′ and d ′ must be on arc st of π/3 of C γ (b), with chord st parallel to cd, as shown in Fig.

  (d), which depicts Q a (dir, dir ′ ) in a generic way. Vertices b and c must be located separately in the two shaded areas, as we analyzed in the proof of Lemma 9. Without loss of generality, let b be in zone 1 and c in zone 2. Obviously, ∠uab < π/4. To ensure the blocking relation R dir a u R dir d , we must have R dir a u ′ R dir ′ b with u ′ ∈ au (not shown in the figure). This implies |bu ′ | ≥ |au ′ | and thus ∠uab ≥ π/4. A contradiction is reached. Theorem 9: ∆(LHC(V )) ≤ 8

  Thus we perform further check on its this blocking potential. In Lines 7 and 8 we compute the sets S 1 and S 2 of vertices (together with the corresponding blocking points) that have the potential to orthogonally block R dir c . The computation can be finished in O(|V |) time. In the light of Lemma 1, we in Line 9 reduce S 1 to S ′ 1 by removing the vertices that are not able to block R dir Lines 10 reduces S 2 to S ′ 2 in a similar way in O(|V | 2 ) time. If the union of S ′ 1 and S ′

	is empty (namely, no vertex blocks R dir c	2 orthogonally), then
	we can conclude that R dir c	is able to block R dir a . Otherwise,
	R dir c	will not block R dir a , and R dir a will reach the point where
	R dir c

3, we find the vertex c that is located on R dir a and nearest to a in O(|V |) time. Assume that R dir a is not blocked by anybody else. This vertex c will collinearly block R dir a at the mid point d of a and c if R dir c is not orthogonally blocked before reaching d.

c before d, due to being blocked by other vertices in S 1 . The computation time is at most O(|V | 2 ).

  d) blocks a such that this blocking relation does not exist. However, in this case, m will block R dir ′ c if no other vertex blocks m by Lemma 1. The same argument can be made iteratively. Since we have a finite number of vertices in V N BR (c, d), finally a vertex will block R dir ′

	c

  Graphs over a same node distribution and ad∈ Q a (dir, dir ′ ). Then b / ∈ D γ (d) and c / ∈ D γ (d).That is, vertex d must be located in the intersection area of the two shaded areas, which however does not exist.

	92	95			92	95			92	95
	82	83	88	89	82	83	88	89	82	83	88	89
		75	76			75	76			75	76
			66				66				66
	42	45	48		42	45	48		42	45	48
	31	34	37	39	31	34	37	39	31	34	37	39
		23				23				23
	11				11				11
		4		9		4		9		4		9
		(a) Del			(b) UDG			(c) GG
	92	95			92	95			92	95
	82	83	88	89	82	83	88	89	82	83	88	89
		75	76			75	76			75	76
			66				66				66
	42	45	48		42	45	48		42	45	48
	31	34	37	39	31	34	37	39	31	34	37	39
		23				23				23
	11				11				11
		4		9		4		9		4		9
		(d) HC			(e) RHC			(f) LHC
	Fig. 3. Lemma 10: ∀ad ∈ LHC(V ), if the creation of ad is due to R dir a u R dir d


Mohammed Daoud
email: mdaoud@hotmail.com

Maurice Robert Kibler
email: m.kibler@ipnl.in2p3.fr

Phase operators, phase states and vector phase states for SU 3 and SU 2,1

come

Phase operators, phase states and vector phase states

for SU(3) and SU(2,1)

INTRODUCTION AND MOTIVATIONS

It is well known that defining an Hermitian (or unitary, by means of an exponentiation trick) phase operator for the Fock space of the isotropic harmonic oscillator and, more generally, for an infinite-dimensional Hilbert space is not an easy problem. [START_REF] Susskind | Quantum mechanical phase and time operator[END_REF] Pegg and Barnett [START_REF] Pegg | Phase properties of the quantized single-mode electromagnetic field[END_REF] solved it by replacing the oscillator algebra by a truncated oscillator algebra and thus were able to give a description of the phase properties of quantum states for the single modes of the electromagnetic field. In this spirit, Vourdas [START_REF] Vourdas | SU(2) and SU(1, 1) phase states[END_REF] introduced phase operators and phase states (i.e., eigenvectors of a phase operator) for su 2 and su 1,1 ; for the su 1,1 Lie algebra, he noticed that the infinite-dimensional character of the representation space prevents to define a unitary phase operator. Phase operators and phase states for other symmetries were also studied. In particular, Klimov et al. [START_REF] Klimov | Quantum phases of a qutrit[END_REF] obtained phase states for some specific representations of su 3 .

Recently, a generalized oscillator algebra A κ , depending on a real parameter κ, was introduced to cover the cases of Lie algebras su 2 (for κ < 0) and su 1,1 (for κ > 0) as well as Weyl-Heisenberg algebra h 4 (for κ = 0). [START_REF] Atakishiyev | SU(2) and SU(1,1) approaches to phase operators and temporally stable phase states: Applications to mutually unbiased bases and discrete Fourier transforms[END_REF][START_REF] Daoud | Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems[END_REF] Temporally stable phase states were defined as eigenstates of phase operators for finite-dimensional (κ < 0) and infinitedimensional representations (κ ≥ 0) of the A κ algebra. [START_REF] Daoud | Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems[END_REF] In the finite-dimensional case, corresponding either to κ < 0 or to κ ≥ 0 with truncation, temporally stable phase states proved to be useful for deriving mutually unbiased bases. [START_REF] Atakishiyev | SU(2) and SU(1,1) approaches to phase operators and temporally stable phase states: Applications to mutually unbiased bases and discrete Fourier transforms[END_REF][START_REF] Daoud | Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems[END_REF] Such bases play an important role in quantum information and quantum cryptography.

In this paper, we introduce an algebra, noted A κ (2), which generalizes the A κ algebra. For κ < 0, this new algebra is similar to that considered in the seminal work of Palev [START_REF] Palev | Lie algebraical aspects of quantum statistics. Unitary quantization (Aquantization)[END_REF][START_REF] Palev | Jacobson generators, Fock representations and statistics of sl(n + 1)[END_REF] in the context of A n -statistics. The A κ (2) algebra allows to give an unified treatment of algebras su 3 (for κ < 0), su 2,1 (for κ > 0) and h 4 ⊗ h 4 (for κ = 0). When we started this work, our aim was to study in an unified way: (i) phase operators for su 3 , su 2,1 and h 4 ⊗ h 4 and (ii) the corresponding phase states. We discovered, for κ < 0, that phase states can be defined only for partitions of the relevant Hilbert spaces and that a global definition of phase states requires the introduction of vector phase states, a concept that is closely related to that of vector coherent states. The notion of vector coherent states was strongly investigated by Hecht [START_REF] Hecht | The vector coherent state method and its application to problems of higher symmetries[END_REF] and Zhang et al. [START_REF] Zhang | Coherent states: theory and some applications[END_REF] at the end of the nineties. This notion was subsequently developed in Refs. [1, 2, 7, 22] with applications to quantum dynamical systems presenting degeneracies. In particular, the authors of Ref. [2] defined a vectorial generalization of the Gazeau-Klauder coherent states [START_REF] Gazeau | Coherent states for systems with discrete and continuous spectrum[END_REF] leading to vector coherent states. Recently, this notion of vector coherent states was extensively investigated (see the works in Refs. [1, 22]).

This paper is organized as follows. The A κ (2) generalized algebra is introduced in Section 2. We then define a quantum system associated with this algebra and generalizing the two-dimensional harmonic oscillator. In Section 3, phase operators and temporally stable vector phase states for the A κ (2) algebra with κ < 0 are constructed. The phase operators and the corresponding temporally stable phase states for A κ (2) with κ ≥ 0 are presented in Section 4. Section 5 deals with a truncation of the A κ (2) algebra, with κ ≥ 0, necessary in order to get unitary phase operators. In Section 6, we show how a quantization of the temporality parameter occurring in the phase states for A κ (2) with κ < 0 can lead to mutually unbiased bases.

GENERALIZED OSCILLATOR ALGEBRA

A κ (2)

The algebra

We first define the A κ (2) algebra. This algebra is generated by six linear operators a - i , a + i and N i with i = 1, 2 satisfying the commutation relations

[a - i , a + i ] = I + κ(N 1 + N 2 + N i ), [N i , a ± j ] = ±δ i,j a ± i , i, j = 1, 2 (1)
and

[a ± i , a ± j ] = 0, i = j, (2)
complemented by the triple relations

[a ± i , [a ± i , a ∓ j ]] = 0, i = j. (3)
In Eq. ( 1), I denotes the identity operator and κ is a deformation parameter assumed to be real.

Note that the A κ algebra introduced in Ref. [5] formally follows from

A κ (2) by omit- ting the relation [a - 2 , a + 2 ] = I + κ(N 1 + 2N 2 )
and by taking

a - 2 = a + 2 = N 2 = 0, a - 1 = a -, a + 1 = a + , N 1 = N
in the remaining definitions of A κ (2). Therefore, generalized oscillator algebra A κ in

Ref. [5] should logically be noted A κ (1).

For κ = 0, the A 0 (2) algebra is nothing but the algebra for a two-dimensional isotropic harmonic oscillator and thus corresponds to two commuting copies of the Weyl-Heisenberg algebra h 4 .

For κ = 0, the A κ (2) algebra resembles the algebra associated with the so-called A n -statistics (for n = 2) which was introduced by Palev [START_REF] Palev | Lie algebraical aspects of quantum statistics. Unitary quantization (Aquantization)[END_REF] and further studied from the microscopic point of view by Palev and Van der Jeugt. [START_REF] Palev | Jacobson generators, Fock representations and statistics of sl(n + 1)[END_REF] In this respect, let us recall that A n -statistics is described by the sl n+1 Lie algebra generated by n pairs of creation and annihilation operators (of the type of the a + i and a - i operators above) satisfying usual commutation relations and triple commutation relations. Such a presentation of sl n+1 is along the lines of the Jacobson approach according to which the A n Lie algebra can be defined by means of 2n, rather than n(n + 2), generators satisfying commutation relations and triple commutation relations. [START_REF] Jacobson | Lie and Jordan triple systems[END_REF] These 2n Jacobson generators correspond to n pairs of creation and annihilation operators. In our case, the A κ (2) algebra for κ = 0, with two pairs of Jacobson generators ((a + i , a - i ) for i = 1, 2), can be identified to the Lie algebras su 3 for κ < 0 and su 2,1 for κ > 0. This can be seen as follows.

Let us define a new pair (a + 3 , a - 3 ) of operators in terms of the two pairs (a + 1 , a - 1 ) and (a + 2 , a - 2 ) of creation and annihilation operators through

a + 3 = [a + 2 , a - 1 ], a - 3 = [a + 1 , a - 2 ]. (4)
Following the trick used in Ref. [3] for the A κ (1) algebra, let us introduce the operators

E +α = 1 |κ| a + α , E -α = 1 |κ| a - α , α = 1, 2, 3 H 1 = 1 2κ [I + κ(2N 1 + N 2 )], H 2 = 1 2κ [I + κ(2N 2 + N 1 )]
with κ = 0. It can be shown that the set {E ±α ; H i : α = 1, 2, 3; i = 1, 2} spans su 3 for κ < 0 and su 2,1 for κ > 0.

Representation of A κ (2)

We now look for a Hilbertian representation of the A κ (2) algebra on a Hilbert-Fock space F κ of dimension d with d finite or infinite. Let

{|n 1 , n 2 : n 1 , n 2 = 0, 1, 2, . . .} be an orthonormal basis of F κ with n 1 , n 2 |n ′ 1 , n ′ 2 = δ n 1 ,n ′ 1 δ n 2 ,n ′ 2 .
Number operators N 1 and N 2 are supposed to be diagonal in this basis, i.e.,

N i |n 1 , n 2 = n i |n 1 , n 2 , i = 1, 2 (5)
while the action of the creation and annihilation operators a ± 1 and a ± 2 is defined by

a + 1 |n 1 , n 2 = F 1 (n 1 + 1, n 2 )e -i[H(n 1 +1,n 2 )-H(n 1 ,n 2 )]ϕ |n 1 + 1, n 2 , (6)
a - 1 |n 1 , n 2 = F 1 (n 1 , n 2 )e +i[H(n 1 ,n 2 )-H(n 1 -1,n 2 )]ϕ |n 1 -1, n 2 , a - 1 |0, n 2 = 0 (7)
and

a + 2 |n 1 , n 2 = F 2 (n 1 , n 2 + 1)e -i[H(n 1 ,n 2 +1)-H(n 1 ,n 2 )]ϕ |n 1 , n 2 + 1 , (8)
a - 2 |n 1 , n 2 = F 2 (n 1 , n 2 )e +i[H(n 1 ,n 2 )-H(n 1 ,n 2 -1)]ϕ |n 1 , n 2 -1 , a - 2 |n 1 , 0 = 0. (9)
In Eqs. ( 6)-( 9), ϕ is an arbitrary real parameter and the positive valued functions F 1 :

N 2 → R + , F 2 : N 2 → R + and H : N 2 → R + are such that H = F 1 + F 2 .
It is a simple matter of calculation to check that ( 5)-( 9) generate a representation of the A κ (2) algebra defined by ( 1)-(3) provided that F 1 (n 1 , n 2 ) and F 2 (n 1 , n 2 ) satisfy the recurrence relations

F 1 (n 1 + 1, n 2 ) -F 1 (n 1 , n 2 ) = 1 + κ(2n 1 + n 2 ), F 1 (0, n 2 ) = 0 (10)
F 2 (n 1 , n 2 + 1) -F 2 (n 1 , n 2 ) = 1 + κ(2n 2 + n 1 ), F 2 (n 1 , 0) = 0. ( 11
)
The solutions of Eqs. (10) and (11) are

F i (n 1 , n 2 ) = n i [1 + κ(n 1 + n 2 -1)], i = 1, 2. (12)
To ensure that the structure functions F 1 and F 2 be positive definite, we must have

1 + κ(n 1 + n 2 -1) > 0, n 1 + n 2 > 0, (13)
a condition to be discussed according to the sign of κ. In the representation of A κ (2) defined by Eqs. ( 5)- (13), creation (annihilation) operators a + i (a - i ) and number operators N i satisfy the Hermitian conjugation relations

a - i † = a + i , (N i ) † = N i , i = 1, 2
as for the two-dimensional oscillator. The d dimension of the representation space F κ can be deduced from condition (13).

Two cases need to be considered according to as κ ≥ 0 or κ < 0.

• For κ ≥ 0, Eq. ( 13) is trivially satisfied so that the d dimension of F κ is infinite. This is well known in the case κ = 0 which corresponds to a two-dimensional isotropic harmonic oscillator. For κ > 0, the representation corresponds to the symmetric discrete (infinite-dimensional) irreducible representation of the SU 2,1 group.

• For κ < 0, there exists a finite number of states satisfying condition (13). Indeed, we have

n 1 + n 2 = 0, 1, . . . , E(- 1 κ ),
where E(x) denotes the integer part of x. In the following, we shall take -1/κ integer when κ < 0. Consequently, for κ < 0 the d dimension of the finite-dimensional space

F κ is d = 1 2 (k + 1)(k + 2), k = - 1 κ ∈ N * . ( 14
)
We know that the dimension d(λ, µ) of the irreducible representation (λ, µ) of SU 3 is given by

d(λ, µ) = 1 2 (λ + 1)(µ + 1)(λ + µ + 2), λ ∈ N, µ ∈ N.
Therefore, the finite-dimensional representation of A κ (2) defined by ( 5)-( 13) with -1/κ = k ∈ N * corresponds to the irreducible representation (0, k) or its adjoint (k, 0) of SU 3 .

Generalized oscillator Hamiltonian

Since the A κ (2) algebra can be viewed as an extension of the two-dimensional oscillator algebra, it is natural to consider the a + 1 a - 1 + a + 2 a - 2 operator as an Hamiltonian associated with A κ (2). The action of this operator on the space F κ is given by

(a + 1 a - 1 + a + 2 a - 2 )|n 1 , n 2 = [F 1 (n 1 , n 2 ) + F 2 (n 1 , n 2 )]|n 1 , n 2 = (n 1 + n 2 )[1 + κ(n 1 + n 2 -1)]|n 1 , n 2 or (a + 1 a - 1 + a + 2 a - 2 )|n 1 , n 2 = H(n 1 , n 2 )|n 1 , n 2 .
Thus, the a + 1 a - 1 + a + 2 a - 2 Hamiltonian can be written

a + 1 a - 1 + a + 2 a - 2 = H, with H ≡ H(N 1 , N 2 ) = (N 1 + N 2 )[1 + κ(N 1 + N 2 -1)]
modulo its action on F κ . The H Hamiltonian is clearly a nonlinear extension of the Hamiltonian for the twodimensional isotropic harmonic oscillator. The eigenvalues

λ n = n[1 + κ(n -1)], n = n 1 + n 2 , n 1 ∈ N, n 2 ∈ N
of H can be reduced for κ = 0 to the energies n of the two-dimensional oscillator (up to additive and multiplicative constants). For κ = 0, the degeneracy of the λ n level is n + 1

and coincides with the degeneracy of the n level corresponding to κ = 0.

PHASE OPERATORS AND PHASE STATES FOR A κ (2)

WITH κ < 0

Phase operators in finite dimension

The E 1d and E 2d phase operators

For κ < 0 the finite-dimensional space F κ is spanned by the basis

{|n 1 , n 2 : n 1 , n 2 ranging | n 1 + n 2 ≤ k}.
This space can be partitioned as

F κ = k l=0 A κ,l ,
where A κ,l is spanned by {|n, l : n = 0, 1, . . . , k -l}.

We have

dim A κ,l = k -l + 1
so that ( 6)-( 7) must be completed by

a + 1 |k -l, l = 0,
which can be deduced from the calculation of k -l, l|a - 1 a + 1 |k -l, l . The operators a + 1 and a - 1 leave each subspace A κ,l invariant. Then, it is convenient to write

a ± 1 = k l=0 a ± 1 (l),
with the actions

a + 1 (l)|n, l ′ = δ l,l ′ F 1 (n + 1, l)e -i[H(n+1,l)-H(n,l)]ϕ |n + 1, l , a + 1 (l)|k -l ′ , l ′ = 0, a - 1 (l)|n, l ′ = δ l,l ′ F 1 (n, l)e +i[H(n,l)-H(n-1,l)]ϕ |n -1, l , a - 1 (l)|0, l ′ = 0,
which show that a + 1 (l) and a - 1 (l) leave A κ,l invariant. Let us now define the E 1d operator by

E 1d |n 1 , n 2 = e i[H(n 1 ,n 2 )-H(n 1 -1,n 2 )]ϕ |n 1 -1, n 2 , 0 ≤ n 1 + n 2 ≤ k, n 1 = 0 and E 1d |0, n 2 = e i[H(0,n 2 )-H(k-n 2 ,n 2 )]ϕ |k -n 2 , n 2 , 0 ≤ n 2 ≤ k, n 1 = 0.
Thus, it is possible to write

a - 1 = E 1d F 1 (N 1 , N 2 ) ⇔ a + 1 = F 1 (N 1 , N 2 )(E 1d ) † . ( 15
)
The E 1d operator can be developed as

E 1d = k l=0 E 1d (l), with E 1d (l)|n, l ′ = δ l,l ′ e i[H(n,l)-H(n-1,l)]ϕ |n -1, l , n = 0, (16)
E 1d (l)|0, l ′ = δ l,l ′ e i[H(0,l)-H(k-l,l)]ϕ |k -l, l , n = 0. (17)
Operator E 1d (l) leaves A κ,l invariant and satisfies

E 1d (l)(E 1d (l ′ )) † = (E 1d (l ′ )) † E 1d (l) = δ l,l ′ k-l n=0 |n, l n, l|.
Consequently, we obtain

E 1d (E 1d ) † = (E 1d ) † E 1d = k l=0 (E 1d (l)) † E 1d (l) = k l=0 k-l n=0
|n, l n, l| = I, which shows that E 1d is unitary. Therefore, Eq. ( 15) constitutes a polar decomposition of a - 1 and a + 1 . Similar developments can be obtained for a - 2 and a + 2 . We limit ourselves to the main results concerning the decomposition

a - 2 = E 2d F 2 (N 1 , N 2 ) ⇔ a + 2 = F 2 (N 1 , N 2 )(E 2d ) † .
In connection with this decomposition, we use the partition

F κ = k l=0 B κ,l ,
where the B κ,l subspace, of dimension k -l + 1, is spanned by the basis {|l, n : n = 0, 1, . . . , k -l}.

We can write

E 2d = k l=0 E 2d (l),
where the E 2d (l) operator satisfies

E 2d (l)|l ′ , n = δ l,l ′ e i[H(l,n)-H(l,n-1)]ϕ |l, n -1 , n = 0, (18)
E 2d (l)|l ′ , 0 = δ l,l ′ e i[H(l,0)-H(l,k-l)]ϕ |l, k -l , n = 0, (19)
and

E 2d (l)(E 2d (l ′ )) † = (E 2d (l ′ )) † E 2d (l) = δ l,l ′ k-l n=0 |l, n l, n|.
This yields

E 2d (E 2d ) † = (E 2d ) † E 2d = k l=0 (E 2d (l)) † E 2d (l) = I
and the operator E 2d , like E 1d , is unitary.

The E 3d phase operator

Let us go back to the pair (a + 3 , a - 3 ) of operators defined by (4) in terms of the pairs (a + 1 , a - 1 ) and (a + 2 , a - 2 ). The action of a + 3 and a - 3 on F κ follows from ( 6)-( 9). We get

a + 3 |n 1 , n 2 = -κ n 1 (n 2 + 1)|n 1 -1, n 2 + 1 , a - 3 |n 1 , n 2 = -κ (n 1 + 1)n 2 |n 1 + 1, n 2 -1 .
From Eqs. ( 3) and ( 4), it is clear that the two pairs (a + 1 , a - 1 ) and (a + 2 , a - 2 ) commute when κ = 0. We thus recover that the A 0 (2) algebra corresponds to a two-dimensional harmonic oscillator.

Here, it is appropriate to use the partition

F κ = k l=0 C κ,l , (20)
where the subspace C κ,l , of dimension l + 1 (but not k -l + 1 as for A κ,l and B κ,l ), spanned by the basis {|l -n, n : n = 0, 1, . . . , l} is left invariant by a + 3 and a - 3 . Following the same line of reasoning as for E 1d and E 2d , we can associate an operator E 3d with the ladder operators a + 3 and a - 3 . We take operator E 3d associated with the partition (20) such that

a - 3 = E 3d F 3 (N 1 , N 2 ) ⇔ a + 3 = F 3 (N 1 , N 2 )(E 3d ) † ,
where

F 3 (N 1 , N 2 ) = -κ (N 1 + 1)N 2 .
The E 3d operator reads

E 3d = k l=0 E 3d (l),
where E 3d (l) can be taken to satisfy

E 3d (l)|l ′ -n, n = δ l,l ′ |l -n + 1, n -1 , n = 0, E 3d (l)|l ′ , 0 = δ l,l ′ |0, l , n = 0.
Finally, we have

E 3d (l)(E 3d (l ′ )) † = (E 3d (l ′ )) † E 3d (l) = δ l,l ′ l n=0 |l -n, n l -n, n|.
As a consequence, we obtain

E 3d (E 3d ) † = (E 3d ) † E 3d = k l=0 (E 3d (l)) † E 3d (l) = I,
a result that reflects the unitarity property of E 3d .

The E d phase operator

Operators E 1d (l), E 2d (l) and E 3d (l), defined for κ < 0 as components of the operators E 1d , E 2d and E 3d , leave invariant the sets A κ,l , B κ,l and C κ,l , respectively. Therefore, operators E 1d , E 2d and E 3d do not connect all elements of F κ , i.e., a given element of F κ cannot be obtained from repeated applications of E 1d , E 2d and E 3d on an arbitrary element of F κ .

We now define a new operator E d which can connect (by means of repeated applications) any couple of elements in the d-dimensional space F κ corresponding to κ < 0. Let this global operator be defined via the action E d |n, l = e i[H(n,l)-H(n-1,l)]ϕ |n -1, l , n = 1, 2, . . . , k -l, l = 0, 1, . . . , k (21) and the boundary actions

E d |0, l = e i[H(0,l)-H(k-l+1,l-1)]ϕ |k -l + 1, l -1 , l = 1, 2, . . . , k (22)
E d |0, 0 = e i[H(0,0)-H(0,k)]ϕ |0, k . (23)
The E d operator is obviously unitary.

By making the identification

Φ1 2 l(2k-l+3)+n ≡ |n, l , n = 0, 1, . . . , k -l, l = 0, 1, . . . , k,
the set

{Φ j : j = 0, 1, . . . , d -1}
constitutes a basis for the d-dimensional Fock space F κ . Then, the various sets A κ,l can be rewritten as

A κ,0 : {Φ 0 , Φ 1 , . . . , Φ k-1 , Φ k } A κ,1 : {Φ k+1 , Φ k+2 , . . . , Φ 2k } . . . A κ,k : {Φ d-1 }.
Repeated applications of E d on the vectors Φ j with j = 0, 1, . . . , d-1 can be summarized by the following cyclic sequence

E d : Φ d-1 → Φ d-2 → . . . → Φ 1 → Φ 0 → Φ d-1 → etc.
The E d operator thus makes it possible to move inside each A κ,l set and to connect the various sets according to the sequence

E d : A κ,k → A κ,k-1 → . . . → A κ,0 → A κ,k → etc.
Similar results hold for the partitions of F κ in B κ,l or C κ,l subsets.

Phase states in finite dimension

3.2.1 Phase states for E 1d (l) and E 2d (l)

We first derive the eigenstates of E 1d (l). For this purpose, let us consider the eigenvalue equation

E 1d (l)|z l = z l |z l , |z l = k-l n=0 a n z n l |n, l , z l ∈ C.
Using definition ( 16)-( 17), we obtain the following recurrence relation for the coefficients a n

a n = e -i[H(n,l)-H(n-1,l)]ϕ a n-1 , n = 1, 2, . . . , k -l with a 0 = e -i[H(0,l)-H(k-l,l)]ϕ a k-l
and the condition

(z l ) k-l+1 = 1.
Therefore, we get a n = e -i[H(n,l)-H(0,l)]ϕ a 0 , n = 0, 1, . . . , k -l and the complex variable z l is a root of unity given by

z l = q m l , m = 0, 1, . . . , k -l,
where

q l = exp 2πi k -l + 1 ( 24
)
is reminiscent of the deformation parameter used in the theory of quantum groups. The a 0 constant can be obtained, up to a phase factor, from the normalization condition z l |z l = 1. We take

a 0 = 1 √ k -l + 1 e -iH(0,l)ϕ , (25)
where the phase factor is chosen in order to ensure temporal stability of the |z l state. Finally, we arrive at the following normalized eigenstates of E 1d (l)

|z l ≡ |l, m, ϕ = 1 √ k -l + 1 k-l n=0
e -iH(n,l)ϕ q mn l |n, l .

The |l, m, ϕ states are labeled by the parameters l ∈ {0, 1, . . . , k}, m ∈ Z/(k -l + 1)Z and ϕ ∈ R. They satisfy

E 1d (l)|l, m, ϕ = e iθm |l, m, ϕ , θ m = m 2π k -l + 1 , m = 0, 1, . . . , k -l, (27)
which shows that E 1d (l) is a phase operator.

The phase states |l, m, ϕ have remarkable properties:

• They are temporally stable with respect to the evolution operator associated with the H Hamiltonian. In other words, they satisfy

e -iHt |l, m, ϕ = |l, m, ϕ + t
for any value of the real parameter t.

• For fixed ϕ and l, they satisfy the equiprobability relation

| n, l|l, m, ϕ | = 1 √ k -l + 1
and the property

k-l m=0 |l, m, ϕ l, m, ϕ| = k-l n=0
|n, l n, l|.

• The overlap between two phase states |l ′ , m ′ , ϕ ′ and |l, m, ϕ reads

l, m, ϕ|l ′ , m ′ , ϕ ′ = δ l,l ′ 1 k -l + 1 k-l n=0 q ρ(m-m ′ ,ϕ-ϕ ′ ,n) l , where ρ(m -m ′ , ϕ -ϕ ′ , n) = -(m -m ′ )n + k -l + 1 2π (ϕ -ϕ ′ )H(n, l)
with q l defined in (24). As a particular case, for fixed ϕ we have the orthonormality relation

l, m, ϕ|l ′ , m ′ , ϕ = δ l,l ′ δ m,m ′ .
However, not all temporally stable phase states are orthogonal.

Similar results can be derived for the E 2d (l) operator by exchanging the roles played by n and l. It is enough to mention that the |z l eigenstates of E 2d (l) can be taken in the form

|z l ≡ |l, m, ϕ = 1 √ k -l + 1 k-l n=0
e -iH(l,n)ϕ q mn l |l, n and present properties identical to those of the states in (26).

Phase states for E 3d (l)

The eigenstates of the E 3d (l) operator are given by

E 3d (l)|w l = w l |w l , |w l = l n=0 c n w n l |l -n, n , w l ∈ C.
The use of ( 18)-( 19) leads to the recurrence relation

c n+1 = c n , n = 0, 1, . . . , l -1
with the condition

c 0 = c l (w l ) l+1 .

It follows that

c n = c 0 , n = 0, 1, . . . , l and the w l eigenvalues satisfy

(w l ) l+1 = 1.
Therefore, the admissible values for w l are

w l = ω m l , m = 0, 1, . . . , l,
with

ω l = exp 2πi l + 1 .
As a result, the normalized eigenstates of E 3d (l) can be taken in the form

|w l ≡ l, m, ϕ = 1 √ l + 1 e -iH(0,l)ϕ l n=0 ω mn l |l -n, n . (28)
The l, m, ϕ states depend on the parameters l ∈ {0, 1, . . . , k}, m ∈ Z/(l + 1)Z and ϕ ∈ R. They satisfy

E 3d (l) l, m, ϕ = e iθm l, m, ϕ , θ m = m 2π l + 1 , (29)
so that E 3d (l) is a phase operator.

For fixed l, the set { l, m, 0 : m = 0, 1, . . . , l}, corresponding to ϕ = 0, follows from the set {|l -n, n : n = 0, 1, . . . , l} by making use of a (quantum) discrete Fourier transform. [START_REF] Vourdas | Quantum systems with finite Hilbert space[END_REF] Note that for ϕ = 0, the l, m, 0 phase states have the same form as the phase states for SU 2 derived by Vourdas. [START_REF] Vourdas | SU(2) and SU(1, 1) phase states[END_REF] In the case where ϕ = 0, the l, m, ϕ phase states for E 3d (l) satisfy properties similar to those of the |l, m, ϕ phase states for E 1d (l) and for E 2d (l) modulo the substitutions (n, l) → (l -n, n), q l → ω l and k -l → l.

Phase states for E d

We are now in a position to derive the eigenstates of the E d operator. They are given by the following eigenvalue equation

E d |ψ = λ|ψ , (30)
where

|ψ = k l=0 k-l n=0 C n,l |n, l . (31)
Introducing (31) into (30) and using the definition in ( 21)-( 23) of the E d operator, a straightforward but long calculation leads to following recurrence relations

C n+1,l e i[H(n+1,l)-H(n,l)]ϕ = λC n,l (32)
C 0,l+1 e i[H(0,l+1)-H(k-l,l)]ϕ = λC k-l,l (33)
for l = 0, 1, . . . , k -1. For l = k, we have

C 0,0 e i[H(0,0)-H(0,k)]ϕ = λC 0,k . (34)
(Note that (33) with l = k yields (34) if C 0,k+1 is identified to C 0,0 .) From the recurrence relation (32), it is easy to get

C n,l = λ n e -i[H(n,l)-H(0,l)]ϕ C 0,l , (35)
which, for n = k -l, gives

C k-l,l = λ k-l e -i[H(k-l,l)-H(0,l)]ϕ C 0,l (36)
in terms of C 0,l . By introducing (36) into (33), we obtain the recurrence relation

C 0,l+1 e i[H(0,l+1)-H(0,l)]ϕ = λ k-l+1 C 0,l (37)
that completely determines the C 0,l coefficients and subsequently the C n,l coefficients owing to (35). Indeed, the iteration of Eq. (37) gives

C 0,l = λ 1 2 l(2k-l+3) e -i[H(0,l)-H(0,0)]ϕ C 0,0 . (38)
Note that for l = k (⇒ n = 0), Eq. (39) becomes

C 0,k = λ 1 2 k(k+3) e -iH(0,k)ϕ C 0,0 . (40)
The introduction of (40) in (34) produces the condition

λ d = 1.
Consequently, the λ eigenvalues are

λ = exp 2πi d m , m = 0, 1, . . . , d -1.
Finally, the normalized eigenvectors of the E d operator read

|ψ ≡ |m, ϕ = 1 √ d k l=0 q 1 2 ml(2k-l+3) k-l n=0
q mn e -iH(n,l)ϕ |n, l ,

where

q = exp 2πi d . (42)
The |m, ϕ states are labeled by the parameters m ∈ Z/dZ and ϕ ∈ R. They satisfy

E d |m, ϕ = e iθm |m, ϕ , θ m = m 2π d , m = 0, 1, . . . , d -1.
As a conclusion, E d is a unitary phase operator. The |m, ϕ phase states satisfy interesting properties:

• They are temporally stable under time evolution, i.e., e -iHt |m, ϕ = |m, ϕ + t for any value of the real parameter t.

• For fixed ϕ, they satisfy the relation

| n, l|m, ϕ | = 1 √ d
and the closure property • The overlap between two phase states |m ′ , ϕ ′ and |m, ϕ reads

m, ϕ|m ′ , ϕ ′ = 1 d k l=0 k-l n=0 q τ (m ′ -m,ϕ-ϕ ′ ,n,l) ,
where

τ (m ′ -m, ϕ -ϕ ′ , n, l) = (m ′ -m) 1 2 l(2k -l + 3) + n + d 2π (ϕ -ϕ ′ )H(n, l)
with q defined in (42). As a particular case, we have the orthonormality relation

m, ϕ|m ′ , ϕ = δ m,m ′ .
However, the temporally stable phase states are not all orthogonal.

The k = 1 particular case

To close Section 3.2, we now establish a contact with the results of Klimov et al. [START_REF] Klimov | Quantum phases of a qutrit[END_REF] which correspond to k = 1 (i.e., κ = -1). In this particular case, the F κ Fock space is three-dimensional (d = 3). It corresponds to the representation space of SU 3 relevant for ordinary quarks and antiquarks in particle physics and for qutrits in quantum information.

For the purpose of comparison, we put

|φ 1 ≡ |0, 0 , |φ 2 ≡ |1, 0 , |φ 3 ≡ |0, 1 .
Then, the operators E 13 , E 23 , E 33 and E 3 assume the form

E 13 = e iϕ |φ 1 φ 2 | + e -iϕ |φ 2 φ 1 | + |φ 3 φ 3 | E 23 = e iϕ |φ 1 φ 3 | + e -iϕ |φ 3 φ 1 | + |φ 2 φ 2 | E 33 = |φ 2 φ 3 | + |φ 3 φ 2 | + |φ 1 φ 1 | E 3 = e iϕ |φ 1 φ 2 | + |φ 2 φ 3 | + e -iϕ |φ 3 φ 1 |.
Operators E 13 , E 23 and E 33 have a form similar to that of the phase operators

Ê12 = |φ 1 φ 2 | -|φ 2 φ 1 | + |φ 3 φ 3 | Ê13 = |φ 1 φ 3 | -|φ 3 φ 1 | + |φ 2 φ 2 | Ê23 = |φ 2 φ 3 | -|φ 3 φ 2 | + |φ 1 φ 1 |
introduced in Ref. [15] in connection with qutrits. Although the E 13 , E 23 and E 33 operators derived in the present work cannot be deduced from the Ê12 , Ê13 and Ê23 operators of Ref. [15] by means of similarity transformations, the two sets of operators are equivalent in the sense that their action on the |φ 1 , |φ 2 and |φ 3 vectors are identical up to phase factors. In addition, in the case where we do not take into account the spectator state (|φ 3 , |φ 2 or |φ 1 for E 13 , E 23 or E 33 , respectively), our SU 3 phase operators are reduced to SU 2 phase operators which present the same periodicity condition (i.e., their square is the identity operator) as the SU 2 phase operators of Ref. [23]. In the ϕ = 0 case, our SU 2 phase states turn out to be identical to the phase states derived by Vourdas. [START_REF] Vourdas | SU(2) and SU(1, 1) phase states[END_REF] Finally, note that the E 3 (and, more generally, E d ) operator is new; it has no equivalent in Ref. [15].

Vector phase states in finite dimension

We have now the necessary tools for introducing vector phase states associated with the unitary phase operators E 1d , E 2d and E 3d . We give below a construction similar to the one discussed in Ref. [2].

Vector phase states for E 1d and E 2d

To define vector phase states, we introduce the (k + 1) × (k + 1)-matrix

Z = diag(z 0 , z 1 , . . . , z k ), z l = q m l and the (k + 1) × 1-vector [n, l] =        0 . . . |n, l . . . 0       
, where the |n, l entry appears on the l-th line (with l = 0, 1, . . . , k). Then, let us define

[l, m, ϕ] = 1 √ k -l + 1 k-l n=0 e -iH(n,l)ϕ Z n [n, l]. ( 43
)
From Eq. ( 26), we have

[l, m, ϕ] =        0 . . . |l, m, ϕ . . . 0        , ( 44
)
where |l, m, ϕ occurs on the l-th line.

We shall refer the states (44) to as vector phase states. In this matrix presentation, it is useful to associate the matrix

E 1d = diag (E 1d (0), E 1d (1), ..., E 1d (k))
with the unitary phase operator E 1d . It is easy to check that E 1d satisfies the matrix eigenvalue equation

E 1d [l, m, ϕ] = e iθm [l, m, ϕ], θ m = m 2π k -l + 1
(cf. Eq. ( 27)).

Other properties of vector phase states [l, m, ϕ] can be deduced from those of phase states |l, m, ϕ . For instance, we obtain

• The temporal stability condition

e -iHt [l, m, ϕ] = [l, m, ϕ + t]
for t real.

• The closure relation

k l=0 k-l m=0 [l, m, ϕ][l, m, ϕ] † = I d ,
where I d is the unit matrix of dimension d × d with d given by (14).

Similar vector phase states can be obtained for E 2d by permuting the n and l quantum numbers occurring in the derivation of the vector phase states for E 1d .

Vector phase states for E 3d

Let us define the diagonal matrix of dimension (k + 1) × (k + 1)

W = diag(w 0 , w 1 , . . . , w k ), w l = ω m l and the column vector of dimension (k + 1) × 1 [[n -l, n]] =        0 . . . |l -n, n . . . 0       
, where the |l -n, n state occurs on the l-th line (with l = 0, 1, . . . , k). By defining

[[l, m, ϕ]] = 1 √ l + 1 e -iH(l,0)ϕ l n=0 W n [[l -n, n]],
we obtain

[[l, m, ϕ]] =        0 . . . l, m, ϕ . . . 0        , (45)
where the l, m, ϕ phase state appears on the l-th line. Equation (45) defines vector phase states associated with the E 3d phase operator.

These states satisfy the eigenvalue equation

E 3d [[l, m, ϕ]] = e iθm [[l, m, ϕ]], θ m = m 2π l + 1
,

where

E 3d = diag (E 3d (0), E 3d (1), ..., E 3d (k)) .
The [[l, m, ϕ]] vector phase states satisfy properties which can be deduced from those of the [l, m, ϕ] vector phase states owing to simple correspondence rules.

PHASE OPERATORS AND PHASE STATES FOR A κ (2)

WITH κ ≥ 0

Phase operators in infinite dimension

In the case κ ≥ 0, we can decompose the Jacobson operators a - i and a + i as

a - i = E i∞ F i (N 1 , N 2 ), a + i = F i (N 1 , N 2 ) (E i∞ ) † , i = 1, 2, (46)
where

E 1∞ = ∞ n 1 =0 ∞ n 2 =0 e i[H(n 1 +1,n 2 )-H(n 1 ,n 2 )]ϕ |n 1 , n 2 n 1 + 1, n 2 | (47) E 2∞ = ∞ n 1 =0 ∞ n 2 =0 e i[H(n 1 ,n 2 +1)-H(n 1 ,n 2 )]ϕ |n 1 , n 2 n 1 , n 2 + 1|. (48)
The operators E i∞ , i = 1, 2, satisfy

E 1∞ (E 1∞ ) † = I, (E 1∞ ) † E 1∞ = I - ∞ n 2 =0 |0, n 2 0, n 2 | (49) E 2∞ (E 2∞ ) † = I, (E 2∞ ) † E 2∞ = I - ∞ n 1 =0 |n 1 , 0 n 1 , 0|. (50)
Equations ( 49) and (50) show that E i∞ , i = 1, 2, are not unitary operators.

In a similar way, operators a + 3 and a - 3 can be rewritten

a - 3 = -κE 3∞ (N 1 + 1)N 2 , a + 3 = -κ (N 1 + 1)N 2 (E 3∞ ) † ,
where

E 3∞ = ∞ n 1 =0 ∞ n 2 =0 |n 1 + 1, n 2 n 1 , n 2 + 1|.
The E 3∞ operator is not unitary since

E 3∞ (E 3∞ ) † = I - ∞ n 2 =0 |0, n 2 0, n 2 |, (E 3∞ ) † E 3∞ = I - ∞ n 1 =0 |n 1 , 0 n 1 , 0|,
to be compared with (49) and (50).

The E 3∞ operator is not independent of E 1∞ and E 2∞ . Indeed, it can be expressed as

E 3∞ = (E 1∞ ) † E 2∞ , (51)
a relation of central importance for deriving its eigenvalues (see Section 4.2).

Phase states in infinite dimension

It is easy to show that operators E 1∞ and E 2∞ commute. Hence, that they can be simultaneously diagonalized. In this regard, let us consider the eigenvalue equations

E 1∞ |z 1 , z 2 ) = z 1 |z 1 , z 2 ), E 2∞ |z 1 , z 2 ) = z 2 |z 1 , z 2 ), (z 1 , z 2 ) ∈ C 2 , (52)
where

|z 1 , z 2 ) = ∞ n 1 =0 ∞ n 2 =0 D n 1 ,n 2 |n 1 , n 2 .
By using the definitions of the nonunitary phase operators (47) and (48), it is easy to check from the eigenvalue equations (52) that the complex coefficients D n 1 ,n 2 satisfy the following recurrence relations

D n 1 +1,n 2 e iH(n 1 +1,n 2 )ϕ = z 1 D n 1 ,n 2 e iH(n 1 ,n 2 )ϕ (53) D n 1 ,n 2 +1 e iH(n 1 ,n 2 +1)ϕ = z 2 D n 1 ,n 2 e iH(n 1 ,n 2 )ϕ , (54)
which lead to

D n 1 ,n 2 = e -iH(n 1 ,n 2 )ϕ z n 1 1 z n 2 2 D 0,0 .
It follows that the normalized common eigenstates of the operators E 1∞ and E 2∞ are given by

|z 1 , z 2 ) = (1 -|z 1 | 2 )(1 -|z 2 | 2 ) ∞ n 1 =0 ∞ n 2 =0 z n 1 1 z n 2 2 e -iH(n 1 ,n 2 )ϕ |n 1 , n 2 on the domain {(z 1 , z 2 ) ∈ C 2 : |z 1 | < 1, |z 2 | < 1}.
Following the method developed in Refs. [24, 26] for the Lie algebra su 1,1 and in Ref. [5] for the algebra A κ (1), we define the states

|θ 1 , θ 2 , ϕ) = lim z 1 →e iθ 1 lim z 2 →e iθ 2 1 (1 -|z 1 | 2 )(1 -|z 2 | 2 ) |z 1 , z 2 ),
where θ 1 , θ 2 ∈ [-π, +π]. We thus get

|θ 1 , θ 2 , ϕ) = ∞ n 1 =0 ∞ n 2 =0
e in 1 θ 1 e in 2 θ 2 e -iH(n 1 ,n 2 )ϕ |n 1 , n 2 .

These states, defined on S 1 × S 1 , turn out to be phase states since we have

E 1∞ |θ 1 , θ 2 , ϕ) = e iθ 1 |θ 1 , θ 2 , ϕ), E 2∞ |θ 1 , θ 2 , ϕ) = e iθ 2 |θ 1 , θ 2 , ϕ).
Hence, the operators E i∞ , i = 1, 2, are (nonunitary) phase operators.

The main properties of the |θ 1 , θ 2 , ϕ) states are the following.

• They are temporally stable in the sense that

e -iHt |θ 1 , θ 2 , ϕ) = |θ 1 , θ 2 , ϕ + t),
with t real.

• They are not normalized and not orthogonal. However, for fixed ϕ, they satisfy the closure relation

1 (2π) 2 +π -π dθ 1 +π -π dθ 2 |θ 1 , θ 2 , ϕ)(θ 1 , θ 2 , ϕ| = I.
In view of Eq. (51), we have

E 3∞ |θ 1 , θ 2 , ϕ) = e i(θ 2 -θ 1 ) |θ 1 , θ 2 , ϕ),
so that the |θ 1 , θ 2 , ϕ) states are common eigenstates to E 1∞ , E 2∞ and E 3∞ .

To close, a comparison is in order. For ϕ = 0, the |θ 1 , θ 2 , 0) states have the same form as the phase states derived in Ref. [6] which present the closure property but are not temporally stable.

TRUNCATED GENERALIZED OSCILLATOR ALGE-BRA

For κ ≥ 0 the F κ Hilbert space associated with A κ (2) is infinite-dimensional and it is thus impossible to define a unitary phase operator. On the other hand, for κ < 0 the F κ space is finite-dimensional and there is no problem to define unitary phase operators.

Therefore, for κ ≥ 0 it is appropriate to truncate the F κ space in order to get a subspace F κ,σ of dimension (σ + 1)(σ + 2)/2 with σ playing the role of k. Then, it will be possible to define unitary phase operators and vector phase vectors for the F κ,σ truncated space with κ ≥ 0. To achieve this goal, we shall adapt the truncation procedure discussed in Ref. [20] for the h 4 Weyl-Heisenberg algebra and in Ref. [3, 5] for the A κ (1) algebra with κ ≥ 0.

The restriction of infinite-dimensional space F κ (κ ≥ 0) to finite-dimensional space F κ,σ with basis

{|n 1 , n 2 : n 1 , n 2 ranging | n 1 + n 2 ≤ σ}
can be done by means of the projection operator

Π σ = σ n 1 =0 σ-n 1 n 2 =0 |n 1 , n 2 n 1 , n 2 | = σ n 2 =0 σ-n 2 n 1 =0 |n 1 , n 2 n 1 , n 2 |.
Let us then define the four new ladder operators

b ± i = Π σ a ± i Π σ , i = 1, 2.
We may ask what is the algebra generated by operators b ± i and N i (i = 1, 2)? Indeed, the latter operators satisfy the following algebraic relations when acting on the F κ,σ space

[b - 1 , b + 1 ] = I + κ(2N 1 + N 2 ) - σ l=0 F 1 (σ -l + 1, l)|σ -l, l σ -l, l| [b - 2 , b + 2 ] = I + κ(2N 2 + N 1 ) - σ l=0 F 2 (l, σ -l + 1)|l, σ -l l, σ -l| [N i , b ± j ] = ±δ i,j b ± i , i, j = 1, 2 [b ± i , b ± j ] = 0, [b ± i , [b ± i , b ∓ j ]] = 0, i = j.
Operators b ± i and N i (i = 1, 2) acting on F κ,σ generate an algebra, noted A κ,σ (2). The A κ,σ (2) algebra generalizes A κ,s (1) which results from the truncation of the A κ (1) algebra. [START_REF] Daoud | Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems[END_REF] By using the trick to pass from A κ (2) to A κ (1), see section 2.1, we get A κ,s-1 (2) → A κ,s (1). The A κ,s (1) truncated algebra gives in turn the Pegg-Barnett truncated algebra [START_REF] Pegg | Phase properties of the quantized single-mode electromagnetic field[END_REF] when κ → 0.

As a conclusion, the action of b ± i (i = 1, 2) on the complement of F κ,σ with respect to F κ leads to the null vector while the action of these operators on the F κ,σ space with κ ≥ 0 is the same as the action of a ± i (i = 1, 2) on the F κ space with κ < 0 modulo some evident changes of notations. It is thus possible to apply the procedure developed for F κ space with κ < 0 in order to obtain unitary phase operators on F κ,σ with κ ≥ 0 and the corresponding vector phase states. The derivation of the vector phase states for the A κ,σ (2) truncated algebra can be done simply by replacing k by σ. In this respect, the σ truncation index can be compared to the k quenching index (or Chen index) used for characterizing the finite-dimensional representation (0, k) or (k, 0) of SU 3 . [START_REF] Zhang | Quantum nonintegrability in finite systems[END_REF]

APPLICATION TO MUTUALLY UNBIASED BASES

We now examine the possibility to produce specific bases, known as mutually unbiased bases (MUBs) in quantum information, for finite-dimensional Hilbert spaces from the phase states of E 1d (l), E 2d (l) and E 3d (l). Let us recall that two distinct orthonormal (cf. Refs. [9, 16, 17, 27]).

We begin with the |l, m, ϕ phase states associated with the E 1d (l) phase operator (see (26) and ( 27)). In Eq. ( 26), l can take the values 0, 1, . . . , k. Let us put l = 0 and switch to the notations

k ≡ N -1, m ≡ α, |n, 0 ≡ |N -1 -n
(with α, n = 0, 1, . . . , N -1) for easy comparison with some previous works. Then, Eq. ( 26) becomes

|0, α, ϕ = 1 √ N N -1 n=0 exp - i N -1 n(N -n)ϕ + 2πi N nα |N -1 -n . (55)
For ϕ = 0, Eq. (55) describes a (quantum) discrete Fourier transform [START_REF] Vourdas | Quantum systems with finite Hilbert space[END_REF] that allows to pass from the set {|N -1 -n : n = 0, 1, . . . , N -1} of cardinal N to the set {|0, α, 0 : α = 0, 1, . . . , N -1} of cardinal N too. In the special case where ϕ is quantized as

ϕ = -π N -1 N a, a = 0, 1, . . . , N -1, (56)
where

q 0 = exp 2πi N .
Equation (57) with a = 0 corresponds to a (quantum) quadratic discrete Fourier transform. [START_REF] Kibler | Angular momentum and mutually unbiased bases[END_REF][START_REF] Kibler | Variations on a theme of Heisenberg, Pauli and Weyl[END_REF][START_REF] Kibler | An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group[END_REF] In this regard, note that the |aα state in (57) can be identified with the |aα; r state with r = 0 discussed recently in the framework of the quadratic discrete Fourier transform. [START_REF] Kibler | Quadratic discrete Fourier transform and mutually unbiased bases[END_REF] Following Ref. [14], we consider the set B N = {|N -1 -n : n = 0, 1, 2, . . . , N -1} = {|n : n = 0, 1, 2, . . . , N -1} E 3d (l) (see (28) and ( 29)). We put ϕ = 0 in (28) and apply the e -iF 3 (N 1 ,N 2 )ϕ operator on the resultant state. This gives Hence, the vector e -iF 3 (N 1 ,N 2 )ϕ l, m, 0 ≡ |aα

reads |aα = 1 √ N N -1 n=0 ω n(N -n)a/2+nα N -1 |N -1 -n , (61)
which bears the same form as (57). Consequently for N a prime integer, Eq. (61) generates N MUBs B 0a with a = 0, 1, . . . , N -1 which together with the computational basis B N form a complete set of N + 1 MUBs.

CONCLUDING REMARKS

The main results of this work are the following. The su 3 , su 2,1 and h 4 ⊗ h 4 algebras can be described in an unified way via the introduction of the A κ (2) algebra. A quantum system with a quadratic spectrum (for κ = 0) is associated with A κ (2) ; for κ = 0, this system coincides with the two-dimensional isotropic harmonic oscillator.

In the case κ < 0, the unitary phase operators (E 1d , E 2d and E 3d ) defined in this paper generalize those constructed in Ref. [15] for an su 3 three-level system (corresponding to d = 3) ; they give rise to new phase states, namely, vector phase states which are eigenstates obtained along lines similar to those developed in Ref. [2, 22] for obtaining a vectorial generalization of the coherent states introduced in Ref. [7]. Still for κ < 0, a new type of unitary phase operator (E d ) can be defined ; it specificity is to span all vectors of the d-dimensional representation space of A κ (2) from any vector of the space.

In the case κ ≥ 0, it is possible to define nonunitary phase operators. They can be turned to unitary phase operators by truncating (to some finite but arbitrarily large order) the representation space of A κ (2). This leads to a truncated generalized oscillator algebra (A κ,σ (2)) that can be reduced to the Pegg-Barnett truncated oscillator algebra [START_REF] Pegg | Phase properties of the quantized single-mode electromagnetic field[END_REF] through an appropriate limiting process where κ → 0.

Among the various properties of the phase states and vector phase states derived for κ < 0 and κ ≥ 0, the property of temporal stability is essential. It has no equivalent in Ref. [23]. In last analysis, this property results from the introduction of a phase factor (ϕ) in the action of the annihilation and creation operators of A κ (2). As an unexpected result, the quantization of this phase factor allows to derive mutually unbiased bases from temporally stable phase states for κ < 0. This is a further evidence that "phases do matters after all" [START_REF] Sánchez-Soto | Multipartite quantum systems: phases do matter after all[END_REF] and are important in quantum mechanics.

  n, l| = I.

  bases {|aα : α = 0, 1, . . . , N -1} and {|bβ : β = 0, 1, . . . , N -1} of the N-dimensional Hilbert spaces C N are said to be unbiased if and only if ∀α = 0, 1, . . . , N -1, ∀β = 0, 1, . . . , N -1 : | aα|bβ | = 1 √ N

  -n)a/2+nα 0 |N -1 -n ,

e -iF 3 (

 3 N 1 ,N 2 )ϕ l, m, (l + 1 -n)ϕ ω mn l |l -n, n .For the sake of comparison, we introducel ≡ N -1, m ≡ α, ω N -1 ≡ exp 2πi N , |l -n, n ≡ |N -1 -nand we quantize ϕ via ϕ = -π k 2 N a, a = 0, 1, . . . , N -1.

l(2k-l+3)+n e -iH(n,l)ϕ C 0,0 . (39)

ACKNOWLEDGMENTS

MD would like to thank the hospitality and kindness of the Service de physique théorique de l'Institut de Physique Nucléaire de Lyon where this work was done.

They can be rewritten as

A straightforward calculation shows that the action of b ± 1 on F κ is given by

Therefore, the action of operators b

as an orthonormal basis for the N-dimensional Hilbert space. This basis is called computational basis in quantum information. Then, the sets B 0a = {|aα : α = 0, 1, 2, . . . , N -1}, a = 0, 1, 2, . . . , N -1 constitute N new orthonormal bases of the space. The B 0a basis is a special case, corresponding to r = 0, of the B ra bases derived in Ref. [14] from a polar decomposition of the su 2 Lie algebra. The overlap between two bases B 0a and B 0b is given by

, a relation which can be expressed in term of the generalized Gauss sum [START_REF] Berndt | Gauss and Jacobi Sums[END_REF] S(u, v, w) = |w|-1 k=0 e iπ(uk 2 +vk)/w .

In fact, we obtain

with

In the case where N is a prime integer, the calculation of S(u, v, w) in (58) yields

On the other hand, it is evident that

holds for any strictly positive value of N. As a result, Eqs. ( 59) and (60) shows that bases B N and B 0a with a = 0, 1, . . . , N -1 provide a complete set of N + 1 MUBs when N is a prime integer.

A similar result can be derived by quantizing, according to (56), the ϕ parameter occurring in the eigenstates of E 2d (0).

The form of the E 3d (l) phase operator being different from those of E 1d (l) and E 2d (l),

we proceed in a different way for obtaining MUBs from the l, m, ϕ eigenstates of