Michel Planat
email: planat@femto-st.fr

Clifford group dipoles and the enactment of Weyl/Coxeter group W (E 8 ) by entangling gates

Keywords: numbers: 03.67.Pp, 03.67.Pp, 02.20.-a, 03.65.Ud

Peres/Mermin arguments about no-hidden variables in quantum mechanics are used for displaying a pair (R, S) of entangling Clifford quantum gates, acting on two qubits. From them, a natural unitary realization of Coxeter/Weyl groups W (D 5 ) and W (F 4 ) emerges, which is also reflected into the splitting of the n-qubit Clifford group C n into group dipoles C ± n . The union of the three-qubit real Clifford group C + 3 and the Toffoli gate ensures a orthogonal realization of the Weyl/Coxeter group W (E 8 ), and of its relatives. Other concepts involved are complex reflection groups, BN pairs, unitary group designs and entangled states of the GHZ, W and chain families.

Introduction

One important feature of quantum mechanics, not present in classical physics, is the possible non-commutativity of observables. Many peculiarities and paradoxes encountered in quantum mechanical measurements may be discussed in a discrete setting in terms of tensor products of Pauli spin matrices σ x , σ y and σ z , and the identity matrix I, which obey non-trivial commutation relations [START_REF] Planat | On the Pauli graphs of N -qudits Quant[END_REF]. In essence, the peculiarities carried by the Bell-Kochen-Specker theorem deal about the algebraic structure of eigenvalues/measurements, that contradicts that of the eigenstates [START_REF] Mermin | Hidden variables and the two theorems of John Bell[END_REF]. In this note, the ingredients of this discussion are given a quantum computing setting.

We first exhibit a pair of two-qubit entangling gates R (a braiding matrix) and S (a non-braiding matrix), that encapsulate Mermin's discussion about quantum paradoxes. The octahedral geometry of the group R, S is investigated. Then, the natural decomposition of the symplectic group Sp(2n, 2) into its orthogonal subgroups [START_REF] Bannai | Modular invariance property of association schemes, type II codes over finite rings and finite abelian groups and reminiscences of Franois Jaeger (a survey)[END_REF] is reflected in the splitting of the Clifford group C n on n qubits into group dipoles indexed by R and S. The relevant Coxeter groups, BN pairs and unitary group designs rise up in the calculations.

Finally, it is shown that the real three-qubit group dipole, when complemented by the Toffoli gate, enact (up to isomorphism) the largest Weyl/Coxeter group W (E 8 ) and its reflection subgroups. The various entangling gates spanning them are displayed.

All group theoretical calculations are performed in Magma [START_REF] Bosma | The Magma algebra system[END_REF]. In the present paper, the term representation, or realization, of a group G denotes a group homomorphism from G to the general linear group GL(n, C) of n × n matrices with complex entries. In Sec. 4, real representations, i.e. group homomorphisms from G to GL(n, R) will occur. The term enactment is used to contrast the new quantum gate presentation of W (E 8 ), found in Sec. [START_REF] Bosma | The Magma algebra system[END_REF], with the standard realization, that makes use of the roots of the exceptional algebra E 8 . We also coin the word Clifford group dipole, in Sec. 3, for featuring the splitting of the Clifford group into its subgroups of positive/negative index. To our knowledge, this terminology was not used before.

Appendices are devoted to the fundamental concepts underlying many aspects of the paper: complex reflection groups [START_REF] Planat | Unitary reflection groups for quantum fault tolerance[END_REF], BN pairs [START_REF] Planat | Clifford groups of quantum gates, BN -pairs and smooth cubic surfaces[END_REF], unitary designs [START_REF] Scott | Unitary designs and codes[END_REF] and the classification of entanglement [START_REF] Coffman | Distributed entanglement[END_REF].

From Mermin's array to octahedral symmetry

The basic pieces of the proof of Kochen-Specker theorem in a four-dimensional space are two triples of (mutually commuting and real) two-qubit observables

{σ x ⊗ σ x , σ y ⊗ σ y , σ z ⊗ σ z } and {σ x ⊗ σ z , σ z ⊗ σ x , σ y ⊗ σ y } . (1)
See [START_REF] Mermin | Hidden variables and the two theorems of John Bell[END_REF] for a full discussion based on the 3×3 Mermin's array and [START_REF] Planat | On the Pauli graphs of N -qudits Quant[END_REF] for the interpretation of the array as a specific hyperplane of the generalized quadrangle of order two. The joined eigenstates of the first triple of mutually commuting observables in (1) are made explicit in [START_REF] Planat | Quantum Entanglement and Projective Ring Geometry SIGMA 2 Paper[END_REF]. They may be casted as the rows of the orthogonal matrix R as below

R = 1 √ 2      1 0 0 1 0 1 -1 0 0 1 1 0 -1 0 0 1      ,      + + - --- -+ + + -+      . ( 2
)
Rows of the second matrix contain the sign of eigenvalues ±1, and the action of the transpose R tr of matrix R on the computational basis leads to the entangled states shared by the triple, i.e.

R tr |00 = 1 √ 2 (|00 + |11 ), • • •.
Matrix R is known as the Bell basis change matrix. It occurs in the braiding approach of quantum computing [START_REF] Kauffman | Braiding operators are universal quantum gates[END_REF][START_REF] Ahlbrecht | Implementation of Clifford gates in the Isinganyon topological quantum computer[END_REF] and is also encountered in our recent papers [START_REF] Planat | On group theory for quantum gates and quantum coherence[END_REF]- [START_REF] Planat | Unitary reflection groups for quantum fault tolerance[END_REF].

The joined eigenstates of the second triple of mutually commuting observables in (1) may be similarly casted as the rows of the entangling orthogonal matrix

S = 1 2      1 -1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1      ,      + -- -+ - --+ + + +      . ( 3
)
Thus the two matrices R and S capture, in a very compact and yet unnoticed form, the ingredients contained in the Mermin's proof of Kochen-Specker theorem. As any entangling matrix associated with local unitary transformations is universal for quantum [START_REF] Jozsa | Matchgates and classical simulation of quantum circuits[END_REF] for details about the Clifford group and [START_REF] Planat | Clifford groups of quantum gates, BN -pairs and smooth cubic surfaces[END_REF] for preliminary observations about its decomposition into BN-pairs [START_REF] Tits | Buildings of Spherical Type and Finite BN-Pairs Lect[END_REF]). Beforehand, the finite group G 96 generated by R and S is of order 96 and satisfies the group isomorphisms

G 96 = R, S ∼ = U 13 ∼ = Z 4 .S 4 , (5)
where U 13 is the complex reflection group No 13 in the Shephard-Todd sequence [START_REF] Kane | Reflection groups and invariant theory[END_REF] (see appendix A for a definition and [START_REF] Planat | Unitary reflection groups for quantum fault tolerance[END_REF] for a recent essay about the relevance of complex reflection groups for quantum information). From [START_REF] Planat | Quantum Entanglement and Projective Ring Geometry SIGMA 2 Paper[END_REF] it is clear that the group G 96 possesses a normal subgroup isomorphic to the cyclic group Z 4 . The dot product of groups means that the extension is not split and that its quotient by Z 4 is the symmetric group on four letters S 4 . The single qubit Clifford group C 1 contains the reflection group U 13 as a maximal subgroup of index two. The reflection group U 8 , No 8 in the Shephard-Todd sequence, is also a maximal subgroup of C 1 , has the same order, and is isomorphic to Z 4 .S 4 as well. The latter is studied in [START_REF] Bannai | Modular invariance property of association schemes, type II codes over finite rings and finite abelian groups and reminiscences of Franois Jaeger (a survey)[END_REF] in connection to the modular invariance property of association schemes and the related self-dual classical codes [START_REF] Nebe | The Invariants of the Clifford Groups Designs[END_REF].

The smallest degree invariant of the Clifford group C 1 is shared by its reflection subgroups U 8 and U 13 as

W := α 8 + 14α 4 β 4 + β 8 . ( 6
)
It corresponds to the octahedral invariance. It was derived for the first time in 1913 Klein's lectures [START_REF] Klein | Lectures on the icosahedron and the solutions of equations of the fifth degree[END_REF]. The octic invariant (6) is obtained by exploiting the invariance under SU(2) transformations of the center of faces of an octahedron with vertices located on the Riemann sphere. Then, seing the Riemann sphere as the usual Bloch sphere the variables α and β may be interpreted as the amplitudes of a single qubit state. Invariant W may also be seen as the complete weight enumerator of the self dual code e 8 [START_REF] Nebe | The Invariants of the Clifford Groups Designs[END_REF].

The corresponding octic invariant of the (rank four) group G 96 is

W (2) := Σ 8 + 14Σ 4,4 + 168Σ 2,2,2,2 , (7)
in the notations of [START_REF] Nebe | The Invariants of the Clifford Groups Designs[END_REF], i.e. Σ 8 = 4 i=1 α 8 i , Σ 4,4 = j>i α 4 i α 4 j and Σ 2,2,2,2 = 4 i=1 α 2 i . Invariant W (2) represents the complete weight enumerator of the (genus two) code e 8 ⊗F 4 [START_REF] Nebe | The Invariants of the Clifford Groups Designs[END_REF], and indeed generalizes Klein's invariant (6).

Clifford group dipoles

Let us call P n the Pauli error group on n qubits and C n the corresponding Clifford group, i.e. the normalizer of P n in the general 2 n -dimensional complex unitary group U(2 n , C). By definition, P n is mapped to itself under the conjugation action

C n P n C -1
n . The group P n is normal in C n and the map from P n to the factor group C n /P n is an element of the symplectic group Sp(2n, 2) [START_REF] Vourdas | Quantum systems with finite Hilbert space: Galois fields in quantum mechanics[END_REF].

Then, the natural decomposition of Sp(2n, 2) into its orthogonal subgroups Ω ± (2n, 2), studied in [START_REF] Bannai | On some subgroups of the group Sp(2n, 2)[END_REF] ‡, may be used for obtaining a straightforward decomposition of the Clifford group

C n into dipole subgroups C ± n C ± n = E ± (2n + 1).Ω ± (2n, 2). ( 8
)
in which E ± (2n + 1) are the extraspecial groups [START_REF] Nebe | The Invariants of the Clifford Groups Designs[END_REF], of order 2 2n+1 , of the Pauli group P n . The splitting of the Clifford group into its subgroup dipoles is inspired by our recent proposal of decomposing the Clifford group into BN-pairs [START_REF] Planat | Clifford groups of quantum gates, BN -pairs and smooth cubic surfaces[END_REF].

Single qubit group dipoles

The single qubit Clifford group possesses a BN-pair (see the appendix B for the meaning of a BN-pair)

B ≡ C + 1 , N ≡ C - 1 ∼ = SL(2, 3), H 0 ∼ = Z 4 and W ∼ = W (D 2 ) ∼ = Z 2 2 , (9)
corresponding to the group dipoles

C + 1 = E + (8), C - 1 ∼ = E -(8).Ω -(2, 2). ( 10
)
The group dipole C ∞ = C ∞ ′ (with ′ meaning the derived subgroup) is nothing but the octahedral group

O = iσ z , 1 2 1 -i i -1 1 + i 1 + i , (11)
that is not of the reflection type, but isomorphic to the complex reflection group U 4 , No 4 in the Shephard-Todd sequence. In virtue of its isomorphism to SL(2, 3), the group dipole C - 1 also corresponds to the smallest known unitary two-dimensional 2-design (see [START_REF] Gross | Evenly distributed unitaries: on the structure of unitary designs[END_REF][START_REF] Scott | Unitary designs and codes[END_REF] and the appendix C for the definition). ‡ The orthogonal group Ω ± (2n, 2), of even dimension 2n (n ≥ 1), is defined over the field GF (2) in terms of two generating matrices. It is the kernel of the spinor map on the orthogonal group SO ± (2n, 2), and may also be defined as the derived subgroup of the general orthogonal group O ± (2n, 2). It is usually a perfect group.

Two-qubit group dipoles

The relevant groups are the Clifford group C 2 (order 92160) and two subgroups: the local Clifford group C L 2 ⊃ S (order 4608) and the Bell group B 2 ⊃ R (order 15360), which may be generated as

C 2 = C 1 ⊗ C 1 , CZ = H ⊗ H, H ⊗ P, CZ , C L 2 = C 1 ⊗ C 1 = H ⊗ H, H ⊗ P, T , B 2 = H ⊗ H, H ⊗ P, R , (12)
with CZ = diag(1, 1, 1, -1), P is the π/2 phase gate and T is the swap gate [START_REF] Jozsa | Matchgates and classical simulation of quantum circuits[END_REF]. The importance of the match gate R and of the swap gate T , in the context for the BN pair decomposition of the Clifford group, is an important issue of [START_REF] Planat | Clifford groups of quantum gates, BN -pairs and smooth cubic surfaces[END_REF].

B ∼ = W (F 4 ), N ≡ B 2 , H 0 ∼ = Z 8 and W ∼ = W (D 5 ), (13)
in which B is the Coxeter group of type "F 4 " (the symmetry group of the 24-cell), N is the Bell group, and the Weyl group W of the pair is isomorphic to the Coxeter group of type "D 5 ." The two-qubit Clifford group dipoles are

C + 2 ∼ = E + (32).Ω + (4, 2) ∼ = E + (32) ⋊ S 2 3 ∼ = W (F 4 ), C - 2 = B ′ 2 ∼ = E -(32).Ω -(4, 2) ∼ = E -(32).A 5 , (14)
There exists an important maximal subgroup of the group dipole C - 2 , that is isomorphic to the group SL(2, 5) and, as the alternating group A 5 , corresponds to the smallest known 2-dimensional 5-design [START_REF] Scott | Unitary designs and codes[END_REF] with generators

1 2     1 + i 1 -i 0 0 -1 -i 1 -i 0 0 0 0 1 + i 1 -i 0 0 -1 -i 1 -i     , 1 2     0 1 + i 1 + i 0 i -1 0 0 1 -i i -1 0 0 i -1 0 -1 -i 1 + i 0     . ( 15
)
Their action on the computational base is either separable (for the l.h.s. generator) or gives rise to a Bell basis with a phase factor (for the r.h.s. generator).

Three-qubit and higher-order group dipoles

The three-qubit Clifford group dipoles are

C + 3 ∼ = E + (128).Ω + (6, 2) ∼ = E + (128).A 8 , C - 3 = B ′ 3 ∼ = E -(128).Ω -(6, 2) ∼ = E -(128).W ′ (E 6 ), (16)
in which the simple groups A 8 (the eight-letter alternating group) and W ′ (E 6 ) (the derived subgroup of the Coxeter group of type "E 6 " ) are of order 20160 and 25920, respectively. More generally, higher order group dipoles involve the orthogonal groups Ω ± (2n, 2), that identify to D n (2) and 2D n (2) (twisted) Chevalley groups.

The two constitutive real entangling gate S ant swap gate T may be used for generating the group dipole C + 3 , of order 2 580 480, as follows

C + 3 = σ x ⊗ S, S ⊗ σ x , σ x ⊗ T, T ⊗ σ x . (17)
See also [START_REF] Nebe | The Invariants of the Clifford Groups Designs[END_REF] for the relation between real Clifford groups and Barnes-Wall lattices. Among the subgroups of C + 3 having relevance to unitary group designs, there are five subgroups, isomorphic to the complex reflection group U 24 (of order 336). As their central quotient, isomorphic to the simple subgroup G 168 = P SL(2, 7), they define threedimensional 2-designs. The smallest known design with these parameters is of order 72, isomorphic to Z 2 3 ⋊ E -(8) [START_REF] Scott | Unitary designs and codes[END_REF], and is also a subgroup of C + 3 . The simple group G 168 is intimately related to tripartite entanglement (see the appendix D for a reminder about the measures of entanglement). In the representation

G 168 = a = σ 0 ⊗ CZ, b = 1 2              1 1 0 0 0 0 -1 1 1 1 0 0 0 0 1 -1 0 0 1 -1 -1 -1 0 0 0 0 -1 1 -1 -1 0 0 -1 1 0 0 0 0 1 1 -1 1 0 0 0 0 -1 -1 0 0 -1 -1 1 -1 0 0 0 0 1 1 1 -1 0 0              , (18)
with orthogonal matrices a and b one observes that the action of b tr on the computational basis creates a 8-dim basis of GHZ states such as the entangled state

b tr |000 = 1 2 (|000 + |001 -|110 + |111 ). ( 19
)
Using concepts recalled at the appendix D, one easily obtains the residual tangle τ (3) = 1 and the vanishing of all bipartite tangles, that are properties specific to quantum states of the GHZ family. With further insight §, it is found that rows of matrix b correspond the joined eigenstates of the following set of mutually commuting operators

I ⊗ σ z ⊗ σ x , σ x ⊗ σ x ⊗ σ z , σ x ⊗ σ y ⊗ σ y , σ z ⊗ I ⊗ σ x , σ z ⊗ σ z ⊗ I, σ y ⊗ σ x ⊗ σ y , σ y ⊗ σ y ⊗ σ z . ( 20
)
§ The mutually commuting sets encoding the matrices b in ( 18), c and d in [START_REF] Hunt | The geometry of some special arithmetic quotients[END_REF] were discovered by Peter Levay, see also [START_REF] Levay | Three-qubit operators, the split Cayley hexagon of order two, and black holes[END_REF].

The three-qubit representation of the simple group P SL(2, 7) in ( 18) is indeed quite different from the Hurwitz presentation with generators and relations x, y|x 2 = y 2 = (xy) 7 = [x, y] 4 = 1 (in which [x, y] means the group commutator of elements x and y) [START_REF] Conder | Hurwitz groups: a brief survey Bull[END_REF]. The presentation with generators and relations one obtains for G 168 is a 2 = b 4 = (ba -1 ) 7 = (b -2 a) 2 = 1.

Enactment the Weyl/Coxeter group W (E 8 ) and its relatives

The finite Coxeter/Weyl group of the largest cardinality is W (E 8 ) ∼ = Z 2 .O + (8, 2), of order 696 729 600, in which O + (8, 2) is the general eight-dimensional orthogonal group over the field GF (2) and O + (8, 2) ′ = Ω + (8, 2). It may be realized as the complex reflection group U 37 (2 120 reflections), the last group in the Shephard-Todd sequence. Group W (E 8 ) plays a unifying role in physics [START_REF] Koca | Noncrystallographic Coxeter group H4 in E8[END_REF][START_REF] Lisi | An eceptionnally simple theory of everything[END_REF], being the symmetry group of the largest exceptional root system, that of the simple Lie group E 8 .

Until now, we have crossed many important subgroups of the Clifford group that are isomorphic to complex reflection groups and one may legitimately ask whether it is a mere coincidence, or if there is a deeper mechanism relating quantum computing and all the finite reflection groups. Remarkably, W (E 8 ) can be generated by adjoining to the real Clifford group C + 3 the (non-Clifford) Toffoli gate C 2 NOT = TOF, which applies a NOT operation to the (target) third qubit only if the two first (control) qubits are set to |1 . Thus,

C + 3 , TOF = I ⊗ S, S ⊗ I, TOF ∼ = W (E 8 ). (21)
Conversely, it is easy to recognize C + 3 as isomorphic to the second largest maximal subgroup of W ′ (E 8 ) ∼ = O + (8, 2).

The Toffoli gate is a well known universal and reversible logic gate for classical computing. Since any reversible gate may be implemented on a quantum computer, it also serves as a quantum gate. The union of Toffoli and Hadamard gates is universal for quantum computation [START_REF] Shi | Both Toffoli and Controlled-NOT need little help to universal quantum computing[END_REF].

Let us rewrite [START_REF] Dür | Three qubits can be entangled in two inequivalent ways[END_REF]

as b, b = 1 2              1 -1 0 0 0 0 1 1 1 1 0 0 0 0 -1 1 0 0 1 1 -1 1 0 0 0 0 -1 1 -1 -1 0 0 0 0 1 1 1 -1 0 0 0 0 -1 1 1 1 0 0 -1 1 0 0 0 0 1 1 -1 -1 0 0 0 0 -1 1              , TOF ∼ = W (E 8 ), ( 22
)
where C + 3 = b, b is an alternative way to generate the 3-qubit real Clifford group with generators of the GHZ-type. One easily observes that b and b only differs from a reordering of the rows and thus correspond to the same set of eigenstates, already displayed in [START_REF] Coffman | Distributed entanglement[END_REF].

Let us now turn to another set of GHZ-type gates for generating the group W (E 8 )

c = 1 2              1 0 0 -1 0 1 1 0 0 1 -1 0 1 0 0 1 0 1 1 0 1 0 0 -1 1 0 0 1 0 1 -1 0 0 -1 1 0 1 0 0 1 -1 0 0 1 0 1 1 0 1 0 0 1 0 -1 1 0 0 1 1 0 -1 0 0 1              , d = 1 2              1 0 -1 0 0 -1 0 -1 0 1 0 1 1 0 -1 0 1 0 1 0 0 -1 0 1 0 -1 0 1 -1 0 -1 0 0 -1 0 1 1 0 1 0 1 0 1 0 0 1 0 -1 0 1 0 1 -1 0 1 0 1 0 -1 0 0 1 0 1              (25)
that are such that

c, d, TOF ∼ = W (E 8 ) , C + 3 = c, d, σ 0 ⊗ CZ and c, d ∼ = C - 2 . ( 26
)
The second largest complex reflection group U 36 ≡ W (E 7 ), of order 2 903 040 with 2 63 reflections, may be generated as 28) Indeed, the unitary realization of W (E 8 ) with quantum gates of the GHZ type is much different from the Weyl group one gets from the Lie algebra of E 8 . Recall that the gears of this new representation of W (E 8 ), that induce the tripartite entanglement, are simply the real bipartite entangling matrix X and the Toffoli gate, that we used in [START_REF] Dür | Three qubits can be entangled in two inequivalent ways[END_REF]. The Pandora's box of cristallographic groups only arises from these two conclusive players.

One can check that matrix c has rows encoding states shared by the following set of seven mutually commuting operators

σ z ⊗ σ z ⊗ σ z , I ⊗ σ y ⊗ σ y , σ y ⊗ I ⊗ σ y , σ y ⊗ σ y ⊗ I, σ z ⊗ σ x ⊗ σ x , σ x ⊗ σ z ⊗ σ x , σ x ⊗ σ x ⊗ σ z , (23)
and that matrix d has rows encoding the shared eigenspace of the triple of mutually commuting operators

σ y ⊗ σ z ⊗ σ y , σ y ⊗ σ y ⊗ σ x , I ⊗ σ x ⊗ σ z . (24)
Enacting the Weyl/Coxeter group W (E 6 )

The Weyl/Coxeter group W (E 6 ) ∼ = O -(6, 2), of order 51840 with 2 36 reflections, is an important subgroup of W (E 7 ), being the symmetry group of a smooth cubic surface embedded in the three-dimensional complex projective space P 3 (C). The surface contains a maximum of 27 lines in general position and 45 sets of tritangent planes. The group of permutations of the 27 lines is W (E 6 ), the stabilizer of a line is

W (D 5 ) (observe that |W (E 6 )|/|W (D 5 )| = 27
) and the stabilizer of a tritangent plane is W (F 4 ) [START_REF] Hunt | The geometry of some special arithmetic quotients[END_REF]. Thus, the BN-pairs, and the Clifford group dipoles described at the previous section,are reflected into the geometry of such a cubic surface. But W (E 6 ) is not a subgroup of the 3-qubit Clifford group, further gates have to be added to display it. One among many unitary realizations of

W (E 6 ) is e, f, TOF ∼ = W (E 8 ) and e, f ∼ = W (E 6 ), where e = 1 2              0 0 0 -1 -1 -1 0 -1 -1 -1 -1 0 0 0 -1 0 0 0 0 1 1 -1 0 -1 -1 -1 1 0 0 0 1 0 -1 1 1 0 0 0 -1 0 0 0 0 -1 1 -1 0 1 1 -1 1 0 0 0 -1 0 0 0 0 -1 1 1 0 -1              , and f = 1 4              -1 -1 1 -1 -1 1 3 -1 1 1 -1 1 1 3 1 1 1 1 -1 1 1 -1 1 -3 3 -1 1 -1 -1 1 -1 -1 -1 -1 -3 -1 -1 1 -1 -1 -1 3 1 -1 -1 1 -1 -1 -1 -1 1 3 -1 1 -1 -1 -1 -1 1 -1 3 1 -1 -1              . (29)
Similarly to the Toffoli gate, gates e and f do not belong to the real Clifford group C + 3 . If one complements the unitary representation of C + 3 , or the one of W (E 7 ), by gate e, or by gate f , or by both gates e and f , one obtains a representation of

W ′ (E 8 ), of cardinality |W (E 8 )|/2. Another useful expression is b, e ∼ = W (E 7 ).
The entanglement involved in the matrices is peculiar. As shown in the example provided at the appendix D, the entanglement for the states arising from matrix e is a linear chain A -B -C, and a similar calculation for the states arising from the matrix f shows that the entanglement of the W type.

Discussion

I discussed a relationship between Mermin's approach of Kochen-Specker theorem and quantum computation. I introduced a bipolar decomposition of the Clifford group, attached to error correction, and made explicit the corresponding generating gates. A new orthogonal realization of Weyl/Coxeter group W (E 8 ) based on quantum gates has been uncovered. It opens up new vistas for quantum computing by providing optimal sets of gates with a clear group theoretical structure, such as BN-pairs and designs, that may serve for specific purposes. It also adds an alternative to the complex reflection groups of the Shephard-Todd list that often serve as the background of essays about the unification of physics [START_REF] Lisi | An eceptionnally simple theory of everything[END_REF]. The peculiar role of the 2-dimensional 5-design SL [START_REF] Mermin | Hidden variables and the two theorems of John Bell[END_REF][START_REF] Planat | Quantum Entanglement and Projective Ring Geometry SIGMA 2 Paper[END_REF] in the cosmological context [START_REF] Kramer | An invariant operator due to F Klein quantizes H Poincar's dodecahedral 3manifold[END_REF] and its relation to W (E 8 ) is intriguing. All types of three-qubit distributed entanglement arise in our unitary realization of W (E 8 ) and of its reflection subgroups. See also [START_REF] Levay | Three-qubit operators, the split Cayley hexagon of order two, and black holes[END_REF] and references therein for the mathematical analogy between some stringy black hole solutions, quantum entanglement, finite geometries, and [START_REF] Planat | hree-qubit entangled embeddings of CPT and Dirac groups within E8 Weyl group[END_REF] for further developments of the present paper.

Appendix D: Measures of entanglement

The resources needed to create a given entangled state may be quantified, and one can define invariants for discriminating the type of entanglement.

For a pair of quantum systems A and B in a pure state of density matrix |ψ ψ|, the entanglement of formation is defined as the entropy of either of the two subsystems As both ρ and ρ are positive operators, the product ρρ also has only real and nonnegative eigenvalues λ i (ordered in decreasing order) and the concurrence reads

C(ρ) = max 0, λ 1 -λ 2 -λ 3 -λ 4 .
For a two-qubit state |ψ = α |00 + β |01 + γ |10 + δ |11 , the concurrence is C = 2 |αδ -βγ|, and thus satisfies the relation 0 ≤ C ≤ 1, with C = 0 for a separable state and C = 1 for a maximally entangled state.

The entanglement of a triple of quantum systems A, B and C in a pure state may be conveniently described by tracing out over partial subsystems AB, BC, and AC. In this generalized context, one introduces the tangle τ = C 2 . Tangles attached to the bipartite subsystems above satisfy the inequality

τ AB + τ AC ≤ 4detρ A ≡ τ A(BC) .
The right hand side is interpreted as the amount of entanglement shared by the single qubit A with the pair BC, in comparison with the amounts of entanglement shared with qubits B and C taken individually. It is remarkable that, for any value of the tangles satisfying this inequality, one can find a quantum state consistent with those values [START_REF] Coffman | Distributed entanglement[END_REF].

It has been shown that an arbitrary three-qubit state |ψ can be entangled in essentially two inequivalent ways, belonging to the GHZ-class:

|GHZ = 1 √ 2 (|000 + |111 ) or to the W-class: |W = 1 √ 3 (|001 ) + |010 + |100
), according whether ψ can be converted to the state |GHZ or to the state |W , by stochastic local operations and classical communication (SLOCC) [START_REF] Dür | Three qubits can be entangled in two inequivalent ways[END_REF]. The relevant class is determined by computing the bipartite tangles of the reduced subsystems. If they vanish, then the subsystems are separable and |ψ belongs to the GHZ-class, meaning that all the entanglement is destroyed by tracing over one subsystem. If none of the bipartite tangles vanish, then |ψ belongs to the W-class, meaning that it maximally retains bipartite entanglement after tracing over one subsystem. For the GHZ state the 3-tangle becomes maximal: τ (3) = 1 and it vanishes for any factorized state. It also vanishes for states of the W -class. The 3-tangle may be interpreted as the residual tangle τ (3) = τ A(BC) -(τ AB + τ AC ), i.e., the amount of entanglement between subsystems A and BC that cannot be accounted for by the entanglements of A with B and C separately. It is of course independent on which qubit one takes as the reference of the construction. The GHZ state is a true tripartite entangled state so that no amount of entanglement is in the bipartite subsystems, as a result the residual entanglement is maximal. In contrast, for the states of the W-class the entanglement is of a pure bipartite type and τ (3) = 0. Mixtures of GHZ and W states are studied in [START_REF] Lohmayer | Entangled Three-Qubit States without Concurrence and Three-Tangle[END_REF], where it is shown that while the amounts of inequivalent entanglement types strictly add up for pure states, the monogamy is in general lifted for mixed states because the entanglement can arise from different types of locally inequivalent quantum correlations.

τ (3) = 4 |d 1 -2d 2 + 4d 3 | , d 1 =
Apart from pure tripartite entanglement (the GHZ states) and equally distributed bipartite entanglement (the W states), one can obtain a linear chain configuration of entanglement of the type A -B -C, where the two parties (A, B) and (A, C) are both entangled, but the parties (A, C) are not. One example is the state

|ψ = 1 2 (|011 + |100 + |101 + |111 ),
which arises in the context in the unitary realization of the group W (E 6 ) in [START_REF] Levay | Three-qubit operators, the split Cayley hexagon of order two, and black holes[END_REF]. Using the relations above one gets τ (3) = 1/4 (i.e. a non equally distributed entanglement) and the tracing over two qubits may be calculated as The sets of eigenvalues for the first two matrices ρ BC and ρ AB are

1 16 (3 + 2 √ 2), 1 16 (3 -2 √
2), 0, 0 so that the corresponding concurrence for pairs (B, C) and (A, B) is strictly positive. In contrast, the set of square eigenvalues for the matrix ρ AC is 1 16 , 1 16 , 0, 0 and the corresponding concurrence for the pair (A, C) vanishes.

  b, c, TOF ∼ = W (E 7 ) where b, c ∼ = Z 4 2 ⋊ D 4 and b, c ∼ = C - 2 , (27) where D 4 is the dihedral group of order 8. Still another way to realize/enact W (E 8 ) in a unitary way is to complement the 3-qubit representation of SL(2, 5) with the Toffoli gate as follows b, d, TOF ∼ = W (E 8 ) with C + 3 = b, d, σ 0 ⊗ CZ and b, d ∼ = SL(2, 5).(

  A and B E(ψ) = -tr(ρ A log 2 ρ A ) = -tr(ρ B log 2 ρ B ), where ρ A and ρ B are partial traces of ρ over subsystems B and A, respectively. The measure is made explicit by defining the spin-flipped density matrix [20] ρ = (σ y ⊗ σ y )ρ * (σ y ⊗ σ y ), and the concurrence C(ψ) = | ψ| ψ | between the original and flipped state ψ = σ y |ψ * .

Further 1 ψ

 1 discrimination of the entanglement type of a general 3-qubit state |ψ = a,b,c=0,abc |abc , can be obtained by calculating the SLOCC invariant three-tangle[START_REF] Coffman | Distributed entanglement[END_REF]

  computation, one can deduce that matrices R and S, separately, are universal (see Sec. 2.1 in[START_REF] Kauffman | Braiding operators are universal quantum gates[END_REF]). The R matrix satisfies the Yang-Baxter equation (see Sec. 2 of[START_REF] Kauffman | Braiding operators are universal quantum gates[END_REF]), but the S matrix does not. It has a different status, that is used for deriving the largest cristallographic group W (E 8 ) in Sec.[START_REF] Bosma | The Magma algebra system[END_REF]. Both matrices are related by a relation involving the Hadamard matrix H as Matrices R and S are distinguished members of the two-qubit Clifford group C 2 . Recall that the n-qubit Clifford group C n is defined as the normalizer, in the unitary group U(2 n , C), of the n-qubit Pauli group P n . The n-qubit Pauli group P n is generated by the n-fold tensor products of ordinary Pauli spin matrices. Thus, by definition, the two-qubit Pauli group P n maps to itself under the conjugation action of matrices/gates of C n . The distinctive character of R and S leads to a natural splitting of the Clifford group into two poles (see

	RS = H ⊗ I with H =	1 2	1 1 1 -1	.	(4)

  of universal matrices of the preceeding section reflects into pairs of subgroups of the Clifford group. In the sequel, one may see the positive group dipole C + 2 (which contains S and not R) as indexed by S and the negative group dipole C -

The inclusion of matrices R in B 2 and S in C L 2 is exclusive, i.e. R / ∈ C L 2 and S / ∈ B 2 . The pair (R, S) 2 (which contains R and not S) as indexed by R.

More precisely, one can construct a BN-pair of the two-qubit Clifford group

[START_REF] Planat | Clifford groups of quantum gates, BN -pairs and smooth cubic surfaces[END_REF]

  = ψ 000 ψ 111 (ψ 011 ψ 100 + ψ 101 ψ 010 + ψ 110 ψ 001 ) +ψ 011 ψ 100 (ψ 101 ψ 010 + ψ 110 ψ 001 ) + ψ 101 ψ 010 ψ 110 ψ 001 , d 3 = ψ 000 ψ 110 ψ 101 ψ 011 + ψ 111 ψ 001 ψ 010 ψ 100 .

	ψ 2 000 ψ 2 111 + ψ 2 001 ψ 2 110 + ψ 2 010 ψ 2 101 + ψ 2 100 ψ 2 011 ,
	d 2

Acknowledgements

The author thanks Peter Levay for finding the mutually commuting operators occuring in the three-qubit entangled matrices, and Maurice Kibler for his comments on the manuscript.

Bibliography

Appendix A: Complex reflection groups

Basically, reflections are linear transformations that leave invariant a hyperplane of a vector space, while sending vectors orthogonal to the hyperplanes to their negatives. For an Euclidean vector space E, finite groups of reflections possess a Coxeter group structure, i.e. a a presentation in terms of a finite set of involutions with specific relations. There is a formal similarity between quantum errors g of the Pauli group P n and reflections s α acting on the Euclidean space E, and between the Clifford group action on P n and the action of the orthogonal group O(E) on E.

A unitary element of the Clifford group maps P n to itself ∀g ∈ P n and C ∈ C n , CgC -1 = g ′ ∈ P n , and the orthogonal group O(E) map reflections to reflections

in which α denotes the index of a hyperplane of E and t(α) the index of the hyperplane mapped by the action of O(E).

Euclidean reflection groups may be generalized to pseudo-reflection groups by replacing E by a vector space over the complex field C. Finite irreducible unitary reflection groups are classified: They include the (real) Coxeter groups [usually denoted W (X i ) for the Coxeter type X i ], three infinite families Z m = Z/mZ, the symmetric groups S n , the imprimitive reflection groups G(m, p, n) = A(m, p, n) ⋊ S n (that are semidirect products of a specific group of diagonal matrices with S n ), and 34 exceptional (Shephard-Todd) groups U n [START_REF] Kane | Reflection groups and invariant theory[END_REF]. The largest one is U 37 ∼ = W (E 8 ), of cardinality 696 729 600 with 2 120 reflections.

Many of the self-dual codes so far derived rely on the well developed invariant theory of reflection groups and its relevance to Clifford groups [START_REF] Nebe | The Invariants of the Clifford Groups Designs[END_REF][START_REF] Planat | Unitary reflection groups for quantum fault tolerance[END_REF].

Appendix B: BN-pairs

Let consider a finite group G, and two subgroups B and N of G generating G, H 0 = B ∩ N a normal subgroup of N and the quotient group W = N/H 0 generated by a set S ⊂ W of involutions. A group G is said to have a BN-pair iff it is generated as above and two extra relations (i) and (ii) are satisfied by the double cosets (i) For any s ∈ S and w ∈ W, sBw ⊆ (BwB) ∪ (BswB), (ii) For any s ∈ S, sBs B.

The pair (W, S) arising from a BN-pair is a Coxeter system.

One can form BN-pairs of the Clifford group [START_REF] Planat | Clifford groups of quantum gates, BN -pairs and smooth cubic surfaces[END_REF].

Appendix C: Unitary designs

A unitary design is a set of unitary matrices that simulates the entire unitary group. It is a variation of spherical-t designs and of Grassmannian t-designs [START_REF] Gross | Evenly distributed unitaries: on the structure of unitary designs[END_REF][START_REF] Scott | Unitary designs and codes[END_REF].

In a unitary t-design, the integral U (d) U ⊗t ⊗ (U * ) ⊗t dU over all d-dimensional unitary matrices is identical to its restriction to a discrete subset X.

For a finite set X ⊂ U(d) of unitary matrices

with equality if and only if X is a t-design.

Many unitary group designs were constructed as the images of unitary representations of finite groups [START_REF] Gross | Evenly distributed unitaries: on the structure of unitary designs[END_REF][START_REF] Scott | Unitary designs and codes[END_REF] using the following theorem:

Let G be a finite group and ρ : G → U(d) a representation with character κ. Then X = {ρ(g) : g ∈ G} is a unitary t-design iff

The right hand side of the above equation is the moment of order 2t of the trace of a random d-dimensional unitary matrix. There exists a combinatorial interpretation as the number of permutations of length t with no increasing subsequence of length greater than d. If d ≥ t, then the r.h.s. is t!.

Many efficient designs may be harversted using the known character tables of finite groups. But several optimal (i.e. small size) Clifford designs could only be obtained from subgroups of the symplectic group Sp(2n, q) acting transitively on a punctured vector space. The latter are closely related to the Clifford group dipoles investigated in this paper.


J Tabet

S Eden

S Feil

H Abdoul-Carime

B Farizon

M Farizon

S Ouaskit

T D Märk

20-150 keV proton impact induced ionization of uracil: fragmentation ratios and branching ratios for electron capture and direct ionization

Keywords: Uracil, proton impact, ionization, dissociative ionization, electron capture, mass spectrometry

Fragmentation ratios have been measured for ionization and dissociative ionization for 20-150 keV (0.9-2.4 v 0 ) proton collisions with gas-phase uracil molecules. Through event-by-event determination of the post-collision projectile charge, it is possible for the first time for such a key biomolecule to distinguish between electron capture (EC) by the incident proton and direct ionization (DI) without projectile neutralization. While the same fragment ion groups are observed in the mass spectrum for both processes, electron capture induces dissociation with greater efficiency than direct ionization in the impact energy range of 35-150 keV (1.2-2.4 v 0 ). In this range electron capture is also less abundant than direct ionization with a branching ratio for electron capture / total ionization of < 50%. Moreover, whereas fragmentation ratios do not change with energy in case of electron capture, direct ionization mass spectra show a tendency for increased fragmentation at lower impact energies.

Introduction

The exposure of living tissue to ionizing radiation can kill cells and initiate mutations or cancers, effects which have been traced to the structural and chemical modification of DNA (deoxyribonucleic acid) including strand breaks and clustered lesions [START_REF] Tabet | The Chemical Basis for Radiation Biology[END_REF]. In the wake of the pioneering work directly linking specific molecular-scale interactions to DNA strand breaks [e.g. Boudaiffa et al. 2000], the experimental and theoretical study of radiation-induced processes in isolated biomolecules has developed into a significant field at the borderline between physics, chemistry, and biology. A number of recent contributions to the subject have focused upon interactions of relevance to cancer therapy techniques in which beams of accelerated ions are used to deliver localized doses of energy to kill cells within tumors (proton and hadron therapies) [Moretto-Capelle and Le Padellec 2006].

These treatments exploit the Bragg peak maximum for energy deposition by incident ions at velocities around 2.0 v 0 (100 keV for protons), this peak maximum being a product of the interplay between ionization, excitation, and charge exchange processes as the projectiles slow down in a medium [Biaggi et al. 1999[START_REF] Cabrera-Trujillo | Application of Accelerators in Research and Industry International Conference: AIP Conference Proceedings[END_REF].

Uracil (C 4 H 4 N 2 O 2 ) is one of the four nucleobases in ribonucleic acid (RNA), the others being adenine, cytosine, and guanine. RNA plays a key role in the translation of genetic information and includes the same nucleobases as DNA except for uracil which replaces thymine; both nucleobases pair with adenine in the respective nucleic acids. While other tautomeric forms of uracil are possible, the structure shown as an insert in Fig. 1 is the only one which has been identified both in solution and in the gas phase [Marian et al. 2002, Becker andKogan 1980]. The geometrical structure and conformational flexibility of uracil has been studied on the basis of MP2 and DFT calculations by Shishkin et al. [2003]. In addition to its important role in biosynthesis and radiobiology, uracil was chosen for the present experiments due to the relatively large volume of comparable gas-phase ionization data available already in the literature (discussed in section 3).

The present work provides for the first time fragmentation patterns (ratios) for the ionization of a nucleobase as a function of proton impact energy also allowing to distinguish between charge exchange collisions between the projectile ion and target molecule and direct ionization events in the velocity range coinciding with maximum energy deposition (Bragg peak maximum). Beyond their relevance to the development of progressively more refined mechanistic models of ion-induced radiation damage in biological materials [Friedland et al. 2003], the results are of fundamental interest with respect to the production of fragment ions either by electron capture or by direct ionization in the case of an electronically and geometrically complex target molecule such as a nucleic base.

Experimental

The crossed-beam apparatus used for the present experiments is shown schematically in Fig. 2 [ Gobet et al. 2001]. Pure molecular hydrogen is ionized in a standard RF-gas discharge source (80 MHz) typically operated at 30 W with a H 2 pressure of 1 Pa. Beams of singly charged ions extracted from this gas discharge ion source are accelerated to energies between 20 and 150 keV with a resolution (∆E/E) of 0.01. The accelerator system has been described in detail elsewhere [START_REF] Carré | [END_REF]. A first magnetic sector field is used to separate protons from other ions such as H 2 + , H 3 + , and other ions originating from impurities in the source. After collimation by means of two circular apertures of radius 0.5 mm set 1 m apart, the proton beam is crossed at right angles with an effusive beam of uracil molecules. The uracil beam is formed by the sublimation of uracil powder (purchased from Sigma-Aldrich, minimum purity 99%) in a temperature-controlled Knudsen-type oven operated typically at 175 -200 °C. Previous studies indicate that min imal thermal decomposition and isomerization of uracil occurs at these temperatures (Desfrançois et al. 1996). Accordingly, no evidence was observed for temperature-dependence in the present mass spectra for uracil ionization by proton impact. The exit aperture of the oven has a diameter of 1 mm and is positioned 2 mm below the incident proton beam in order to achieve a high-density target beam. The charge state of the projectile after a collision with an uracil molecule is determined using a second magnetic sector field mass analyzer with three channeltron detectors located at the appropriate positions to detect H + , H 0 , and H -. However, due to the low statistics for the coincident detection of an H -projectile with a product ion (e.g. less than 0.2% of all coincidence events at 80 keV), double electron capture results are not presented in this paper.

A customs-built linear time-of-flight (TOF) mass spectrometer is used to analyze the uracil product ions formed by the impact of a proton with an uracil molecule. The instrument comprises an extraction region defined by parallel plates (±150 V, 10 mm apart) on either side of the uracil beam, an acceleration region, a drift tube of 120 mm length, and a channeltron detector. The positive product ions are extracted from the interaction region perpendicularly to both the proton and uracil beam, the extraction and acceleration fields are set following the conditions defined by Wiley and McLaren [1955] in order to focus ions selected precisely at the detector entrance.

It is important that the proton projectile beam does not contain fast hydrogen atoms formed by the neutralization of protons in collisions with surfaces or the residual gas. Thus the background vacuum is maintained below 10 -6 Torr and the alignment of the proton beam is verified prior to each experiment. Furthermore, single collision conditions are necessary to guarantee the unambiguous identification of the ionization processes. Both was checked with 80 keV protons by varying the target jet density by a factor of five changing the oven temperature accordingly. As changing the oven temperature in this way did not affect the measured branching ratio (25.4% ± 2) for electron capture (electron capture events divided by the sum of EC and DI ecents), it can be concluded that interactions between uracil molecules and hydrogen atoms neutralized in the jet did not contaminate the product ion signal because it is know from earlier measurements that for instance for 80 keV neutral hydrogen atom collisions with water molecules [Gobet et al. 2006] about 70% of the collisons led to electron loss reactions.

The time-of-flight (TOF) measurement to allow the determination of the mass-to-charge ratio (m/q in Thomson) of the product ions relies for its starting pulse on the detection of the corresponding proton projectile. As mentioned above each projectile that crosses the interaction region can be detected whatever its post-interaction charge state. The energy transfer during a collision with a target molecule is expected be less than ~ 100 eV by analogy with Cabrera-Trujillo et al.'s [2000] calculations for 25 keV low-impact-parameter proton collisions with nitrogen, oxygen, and fluorine atoms. As this is small in comparison with the incident kinetic energy of the projectiles (20-150 keV),

the precise time at which the proton/uracil interaction takes place can be determined for each detected projectile, and the time difference between a pulse at the product ion channeltron detector of the TOF and the proton/uracil interaction equals the flight time of the product ion. Clearly, the number of projectiles has to be sufficiently low for each product ion signal to be correlated to exactly one projectile. Therefore, only one proton is allowed to cross the interaction region during a time interval equal to twice the flight time of the heaviest conceivable product ion, that is the uracil parent ion. For the present experimental arrangement, this limits the primary ion beam current to 2000 protons per second.

By simultaneously determining the mass-per-charge ratio of the product ions and the postinteraction charge of the projectile, the experiment enables direct ionization (product ion detection with coincident H + detection after the secondary magnetic analyzer) to be distinguished from electron capture (coincident product ion and H 0 detection) for each ionization event. Thus, in the present terminology, direct ionization (DI) describes the removal of an electron from the uracil molecule to the continuum, and electron capture (EC) describes the transfer of an electron from the uracil molecule to the projectile. The fragmentation and branching ratios presented in section 3 correspond to single ion production only; events involving the detection of two or more fragment ions in coincidence with a single projectile (including events involving charge transfer and electron emission) are thus not included. At 80 keV, double ion production represents only about 5% of all observed ionization events.

Due to the relatively poor statistics, double ion production results are not discussed further in the present communication.

Results and Discussion

Branching ratios for electron capture and direct ionization

Fig. 1 shows ions formed by electron capture as a percentage of the total number of ions produced ( that is EC/EC + DI). The errors for this branching ratio estimated on the basis of the variation between repeated measurements for 80-150 keV protons are approximately ±2%.. In this energy range, the projectile detection signals were sufficiently strong to be separated completely from the noise, corresponding to projectile detection efficiencies approaching 100%. The errors are larger for impact energies below 65 keV because the threshold had to be set closer to the noise level.

Branching ratios for the 20 and 23 keV measurements are not included here due to large errors resulting from the low number of observed direct ionization events.

Branching ratios for electron capture in ionizing collisions with protons are available for a number of atomic and molecular targets [e.g. Rudd et al. 1983, Gobet et al. 2001, Gobet et al. 2004, Luna et al. 2007]. In each case, the %EC decreases with increasing impact energy in the present energy range. Fig. 3 shows the present uracil results on a logarithmic impact energy scale with previous electron capture branching ratios measured for proton impact ionization of water [Gobet et al. 2001, Gobet et al. 2004, Luna et al. 2007]. It is worth noting that the branching ratios calculated (assuming negligible double ionization) from Luna et al.'s [2007] recent coincidence data are in good agreement with the water ionization results recorded using the present experimental system [Gobet et al. 2001[Gobet et al. , 2004]]. Fig. 3 also shows %EC calculated from Rudd et al.'s [1983] absolute cross sections for electron emission and total ionization in proton collisions with He, CO 2 , CH 4 , and O 2 .

It is interesting to consider Fig. 3 in the context of the loose trend apparent in Rudd et al.'s data [1983] for atoms with higher ionization energies to demonstrate greater %EC in the lower energy part of the present range. In a simple Bohr-type model, this trend can be rationalized on the basis of approximate equivalent velocities of the bound electron and the incident proton providing favorable conditions for electron capture. As far as we are aware, no previous data is available to derive %EC values for a molecule with a similarly low ionization energy to that of uracil (IE = 9.59 ± 0.08 eV [Denifl et al. 2004]). However, the close agreement of the uracil data with the previous H 2 O (IE = 12.65 ± 0.05

eV [Snow and Thomas 1990]) and CH 4 (IE = 12.61 ± 0.01 eV [Berkowitz et al. 1987]) measurements [Rudd et al. 1983, Gobet et al. 2004, Luna et al. 2007] indicates that the lowest ionization energy is not a sensitive determinant for the relative contributions of EC and DI in 20-150 keV proton collisions with molecules, possibly suggesting that the capture of valence electrons from orbitals other than the HOMO plays a significant role. This interpretation appears to be consistent with uracil + accounting for less than 15% of the presently observed ions formed by electron capture (see section 3.2), while Denifl et al. [2004] reported uracil fragment ion appearance energies in the range 10.89 -14.77 eV.

Proton impact ionization mass spectra

Fig. 4 shows the mass spectrum for single ion production by electron capture and direct ionization in 80 keV (1.8 v 0 ) proton collisions with gas-phase uracil molecules. The histogram includes data which contributed to the summed mass spectrum for 20-150 keV (0.9-2.4 v 0 ) proton impact presented by Coupier et al. [2002]. More recently, a mass spectrum for 100 keV (2.0 v 0 ) proton impact ionization of gas-phase uracil has been reported by Le Padellec et al. [2008]. Schlathölter and co-workers studied uracil ionization in collisions with various ions, including multiply charged species. In particular, complete mass spectra were presented for He 2+ , C 2+ , N 2+ , and O 2+ impact at 0.2 v 0 [Schlathölter et al. 2005], C 1-6+ impact at 0.4 v 0 [Schlathölter et al. 2005, Schlathölter et al. 2006, de Vries et al. 2004, de Vries et al. 2002], C 6+ impact at 0.5 v 0 [Schlathölter et al. 2005], O 5+ impact at 0.5 v 0 [Schlathölter et al. 2006], C + impact at 0.1-0.3 v 0 [Schlathölter et al. 2005, de Vries et al. 2004], and 129 Xe 14+ impact at 0.2 v 0 [de Vries et al. 2003]. The same groups of singly charged product ions were observed in these varied ion impact conditions, with the exception of Schlathölter and co-workers' He 2+ impact result (discussed below). Unlike the present work, the previously presented ion impact mass spectra did not separate ionization processes as a function of charge transfer between the target and the projectile.

Recent electron impact ionization studies of gas-phase uracil have been carried out at incident energies of 200 eV (3.8 v 0 ) [Coupier et al. 2002], 120 eV (3.0 v 0 ) [Feil et al. 2004], and 70 eV (2.3 v 0 ) [Denifl et al. 2004, Imhoff et al. 2007, NIST] using quadrupole mass spectrometers (QMS). Also using a QMS analyzer, Jochims et al. [2005] measured ion yields following 20 eV photo-ionization of uracil.

For gas-phase uracil ionization, table 1 compares the present 80 keV (1.8 v 0 ) proton impact mass spectrum with Le Padellec et al. 's [2008] 100 keV (2.0 v 0 ) proton impact data and with the previous high-resolution electron impact and photo-ionization measurements [Imhoff et al. 2007, Denifl et al. 2004, Jochims et al. 2005]. The desorbed cations observed by Imhoff et al. [2007] following 200 eV (0.01 v 0 ) Ar + impact on condensed uracil are also listed in the table. With reasonable allowance for differences in resolution and background noise, peaks were generally observed at the same m/q values for the different projectiles. Similarly, whereas the relative intensities of the different ion groups differed for direct ionization and electron capture and varied to some extent with impact energy (see section 3.3), no associated variations in peak positions were observed in the present mass spectra.

The previous high-resolution electron impact, ion impact, and photo-ionization mass spectrometric studies of gas-phase uracil have shown the contributions of fragment ions of m/q close to uracil + to be negligible [Denifl et al. 2004, Feil et al. 2004, Imhoff et al. 2007, Jochims et al. 2005, de Vries et al. 2004]. Therefore, while the m/q resolution is insufficient to confirm or discount the production of intact uracil ions stripped of one or more exterior hydrogen atoms, it is reasonable to assume that such channels have a negligible contribution to the present data.

Fig. 4 shows that fragment ion production was significant in the m/q ranges corresponding to ions with 1-5 heavier (C, N, or O, as opposed to H) atoms. By contrast, ions produced by the loss of just one heavier atom have only been observed by 20 eV photo-ionization of gas-phase uracil (a very weak feature at 96 Thomson) [Jochims et al. 2005], 0.2 v 0 He 2+ impact on gas-phase uracil (78 and 94-96 Thomson), and 200 eV (0.01 v 0 ) Ar + irradiation of condensed uracil (95-97 Thomson) [Imhoff et al. 2007]. Jochims et al. [2005] and Imhoff et al. [2007] attributed these weak peaks to oxygen removal (combined with the possible removal of one or two hydrogen atoms) from uracil or protonated uracil, respectively. Schlathölter et al. [2005] rationalized the unusually strong production of fragment ions in this range following 0.2 v 0 He 2+ impact on the basis of the specific interplay between target and projectile electronic levels. Although no corresponding features were observed in the present 20-150 keV (0.9-2.4 v 0 ) proton impact mass spectra, the count rates between 75 and 90 Thomson were slightly higher than the background noise, suggesting very weak ion production. No evidence was observed for fragment ions above 90 Thomson. The weakness of any production of ions with 6 or 7 heavier atoms suggests that the dissociative ionization of gas-phase uracil following proton impact occurs almost exclusively via cleavage of the central aromatic ring. Accordingly, neutral HNCO loss (a retro Diels-Alder reaction) has been widely recognized as the initial step in the dominant fragmentation pathways of (uracil + )*, leading to fragment ion production with m/q ≤ 69 Thomson. Subsequent production of HNCO, HCN, CO, and H production (as well as combinations of these neutrals) are understood to account for the lower m/q fragment ion peaks [Jochims et al. 2005 Schlathölter et al. 2006, Coupier et al. 2002, Imhoff et al. 2007]. Indeed, the only previous mass spectrum showing the absence of H 2 + products from gas-phase uracil was de Vries et al.'s [2003] electron-ion coincidence measurement for 0.2 v 0 129 Xe 14+ impact. To the authors' knowledge, no attempt has been made to pinpoint the dominant fragmentation pathways associated with H + or H 2 + production from uracil.

The present lack of evidence for the production of small doubly charged ions (notably C 2+ , N 2+ , and O 2+ ) is consistent with Feil et al.'s [2004] observation of no signals of appreciable intensity for multiply charged ions following electron impact upon gas-phase uracil at energies from the ionization threshold to 1 keV (8.6 v 0 ). Accordingly, Le Padellec et al. [2008] commented that correlated fragment ion measurements show doubly charged nucleobase parent ions (e.g. uracil 2+ ) produced by proton impact to be scarce. De Vries et al. [2002] described the ratio of doubly to singly charged product ions as surprisingly low at ~0.75% for C 1-6+ impact at velocities in the range 0.1-0.7 v 0 , while the formation of specific multiply charged product ions has been investigated in more detail for 129 Xe 5-25+ impact upon uracil at 0.2 v 0 [de Vries et al. 2003, de Vries et al. 2004], Xe 8+ impact at 0.2 v 0 [Schlathölter et al. 2006], and 129 Xe 25+ impact at 0.6 v 0 [Schlathölter et al. 2004].

Fragment ion production as a function of impact energy and ionization process (EC / DI)

The present data provides an ideal platform to compare direct ionization with electron capture in terms of the branching ratios for fragment ion production against total ionization and their variation with impact energy. Product ion branching ratios calculated separately for EC and DI (e.g. the number of product ions produced by EC in a given mass range / the total number of product ions produced by EC) are presented in Fig. 5 ) and do not take into account the acceptance of the time-of-flight apparatus. Fragment ions are separated into 7 groups corresponding to the clear peaks in the mass spectra (see Fig. 4). Although the groups have been named after the ions associated with the maxima (see table 1), they include counts over the full range of each peak (e.g. 35-47 Thomson for the CNO + / C2H4N + / C2H2O + group). The contribution of background noise could be removed easily as it was observed to be constant across all flight times. Fig. 5 and table 2 do not show any clear evidence for impact energy-dependence in the electron capture product ion branching ratios following proton collisions with gas-phase uracil. It is interesting to contrast these results with the proton -H 2 O collision data recorded by Gobet et al. [2004] using the same apparatus. For electron capture from H 2 O, the branching ratio for fragment ion production increased from 47% at 20 keV to 67% at 150 keV * . This coincided with an approximate 25-fold decrease in the total cross section for electron capture [Gobet et al. 2001[Gobet et al. , 2004]]. Thus the observed product ion branching ratios following EC in proton -H 2 O collisions were broadly consistent with the generalized association of smaller impact parameters (more direct collisions, smaller cross sections)

with greater energy deposition and increased fragmentation [START_REF] Walch | [END_REF], Cabrera-Trujillo et al. 2000]. The cross sections for EC in proton -uracil collisions also decrease significantly from 20 to 150 keV (demonstrated in a forthcoming publication [Tabet et al. unpublished]). Why this does not have a discernable effect on the relative production of fragment ions from uracil is an open question. The * Allowing for errors associated primarily with corrections for ion acceptance, Gobet et al.'s [2004] proton impact data is in good agreement with the subsequent measurements carried out by Luna et al. [2007] in the impact energy range 20-100 keV.

relatively complex electronic configuration of uracil combined with the notoriously difficult theoretical treatment of ion-molecule interactions at intermediate velocities means that modeling the ionization processes observed in the present work represents a major challenge [Wang et al. unpublished].

In contrast to the electron capture results, energy dependence was observed in the relative production of uracil + and fragment ions following direct ionization. In particular, at low impact energies (42-27 keV) table 2 shows a clear reduction in the relative production of uracil + and of fragment ions belonging to the largest m/q group (the C 3 H 3 NO + group), as well as increased relative production of H + . Hence an increase in direct ionization-induced dissociation was observed at lower proton impact energies. Between 50 and 150 keV, however, the present DI results do not provide clear evidence for impact energy dependence in the relative production of the different ion groups. Conversely, the energy dependence of the DI cross section for proton impact upon water [Gobet et al. 2001[Gobet et al. , 2004] ] or uracil [Tabet et al. unpublished] is weak in the lower energy part of the present range, whereas it becomes progressively more significant from 50 to 150 keV. Therefore, as with the electron capture data, the branching ratios shown in Fig. 5 for direct ionization cannot be explained adequately by a simple association of increased fragmentation with smaller impact parameters.

Conclusions

The first branching ratios for electron capture and direct ionization in proton-uracil collisions are presented as a function of impact energy in the range 20-150 keV (0.9-2.4 v 0 ). The impact energy dependence of the percentage of ionization events occurring through electron capture as opposed to direct ionization shows the same broad characteristics as observed for smaller molecules [Gobet et al. 2004, Luna et al. 2007, Rudd et al. 1983]. To the authors' knowledge, the present work provides the first comparison between molecular fragmentation following electron capture and direct ionization in proton collisions with a relatively large and electronically complex molecule; the only previous experiments of this kind were carried out on O 2 [Luna et al. 2005] and H 2 O [Gobet et al. 2004, Luna et al. 2007]. No clear evidence was observed for energy-dependence in the relative production of uracil + and fragment ions following electron capture, whereas a relative increase in fragment ion production was observed for direct ionization in the low impact energy part of the present range.

The Institut de Physique Nucléaire de Lyon is part of IN2P3-CNRS, the French national research institute for nuclear and particle Physics. Financial support was provided by the French, Austrian, and Moroccan governments, and the EU Commission (Brussels), through the Amadee and PICS 2290 programs, the CNRS-CNRST (n°17689) convention. S. E den acknowledges the FP6 Marie Curie Fellowship (IEF RADAM-BIOCLUS).. Fig. 3: Electron capture ionization of uracil as a percentage of total ionization (electron capture + direct ionization) following proton impact in the energy range 27-150 keV. The data is compared to previous results for H 2 O [Gobet et al. 2004, Luna et al. 2007], and for He, CH 4 , CO 2 , and O 2 [Rudd et al. 1983]. H 2 O [Gobet et al. 2004] H 2 O [Luna et al. 2007] He [Rudd et al. 1983] CH 4 [Rudd et al. 1983] CO 2 [Rudd et al. 1983] O 2 [Rudd et al. 1983] EC Branching ratio (%)

Energy in keV

Not available Not available 1

A This column only includes the masses tabulated in the work of Denifl et al. [2004]; other trace ions are visible in the published mass spectrum. B The photo-ionization channels labeled weak correspond to those reported by Jochims et al. [2005] to have intensities ≤ 5% of the maximum peak intensity (42 amu).

C The non-asterisked product ion masses were labeled or mentioned explicitly by Le Padellec et al. [2004]. Conversely, the asterisked masses have been taken from a published figure and are therefore subject to greater uncertainty. D The same product ion groups and peak positions were observed for both direct ionization and electron capture across the full proton impact energy range studied (20-150 keV). E With the exception of 12-18 (the full range of the group), the ranges given in the present data column correspond to the halfmaximum width of the 80 keV DI peaks. F The present data only includes single ion production. G Imhoff et al. [2007] also suggested that C3H3NO + including a 13 C isotope may contribute to this peak. A Unlike the other product ion groups, this group contains more than one peak. The group is named after CH2 + , associated with the lower mass peak in the product ion group. 12-18 Thomson is the full m/q range of the group (see also figure 4 and table 1) B The m/q values in brackets correspond to the half-maximum width of each product ion group. The groups are named after the principle ion associated with the peak [Jochims et al. 2005, Imhoff et al. 2007, Denifl et al. 2004].

The errors given in the table are purely statistical; variations in the detection efficiency of different ions due to the acceptance of the TOF mass spectrometer have not been taken into account.

Fig. 1 :

 1 Fig. 1: Electron capture ionization of uracil as a percentage of total ionization (electron capture + direct ionization) following proton impact in the energy range 27-150 keV. Dark line: exponential fit. Insert: schematic representation of the structure of the uracil.

Fig. 4 :Figure 5 :

 45 Fig.4: Mass spectrum for the proton impact ionization of uracil (C 4 H 4 N 2 O 2 , 112 amu) by electron capture and by direct ionization at 80 keV. The principle ions expected to account for the peaks are listed in table 1.

  ]. It should be noted that bond rearrangements have been shown to occur prior to the fragmentation of quite similar metastable polyatomic cations to (uracil

	fragmentation prior to nuclear rearrangement, are expected to be significant in the present collision
	conditions. Accordingly, we suggest that the present peak at 42 Thomson may contain a relatively
	strong contribution of CNO	+ ions.
		The 12-18 Thomson group is apparent in the electron impact measurements covering this range
	[Coupier et al. 2002, Imhoff et al. 2007, NIST], in the previous ion impact data [Le Padellec et al. 2008,
	Schlathölter et al. 2005, Schlathölter et al. 2006, de Vries et al. 2004], and in Jochims et al.'s [2005]
	photo-ionization mass spectrum. In the present data, particularly strong peaks were observed at 12
	(C	+ ) and 14 Thomson (N	+ or CH 2 + ). For 70 eV electron impact on gas-phase uracil and 200 eV Ar	+
	impact on condensed uracil, Imhoff et al. [2007] assigned cation production in this mass range
	principally to CH 2 + and CH 3 + production. The relatively high intensity of the C	+ peak in the present
	mass spectra may be due to greater energy deposition by 20-150 keV proton impact leading to
	increased multi-fragmentation.
		H
					, C 2 H 4 N	+ , and
	C 2 H 2 O	+ . Jochims et al. [2005] suggested that direct CNO	+ production from uracil + is unlikely as it
	would require the rupture of 3 bonds, while CNO	+ loss from (C 3 H 3 NO	+ )* would involve a complex
	nuclear rearrangement. However it may be countered that the C 3 H 3 NO	+ -HCN → C 2 H 2 O	+ channel
	proposed by Jochims et al. [2005] would also involve a fairly complex rearrangement of the metastable
	precursor. Imhoff et al. [2005] attributed C 2 H 4 N	+ production to cleavage of the N1-C2 and C4-C5
	bonds (see Fig. 2) combined with the translation of the H atom bonded to N3 in (uracil + )*. Although the
	present work does not provide any new evidence to identify the dominant 42 Thomson fragment ion, it
	should be noted that higher energy transfer can be expected for 20-150 keV proton impact than for 70
	eV electron impact or 20 eV photo-ionization. Indeed Moretto-Capelle and Le Padellec [2006] reported
	significant emission of electrons with kinetic energies up to 50 eV following 25-100 keV proton impact
	upon gas-phase uracil, as well as weaker emission of 50 -200 eV electrons. Therefore ionization
	pathways involving high energy deposition, increasing the likelihood of multiple bond cleavage and

+ )* (see, for example,

Imhoff et al.'s [2005]

studies of the 70 eV electron-impact induced dissociative ionization of thymine and deuterated thymine).

The peak structure observed between 20 and 69 Thomson was in close agreement with the previous ion impact [e.g. De

Vries et al. 2004]

, electron impact [e.g

Imhoff et al. 2007

] and 20 eV photo-ionization mass spectra

[Jochims et al. 2005]

.

Denifl et al. [2004]

,

Imhoff et al. [2007]

, and

Jochims et al. [2005]

proposed broadly consistent assignments for the various peaks, with the notable exception of the major peak at 42 Thomson which was respectively attributed to CNO + + production was observed in the present work and in all the previous ion and electron impact measurements which covered the full product ion mass range [de

Vries et al. 2003, de Vries et al. 2004, Schlathölter et al. 2004, Schlathölter et al. 2006, Coupier et al. 2002, Imhoff et al. 2007]

. No evidence was observed in the present data for ion production between the strong peaks at 1 and 12

Thomson. Conversely, H 2 + production from gas-phase uracil was observed in

Imhoff et al.'s [2007]

70 eV (2.3 v 0 ) electron impact experiments and in diverse ion impact mass spectra reported by

Schlathölter and co-workers [de Vries et al. 2003, de Vries et al. 2004, Schlathölter et al. 2004,

  and table 2. The errors listed in the table are statistical (n

-½

Table 2 :

 2 Product ion fragmentation ratios (the number of ions detected in a given mass range over the total number of ions detected) in % for 20-150 keV proton collisions with uracil. Background noise has been removed and ions produced by direct ionization and by electron capture are treated separately.

						Fragment ion production / total ionization (%)
	Impact energy (keV)	H	+	CH2 + group (12-18) A	CH2N + group (27-28) B	CNO + / C2H4N + / C2H2O + group (39-43) B	C2H2NO + group (50-56) B	C3H3NO + group (67-69) B	uracil + (111-112) B
		EC	DI	EC	DI	EC	DI	EC	DI	EC	DI	EC	DI	EC	DI
	20	2.1 ± 0.4	-	4.6 ± 0.6	-	25.0 ± 1.4	-	39.1 ± 1.9	-	3.6 ± 0.5	-	13.2 ± 1.0	-	11.9 ± 0.9	-
	23	2.6 ± 0.5	-	5.6 ± 0.8	-	24.9 ± 1.8	-	39.3 ± 2.4	-	3.2 ± 0.6	-	12.9 ± 1.2	-	11.4 ± 1.1	-
	27	2.6 ± 0.4	6.4 ± 2.7	6.2 ± 0.6	9.6 ± 4.4	24.8 ± 1.3	29.9 ± 8.4	39.6 ± 1.7	34.8 ± 7.6	2.5 ± 0.4	4.0 ± 2.8	13.4 ± 0.9	5.9 ± 3.1	10.2 ± 0.7	5.5 ± 2.7
	35	5.0 ± 0.6	6.4 ± 3.4	6.9 ± 0.8	7.7 ± 4.8	27.3 ± 1.6	23.3 ± 8.4	39.0 ± 2.0	37.2 ± 10.2	3.0 ± 0.5	4.5 ± 3.3	11.0 ± 0.9	7.4 ± 4.0	7.7 ± 0.7	12.9 ± 4.7
	42	2.7 ± 0.4	2.4 ± 1.8	7.4 ± 0.6	6.4 ± 3.0	24.8 ± 1.2	22.1 ± 5.5	41.0 ± 1.6	36.8 ± 6.7	3.0 ± 0.4	2.8 ± 2.0	11.2 ± 0.7	13.8 ± 3.6	9.5 ± 0.7	16.0 ± 3.6
	50	1.8 ± 0.4	1.9 ± 0.9	5.5 ± 0.7	4.5 ± 1.3	25.6 ± 1.8	23.1 ± 2.8	40.4 ± 2.2	33.5 ± 2.6	3.1 ± 0.5	2.5 ± 0.9	11.2 ± 1.0	17.1 ± 2.2	11.9 ± 1.1	16.5 ± 2.0
	65	2.4 ± 0.5	2.2 ± 0.6	6.6 ± 0.8	3.9 ± 0.7	27.6 ± 1.7	21.7 ± 1.7	39.4 ± 2.2	37.2 ± 1.6	2.5 ± 0.5	3.3 ± 0.6	11.0 ± 1.0	15.5 ± 1.3	9.9 ± 1.0	15.2 ± 1.2
	80	2.7 ± 0.3	1.5 ± 0.3	6.9 ± 0.6	5.0 ± 0.5	24.9 ± 1.1	22.3 ± 1.0	40.2 ± 1.5	37.5 ± 1.0	3.6 ± 0.4	2.9 ± 0.3	11.2 ± 0.7	14.2 ± 0.7	9.6 ± 0.7	15.5 ± 0.7
	95	1.8 ± 1.0	1.7 ± 0.8	4.8 ± 1.8	4.6 ± 1.2	27.3 ± 4.7	21.3 ± 2.5	43.7 ± 6.3	38.0 ± 2.3	4.2 ± 1.7	2.1 ± 0.8	9.1 ± 2.5	16.4 ± 2.1	7.9 ± 2.3	15.1 ± 1.9
	110	1.9 ± 0.7	1.3 ± 0.5	7.0 ± 1.3	4.5 ± 0.7	29.3 ± 2.9	22.3 ± 1.5	36.7 ± 3.3	38.3 ± 1.5	3.8 ± 0.9	2.7 ± 0.5	12.3 ± 1.7	14.6 ± 1.1	8.7 ± 1.4	15.2 ± 1.1
	125	3.2 ± 0.9	1.8 ± 0.5	7.2 ± 1.4	3.9 ± 0.6	23.9 ± 2.7	20.9 ± 1.4	41.8 ± 3.8	39.3 ± 1.5	2.6 ± 0.8	3.3 ± 0.5	8.3 ± 1.5	14.9 ± 1.1	12.3 ± 1.8	15.0 ± 1.0
	140	2.8 ± 0.8	1.4 ± 0.4	5.6 ± 1.2	3.6 ± 0.6	27.6 ± 2.8	20.4 ± 1.3	37.8 ± 3.4	37.5 ± 1.4	3.6 ± 0.9	2.9 ± 0.5	10.9 ± 1.7	15.7 ± 1.0	11.3 ± 1.7	17.7 ± 1.0
	150	2.1 ± 0.7	1.1 ± 0.3	9.4 ± 1.6	4.1 ± 0.5	20.3 ± 2.5	21.3 ± 1.1	41.8 ± 3.9	39.4 ± 1.2	2.9 ± 0.9	2.7 ± 0.4	11.9 ± 1.9	15.1 ± 0.9	11.4 ± 1.8	15.3 ± 0.8

Acknowledgements

Table 1: Product ions observed following the ionization of gas-phase uracil (C 4 H 4 N 2 O 2 ) by photons [Jochims et al. 2005], by fast incident protons (present work) and by electrons [Imhoff et al. 2007, Denifl et al. 2004]. The desorbed cations observed for slow Ar + impact upon condensed phase uracil are also listed [Imhoff et al. 2007