3 3-Kronecker Pauli Matrices × Christian Rakotonirina Civil Engineering Department, Institut Supérieur de Technologie d'Antananarivo, IST-T, Madagascar [email protected] Simon Rasamizafy Department of Physics University of Toliara, Madagascar [email protected] Abstract : The properties of what we call inverse-symmetric matrices have helped us for ## constructing a basis of ℂ which satisfy four properties of the $ronecker generalize Pauli matrices. The Pauli group of this basis has been efine . In using some properties (( )) of the $ronecker commutation matrices, bases of " an " which share the same properties have also been constructe . $eywor s: $ronecker pro uct, Pauli matrices, $ronecker commutation matrices, $ronecker generalize Pauli matrices. 1 Intro uction This paper tries to solve a problem pose in [1,, of searching a set -# . /01, 02, 3 , 045 of nine 33-matrices satisfy the following relation with the 373-$ronecker commutation matri8 9$CM), 4 1 =#7# . > 0? A 0? 91) 3 ?@1 The usefulness of the $ronecker permutation matrices, particularly the $ronecker commutation matrices 9$CMs) in mathematical physics can be seen in [2,, [3,, [C,, [5,. In these papers, the 272- $CM is written in terms of the Pauli matrices, which are 22 matrices, by the following way # 1 =2A2 . > E? AE? 2 ?@F The generalization of this formula in terms of generalize Gell-Mann matrices, which are a generalization of the Pauli matrices, is the topic of [6,. Iut there are other generalization of the Pauli matrices in other sense than the generalize Gell-Mann matrices, among others the $ibler matrices [7,, the $ronecker generalize Pauli matrices, see for e8ample [8,. These last are obtaine by $ronecker pro uct of the Pauli k k matrices. The more general relation giving the 2 72 -$CM in terms of the $ronecker generalize Pauli matrices may be seen in [1,. That makes us search for other generalization of the Pauli matrices in this sense, like the generalization of the Gell-Mann matrices to what we call rectangle Gell-Mann matrices in [9,. That is, we search for set of 33 matrices which have got some properties of the $ronecker generalize Pauli k k matrices, which are 2 2 matrices. Me will call these matrices 33-$ronecker Pauli matrices 9$PMs). These properties of the $ronecker generalize Pauli matrices or N N-$PMs S U 9OP)FQ?QR T1, with N . 2 an O? .E?V7E?S7 3 7E?W, are S RR P i. 9O )FQ?QR T1 is a basis of " ii. S R T1 1 =R7R . > O? 7 OP N ?@F X iii. O? .O? 9hermiticity) 2 iv. O? .YZ 9Square root of unit) X v. ]^_O` Oab . Nc`U 9drthogonality) vi. ]^_O`b . e 9Tracelessness) fowever, there is no 33 matri8, forme by zeros in the iagonal which satisfies both the relations iii. an iv. [1,. Thus, at a first time, for the 33-$PMs we o not eman tracelessness. Me will call 33-$PMs a set of 33 matrices which satisfy, not only the formula 91), but the five properties above, tracelessness vi. move apart. Thus, in this paper we will talk at first about $CMs. In the ne8t section, we will talk about what we call inverse-symmetric matrices. These matrices have got interesting properties for constructing the 33-$PMs. After, we will give the set of 33-$PMs, which are inverse-symmetric matrices. ginally, some way to the generalization will be iscusse . k k Me know that the set of 2 2 -$PMs, up to multiplicative phases: 1, h1, i an hi, is the Pauli group for the usual matri8 pro uct. Thus, we will try to efine the Pauli group of the 33-$PMs. Some calculations such as the e8pression of 373-$CMs request calculations with software. Me have use SCILAI for those calculations. 2 $ronecker Commutation matrices The $ronecker pro uct of matrices is not commutative, but there is a permutation matri8 which, in multiplying to the pro uct, commutes the pro uct. Me call such matri8 $ronecker commutation matri8. ZlZl Z1 Defi nition 1 The permutation matri8 =Z7l m " , such that for any matrices n m " , l1 o m " =Z7l9n7o) . o7n is calle p7q-$ronecker commutation matri8, p7q-$CM. 1 e e e e e e e e e e e 1 e e e e e ve e e e e e 1 e ey 1 e e e ue 1 e e e e e e ex u x =2 7 2 . re e 1 es, =#7# . e 1 e e e e e 1 e e ue e e x ue e e e e e e 1 ex e e e 1 ue e 1 e e e e e ex e e e e e 1 e e e te e e e e e e e 1w gor constructing p7q z$CM, we can use the following rule [13,. Rule 2 Let us start in putting 1 at first row, first column, then let us pass into the secon column in going own at the rate of n rows an put 1 at this place, then pass into the thir column in going own at the rate of n rows an put 1, an so on until there is only for us n-1 rows for going own 9then we have obtaine as number of 1 : p). Then pass into the ne8t column which is the 9p + 1)-th column, put 1 at the secon row of this column an repeat the process until we have only n-2 rows for going own 9then we have obtaine as number of 1 : 2p). After that pass into the ne8t column which is the 92p + 2)-th column,- put 1 at the thir row of this column an repeat the process until we have only n 3 rows for going own 9then we have obtaine as number of 1 : 3p). Continuing in this way we will have that the element at np-th row, np-th column is 1. Proposition 3 Suppose ~ =Z7| . > }? 7•` ?,`@1 an =l7€ . > •U 7 U,@1 |Z Z| with the }?’s are elements of " , the •`’s are elements of " , the •U’s are elements €l l€ of " an the ’s are elements of " . Then, ~ =Zl7|€ . > > }?7•U7•`7 ?,`@1 U,@1 Proof Z1 l1 |1 €1 Let 9n), 9), 9o) an 9) be, respectively, bases of " , " , " an " . Then, Zl|€1 _n77o7b is a basis of " . It is enough to prove that ~ > > }?7•U7•`7 _n77o7b . o77n7 ?,`@1 U,@1 Me use the proposition 11. grom ~ > }? 7•`_n7ob . o7n ?,`@1 we have ~ > > }?n7•U7•`o7 . > o7•U 7n7 ?,`@1 U,@1 U,@1 . o7 > •U 7n7 U,@1 Moreover > •U 7n7 . 7n7 U,@1 an that en s the proof. 3 Inverse -symmetric matrices In this section, we intro uce what we call inverse-symmetric matrices. Me think that this term will be useful for the continuation. ? Defi nition C Let us call inverse-symmetric matri8 an invertible comple8 matri8 } . _}`b ` 1 ? ` such that }? . • if } e. Unlike an antisymmetric matri8, for the non-zero element of the matri8, its symmetric with respect to the iagonal is its inverse. 1 e e 1 E8ample 5 The 22 unit matri8 Y2 .’ “ an the Pauli matrices E1 .’ “, e 1 1 e e hi 1 e E2 .’ “ an E# .’ “ are inverse-symmetric matrices. i e e h1 ? ? Proposition 6 Let } . _}`b an • . _•` b be inverse-symmetric matrices. Then, }7• is an inverse-symmetric matri8. ?U ? U Proof . }7• is an invertible matri8. If 9}7•)` e if, only if }` e an • e. Its ` ` 1 1 1 U W W symmetric with respect to the iagonal is 9}7•)?U .}? • . • ”• . 97”)•• . Proposition 7 gor any p p inverse-symmetric matri8 }, with only n non zero elements, 2 } .YZ. ? 2 Z ? U Proof . Let } . _}`b1Q?,`QZan then } ._?@1 }U}` b1Q?,`QZ. In or er that } is invertible, ? there must be only a non zero element in each row an in each column. Let }| be the l 2 ? non zero element in the row i an }` the non zero element in the column —. 9} )` . ? | ? l }|}` + }l}` . | 1 l | ` 2 ? If i —, }? . e an }` e, thus }` . e an }l . e. fence, 9} )` . e. 2 ? Z ? U ? | If i . —, 9} )` . ?@1 }U}` .}|}` . 1. Therefore, let us take some inverse-symmetric matrices forme by only three non zero elements for the nine 33 matrices we woul like to search for, in or er that iv. is satisfie . C $ronecker generalization of the Pauli ma trices The $ronecker generalize Pauli matrices are the matrices _E?7E`bFQ?,`Q#[1e,, [11,, V S ™ _E?7E`7EUbFQ?,`,UQ#, _E? 7E? 7 3 7E? bFQ?V,?S,3,?™Q# [8, obtaine by $ronecker pro uct of the Pauli matrices an the 22 unit matri8. Accor ing to the propositions above, they are inverse-symmetric matrices an share many of the properties of the Pauli matrices: ™ ™ 2 2 basis of " , ii., iii., iv., v. an vi. in the intro uction, for [8,. ™ V S ™ Denote the set of _E? 7E? 7 3 7E? bFQ?V,?S,3,?™Q# by -2 . ™ 7Z -2 . /EF,E1,E2,E#5 . E?V7E?S7 3 7E?™/e ≤ i1, i2, 3 , iZ ≤ 3• The set ™ ™ 2 .-2 7/1, h1, i, hi5 V S ™ is a group calle the Pauli group of _E? 7E? 7 3 7E? bFQ?V,?S,3,?™Q# [8,. # Me can check easily, in using the relation E`EU . c`U EF + i ?@1 `U E, for —, m /1, 2, 35 ™ ™ that 2 is equal to the set of the pro ucts of two elements of -2 , up to multiplicative phases, which are elements of /1, h1, i, hi5.