Matrix exponentials, SU(N) group elements, and real polynomial roots

Abstract

The exponential of an N × N matrix can always be expressed as a matrix polynomial of order N − 1. In particular, a general group element for the fundamental representation of SU (N ) can be expressed as a matrix polynomial of order N −1 in a traceless N ×N hermitian generating matrix, with polynomial coefficients consisting of elementary trigonometric functions dependent on N − 2 invariants in addition to the group parameter. These invariants are just angles determined by the direction of a real N -vector whose components are the eigenvalues of the hermitian matrix. Equivalently, the eigenvalues are given by projecting the vertices of an (N − 1)-simplex onto a particular axis passing through the center of the simplex. The orientation of the simplex relative to this axis determines the angular invariants and hence the real eigenvalues of the matrix. "Let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused." -H S M Coxeter Curtright and Zachos (CZ) wrote a brief summary [1] of essential elementary features for the triplet representation [2] of SU (3), thereby distilling some older results [3,4,5]. Here, I show how their main results may be extended to almost any N × N matrix, thereby embellishing [3,6,7]. A matrix polynomial form for an exponentiated matrix can always be expressed succinctly in terms of a "response function" that encodes the eigenvalues of the matrix.In particular, I show how the CZ results may be extended to the fundamental representation of SU (N ) for any N . I show that a polynomial form for any group element in the fundamental representation can be expressed in terms of a response function that encodes the real eigenvalues of the hermitian matrix that generates the group element. In addition, I provide a clear geometrical picture of the relevant group invariants in terms of the elementary properties of an (N − 1)-simplex [8], as a simple generalization of Viète's venerable results for the real roots of a cubic equation [9]. I give specific results for SU (N ≤ 5).The exponential of an N × N matrix M can be written as a matrix polynomial as a consequence of the CayleyHamilton theorem [10], or more directly, as the result of the Lagrange-Sylvester projection matrix method [11]. A reasonably compact polynomial form is given byThe invariant functions appearing in the E n coefficients are given by symmetric polynomials,where λ k for k = 1, · · · , N are the eigenvalues of M , and by various derivatives of the response function for the matrix, F (t). The latter is defined in terms of the characteristic function C (z) aswhere I have assumed the eigenvalues are non-degenerate. For degenerate eigenvalues, appropriate limits must be taken. Alternatively, the symmetric polynomials can be written as invariant traces, hence computed directly from M without knowing the individual eigenvalues [12].1

A quantum number theory

“…Algebraic structures in quantum mechanics are of two-fold interest, namely for explaining microscopic physical phenomena and for applications in quantum computing/information [1][2][3][4][5][6] . An interesting example of this successful mathematical structure refers to the problem of universal quantum gates [7][8][9] , which is recognized as a particular case of the Lie algebras applied to quantum computers 10 . A popular and useful representation of this algebraic structure is related to the Pauli matrices, particularly applied to quantum logic gates 11 for qubits.…”

Geometric invariant decomposition of SU(3)

“…where e bi ∈ U(3), but the product e b1 e b2 e b3 ∈ SU (3). As each b i is invariant under the transformation Ub i U † , the decomposition of eq.…”

“…Closed forms for the exponential function of su(3) elements have been published before, see e.g. [2,3]. However, the invariant decomposition of eq.…”

Parameterizing qudit states

“…Among the important contributions to the problem of parameterizing 𝑆𝑈(𝑁), we would like to mention the following publications that influenced the present work:[31]-[34].…”