More on rotations as spin matrix polynomials

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

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“…2 , and diagonal J (n) 3 . We mention in passing that calculating the time evolution operator of such Hamiltonians becomes much easier when one uses the Cayley-Hamilton theorem [33,34].…”

Section: Generalizing To Other Representationsmentioning

confidence: 99%

Complete Population Transfer of Maximally Entangled States in $2^{2N}$-level Systems via Pythagorean Triples Coupling

Erew,

Goldstein,

Suchowski

2020

Preprint

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Maximally entangled states play a central role in quantum information processing. Despite much progress throughout the years, robust protocols for manipulations of such states in many-level systems are still scarce. Here we present a control scheme that allow efficient manipulation of complete population transfer between two maximally entangled states. Exploiting the self-duality of SU (2), we present in this work a family of 2 2N -level systems with couplings related to Pythagorean triples that make a complete population transfer from one state to another (orthogonal) state, using very few couplings and generators. We relate our method to the recently-developed retrograde-canon scheme and derive a more general complete transfer recipe. We also discuss the cases of (2n) 2 -level systems, (2n + 1) 2 -level systems and other unitary groups.

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More on rotations as spin matrix polynomials

Cited by 4 publications

References 26 publications

Elementary Results for the Fundamental Representation of SU(3)

Elementary Results for the Fundamental Representation of SU(3)

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

Complete Population Transfer of Maximally Entangled States in $2^{2N}$-level Systems via Pythagorean Triples Coupling

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