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A systematic and compact treatment of arbitrary $su(2)$ invariant spin-$s$quantum chains with nearest-neighbour interactions is presented. Theground-state is derived in terms of matrix product states (MPS). Thefundamental MPS calculations consist of taking products of basic tensors ofrank 3 and contractions thereof. The algebraic $su(2)$ calculations are carriedout completely by making use of Wigner calculus. As an example of application,the spin-1 bilinear-biquadratic quantum chain is investigated. Various physicalquantities are calculated with high numerical accuracy of up to 7 digits. Weobtain explicit results for the ground-state energy, entanglement entropy,singlet operator correlations and the string order parameter. We findinteresting crossover phenomena in the correlation lengths.

Matrix product states for $su(2)$ invariant quantum spin chains