A remark on the unitary group of a tensor product of n finite-dimensional Hilbert spaces

Abstract

Let H i, 1 ≤ i ≤n be complex finite-dimensional Hilbert spaces of dimension di,1 ≤ i ≤ n respectively with d_i ≥ 2 for every i. By using the method of quantum circuits in the theory of quantum computing as outlined in Nielsen and Chuang [2] and using a key lemma of Jaikumar [1] we show that every unitary operator on the tensor product H =H₁ ⊗ H₂ ⊗... ⊗ H_n can be expressed as a composition of a finite number of unitary operators living on pair products H_i ⊗ H_j,1 ≤i,j ≤ n. An estimate of the number of operators appearing in such a composition is obtained.