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

Abstract

LetHi, 1 ≤ i ≤n be complex finite-dimensional Hilbert spaces of dimension di,1 ≤ i ≤n respectively withdi ≥ 2 for everyi. 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 productH =H1 ⊗H2 ⊗... ⊗Hn can be expressed as a composition of a finite number of unitary operators living on pair productsHi ⊗Hj,1 ≤i,j ≤n. An estimate of the number of operators appearing in such a composition is obtained.