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A remark on the unitary group of a tensor product of \(n\) finite-dimensional Hilbert spaces. (English) Zbl 1025.47047

Introduction: From the theory of quantum computing and quantum circuits (as outlined, for example, in [M. A. Nielsen and I. L. Chuang, “Quantum computation and quantum information” (Cambridge Univ. Pr.) (2000; Zbl 1049.81015)]) it is now well-known that every unitary operator on the \(n\)-fold tensor product \((\mathbb{C}^2)^{\otimes^n}\) of copies of the two-dimensional Hilbert space \(\mathbb{C}^2\) can be expressed as a composition of a finite number of unitary operators living on pair products \(H_i\otimes H_j\) where \(H_i\) and \(H_j\) denote the \(i\)th and \(j\)th copies of \(\mathbb{C}^2\). The proof outlined in [loc. cit.] also yields an upper bound on the number of such ‘pair product’ operators as a function of \(n\). Following more or less their lines of proof and using a key lemma suggested to me by R. Jaikumar, we present a generalization when copies of \(\mathbb{C}^2\) are replaced by arbitrary finite-dimensional complex Hilbert spaces. Thus the present note is of a pedagogical and expository nature.

MSC:

47N50 Applications of operator theory in the physical sciences
81P68 Quantum computation
46M05 Tensor products in functional analysis

Citations:

Zbl 1049.81015
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References:

[1] Jaikumar Radhakrishnan: Private communication, April 2001
[2] Nielsen, M. A.; Chuang, I. L., Quantum Computation and Quantum Information (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 1049.81015
[3] Parthasarathy, K. R., Lectures on quantum computation and quantum error correcting codes, Notes by Amitava Bhattacharya (2001), Mumbai: Tata Institute of Fundamental Research, Mumbai
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