Characterizing the entangling capacity of n-qubit computations

The state space script H sign_n of n quantum bits of data is exponentially large, having dimension 2ⁿ. The (pure) local states which correspond to each individual quantum bit being in an isolated one-qubit state, i.e. those which are tensor products, form a much smaller orbit of ⊗₁ ⁿU(2) · |00 ⋯ 0〉 of linear dimension within the state space. Hence most states are non-local, or entangled. The concurrence function on quantum data states is one measure of entanglement, intuitively capturing an exponentially small fraction of the phenomenon. This paper reports numerical tests of how concurrence changes as one applies a quantum computation u to a pure n quantum-bit data state |Ψ〉. We make strong use of a mathematical tool for factoring u = k₁ k₂ into subcomputations, namely the CCD matrix decomposition. The concurrence dynamics of a computation |Ψ〉 → u|Ψ〉 are in a certain sense localized to the a factor, and so our actual numerics concentrate on |Ψ〉 → a|Ψ〉. This is a great simplification, since an arbitrary unitary evolution may vary over 4ⁿ - 1 real degrees of freedom, while the a ∈ A of the appropriate form for the CCD matrix decomposition may vary over 2ⁿ - 1 or 2^n/2 - 1 as n = 2p, 2p - 1.