1997/10/02 by N. J. Cerf, C. Adami, R. M. Gingrich
Physics and Astronomy · #quant-ph
paper · pdf · doi:10.1103/physreva.60.893
published as Phys.Rev.A60:893-898,1999 · 19 pages, RevTeX
arxiv created 1997/10/02 · arxiv updated 2011/07/19
We analyze the properties of the conditional amplitude operator, the quantum analog of the conditional probability which has been introduced in [quant-ph/9512022]. The spectrum of the conditional operator characterizing a quantum bipartite system is invariant under local unitary transformations and reflects its inseparability. More specifically, it is shown that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding 1, which results in a necessary condition for separability. This leads us to consider a related separability criterion based on the positive map Γ:ρ→ (Tr ρ) - ρ, where ρ is an Hermitian operator. Any separable state is mapped by the tensor product of this map and the identity into a non-negative operator, which provides a simple necessary condition for separability. In the special case where one subsystem is a quantum bit, Γ reduces to time-reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for 2× 2 and 2× 3 systems. Finally, a simple connection between this map and complex conjugation in the "magic" basis is displayed.