Erratum: Numerical estimation of the relative entropy of entanglement [Phys. Rev. A82, 052336 (2010)

Abstract

It has come to the attention of the authors that the MATLAB code1 provided in the original paper does not work as stated. Although the code presented in the original paper was only intended to act as a guide that other researchers might follow for writing their own codes, for the sake of completeness, the present Erratum presents a fully functional code for correctly estimating the relative entropy of entanglement (REE). The original code relies on SEDUMI [1] as its internal solver for semidefinite programming. Although the code was intended to be used with SEDUMI version 1.1R3, this version is no longer available on the SEDUMI web site. Unfortunately, newer versions of SEDUMI (after version 1.1R3) that are available do not correctly handle Hermitian matrices and complex variables. As a result of this bug, the code in the original paper only works properly for input matrices whose entries are real valued. To correct these shortcomings, the code for numerical estimation of the REE has been revised to implement the algorithm using CVX [2] (a MATLAB-based modeling system for convex optimization) instead of SEDUMI. The new MATLAB code may be found in Ref. [3]. Other modifications have been made to the code that make it easier to understand. In particular, CVX allows one to more easily define the positive partial transpose (PPT) constraint when setting up a semidefinite program. The requirement X 0 may be given directly as a constraint, rather than manually defining the semidefinite cone to include only matrices that are positive under partial transpose as is performed in the original paper. The PartialTranspose routine [4] from QETLAB [5] is necessary for the code presented here. It must be shown that the present code does, in fact, correctly compute the REE of input density matrices. One way to check the validity of the numerical estimates is to compare the computed value of the REE to the negativity, another well-known entanglement monotone. This is defined as N (ρ) = Tr |ρ | − 1 2 .