Entanglement cost of quantum state preparation and channel simulation*

We study various aspects of the entanglement cost of quantum state preparation and quantum channel simulation. First, we establish that the exact entanglement cost of any bipartite quantum state under PPT-preserving operations is given by a single-letter formula, here called the κ-entanglement of a quantum state. This formula is calculable by a semidefinite program, thus allowing for an efficiently computable solution for general quantum states. Notably, this is the first time that an entanglement measure for general bipartite states has been proven not only to possess a direct operational meaning but also to be efficiently computable, thus solving a question that has remained open since the inception of entanglement theory over two decades ago. Second, we study the exact entanglement cost of quantum channel simulation in the parallel setting. In particular, the largest κ-entanglement that can be established via a quantum channel remarkably gives a single-letter formula for the exact parallel PPT-entanglement cost of simulating this channel, and it is also efficiently computable by a semidefinite program.