Random Witnesses and the Classical Character of Macroscopic Objects

An observableW that distinguishes all the unentangled states from some entangled states is called a witness. We consider witnesses on n qbits, and use the following normalization: A witness W satisfies |tr(Wρ)| ≤ 1 for all separable states ρ, while kWk > 1, with the norm being the maximum among the absolute values of the eigenvalues of W . Although there are n-qbit witnesses whose norm is exponential in n, we conjecture that for a large majority of n-qbit witnesses kWk ≤ O(√n logn). We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf (Phys. Rev. A 64, 032112 (2001)). Assuming the conjecture is valid we argue that multiparticle entanglement can be detected only if a system has been carefully prepared in a very special state. Otherwise, multiparticle entanglement lies below the possibility of detection, even if it exists, and even if decoherence has been “turned off”.