Macroscopic objects in quantum mechanics: A combinatorial approach

Why do we not see large macroscopic objects in entangled states? There are two ways to approach this question. The first is dynamic. The coupling of a large object to its environment cause any entanglement to decrease considerably. The second approach, which is discussed in this paper, puts the stress on the difficulty of observeing a large-scale entanglement. As the number of particles $n$ grows we need an ever more precise knowledge of the state and an ever more carefully designed experiment, in order to recognize entanglement. To develop this point we consider a family of observables, called witnesses, which are designed to detect entanglement. A witness $W$ distinguishes all the separable (unentangled) states from some entangled states. If we normalize the witness $W$ to satisfy $\mid \mathrm{tr}\left(W\rho \right)\mid ⩽1$ for all separable states $\rho$, then the efficiency of $W$ depends on the size of its maximal eigenvalue in absolute value; that is, its operator norm $\parallel W\parallel$. It is known that there are witnesses on the space of $n$ qubits for which $\parallel W\parallel$ is exponential in $n$. However, we conjecture that for a large majority of $n$-qubit witnesses $\parallel W\parallel ⩽O\left(\sqrt{n\log n}\right)$. Thus, in a nonideal measurement, which includes errors, the largest eigenvalue of a typical witness lies below the threshold of detection. We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf [Phys. Rev. A 64, 032112 (2001)].