Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 57 / 1 / c

The state in part (b) is symmetric in the receiving qubits. Single out Bob using the Dicke state recursion. For $k\ge 1$, the two vectors areDistinct excitation counts give orthogonal states of the other recipients. Taking their partial trace, the only surviving off-diagonal term comes from the common rest vector $|D_k^{N-1}⟩$. The one-qubit reduction of Dicke-state superpositions is consequentlyPut $p=|\alpha {|}^2$ and $q=|\beta {|}^2$, so $p+q=1$. The overlap with the input isFor $N=1$ there is no system to trace out, and the same matrix is $|\psi ⟩⟨\psi |$, with overlap one. For every $N>1$ the overlap is below one, even at its maximum $p=q=1/2$, so these are imperfect copies.