Deterministic generation of arbitrary symmetric states and entanglement classes

We propose a method to generate arbitrary symmetric states of $N$ qubits, which can be easily associated with their entanglement classes. It is particularly suited to quantum optics systems like trapped ions or superconducting circuits. We encode each qubit in two metastable levels of the system and use a bosonic quantum bus for creating the states. The method is deterministic and relies on a sequence of selective unitary gates upon the qubits within the system coherence time.

Figure 1

(a) Scheme of multiqubit system as proposed in the text. In one step of the proposed protocol, the first $n$ qubits are addressed with a global red-sideband interaction of Rabi frequency $g_1$ and detuning ${\Delta }_1$, while the last qubit (first one from the right in the figure) is addressed with a second red-sideband driving of Rabi frequency $g_2$ and detuning ${\Delta }_2$. Here $|e,0\rangle$ (respectively, $|g,1\rangle$) is the excited internal state with no excitations in the bosonic mode (respectively, ground internal state with one excitation in the bosonic mode). (b)–(d) Probabilities for the evolved state of being in the corresponding states shown in the curves, as a function of time and the respective step, in the protocol for obtaining state $|{\Psi }_4{\rangle }_S$ in Eq. (1), with $c_k=\{1,2,3,4,5\}/\sqrt{55}$. Here ${\tau }_1=\pi /\left(4\beta \right)$. Depicted is the time evolution of the probabilities for the three states appearing in Eq. (4), at each step: $|D_{N,k}\rangle$, $|D_{N-1,k-1}\rangle |e\rangle$, and $|D_{N-1,k}\rangle |g\rangle$.