Efficient Verification of Dicke States

Among various multipartite entangled states, Dicke states stand out because their entanglement is maximally persistent and robust under particle losses. Although they have attracted much attention for their potential applications in quantum information processing and foundational studies, the characterization of Dicke states remains as a challenging task in experiments. Here we propose efficient and practical protocols for verifying arbitrary $n$-qubit Dicke states in both adaptive and nonadaptive ways. Our protocols require only two distinct settings based on Pauli measurements besides permutations of the qubits. To achieve infidelity $ϵ$ and confidence level $1-\delta$, the total number of tests required is only $O(n{ϵ}^{-1}\ln {\delta }^{-1})$. This performance is exponentially more efficient than all previous protocols based on local measurements, including quantum state tomography and direct fidelity estimation, and is comparable to the best global strategy. Our protocols are readily applicable with current experimental techniques and are able to verify Dicke states of hundreds of qubits.

Figure 1

The adaptive protocol for verifying $|W_n⟩$ as in Theorem t1. The vertical dashed line indicates that the protocol is two-step adaptive. For any two qubits $i$ and $j$ (Ida and Jim) chosen a priori, the measurement outcomes of the other $n-2$ qubits determine which measurements on them to perform.

Figure 2

Comparison between our efficient adaptive and nonadaptive protocols and the DFE protocol [8] and the optimal global protocol. For a given number of qubits $n=10$ and confidence level $1-\delta =0.95$, the plot shows the number of tests $N$ required to verify $|W_{10}⟩$ and $|D_{10}^5⟩$ within infidelity $ϵ$ for each protocol.

Figure 3

Simulation results on the adaptive verification of $|W_8⟩$. The two axes denote the number of tests $N$ and the reciprocal of the infidelity ${ϵ}^{-1}$. For each $ϵ$, the simulation is performed $M=10000$ times, but only 500 points are shown in the plot for clarity. The open circles denote the minimum number of tests required to achieve infidelity $ϵ$ and confidence level $1-\delta$, and the four blue lines are fitted for $\delta =0.01$, $0.05$, $0.1$, and $0.2$ (from top to bottom), respectively. Here the number of tests can be approximated by the formula $N\approx 7.0306(\pm 0.0188){ϵ}^{-1}\ln {\delta }^{-1}$, which is very close to the theoretical prediction.

Figure 4

The adaptive protocol for verifying the Dicke state $|D_n^k⟩$ as in Theorem t2. The protocol is two-step adaptive as indicated by the vertical dashed line. For any two qubits $i$ and $j$ (Ida and Jim) chosen a priori, the outcomes of $Z$ measurements on the other $n-2$ qubits determine which measurements on them to perform. Recall that $B_{n,k}$ denotes the set of all strings in $\{0,1{\}}^n$ that have Hamming weight $k$, where 0 and 1 correspond to eigenvalues 1 and $-1$ of $Z$, respectively.

Figure 5

Simulated experiments on quantum state verification.