Extracting perfect GHZ states from imperfect weighted graph states via entanglement concentration

TL;DR

The paper tackles the challenge of generating perfect GHZ states from imperfect photonic weighted graph states by introducing an entanglement-concentration protocol that operates with only local single-qubit gates and measurements on a 1D uniform WGS to probabilistically yield a GHZ state. The method measures all even sites in the basis $\hat{M}_{\phi}=R_{z}(\phi)\hat{X}R_{z}^{\dagger}(\phi)$, projecting the remaining qubits into an $(n+1)$-qubit GHZ when all outcomes are $-1$, with a success probability $P_{s,n}=\frac{1}{2^{n}}|\sin(\phi/2)|^{2n}$; the special case $\phi=\pi$ makes the process deterministic. The authors analyze robustness to coherent CP-gate phase errors and to depolarizing noise, showing improved entanglement fidelity and resilience for moderate errors and larger GHZ sizes, relative to direct CP-gate generation or linear-optics methods. This approach provides a scalable, resource-efficient route to high-fidelity photonic GHZ resources compatible with existing technologies. Overall, the work highlights a practical path to robust GHZ preparation in photonic platforms by concentrating entanglement from imperfect weighted graph states using local operations.

Abstract

Photonic GHZ states serve as the central resource for a number of important applications in quantum information science, including secret sharing, sensing, and fusion-based quantum computing. The use of photon-emitter entangling gates is a promising approach to creating these states that sidesteps many of the difficulties associated with intrinsically probabilistic methods based on linear optics. However, the efficient creation of high-fidelity GHZ states of many photons remains an outstanding challenge due to both coherent and incoherent errors during the generation process. Here, we propose an entanglement concentration protocol that is capable of generating perfect GHZ states using only local gates and measurements on imperfect weighted graph states. We show that our protocol is both efficient and robust to incoherent noise errors.

Figures (8)

• Figure 1: (a) Uniform WGS with weights $\phi$. (b),(c) The entanglement concentration protocol: (b) Start with a 1D uniform WGS containing $2n+1$ qubits and perform optimized single-qubit measurements $\hat{M}_{\phi}$ (see Eq. \ref{['eq:measeq']}) on all even qubits. (c) After performing single-qubit measurements, apply a Pauli-$Z$ rotation on qubit number $2n+1$. The final graph state is a $(n+1)$-qubit GHZ state.
• Figure 2: The success probability of constructing the 3-qubit GHZ state from a 5-qubit uniform WGS. Here, the horizontal red dashed line marks the 1/32 probability of getting a 3-qubit GHZ state using linear optical methods. When $\phi = \pi$, the success probability is unity (not shown in the plot).
• Figure 3: Constructing a 2-qubit entangled state from a 3-qubit uniform WGS with coherent error that causes the weights to differ: $\phi_{1,2} \neq \phi_{2,3}$. (a) The concurrence of the resulting two-qubit state when our protocol succeeds. (b) The success probability $P_s$ of our protocol. (c) Concurrence (blue) of the two-qubit state after a successful measurement in our protocol in the special case of a 3-qubit WGS where $\phi_{1,2}$ is fixed to $0.55 \pi$, and $\phi_{2,3}$ can vary from $0.3\pi$ to $0.9\pi$. For comparison, we also show the concurrence of a 2-qubit WGS with weight $\phi_{\text{ref}} = \text{Max}(0.6 \pi, \phi_{2,3})$ (orange).
• Figure 4: Effect of correlated errors on the entanglement and fidelity of concentrated GHZ states containing up to 16 qubits. (a) $ZZ$ correlation functions of 16-qubit GHZ states obtained from measuring the even qubits of 31-qubit linear WGSs with alternating CP weights $\phi_{\text{mean}} \pm \Delta \phi$ with $\Delta\phi=0.1 \pi$. $\langle Z_1 Z_j \rangle$ correlation functions with $j=2,...,16$ are shown for three different values of $\phi_{\text{mean}}$. The solid lines are fits of the data to exponential decays. For comparison, the $\langle Z_1 Z_j \rangle$ correlation functions of GHZ-like states directly generated by starting from a linear array of 16 qubits in the state $\ket{+}^{\otimes16}$ and applying CP gates with angle $\phi_{\text{mean}}$ between neighboring qubits and Hadamard gates are also shown (dashed lines). (b) 3-qubit example circuit used to create the directly generated reference states in (a) and (d). (c) The decay lengths of the correlation functions $\langle Z_1 Z_j \rangle$ with $j=2,...,16$ as a function of $\phi_{\text{mean}}$ and $\Delta \phi$ obtained from fits to exponential decays like those shown in (a). (d) Fidelity of concentrated GHZ states containing $n=2$ to $6$ qubits (blue dots) relative to a perfect GHZ state. Here, the initial linear WGS has alternating weights with $\phi_{\text{mean}}=0.55\pi$ and $\Delta\phi=0.05\pi$, which are values quoted in Ref. jeannic2021dynamical. For comparison, fidelities of imperfect GHZ states directly constructed using circuits as in (b) with $\phi = 0.55 \pi$ and initial states $\ket{+}^{\otimes n}$ for $n=2$ to $6$ are also shown (orange squares). Each point is obtained by applying arbitrary single-qubit gates to all qubits and adjusting gate parameters until the fidelity of the resulting state relative to a perfect GHZ state is maximized.
• Figure 5: The performance of our protocol in the presence of depolarizing errors on the initial linear WGS. (a) The concurrence $C$ of the 2-qubit state after successfully applying our protocol on a 3-qubit WGS with depolarizing error. The concurrence is shown as a function of CP weight $\phi$ and depolarizing error probability $p$. (b) The success probability of our protocol as a function of $\phi$ and $p$. (c) Comparison of the concurrences of the state from our protocol with that of a 2-qubit uniform WGS in the presence of the same depolarizing error. The concurrence advantage ($\Delta C$) is shown as a function of $\phi$ and $p$.