Entanglement-efficiency trade-offs in the fusion-based generation of photonic GHZ-like states

TL;DR

This paper presents linear-optical strategies to generate and fuse GHZ-like states with tunable entanglement, defined by $|G_n(\alpha)\rangle = \cos(\alpha)|0\rangle^{\otimes n} + \sin(\alpha)|1\rangle^{\otimes n}$. It introduces two fusion-based protocols using standard and modified fusion gates (with a variable beam splitter $\text{VBS}(\theta)$) to trade entanglement for higher generation efficiency, enabling success probabilities that can exceed the traditional $1/2$ bound without extra photonic resources. A general and a more efficient method describe how to assemble large $N$-qubit GHZ-like states from small $|G_2(\alpha)\rangle$ resources, with total success probabilities $P_{gen}^{(1)}$ and $P_{gen}^{(2)}$ respectively and detailed dependence on input Schmidt angles. The work analyzes entanglement-capability trade-offs via fusion entropy, demonstrates tunable output entanglement via modified gates, and discusses resource requirements and multiplexing to approach near-deterministic generation, highlighting significant potential for scalable quantum computing and communication using photonic GHZ-like states.

Abstract

Probabilistic entangling measurements are key operations in linear-optical quantum technologies, enabling the generation and manipulation of high-dimensional quantum states. While prior research has focused predominantly on specific entangled states, notably graph states and Greenberger-Horne-Zeilinger (GHZ) states, broader classes of states with variable entanglement remain underexplored. In this work, we present a linear-optical approach for generating and fusing GHZ-like states, which generalize standard GHZ states to include variable entanglement degrees. We introduce two schemes based on modified fusion gates that allow flexible control over generation efficiency and the entanglement of the output states. These results offer a promising pathway toward resource-efficient entangled-state generation for scalable quantum computing and communication.

Figures (11)

• Figure 1: Linear-optical generation of multiqubit GHZ-like states. a) The scheme employs sequential probabilistic entangling fusion operations applied to constant-size resource states $|G_{r}(\alpha)\rangle$ produced by resource state generators (RSGs). Each N-qubit state is represented as N connected nodes labeled with the corresponding Schmidt angles. b) The fusion success probability depends on the Schmidt angles $\alpha$ and $\beta$ of the fused states. The Schmidt angle $\delta$ of the resulting state can be controlled via the fusion gate parameter $\theta$.
• Figure 2: Linear-optical circuits realizing modified fusion gates: a) Circuit implementing the modified type-I fusion gate; b) circuit implementing the modified type-II fusion gate. Both gates act on two input qubits $a$ and $b$ from two initially separated states. The specific fusion operation is determined by the splitting ratio of the variable beam-splitter (VBS), defined by the parameter $\theta$. Standard fusion gates correspond to the special case $\theta = \pi/4$.
• Figure 3: Fusion success probability as a function of the degrees of entanglement of the input states: the trade-off between the success probability of the standard type-I and type-II fusion gates and the Schmidt angles $\alpha$ and $\beta$ of the input states (\ref{['Type_1_prob']}).
• Figure 4: Integrated photonic implementation of a variable beam splitter using two 50:50 beam splitters and a phase shift.
• Figure 5: Fusion entropy as a function of the total success probability for the initial states with entanglement parameters $\alpha, \beta \in [0,\pi/4]$ using standard fusion gates. The solid line indicates the maximal entropy achievable for a given success probability.