Generalized Fusion of Qudit Graph States
N. Rimock, Y. Oz
TL;DR
This work addresses whether generalized type-II fusion of qudit cluster states can be realized with passive linear optics and heralded detection. It formalizes a generalized fusion protocol with ancilla qudits, deriving a Schmidt-rank bound across the fused cut, showing the rank is limited by the number of measured qudits $M$. The key result is that fusing to a $d$-dimensional cluster requires at least $d-2$ ancilla qudits, extending previous qubit no-go results to qudit and non-Bell projections, with a maximally entangled fused state possible only in the $k_1=k_2=2$ case. The findings establish a concrete resource threshold for high-dimensional, fusion-based photonic MBQC and guide design toward ancilla-assisted fusion strategies with potential implications for percolation and fault-tolerant encodings.
Abstract
We formalize a generalized type-II fusion operation for qudit cluster states within linear optics. Two designated qudits, one from each input cluster, interfere with optional ancilla qudits via a passive linear-optical network, followed by number-resolving detection; conditioned on %a two-click outcome, measurement outcome, the remaining qudits form the post-selected fused state. We prove a general rank bound: for any such interferometer and outcome, the reduced density matrix across the two parent clusters has Schmidt rank at most $M$, the total number of measured qudits including ancillae. Consequently, a correct qudit fusion which requires rank $d$ is impossible without ancillae and requires at least $d-2$ ancilla qudits. Our analysis extends previous no-go results for Bell-type qubit fusion to the qudit setting and to generalized, non-Bell projections. We analyze the probabilities and entanglement of the relevant measurement outcomes, and discuss how our lower bound aligns with existing constructive schemes. These results set a clear resource threshold for high-dimensional, fusion-based photonic MBQC.