Generalized Fusion of Qudit Graph States

• URL: http://arxiv.org/abs/2510.16623v1
• Date: Sat, 18 Oct 2025 19:25:58 GMT
• Title: Generalized Fusion of Qudit Graph States
• Abstract summary: We formalize a type-II fusion operation for qudit cluster states within linear optics.<n>Two designated qudits, one from each input cluster, interfere with optional ancilla qudits via a passive linear-optical network.<n>We analyze the probabilities and entanglement of the relevant measurement outcomes, and discuss how our lower bound aligns with existing constructive schemes.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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