2026/07/20 by Maximilian Illmer, Tim Netzer, Michael Wolf · 1 voice
#quant-ph #math.AT
We investigate the topology of the set \mathsf E of entangled bipartite density operators acting on ℂn1⊗ℂn2. We start by showing that \mathsf E is path-connected, and even simply connected except in the two-qubit case. In this exceptional case \mathsf E turns out to be homotopy equivalent to the set of maximally entangled states, which itself is homeomorphic to \mathbbRP3. Here we also compute the complete homology of the closure and interior of \mathsf E. In all larger dimensions, we show that the homology and homotopy groups of \mathsf E vanish in degrees 1≤ k≤ 2(n1-1)(n2-1)-2, and all homology groups of degree k≥ (n1n2)2-3 also vanish. This range is controlled by the space \mathsf W of entanglement witnesses, which we show is highly connected beyond two qubits and homotopy equivalent to \mathsf E. By computing the Euler characteristic, using a torus-action fixed point argument together with Alexander duality, we show that \mathsf E nevertheless has non-trivial reduced homology over every field for all n1, n2 ≥ 2.