Entanglement of spin waves among four quantum memories SUPPLEMENTARY INFORMATION

2 (  Ec + e (w) 3  Ed). We control their relative intensities using the two waveplates (λ/2) near {BD1,BD2}, with writing phases φ i ∈ {φ 1 , φ (w) 2 , φ (w) 3 } set by the tilting angles of {BD1,BD2}. In turn, the heralding fields γ1 = {a1, b1, c1, d1} emitted from the writing process are combined into two spatial modes at BD3, with each mode carrying polarizations {|H, |V } to accommodate the fields γ 1 = {a1, b1} and γ (cd) 1 = {c1, d1}. We then interfere the polarization components of the spatial modes {γ 1 , γ (cd) 1 } by way of BD4, whose output modes experience polarization interference at the polarizing beamsplitter PBSh, with one output monitored by the heralding detector Dh. Here,