n psd-rank m ax rank(M)=minfr:Mxy (ax,by),ax,by E R"} rankpsd(M)=minfr:Mxy =(Ax,By),Ax,By E PSDrxr} o Defined in [FMPTW'12],and showed to characterize the communication complexity of the following task: Px POVM {E} Requirement: E[]=p(x,y) quantum comm flogrankpsa(P)]psd-rank 𝑟𝑎𝑛𝑘 𝑀 = min 𝑟: 𝑀𝑥𝑦 = 𝑎𝑥 , 𝑏𝑦 , 𝑎𝑥 , 𝑏𝑦 ∈ 𝐑 𝑟 𝑟𝑎𝑛𝑘𝑝𝑠𝑑 𝑀 = min 𝑟:𝑀𝑥𝑦 = 𝐴𝑥,𝐵𝑦 , 𝐴𝑥 ,𝐵𝑦 ∈ 𝐏𝐒𝐃𝑟×𝑟 • Defined in [FMPTW’12], and showed to characterize the communication complexity of the following task: quantum comm ≈ ⌈log 𝑟𝑎𝑛𝑘𝑝𝑠𝑑(𝑃)⌉ 𝑥 𝜌𝑥 𝑃𝑂𝑉𝑀 { 𝐸𝜃 𝑦 } Requirement: 𝐄 𝜃 = 𝑝(𝑥, 𝑦) m r r n 𝑎𝑥 𝑏𝑦