Consider an arbitrary set of rank-1 projectors ( Π Π i ) i = 1 d 2 such that F i = Π Π i / d is a POVM, and thus 1 d ∑ ∑ i Π Π i = I . Asking the projectors to have equal pairwise inner products, T r ( Π Π i Π Π j ) = c for all i ≠ ≠ j , fixes the value of c . To see this, observe that d = T r ( I 2 ) = 1 d 2 ∑ ∑ i , j T r ( Π Π i Π Π j ) = 1 d 2 ( d 2 + c d 2 ( d 2 − − 1 ) ) implies that c = 1 d + 1 . Thus, T r ( Π Π i Π Π j ) = d δ δ i j + 1 d + 1 . This property is what makes SIC-POVMs symmetric ; with respect to the Hilbert–Schmidt inner product, any pair of elements is equivalent to any other pair.