  A conjecture arising naturally in the investigation of additivity of
classical information capacity of quantum channels states that the maximal
purity of outputs from a quantum channel, as measured by the p-norm, should be
multiplicative with respect to the tensor product of channels. We disprove this
conjecture for p>4.79. The same example (with p=infinity) also disproves a
conjecture for the multiplicativity of the injective norm of Hilbert space
tensor products.
