Department of Mathematics

The general stable quantum memory unit is a hybrid consisting of a classical digit with a quantum digit (qudit) assigned to each classical state. The shape of the memory is the vector of sizes of these qudits, which may differ. We determine when N copies of a quantum memory A embed in N(1+o(1)) copies of another quantum memory B. This relationship captures the notion that B is as at least as useful as A for all purposes in the bulk limit. We show that the embeddings exist if and only if for all p >= 1, the p-norm of the shape of A does not exceed the p-norm of the shape of B. The log of the p-norm of the shape of A can be interpreted as the maximum of S(\rho) + H(\rho)/p (quantum entropy plus discounted classical entropy) taken over all mixed states \rho on A. We also establish a noiseless coding theorem that justifies these entropies. The noiseless coding theorem and the bulk embedding theorem together say that either A blindly bulk-encodes into B with perfect fidelity, or A admits a state that does not visibly bulk-encode into B with high fidelity. In conclusion, the utility of a hybrid quantum memory is determined by its simultaneous capacity for classical and quantum entropy, which is not a finite list of numbers, but rather a convex region in the classical-quantum entropy plane.