The capacity of hybrid quantum memory

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+&ogr;(1)) copies of another quantum memory ℬ. This relationship captures the notion thatℬ 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(ρ)+H(ρ)/p (quantum entropy plus discounted classical entropy) taken over all mixed states ρ 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 intoℬwith 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.