The capacity of hybrid quantum memory
Greg Kuperberg · IEEE Transactions on Information Theory · 2003
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 /spl Ascr/ embed in N(1+/spl ogr/(1)) copies of another quantum memory /spl Bscr/. This relationship captures the notion that/spl Bscr/ is as at least as useful as /spl Ascr/ for all purposes in the bulk limit. We show that the embeddings exist if and only if for all p/spl ges/1, the p-norm of the shape of /spl Ascr/ does not exceed the p-norm of the shape of /spl Bscr/. The log of the p-norm of the shape of /spl Ascr/ can be interpreted as the maximum of S(/spl rho/)+H(/spl rho/)/p (quantum entropy plus discounted classical entropy) taken over all mixed states /spl rho/ on /spl Ascr/. We also establish a noiseless coding theorem that justifies these entropies. The noiseless coding theorem and the bulk embedding theorem together say that either /spl Ascr/ blindly bulk-encodes into /spl Bscr/ with perfect fidelity, or A admits a state that does not visibly bulk-encode into/spl Bscr/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.