dorsal/arxiv
View SchemaThe capacity of hybrid quantum memory
| Authors | Greg Kuperberg |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0203105 |
| URL | https://arxiv.org/abs/quant-ph/0203105 |
| Journal | IEEE Trans. Inform. Theory 49 (2003), 1465-1473 |
Abstract
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.
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"abstract": "The general stable quantum memory unit is a hybrid consisting of a classical\ndigit with a quantum digit (qudit) assigned to each classical state. The shape\nof the memory is the vector of sizes of these qudits, which may differ. We\ndetermine when N copies of a quantum memory A embed in N(1+o(1)) copies of\nanother quantum memory B. This relationship captures the notion that B is as at\nleast as useful as A for all purposes in the bulk limit. We show that the\nembeddings exist if and only if for all p \u003e= 1, the p-norm of the shape of A\ndoes not exceed the p-norm of the shape of B. The log of the p-norm of the\nshape of A can be interpreted as the maximum of S(\\rho) + H(\\rho)/p (quantum\nentropy plus discounted classical entropy) taken over all mixed states \\rho on\nA. We also establish a noiseless coding theorem that justifies these entropies.\nThe noiseless coding theorem and the bulk embedding theorem together say that\neither A blindly bulk-encodes into B with perfect fidelity, or A admits a state\nthat does not visibly bulk-encode into B with high fidelity.\n In conclusion, the utility of a hybrid quantum memory is determined by its\nsimultaneous capacity for classical and quantum entropy, which is not a finite\nlist of numbers, but rather a convex region in the classical-quantum entropy\nplane.",
"arxiv_id": "quant-ph/0203105",
"authors": [
"Greg Kuperberg"
],
"categories": [
"quant-ph",
"cs.IT",
"math-ph",
"math.IT",
"math.MP",
"math.OA"
],
"journal_ref": "IEEE Trans. Inform. Theory 49 (2003), 1465-1473",
"title": "The capacity of hybrid quantum memory",
"url": "https://arxiv.org/abs/quant-ph/0203105"
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