dorsal/arxiv
View SchemaOptimal Tight Frames and Quantum Measurement
| Authors | Y. C. Eldar, G. David Forney Jr |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0106070 |
| URL | https://arxiv.org/abs/quant-ph/0106070 |
| Journal | IEEE Trans. Inform. Theory, vol. 48, pp. 599-610, Mar. 2002 |
Abstract
Tight frames and rank-one quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rank-one generalized quantum measurements (POVMs) on that space. Using this relationship, frame-theoretical analogues of various quantum-mechanical concepts and results are developed. The analogue of a least-squares quantum measurement is a tight frame that is closest in a least-squares sense to a given set of vectors. The least-squares tight frame is found for both the case in which the scaling of the frame is specified (constrained least-squares frame (CLSF)) and the case in which the scaling is free (unconstrained least-squares frame (ULSF)). The well-known canonical frame is shown to be proportional to the ULSF and to coincide with the CLSF with a certain scaling. Finally, the canonical frame vectors corresponding to a geometrically uniform vector set are shown to be geometrically uniform and to have the same symmetries as the original vector set.
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"abstract": "Tight frames and rank-one quantum measurements are shown to be intimately\nrelated. In fact, the family of normalized tight frames for the space in which\na quantum mechanical system lies is precisely the family of rank-one\ngeneralized quantum measurements (POVMs) on that space. Using this\nrelationship, frame-theoretical analogues of various quantum-mechanical\nconcepts and results are developed.\n The analogue of a least-squares quantum measurement is a tight frame that is\nclosest in a least-squares sense to a given set of vectors. The least-squares\ntight frame is found for both the case in which the scaling of the frame is\nspecified (constrained least-squares frame (CLSF)) and the case in which the\nscaling is free (unconstrained least-squares frame (ULSF)). The well-known\ncanonical frame is shown to be proportional to the ULSF and to coincide with\nthe CLSF with a certain scaling.\n Finally, the canonical frame vectors corresponding to a geometrically uniform\nvector set are shown to be geometrically uniform and to have the same\nsymmetries as the original vector set.",
"arxiv_id": "quant-ph/0106070",
"authors": [
"Y. C. Eldar",
"G. David Forney Jr"
],
"categories": [
"quant-ph"
],
"journal_ref": "IEEE Trans. Inform. Theory, vol. 48, pp. 599-610, Mar. 2002",
"title": "Optimal Tight Frames and Quantum Measurement",
"url": "https://arxiv.org/abs/quant-ph/0106070"
},
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