dorsal/arxiv
View SchemaDesigning Optimal Quantum Detectors Via Semidefinite Programming
| Authors | Yonina C. Eldar, Alexandre Megretski, George C. Verghese |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0205178 |
| URL | https://arxiv.org/abs/quant-ph/0205178 |
| DOI | 10.1109/TIT.2003.809510 |
| Journal | IEEE Trans. Inform. Theory, vol. 49, pp. 1017-1012, Apr. 2003. |
Abstract
We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing between a collection of quantum states, represented by a set of density operators. We show that the design of the optimal detector can be formulated as a semidefinite programming problem. Based on this formulation, we derive a set of necessary and sufficient conditions for an optimal quantum measurement. We then show that the optimal measurement can be found by solving a standard (convex) semidefinite program followed by the solution of a set of linear equations or, at worst, a standard linear programming problem. By exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum, the optimal measurement can be computed very efficiently in polynomial time. Using the semidefinite programming formulation, we also show that the rank of each optimal measurement operator is no larger than the rank of the corresponding density operator. In particular, if the quantum state ensemble is a pure-state ensemble consisting of (not necessarily independent) rank-one density operators, then we show that the optimal measurement is a pure-state measurement consisting of rank-one measurement operators.
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"abstract": "We consider the problem of designing an optimal quantum detector to minimize\nthe probability of a detection error when distinguishing between a collection\nof quantum states, represented by a set of density operators. We show that the\ndesign of the optimal detector can be formulated as a semidefinite programming\nproblem. Based on this formulation, we derive a set of necessary and sufficient\nconditions for an optimal quantum measurement. We then show that the optimal\nmeasurement can be found by solving a standard (convex) semidefinite program\nfollowed by the solution of a set of linear equations or, at worst, a standard\nlinear programming problem. By exploiting the many well-known algorithms for\nsolving semidefinite programs, which are guaranteed to converge to the global\noptimum, the optimal measurement can be computed very efficiently in polynomial\ntime.\n Using the semidefinite programming formulation, we also show that the rank of\neach optimal measurement operator is no larger than the rank of the\ncorresponding density operator. In particular, if the quantum state ensemble is\na pure-state ensemble consisting of (not necessarily independent) rank-one\ndensity operators, then we show that the optimal measurement is a pure-state\nmeasurement consisting of rank-one measurement operators.",
"arxiv_id": "quant-ph/0205178",
"authors": [
"Yonina C. Eldar",
"Alexandre Megretski",
"George C. Verghese"
],
"categories": [
"quant-ph"
],
"doi": "10.1109/TIT.2003.809510",
"journal_ref": "IEEE Trans. Inform. Theory, vol. 49, pp. 1017-1012, Apr. 2003.",
"title": "Designing Optimal Quantum Detectors Via Semidefinite Programming",
"url": "https://arxiv.org/abs/quant-ph/0205178"
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