dorsal/arxiv
View SchemaOptimizing Completely Positive Maps using Semidefinite Programming
| Authors | K. Audenaert, B. De Moor |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0109155 |
| URL | https://arxiv.org/abs/quant-ph/0109155 |
| DOI | 10.1103/PhysRevA.65.030302 |
| Journal | Phys.Rev.A (65), 030302 (2002) |
Abstract
Recently, a lot of attention has been devoted to finding physically realisable operations that realise as closely as possible certain desired transformations between quantum states, e.g. quantum cloning, teleportation, quantum gates, etc. Mathematically, this problem boils down to finding a completely positive trace-preserving (CPTP) linear map that maximizes the (mean) fidelity between the map itself and the desired transformation. In this note we want to draw attention to the fact that this problem belongs to the class of so-called semidefinite programming (SDP) problems. As SDP problems are convex, it immediately follows that they do not suffer from local optima. Furthermore, this implies that the numerical optimization of the CPTP map can, and should, be done using methods from the well-established SDP field, as these methods exploit convexity and are guaranteed to converge to the real solution. Finally, we show how the duality inherent to convex and SDP problems can be exploited to prove analytically the optimality of a proposed solution. We give an example of how to apply this proof method by proving the optimality of Hardy and Song's proposed solution for the universal qubit $\theta$-shifter (quant-ph/0102100).
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"abstract": "Recently, a lot of attention has been devoted to finding physically\nrealisable operations that realise as closely as possible certain desired\ntransformations between quantum states, e.g. quantum cloning, teleportation,\nquantum gates, etc. Mathematically, this problem boils down to finding a\ncompletely positive trace-preserving (CPTP) linear map that maximizes the\n(mean) fidelity between the map itself and the desired transformation. In this\nnote we want to draw attention to the fact that this problem belongs to the\nclass of so-called semidefinite programming (SDP) problems. As SDP problems are\nconvex, it immediately follows that they do not suffer from local optima.\nFurthermore, this implies that the numerical optimization of the CPTP map can,\nand should, be done using methods from the well-established SDP field, as these\nmethods exploit convexity and are guaranteed to converge to the real solution.\nFinally, we show how the duality inherent to convex and SDP problems can be\nexploited to prove analytically the optimality of a proposed solution. We give\nan example of how to apply this proof method by proving the optimality of Hardy\nand Song\u0027s proposed solution for the universal qubit $\\theta$-shifter\n(quant-ph/0102100).",
"arxiv_id": "quant-ph/0109155",
"authors": [
"K. Audenaert",
"B. De Moor"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.65.030302",
"journal_ref": "Phys.Rev.A (65), 030302 (2002)",
"title": "Optimizing Completely Positive Maps using Semidefinite Programming",
"url": "https://arxiv.org/abs/quant-ph/0109155"
},
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