dorsal/arxiv
View SchemaEntanglement cost of generalised measurements
| Authors | Richard Jozsa, Masato Koashi, Noah Linden, Sandu Popescu, Stuart Presnell, Dan Shepherd, Andreas Winter |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0303167 |
| URL | https://arxiv.org/abs/quant-ph/0303167 |
| Journal | Quantum Inf. Comput., vol. 3, no. 5, 405-422 (2003) |
Abstract
Bipartite entanglement is one of the fundamental quantifiable resources of quantum information theory. We propose a new application of this resource to the theory of quantum measurements. According to Naimark's theorem any rank 1 generalised measurement (POVM) M may be represented as a von Neumann measurement in an extended (tensor product) space of the system plus ancilla. By considering a suitable average of the entanglements of these measurement directions and minimising over all Naimark extensions, we define a notion of entanglement cost E_min(M) of M. We give a constructive means of characterising all Naimark extensions of a given POVM. We identify various classes of POVMs with zero and non-zero cost and explicitly characterise all POVMs in 2 dimensions having zero cost. We prove a constant upper bound on the entanglement cost of any POVM in any dimension. Hence the asymptotic entanglement cost (i.e. the large n limit of the cost of n applications of M, divided by n) is zero for all POVMs. The trine measurement is defined by three rank 1 elements, with directions symmetrically placed around a great circle on the Bloch sphere. We give an analytic expression for its entanglement cost. Defining a normalised cost of any d-dimensional POVM by E_min(M)/log(d), we show (using a combination of analytic and numerical techniques) that the trine measurement is more costly than any other POVM with d>2, or with d=2 and ancilla dimension 2. This strongly suggests that the trine measurement is the most costly of all POVMs.
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"abstract": "Bipartite entanglement is one of the fundamental quantifiable resources of\nquantum information theory. We propose a new application of this resource to\nthe theory of quantum measurements. According to Naimark\u0027s theorem any rank 1\ngeneralised measurement (POVM) M may be represented as a von Neumann\nmeasurement in an extended (tensor product) space of the system plus ancilla.\nBy considering a suitable average of the entanglements of these measurement\ndirections and minimising over all Naimark extensions, we define a notion of\nentanglement cost E_min(M) of M.\n We give a constructive means of characterising all Naimark extensions of a\ngiven POVM. We identify various classes of POVMs with zero and non-zero cost\nand explicitly characterise all POVMs in 2 dimensions having zero cost. We\nprove a constant upper bound on the entanglement cost of any POVM in any\ndimension. Hence the asymptotic entanglement cost (i.e. the large n limit of\nthe cost of n applications of M, divided by n) is zero for all POVMs.\n The trine measurement is defined by three rank 1 elements, with directions\nsymmetrically placed around a great circle on the Bloch sphere. We give an\nanalytic expression for its entanglement cost. Defining a normalised cost of\nany d-dimensional POVM by E_min(M)/log(d), we show (using a combination of\nanalytic and numerical techniques) that the trine measurement is more costly\nthan any other POVM with d\u003e2, or with d=2 and ancilla dimension 2. This\nstrongly suggests that the trine measurement is the most costly of all POVMs.",
"arxiv_id": "quant-ph/0303167",
"authors": [
"Richard Jozsa",
"Masato Koashi",
"Noah Linden",
"Sandu Popescu",
"Stuart Presnell",
"Dan Shepherd",
"Andreas Winter"
],
"categories": [
"quant-ph"
],
"journal_ref": "Quantum Inf. Comput., vol. 3, no. 5, 405-422 (2003)",
"title": "Entanglement cost of generalised measurements",
"url": "https://arxiv.org/abs/quant-ph/0303167"
},
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