dorsal/arxiv
View SchemaConstruction of extremal local positive operator-valued measures under symmetry
| Authors | S. Virmani, M. B. Plenio |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0212020 |
| URL | https://arxiv.org/abs/quant-ph/0212020 |
| DOI | 10.1103/PhysRevA.67.062308 |
| Journal | Phys. Rev. A. 67, 062308 (2003) |
Abstract
We study the local implementation of POVMs when we require only the faithful reproduction of the statistics of the measurement outcomes for all initial states. We first demonstrate that any POVM with separable elements can be implemented by a separable super-operator, and develop techniques for calculating the extreme points of POVMs under a certain class of constraint that includes separability and PPT-ness. As examples we consider measurements that are invariant under various symmetry groups (Werner, Isotropic, Bell-diagonal, Local Orthogonal), and demonstrate that in these cases separability of the POVM elements is equivalent to implementability via LOCC. We also calculate the extrema of these classes of measurement under the groups that we consider, and give explicit LOCC protocols for attaining them. These protocols are hence optimal methods for locally discriminating between states of these symmetries. One of many interesting consequences is that the best way to locally discriminate Bell diagonal mixed states is to perform a 2-outcome POVM using local von Neumann projections. This is true regardless of the cost function, the number of states being discriminated, or the prior probabilities. Our results give the first cases of local mixed state discrimination that can be analysed quantitatively in full, and may have application to other problems such as demonstrations of non-locality, experimental entanglement witnesses, and perhaps even entanglement distillation.
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"abstract": "We study the local implementation of POVMs when we require only the faithful\nreproduction of the statistics of the measurement outcomes for all initial\nstates. We first demonstrate that any POVM with separable elements can be\nimplemented by a separable super-operator, and develop techniques for\ncalculating the extreme points of POVMs under a certain class of constraint\nthat includes separability and PPT-ness. As examples we consider measurements\nthat are invariant under various symmetry groups (Werner, Isotropic,\nBell-diagonal, Local Orthogonal), and demonstrate that in these cases\nseparability of the POVM elements is equivalent to implementability via LOCC.\nWe also calculate the extrema of these classes of measurement under the groups\nthat we consider, and give explicit LOCC protocols for attaining them. These\nprotocols are hence optimal methods for locally discriminating between states\nof these symmetries. One of many interesting consequences is that the best way\nto locally discriminate Bell diagonal mixed states is to perform a 2-outcome\nPOVM using local von Neumann projections. This is true regardless of the cost\nfunction, the number of states being discriminated, or the prior probabilities.\nOur results give the first cases of local mixed state discrimination that can\nbe analysed quantitatively in full, and may have application to other problems\nsuch as demonstrations of non-locality, experimental entanglement witnesses,\nand perhaps even entanglement distillation.",
"arxiv_id": "quant-ph/0212020",
"authors": [
"S. Virmani",
"M. B. Plenio"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.67.062308",
"journal_ref": "Phys. Rev. A. 67, 062308 (2003)",
"title": "Construction of extremal local positive operator-valued measures under symmetry",
"url": "https://arxiv.org/abs/quant-ph/0212020"
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