## 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(2) 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.

Translated title of the contribution | Entanglement cost of generalised measurements |
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Original language | English |

Pages (from-to) | 405 - 422 |

Number of pages | 18 |

Journal | Quantum Information and Computation |

Volume | 3 |

Issue number | 5 |

Publication status | Published - Sept 2003 |

### Bibliographical note

Publisher: Rinton Press, IncOther identifier: IDS Number: 718JV