Entanglement cost of generalised measurements

R Jozsa, M Koashi, N Linden, S Popescu, S Presnell, D Shepherd, Alex T Winter

Research output: Contribution to journalArticle (Academic Journal)peer-review

16 Citations (Scopus)


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 contributionEntanglement cost of generalised measurements
Original languageEnglish
Pages (from-to)405 - 422
Number of pages18
JournalQuantum Information and Computation
Issue number5
Publication statusPublished - Sept 2003

Bibliographical note

Publisher: Rinton Press, Inc
Other identifier: IDS Number: 718JV


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