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.Comment: 20 pages, plain late
