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arXiv:1703.01835 · International Journal of Quantum Information

An effective iterative method to build the Naimark extension of rank-n POVMs

Nicola Dalla Pozza, Matteo G. A. Paris

Published Jun 1, 2017• 5 citations
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Abstract

We revisit the problem of finding the Naimark extension of a probability operator-valued measure (POVM), i.e. its implementation as a projective measurement in a larger Hilbert space. In particular, we suggest an iterative method to build the projective measurement from the sole requirements of orthogonality and positivity. Our method improves existing ones, as it may be employed also to extend POVMs containing elements with rank larger than one. It is also more effective in terms of computational steps.

DOI
10.1142/s0219749917500290

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We revisit the problem of finding the Naimark extension of a probability operator-valued measure (POVM), i.e. its implementation as a projective measurement in a larger Hilbert space. In particular, we suggest an iterative method to build the projective measurement from the sole requirements of orthogonality and positivity. Our method improves existing ones, as it may be employed also to extend POVMs containing elements with rank larger than one. It is also more effective in terms of computational steps.
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Abstract In this chapter we first review projective or von Neumann measurements from the point of view of the projection postulate and the Lüders rule, before turning to more general measurements modelled using positive operator-valued measures (POVMs). After examining their mathematical description, we discuss symmetric informationally complete (SIC) POVMs and Zauner’s conjecture, as well as the representation of POVMs as projective measurements on a larger Hilbert space, a result called Naimark dilation. We then a model for non-ideal projective measurements and analyse the properties of such measurements in relation to the properties unbiasedness, faithfulness, and non-invasiveness of ideal projective measurements. Finally, we consider the problem of distinguishing non-orthogonal quantum states and examine how the use of POVMs can be beneficial for this task.
We obtain a formal characterization of the compatibility or otherwise of a set of positive-operator-valued measures (POVMs) via their Naimark extensions. We show that a set of POVMs is jointly measurable if and only if there exists a single Naimark extension, specified by a fixed ancilla state on the same ancilla Hilbert space, that maps them to a set of commuting projective measurements (PVMs). We use our result to obtain an easily checkable sufficient condition for the compatibility of a pair of dichotomic observables in any dimension. This in turn leads to a characterization of the compatibility regions for some important classes of observables including a pair of unsharp qubit observables. Finally, we also outline as to how our result provides an alternate approach to quantifying the incompatibility of a general set of quantum measurements.
Abstract After a short review of the main properties of unsharp observables we provide examples of commutative unsharp observables obtained as the projection of sharp observables. In particular, we consider a family of PVMs defined on the tensor product Hilbert space H ⊗ K which include the one introduced by Ozawa in his modification of the von Neumann position measurement model and show that their projections onto H are commutative POVMs. Then, we make some observations on the concept of Naimark dilation and its connections with the integral representation of commutative POVMs. Finally, we introduce a partial order relation in the set of Naimark dilations, show that there is a minimal element and that it is unique. Minimality is a consequence of the partial ordered structure and coincides with the usual definition of minimal Naimark dilation. In passing, we use Naimark’s theorem in order to prove some well known properties of positive operator valued measures.
In the quantum theory of measurement, the positive operator-valued measure (POVM) is an important concept, and its implementation can be useful. A POVM consists of a set of non-negative quantum-mechanical Hermitian operators that add up to the identity. The probability that a quantum system is in a particular state is given by the expectation value of the POVM operator corresponding to that state. Following a brief review of the mathematics and mention of the history of POVMs in quantum theory, a particular implementation of a POVM for use in the measurement of nonorthogonal photon polarization states is reviewed. The implementation consists simply of a Wollaston prism, a mirror, two beam splitters, a polarization rotator and three phototubes arranged in an interferometric configuration, and it is shown analytically that the device faithfully represents the POVM. Based on Neumark's extension theorem, the two-dimensional Hilbert space of the POVM implementation can be embedded in the three-dimensional Hilbert space of an ordinary projective-valued measure. Also, analytical expressions are given for the maximum Renyi information loss from the device to a disturbing probe, and for the error and inconclusive rates induced by the probe. Various aspects of the problem of probe optimization are elaborated.
We propose a scheme that can realize a class of positive-operator-valued measures (POVMs) by performing a sequence of projective measurements on the original system, in the sense that for an arbitrary input state the probability distribution of the measurement outcomes is faithfully reproduced. A necessary and sufficient condition for a POVM to be realizable in this way is also derived. In contrast to the canonical approach provided by Neumark's theorem, our method has the advantage of requiring no auxiliary system. Moreover, an arbitrary POVM can be realized by utilizing our protocol on an extended space which is formed by adding only a single extra dimension.