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arXiv:quant-ph/0505095 · Journal of Mathematical Physics

Clean positive operator valued measures

Francesco Buscemi, Michael Keyl, Giacomo Mauro D’Ariano, Paolo Perinotti, Reinhard F. Werner

Published Aug 1, 2005• 91 citations
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Abstract

In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVMs into POVMs, generally irreversibly, thus losing some of the information retrieved from the measurement. This poses the problem of which POVMs are “undisturbed,” i.e., they are not irreversibly connected to another POVM. We will call such POVMs clean. In a sense, the clean POVMs would be “perfect,” since they would not have any additional “extrinsical” noise. Quite unexpectedly, it turns out that such a “cleanness” property is largely unrelated to the convex structure of POVMs, and there are clean POVMs that are not extremal and vice versa. In this article we solve the cleannes classification problem for number n of outcomes n⩽d (d dimension of the Hilbert space), and we provide a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVMs for n=d2.

DOI
10.1063/1.2008996

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In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVMs into POVMs, generally irreversibly, thus losing some of the information retrieved from the measurement. This poses the problem of which POVMs are “undisturbed,” i.e., they are not irreversibly connected to another POVM. We will call such POVMs clean. In a sense, the clean POVMs would be “perfect,” since they would not have any additional “extrinsical” noise. Quite unexpectedly, it turns out that such a “cleanness” property is largely unrelated to the convex structure of POVMs, and there are clean POVMs that are not extremal and vice versa. In this article we solve the cleannes classification problem for number n of outcomes n⩽d (d dimension of the Hilbert space), and we provide a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVMs for n=d2.
A quantum probability measure–or quantum measurement–is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. The notion of a clean quantum measure was introduced by Buscemi et al. [“Clean positive operator valued measures,” J. Math. Phys. 46(8), 082109 (2005)10.1063/1.2008996] for finite-dimensional Hilbert space, and was studied subsequently by Kahn [“Clean positive operator-valued measures for qubits and similar cases,” J. Phys. A 40(18), 4817–4832 (2007)10.1088/1751-8113/40/18/009] and Pellonpää [“Complete characterization of extreme quantum observables in finite dimensions,” J. Phys. A 44(8), 085304 (2011)10.1088/1751-8113/44/8/085304]. The present paper provides new descriptions of clean quantum probability measures in the case of finite-dimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of “approximately clean quantum probability measures” and characterise this property for measures whose range determines a finite-dimensional operator system.
Similarly to quantum states, also quantum measurements can be 'mixed', corresponding to a random choice within an ensemble of measuring apparatuses. Such mixing is equivalent to a sort of hidden variable, which produces a noise of purely classical nature. It is then natural to ask which apparatuses are indecomposable , i.e. do not correspond to any random choice of apparatuses. This problem is interesting not only for foundations, but also for applications, since most optimization strategies give optimal apparatuses that are indecomposable. Mathematically the problem is posed describing each measuring apparatus by a positive operator-valued measure (POVM), which gives the statistics of the outcomes for any input state. The POVMs form a convex set, and in this language the indecomposable apparatuses are represented by extremal points—the analogous of 'pure states' in the convex set of states. Differently from the case of states, however, indecomposable POVMs are not necessarily rank-one, e.g. von Neumann measurements. In this paper we give a complete classification of indecomposable apparatuses (for discrete spectrum), by providing different necessary and sufficient conditions for extremality of POVMs, along with a simple general algorithm for the decomposition of a POVM into extremals. As an interesting application, 'informationally complete' measurements are analysed in this respect. The convex set of POVMs is fully characterized by determining its border in terms of simple algebraic properties of the corresponding POVMs.
In a recent paper (Buscemi et al 2005 J. Math. Phys. 46 082109) a notion of clean positive operator-valued measures (POVMs) was defined. We characterize those POVMs which are clean in a class that we call quasi-qubit POVMs, namely POVMs whose elements are all rank-1 or full-rank elements. We give an algorithm to check whether a given quasi-qubit POVM satisfies this condition. We describe explicitly all the POVMs that are clean for the qubit. Meanwhile, we give a sufficient condition for a general POVM to be clean.
Measurements on quantum channels are described by so-called process positive operator valued measures, or process POVMs. We study implementing schemes of extremal process POVMs. As it turns out, the corresponding measurement must satisfy certain extremality property, which is stronger than the usual extremality given by the convex structure. This property motivates the introduction and investigation of the A-convex structure of POVMs, which generalizes both the usual convex and C*-convex structures. We show that extremal points and faces of the set of process POVMs are closely related to A-extremal points and A-faces of POVMs, for a certain subalgebra A. We also give a characterization of A-extremal and A-pure POVMs.
Various forms of optimality for quantum observables described as normalized positive-operator-valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables that are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient conditions are imposed on informationally complete POVMs within the set of pure states. We also discuss joint and sequential measurements of optimal quantum observables.