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arXiv:quant-ph/0310013 · Journal of Optics B Quantum and Semiclassical Optics

Informationally complete measurements and group representation

Giacomo Mauro D’Ariano, Paolo Perinotti, Massimiliano Federico Sacchi

Published May 29, 2004• 123 citations
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Abstract

Informationally complete measurements on a quantum system allow one to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kinds of measurement can be achieved through positive-operator valued measures (POVMs) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help of frame theory we provide a constructive way to evaluate the data-processing function for arbitrary operators.

DOI
10.1088/1464-4266/6/6/005

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Informationally complete measurements on a quantum system allow one to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show that such kinds of measurement can be achieved through positive-operator valued measures (POVMs) related to unitary irreducible representations of a group on the Hilbert space of the system. With the help of frame theory we provide a constructive way to evaluate the data-processing function for arbitrary operators.
The expectation value of an arbitrary operator O can be obtained via a universal measuring apparatus that is independent of O, by changing only the data-processing of the outcomes. Such a ``universal detector'' performs a joint measurement on the system and on a suitable ancilla prepared in a fixed state, and is equivalent to a positive operator valued measure (POVM) for the system that is ``informationally complete''. The data processing functions generally are not unique, and we pose the problem of their optimization, providing some examples for covariant POVM's, in particular for SU(d) covariance group.
Generalized quantum measurements (also known as positive operator-valued measures or POVMs) are of great importance in quantum information and quantum foundations, but often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the symmetric, informationally complete-POVM (or SIC-POVM), is the most compact set of measurements that can be used to fully describe a quantum state. We use our technique to carry out the first experimental characterization of the state of a qutrit using SIC-POVMs. Because of the highly symmetric nature of this measurement, such a representation has the unique property that it permits all other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states encoded in single photons.
Generalized quantum measurements [also known as positive operator-valued measures (POVMs)] are of great importance in quantum information and quantum foundations but are often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the symmetric informationally complete (SIC) POVM, is the most compact set of measurements that can be used to fully describe a quantum state. We use our technique to carry out the first experimental characterization of the state of a qutrit using SIC POVMs. Because of the highly symmetric nature of this measurement, such a representation has the unique property that it permits all other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states encoded in single photons.