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Complete Measurements of Quantum Observables

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Abstract

We define a complete measurement of a quantum observable (POVM) as a measurement of the maximally refined (rank-1) version of the POVM. Complete measurements give information on the multiplicities of the measurement outcomes and can be viewed as state preparation procedures. We show that any POVM can be measured completely by using sequential measurements or maximally refinable instruments. Moreover, the ancillary space of a complete measurement can be chosen to be minimal.

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Notes

  1. For example, the measurement of the position Q of a spinless particle is complete since the spectral measure of Q is of rank-1, but Q is not IC since one cannot reconstruct the state of the system from the position measurement statistics [1–3]. Conversely, an IC observable is not necessarily rank-1, i.e. complete, so that one cannot directly get information about the multiplicities of the measurement outcomes.

  2. On the one hand, for any self-adjoint operator S, there exists a unique real spectral measure M such that \(S=\int_{\mathbb{R}}x\,{\mathrm{d}}\mathsf {M}(x)\). On the other hand, any real spectral measure M defines a unique self-adjoint operator \(\int_{\mathbb{R}}x\,{\mathrm{d}}\mathsf {M}(x)\).

  3. A quantum channel is a unital normal completely positive linear map on \(\mathcal {L(H)}\).

  4. That is, for each i, operators A is , s=1,2,… , are linearly independent, i.e. the conditions ∑ s c s φ iks =0 for all k implies that the complex numbers c s are 0. (Hence, the rank r i is minimal.)

  5. If m(x)=0 then the sum \(\sum_{k=1}^{0}(\cdots )\) in (9) is interpreted to be 0 [9].

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Acknowledgements

This work was supported by the Academy of Finland grant No. 138135.

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  1. Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, 20014, Turku, Finland

    Juha-Pekka Pellonpää

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  1. Juha-Pekka Pellonpää
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Correspondence to Juha-Pekka Pellonpää.

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Pellonpää, JP. Complete Measurements of Quantum Observables. Found Phys 44, 71–90 (2014). https://doi.org/10.1007/s10701-013-9764-y

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