Characterizing incompatibility of quantum measurements via their Naimark extensions

• URL: http://arxiv.org/abs/2011.11364v2
• Date: Sun, 17 Oct 2021 06:16:04 GMT
• Title: Characterizing incompatibility of quantum measurements via their Naimark extensions
• Authors: Arindam Mitra, Sibasish Ghosh, Prabha Mandayam
• Abstract summary: We show that a set of POVMs is jointly measurable if and only if there exists a single Naimark extension. We also outline as to how our result provides an alternate approach to quantifying the incompatibility of a general set of quantum measurements.
• Score: 5.759887284136111
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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