A simple realization of Weyl-Heisenberg covariant measurements

• URL: http://arxiv.org/abs/2512.22111v1
• Date: Fri, 26 Dec 2025 18:50:53 GMT
• Title: A simple realization of Weyl-Heisenberg covariant measurements
• Authors: Sachin Gupta, Matthew B. Weiss,
• Abstract summary: Informationally complete (IC) measurements are fundamental tools in quantum information processing.<n>We elaborate on a simple algorithm for realizing Naimark extensions for rank-one Weyl-Heisenberg covariant informationally complete measurements.
• Score: 0.8594140167290097
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Informationally complete (IC) measurements are fundamental tools in quantum information processing, yet their physical implementation remains challenging. By the Naimark extension theorem, an IC measurement may be realized by a von Neumann measurement on an extended system after a suitable interaction. In this work, we elaborate on a simple algorithm for realizing Naimark extensions for rank-one Weyl-Heisenberg covariant informationally complete measurements in arbitrary finite dimensions. Exploiting Weyl-Heisenberg covariance, we show that the problem reduces to determining a $d \times d$ unitary from which the full $d^2 \times d^2$ unitary interaction can be constructed. The latter unitary enjoys a block-circulant structure which allows e.g., for an elegant optical implementation. We illustrate the procedure with explicit calculations for qubit, qutrit, and ququart SIC-POVMs. Finally, we show that from another point of view, this method amounts to preparing an ancilla system according to a so-called fiducial state, followed by a generalized Bell-basis measurement on the system and ancilla. These results provide a straightforward framework for implementing informationally complete measurements in the laboratory suitable for both qubit and qudit based systems.

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