Theory and experiment for resource-efficient joint weak-measurement

Aldo C. Martinez-Becerril1, Gabriel Bussières1, Davor Curic2, Lambert Giner1,3, Raphael A. Abrahao1,4, and Jeff S. Lundeen1,4

1Department of Physics and Centre for Research in Photonics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada
2Complexity Science Group, Department of Physics and Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4
3Département de Physique et d’Astronomie, Université de Moncton, 18 Ave. Antonine-Maillet, Moncton, New Brunswick E1A 3E9, Canada
4Joint Centre for Extreme Photonics, University of Ottawa - National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1A 0R6, Canada

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Incompatible observables underlie pillars of quantum physics such as contextuality and entanglement. The Heisenberg uncertainty principle is a fundamental limitation on the measurement of the product of incompatible observables, a 'joint' measurement. However, recently a method using weak measurement has experimentally demonstrated joint measurement. This method [Lundeen, J. S., and Bamber, C. Phys. Rev. Lett. 108, 070402, 2012] delivers the standard expectation value of the product of observables, even if they are incompatible. A drawback of this method is that it requires coupling each observable to a distinct degree of freedom (DOF), i.e., a disjoint Hilbert space. Typically, this 'read-out' system is an unused internal DOF of the measured particle. Unfortunately, one quickly runs out of internal DOFs, which limits the number of observables and types of measurements one can make. To address this limitation, we propose and experimentally demonstrate a technique to perform a joint weak-measurement of two incompatible observables using only one DOF as a read-out system. We apply our scheme to directly measure the density matrix of photon polarization states.

Many phenomena in quantum mechanics involve the measurement of the product of two incompatible observables of a quantum system. For example, the fundamental limitation on a joint measurement of two incompatible observables, e.g., position and momentum, is set by the Heisenberg uncertainty principle. Previous work used weak measurement for the realization of joint measurements. Weak measurement is a technique where the disturbance caused by the measurement process is reduced. These joint weak-measurements require a read-out system per measured observable. In this work, we theoretically introduce and experimentally demonstrate a joint weak-measurement technique that uses a single read-out system. This frees useful internal degrees of freedom of the measured quantum system for other tasks. We apply our scheme to directly measure the density matrix of photon polarization states.

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