Theory and experiment for resource-efficient joint weak-measurement

Abstract

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.

Quantum circuits for measuring weak values, Kirkwood–Dirac quasiprobability distributions, and state spectra

“…since, by construction, ψ |ϕ = 0. Other generalizations of weak values, such as sequential weak values or joint weak values, provide valuable information regarding the behavior of a given state with respect to two-possibly incompatible-observables [43][44][45][46]. It so happens that the numerators in the expression for the weak value above, or those of joint weak values, correspond to values of the KD quasiprobability distribution at the phase-space point determined by the pre-and post-selected states [47].…”