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.
Highlights
Modern quantum measurement techniques have pushed forward our understanding and ability to manipulate quantum particles
We propose and experimentally demonstrate a technique to perform a joint weakmeasurement of two incompatible observables using only one degree of freedom (DOF) as a read-out system
The unitary evolution (Eq 5) of the two von Neumann interactions (i.e., Eq 2) is equivalent to a single unitary given by UAUA = eγA(a†+a)eγA(a†+a) = e2γA(a†+a). This corresponds to a unitary of a single von Neumann interaction of the A observable with a doubled interaction strength 2γ
Summary
Modern quantum measurement techniques have pushed forward our understanding and ability to manipulate quantum particles. Weak measurement decreases the disturbance caused by the measurement process and thereby mostly preserves the quantum state of the system, allowing one to obtain correlations between any chosen set of general observables, including incompatible ones [3,4,5,6,7,8,9,10]. To perform such a measurement, the observable is weakly coupled to a separate read-out system (the ‘pointer’) that indicates the average result of the measurement. An overview of the Fractional Fourier Transform (FrFT) is given in Appendix A
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