  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's polarization.
