Direct measurement of the density matrix of a quantum system

TL;DR

This paper introduces an experimental scheme for directly measuring individual elements of a quantum state's density matrix using a sequence of weak and strong measurements, enabling more efficient quantum state characterization.

Contribution

The authors demonstrate a novel direct measurement method for density matrix elements that does not require global reconstruction, using a sequence of complementary measurements on polarized photons.

Findings

Successfully measured density matrices of pure and mixed photon states

Implemented weak measurements with polarization-dependent spatial shifts

The method provides an operational way to access density matrix elements

Abstract

One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements, since it requires a global reconstruction. Here we experimentally demonstrate a scheme that can be used to directly measure individual density matrix elements of general quantum states. The scheme relies on measuring a sequence of three observables, each complementary to the last. The first two measurements are made weak to minimize the disturbance they cause to the state, while the final measurement is strong. We perform this joint measurement on polarized photons in pure and mixed states to directly measure their density matrix. The weak measurements are achieved using two walk-off crystals, each inducing a polarization-dependent spatial shift that couples the spatial and polarization degree of freedom of the photons. This direct measurement method provides an operational meaning to the density matrix and promises to be especially useful for large dimensional states.