1. Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University, Sendai 980-8577, Japan
   
2. CREST-JST, Saitama 332-0012, Japan
   
3. Department of Information Engineering, Sendai National College of Technology, Sendai 989-3128, Japan
   
4. Nanotechnology Research Institute, AIST, Tsukuba 305-8568, Japan

Correspondence to: Hideo Kosaka^1,2 Correspondence and requests for materials should be addressed to H.K. (Email: kosaka@riec.tohoku.ac.jp).

Spin is a fundamental property of electrons, with an important role in information storage^{1, 2, 3, 4}. For spin-based quantum information technology, preparation and read-out of the electron spin state are essential functions^{5, 6, 7, 8, 9, 10, 11, 12, 13}. Coherence of the spin state is a manifestation of its quantum nature, so both the preparation and read-out should be spin-coherent. However, the traditional spin measurement technique based on Kerr rotation, which measures spin population using the rotation of the reflected light polarization that is due to the magneto-optical Kerr effect, requires an extra step of spin manipulation or precession to infer the spin coherence^{14, 15, 16, 17, 18, 19, 20}. Here we describe a technique that generalizes the traditional Kerr rotation approach to enable us to measure the electron spin coherence directly without needing to manipulate the spin dynamics, which allows for a spin projection measurement on an arbitrary set of basis states. Because this technique enables spin state tomography, we call it tomographic Kerr rotation. We demonstrate that the polarization coherence of light is transferred to the spin coherence of electrons, and confirm this by applying the tomographic Kerr rotation method to semiconductor quantum wells with precessing and non-precessing electrons. Spin state transfer and tomography offers a tool for performing basis-independent preparation and read-out of a spin quantum state in a solid.