Quantum state reconstruction made easy: a direct method for tomography

In quantum mechanics, the state of the system, or set of systems, is encoded as a vector in a state space. While this sounds simple, it has several implications that are not limited to the more well known and „spooky“ consequences of quantum physics, such as entanglement. One of these is that we can never directly observe the state of the system itself; we are only able to measure its observable properties, such as position and momentum, that are affected by the system’s state. By making many measurements of this set of observables of the state, we can generate marginal distributions of each. From these distributions, we can reconstruct the original state vector from its associated phase-space Wigner function. It has been shown that such a measurement of the Wigner function is possible for light. Here we give a procedure for the direct measurement and reconstruction of the Wigner function for a series of quantum spin states that should be applicable to any quantum system. We have applied our procedure to IBM’s Quantum Experience five qubit quantum processor to demonstrate that we can directly measure and distinguish two different Bell states – states central to understanding entanglement – via this method. We have also performed direct measurements of the five qubit Greenberger-Horne-Zeilinger (GHZ) state Wigner function.