The formalism for quantum systems developed so far applies to what are called pure states. A system in a pure state is completely specified by the state vector. A complete set of experimental tests will determine the system state fully: we have maximal knowledge of the system. For example, for a spin system, we can find a particular orientation of an SG machine such that the state is in its + or − port. This also means that the state is an eigenvector of some operator, or is always a linear combination of the computational basis states.
The quasiprobability distribution of the discrete Wigner function provides a complete description of a quantum state and is, therefore, a useful alternative to the usual density matrix description. Moreover, the experimental quantum state tomography in discrete Wigner phase space can also be implemented. We observe that for a certain class of states, such as harmonic states, the Wigner matrix is far more sparse compared to the density matrix in the computational basis. Additionally, reading only a small part of the Wigner matrix may suffice to infer certain behavior of quantum dynamics. In such cases, selective Wigner phase space tomography (SWPST) can be more efficient than the usual density matrix tomography (DMT). Employing nuclear magnetic resonance methods on a three-qubit nuclear spin register, we experimentally estimate Wigner matrices of various two-qubit quantum states. As a specific example application of SWPST, we study the evolution of spin coherent states under the quantum chaotic kicked top model and extract signatures of quantum-classical correspondence in the Wigner phase space.
The Husimi function provides a phase-space view of quantum systems. This paper considers a number of its properties, including ways in which it can be expanded in terms of basis functions. A spectral or ‘‘natural’’ expansion and an expansion analogous to the Carlson–Keller expansion in terms of coordinate-density momentum-density products are considered, as is a method for separating the angular dependence of the momentum. There is a set of functions in phase space having the same overlap properties as the initial orbital basis in terms of which the charge density matrix is expressed. A Husimi matrix is defined and a scalar product in the space containing such matrices as elements is introduced. The connection with the vector space of density matrices is examined. The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted. For a harmonic oscillator basis the phase space basis functions, in terms of which the Husimi function is expressed, span a linear space but do not provide a complete set; their products provide a linearly independent set that is complete. It is suggested that similar behavior can be expected for other basis sets.
Presents the basic postulates of quantum mechanics in terms of the density matrix instead of the usual state vector formalism in the case of an isolated system. Extends it to the case of open systems with the help of the reduced density matrix formalism, and to the case of an imperfect state preparation described by a statistical mixture. Introduces the concept of quantum state purity to characterize the degree of mixture of the state, and shows that one can always "purify" a density matrix by going into a Hilbert space of larger dimension.