Reconstructing the density operator via simple projectors.

In the realm of quantum theory a state of a physical system is most generally expressed by its density operator ˆ̺. Knowledge of this operator gives complete information of the quantum state. Schemes have been proposed in a number of fields in quantum physics to determine ˆ̺ from measurements either explicitely [1,2,3,4,5,6,7] or indirectly via quasiprobability distributions [8,9,10,11] for mixed states and also for pure states only [12,13]. In this Letter we describe a general method of representing any density operator ˆ̺ in terms of expectation values of simple projection operators. Since the expectation values of projectors can, in principle, be determined experimentally this approach leads to schemes for experimentally determining the density operator. Our approach differs from previously proposed schemes in quantum optics for determining the density operator in its use of simple projectors which project onto a single or a linear superposition of two basis states [2,5]. We place an emphasis on a ’minimalistic’ representation which comprises the least number of projection operators and thus leads to the most efficient scheme, it is a generalization of the previous considerations in [2,5]. The plan of the paper is as follows. First we introduce the general idea of our approach, then, in section II, we cast it into two specific representations and describe their relative virtues. In section III we describe a quantum optical implementation and we end with a discussion in section IV.