Quantum homodyne tomography with a priori constraints

An intriguing aspect of quantum mechanics is the intricate description of the state of a physical system. Therefore, a great deal of interest has been attracted by the recent experimental demonstration that the quantum state of a simple system, namely a single light mode, can be completely characterized in a feasible scheme [1]. The measurement was based on an observation that marginals of the Wigner function, which contains complete information on the quantum state, can be collected with the help of a balanced homodyne detector [2]. The Wigner function was reconstructed from the statistics of homodyne events using a standard filtered back-projection algorithm developed in image processing. This seminal work initiated an extensive research in the field of quantum state measurement, which has brought beautiful demonstrations of quantum phenomena as well as deeper understanding of the foundations of quantum theory [3]. The Wigner function is particularly well suited for visualization of quantum states, as it represents pictorially quantum coherence in the form of nonclassical phase space structures. The purpose of this Communication is to present a novel method for reconstructing the Wigner function from homodyne statistics. Its essential advantage compared to the standard back-projection algorithm is the capability of compensating, in a numerically stable way, for non-unit efficiency of the homodyne detector. Imperfect detection is well known to have a deleterious effect on nonclassical features of the Wigner function, such as negativities and oscillatory interference patterns [4,5]. Furthermore, an attempt to incorporate compensation into the standard linear reconstruction scheme fails due to rapidly exploding statistical errors and numerical instabilities [6–8]. In the present Communication I show that these difficulties can be effectively overcome by taking into account a priori constraints imposed by the quantum mechanical form of the Wigner function. This new method is derived using the maximum-likelihood estimation [9], and constraints are formulated in a way which provides a convenient algorithm for reconstructing the Wigner function from realistic, finite and imperfect homodyne data. Let us start the considerations from tracing the standard route from raw statistics of homodyne events to the Wigner function. The quantum mechanical probability distribution of measuring the quadrature x in a single run of the homodyne setup with the local oscillator phase θ is given by the expectation value of the following positive operator-valued measure [10]: