2 research outputs found
State estimation in quantum homodyne tomography with noisy data
In the framework of noisy quantum homodyne tomography with efficiency
parameter , we propose two estimators of a quantum state whose
density matrix elements decrease like , for
fixed known and . The first procedure estimates the matrix
coefficients by a projection method on the pattern functions (that we introduce
here for ), the second procedure is a kernel estimator of the
associated Wigner function. We compute the convergence rates of these
estimators, in risk
Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors
We estimate the quantum state of a light beam from results of quantum
homodyne measurements performed on identically prepared pulses. The state is
represented through the Wigner function, a ``quasi-probability density'' on
which may take negative values and must respect intrinsic
positivity constraints imposed by quantum physics. The data consists of
i.i.d. observations from a probability density equal to the Radon transform of
the Wigner function. We construct an estimator for the Wigner function, and
prove that it is minimax efficient for the pointwise risk over a class of
infinitely differentiable functions. A similar result was previously derived by
Cavalier in the context of positron emission tomography. Our work extends this
result to the space of smooth Wigner functions, which is the relevant parameter
space for quantum homodyne tomography.Comment: 15 page