## Theory

**Laser modes**

The propagation of a laser beam is well described by a wave model and diffraction. A laser beam is the output beam from an optical cavity and, for technically well designed lasers, the beam can be described by the Hermitian Gauss modes for the transverse electro-magnetic field with an amplitude distribution given by the function TEM_{nk}.

The fundamental mode TEM_{00} describes a beam with Gaussian intensity distribution propagating along a well defined axis. The higher order modes describe more complex field distributions. All TEM_{nk} modes are immune to diffraction in the sense that they retain their shape. A Gaussian beam for example remains a Gaussian beam. The modes TEM_{nk} are orthogonal and we can describe arbitrary beam shapes using the Hermite Gauss modes as a basis.

We can detect the position of such a laser beam, for example by a split detector which shows an change in the intensity balance when the beam moves. This allows is very sensitive, and allows measurements well below the optical wavelength. After eliminating technical noise sources, such as vibrations or imperfections in the laser, we are then only limited by quantum noise. Even for the perfect laser we have noise in the position of the mode: a laser beam: it can never go in a perfectly straight line.

**Quantum information**

In our work we encode and transfer information in the spatial properties of the beam. The advantage is that simple physical parameters map directly on the specific modes: for example displacement corresponds to the real part of the TEM_{01}, tilt corresponds to the imaginary part of the TEM_{01}, waist size corresponds to the real part of TEM_{02}, waist position to the imaginary part of TEM_{02}. Using mirrors and lenses we can control direction and displacement and the wavefront curvature an in this way write information into few independent modes TEM_{01}, TEM_{10}, TEM_{02}, TEM_{20}, each with one pair of conjugate parameters. At the same time we can sense the effect of a sample, which changes the beam spatially, directly by measuring the modulation of the TEM_{nk} modes.

We have shown that for every optical measurement and detector, we can find an optical mode which contains all the information and noise associated with this measurement. If we can match the mode that contains the information with the mode of detection we have found the best possible detection system, which allows detection with 100% efficiency. By using squeezed light in such a transmission system we can send information that contain quantum correlations. This is an alternative to the well established techniques for single photons.- system. Using the multimode system with several degrees of freedom we can transmit multimode quantum information.

The output of two squeezed laser beams, operating on the same mode, can be combined to a pairs of entangled modes. This is well established for Gaussian beam and has been used in many applications, such as dense coding and teleportation. We can now extend this to spatial systems where the entangled properties are for example position and tilt.