The spatial resolution in photoacoustic imaging is essentially limited by acoustic attenuation, which can be numerically compensated only up to a theoretical limit. The physical background for this “ill-posedness” is the second law of thermodynamics: the loss of information is equal to the entropy production, which is the energy decay of the attenuated wave divided by the temperature. As acoustic attenuation increases with higher frequencies, a cut-off frequency can be determined, where the information content for that frequency gets so low that it cannot be distinguished from equilibrium distribution within a certain statistical significance. This cut-off frequency can be determined also by setting the amplitude of the attenuated signal in frequency domain equal to the noise-level. Compensating for acoustic attenuation requires to solve an ill-posed inverse problem, where an adequate regularization parameter is the cut-off frequency, when the acoustic wave amplitude is damped just below the noise level. If additional information, such as positivity or sparsity is used, this theoretical resolution limit can be overcome. This is experimentally demonstrated for the propagation of planar acoustic waves in fat tissue, which are induced by short laser pulses and measured by piezoelectric transducers. For fatty porcine tissue the frequency dependent acoustic attenuation was measured. This was used to invert the problem and by using additional information, in the form of positivity and sparsity (Douglas-Rachford splitting algorithm) the resolution could be enhanced significantly compared to the limit given by the cut-off frequency from attenuation through 20 mm of porcine fat tissue.