No, two photons emitted from the same source towards two detectors at different distances are not equally likely to be entangled. The entanglement of two particles, such as photons, depends on the specific experimental setup and the physical processes involved.
Entanglement is a phenomenon in quantum mechanics where two or more particles become correlated in such a way that the state of one particle is dependent on the state of the other(s), even when they are physically separated. However, the probability of entanglement occurring between two particles depends on various factors, including the characteristics of the source, the detection process, and the distance between the particles.
In the scenario you described, where two photons are emitted towards two detectors at different distances (1 meter and 10 meters), the probability of entanglement would generally decrease with increasing distance. Entanglement is typically more likely to occur when the particles are in close proximity and experience strong interactions during their creation or subsequent interactions.
However, it is important to note that the precise conditions and experimental setup would determine whether entanglement is possible in a given scenario. Factors such as the specific source of photons, the entanglement generation mechanism, and the detection method used can all influence the likelihood of entanglement. In some cases, it might be possible to design an experimental setup that still allows for entanglement between photons even at different distances, but it would require specific considerations and techniques tailored to that purpose.
In summary, the likelihood of entanglement between two photons emitted from the same source towards two detectors at different distances would depend on the details of the experimental setup and the physical processes involved. Generally, the probability of entanglement decreases with increasing distance, but the specific conditions of the experiment would determine the extent of entanglement.