Why is the Carnot engine the most efficient?

However, isn't any closed loop on a PV diagram reversible? The arrows can simply be drawn in the reverse way to create a refrigerator. If any closed loop is reversible then why does the specific Carnot engine (a specific loop) have the highest efficiency?

This was exactly the question I asked myself ten years ago :-) The problem is that often students do not appreciate the whole statement: Carnot's engine is operating between two temperatures (heat sources). Any circle on the PV-plane is reversible if you have many heat sources. In the case of many heat sources, you may also know that you do not talk about the efficiency of the engine, but you talk about the Clausius' equality: $$\sum_{i} \frac{Q_i}{T_i}= 0.$$ Note that $T_i$ is the temperature of the $i$th heat source (this is a very important point often missed!), which equals the temperatures of the system when they are in reversible contact. This is not true if the process is irreversible: you have heat flow from hot sources to the (colder) engine. Then one has the Clausius' inequality: $$\sum_{i}\frac{Q_i}{T_i}<0.$$

So, in short: Carnot's engine is the only reversible engine operating between two temperatures.

All the answers here are wrong to an extent - or at least very misleading. The Carnot cycle is not the highest efficiency cycle of all possible cycles, it is only one of an infinity of cycles all of which exhibit the highest possible efficiency. There is nothing particularly special about the Carnot cycle, except that it is a simple cycle, that it is relatively easy to conceptualize and thus it makes a good teaching example, and that it is the cycle Carnot chose to use to explain the concept so it has historical precedence. Any cycle made up entirely of reversible processes will be a reversible cycle and all of these will have the same highest possible efficiency - the Carnot efficiency. As an example of one - look at the Stirling Cycle with an ideal regenerator.

Also, the Carnot cycle has relatively little enclosed area on the P-V diagram so it does relatively little work per cycle making it a relatively poor cycle to implement in real machinery. Thus, there are not a lot of intentionally designed Carnot Cycle engines lying about.

We do everyone a disservice propagating the misconception that the Carnot Cycle is the one best cycle of all possible cycles efficiency wise. It is only one of many.