So its a quirk in the measuring rather than output exceeding input so to speak? Would the real "perceived" output also be greater than say if you use a electrical heater with equal input?
More a quirk in definition. Entropy as a differential is macroscopic, and concerns its self with reversible heat flow from first principals. Carnot efficiency is a detailed look at heat flow from one state to another, not necessarily entropic formations within each state.
Electrical heater will have the same problem of carnot efficiency. It obeys the same rules, since heat must flow from the element to the air. But in a heater, you are adding heat to a state (heat energy naturally flows from hot to cold, hot element to cold air). In an aircon, you are removing heat. In order to get heat to flow in the "wrong" direction in an aircon (from cold to hot), you need to generate more entropy in gas expansion than the entropy lost in heat travelling the wrong way.
IMPORTANT DEFINITION OF EFFICIENCY: \eta = W/Q_h, or in words, the fraction of work(in this case, electrical work) as heat flow. This is NOT what fraction of electrical power is in the form of heat/cooling, which is where I think the confusion is being caused.
In a heater, you efficiency is more or less 100%, since the heat flowing from hot element to the air is unhindered. In an aircon, you use electrical energy to generate an entropy differential in the form of compressing the fluid to a gas on the hot side, cooling on a radiator to more or less room temp, then taking that compressed gas/liquid to the cold side, allowing it to expand pulling heat from the room. See here, we need electricity to make the entropy differential to cause the heat to flow the "wrong" way.
It is important to understand here that it is quite easy to generate entropy with a gas expansion/contraction compared with straightforward heating of an element.
If we were to operate the aircon in reverse, (commonly called a heat pump), then yes we could produce heat more easily (electrically efficiently) than using a heating element. Sometimes even over unity. Why? Because rather than generating the heat in an element, we are moving heat between hot and cold places.
It is easier to move the heat than it is to generate the heat.
Just as a thumb suck, say we have an aircon with a cooling duty of 1 ton (1 short ton of ice melting over a 24 hour period =~ 3.5kW or 12 000BTU), we are saying it is moving an equivalent of 3.5kJ of heat energy per second, but to move that much heat energy, the compressor may only need say 1kW of electrical power.
Another analogy: Say I have a tanker truck filled with a million liters of diesel. The truck needs say 100 liters of diesel to move the rest of the diesel out of the depot and to the fuel station. We have moved a million litres of diesel only using 100 litres of diesel.
