EXPERT RESPONSE
There are actually two separate metrics at work here: Transmit Power and Receive Sensitivity. Transmit Power measures the output power generated by the sender, while Receive Sensitivity measures the minimum power required for a receiver to handle arriving frames at a given data rate.
Some power will be lost between sender and receiver, due to free space path loss, attenuation, etc. Conversely, high-gain antennas actually increase power in a certain direction, by focusing available power into a smaller area. This can of course be true at both the sender and receiver.
If you improve the antenna at the AP only, you increase the AP's radiated output power, hopefully making the signal stronger when it finally reaches the client. A client that was not sensitive enough to "hear" the AP before may now receive sufficient signal.
Now, when that client transmits back to the AP, the AP is also likely to hear the client. Why? APs usually have better Receive Sensitivity than clients to start with. But the AP's high-gain antenna also improves that AP's receive sensitivity.
Improving the client's antenna could add further benefit, but is not strictly speaking necessary to get some improvement. Similarly, purchasing more sensitive clients could add further benefit by letting them operate at higher data rates, given the same AP transmit power. For a good illustrated example, see this tutorial on transmission speed and distance relations.
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