Dedicated I missing something?

As I live in an apartment, I don't think that the management would take too kindly of rewiring the place! However, if I am able to unplug/turn-off all other electrical equipment on my sound system's circuit breaker, then haven't I created a dedicated circuit or am I missing something?
You have a good start. The limiting factor now seems to be the physical connections at all the AC outlets between the breaker and your system outlet. When I lived in a townhouse, I had an electrician remake all these connections so that none were using the push-ins on the back of the outlets. We bolted the bare copper wire to the side screws and there was an immediate improvement. I also plugged nothing else into the outlets on the same breaker. I have also discovered that AC dimmers used on halogen lamps and ceiling fans anywhere in the house are a huge source of noise, even when off.
The word on the street is to use the "first" box on the circuit -i.e., the box that, when disconnected, kills every other box fed by the particular fuse or circuit breaker. This is as close to having a "dedicated" circuit in an apartment as you can get.
I, too, am an apartment dweller with a fairly serious stereo (hey, some things come first!) and found that replacing the outlets made a good difference. I used Hubbell hospital grade long before all the specialist outlets came out. Replace the circuit breaker if it is fairly elderly. Another positive was the Audio Prism Quiet Lines along with a Noise Sniffer borrowed from the Cable Company. There was measurable noise on duplex outlets I wasn't even using! I could only blame "the neighbors". Also check out this site for tweaks:
(He's not just trying to sell cables, either.)

Good Luck! It is definitely worth working on!