So I've heard that Bluetooth reduces audio quality, but the loss varies according to Bluetooth versions and profiles that are being used on the player and the headphone/speaker. So my question is, in terms of audio quality, what combination of Bluetooth version and audio profiles can stream audio similar to or better than analog (aux) cable? I bought this headphone, which is Bluetooth version 3.0 with HSP.HFP.A2DP.AVRCP and whatnot. Can its quality be compared with aux cable?
First of all, Bluetooth is only a transmission protocol, and doesn't have anything to do with how the audio is encoded or compressed. Bluetooth itself doesn't reduce audio quality, it only transmits digital audio. Since digital audio is, by definition, compressed, it is impossible for Bluetooth (or any other digital transmission protocol) to perform better than an AUX analog cable.
What reduces the quality of the audio is the compression applied to it before it is sent over to your headset via Bluetooth. Now, with that said, this means the codec used to transform the audio is the crucial part. A really good codec (and maybe also the best), is the Qualcomm aptX HD which supports up to 24bit 44kHz digital audio, which is very good quality, and basically the best you could ask for nowdays. You can google it to find out which headphones support it and how to use it.
Can this be compared with the quality of an AUX cable? Well, depends on what you are playing! If you play any track that is digitally compressed to less than or equal to 24bit at 44kHz then there is no difference, because both AUX and BT are able to reproduce it, but if you play anything with a higher bit rate or a higher sample rate then AUX is always going to win.