Let's look at LP. It has a dynamic range of about 70dB in the best of times and cases (must count surface noise, an unavoidable part, as well as rumble, etc).
If it had two channels and the same bandwidth (and the same SNR at every frequency) that would work out to about 70/6 bits per Hz, that is to say about 12 bits/Hz (11 2/3). That by itself is inferior to the CD's 16 bits/Hz.
However, it is argued, and with some meaning, although how much is actually present in any real recording can be debated, that LP's can go to 30kHz.
So, multiplying by frequency, you see (one sided spectrum, for various reasons at the end we have to double all of this, but we'll compare apples to apples) 20*16 vs. 30 *12. That would seem to put LP with more information, HOWEVER...
Above 5-6kHz, the SNR of an LP goes down rapidly, instead of being 70dB, the actual SNR is more like 20-30 dB. This is now 4-5 bits/Hz. This applies over at least half of the bandwidth, in reality, so now we can treat that as (I'm approximating, but not terribly harmfully so)
lp bits = 12 * 5 +5*25 or about 185 compared to the cd's 320
(note both measures are for one-sided spectrum and one channel)
But that's not all. The LP has distortion. We can't send as much information at lower frequencies, so we'd better use 50dB (.3 % distortion) as an upper figure instead of 70dB.
No, but wait, at high frequencies, channel independence goes to pot, and we only have 10-15dB of channel independence. So really at high frequencies (supra 15kHz for sure, maybe lower) we don't really have 5 bits/Hz/channel, we have more like (5+2,5) bits/Hz/2channels.
In other words, the LP isn't close in terms of total information content.
Point out to your friend that LP's do have some very euphonic kinds of distortion, i.e. distortions (l-r enhancement, noise floor, out of phase rumble, interchannel intermod, etc) that often sound better than the linear system.
Note: the above analysis is very, very rough. It is, however, not too far off the mark.
You may hear people claim 90dB from turntables, ask them to turn off their A-weighting if they do. You will have people who claim less distortion at high frequencies, or higher frequencies, but then we can have several discussions if you'd like, on the realities of getting 30kHz through either a new or old recording chain, as well as the atmosphere.