What are "collective measurements", and how do they differ from "normal" measurements?

Question

I am reading the paper Polar codes for classical-quantum channels by Wilde and Guha, and it is stated the fact that collective measurements are necessary in order to aciheve the Holevo symmetric information as it can be seen from the HSW theorem.

I am wondering what are such collective measurements and why are they different from the normal quantum measurements. Additionally, I was expecting to obtain some insight about some physical realization of such kind of measurements, as it seems that if applied in an optical-fiber link, the usage of such codes would be a way to obtain a tranmission rate equal to the Holevo symmetric information, which would be the ultimate capacity of such a channel.

Answer 1

Collective measurements are normal measurements. You just need to be clear on the setting under which they are operating. I haven't delved deeply into the specific paper you mention (so it's always possible they make marginally different assumptions), but I expect it goes like this:

• You are looking at using many copies of the same channel.

• Encoding will, generically, be an entangled state across the inputs for a quantum-quantum channel. In this case of a classical-quantum channel, the inputs are classical bits, so the inputs can be correlated but not entangled.

• Decoding will, generically, involve measurement of all outputs of the channel simultaneously, in an entangled basis.

It is these measurements across multiple outputs in an entangled basis that are referred to as collective measurements, in comparison to single-system measurements on the outputs of individual channels. In comparison to measurements on the outputs of the individual channels, these collective measurements first involve an entangling unitary between all the outputs.

Now, I said "generically" in the sense that this is the most general case that you should consider. One might hope that the optimum measurement might be simpler than that, e.g. measurements performed on individual channel outputs. Presumably one of the points this paper is making is that this is not the case in their specific setting.