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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.

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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.

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    $\begingroup$ The paper deals with codes for the classical-quantum channel, and so the inputs are purely classical bits. The collective measurements arise considering the decoding, as many copies of the same channel are involved, and the outputs to the channel are quantum states. So as far as I understood in your answer, collective measurement is just the measurement of the composite quantum state form the several uses of the channel, instead of meauring each of the outputs individually, right? Anyway, I don't see where entanglement takes place here,as you state it,and I don't know if it is applicable here. $\endgroup$ – Josu Etxezarreta Martinez Sep 26 '18 at 8:04
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    $\begingroup$ @JosuEtxezarretaMartinez Yes, that's right. If you're performing a collective measurement, unless the measurement operators are of the form $M_1\otimes M_2\otimes\ldots\otimes M_n$ (in which case you can decompose the measurement into local measurements), the basis of the measurement operator contains entangled states. The fact that there's entanglement is not so important, except to emphasise that it's something that cannot be done with just the single copies. $\endgroup$ – DaftWullie Sep 26 '18 at 8:11
  • $\begingroup$ Ok, now I have understood. Thanks a lot. Other thing, how can someone do something like that physically? Is it possible and known yet? $\endgroup$ – Josu Etxezarreta Martinez Sep 26 '18 at 8:14
  • $\begingroup$ One way to do it physically is to make the different output interact with each other, before making individual measurements of the outputs. I all generality, this would imply to essentially have a quantum computer at the output of our channel. Of course, this is generally not possible with current technologies, but some collective measurement are indeed doable by small quantum circuits, like Bell measurements. $\endgroup$ – Frédéric Grosshans Oct 1 '18 at 12:02

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