One for the gurus

[Tarnash]

One for the gurus or, for anyone who has a better understanding of optics and optical design than I (which I suspect is most of you!).  A purely hypothetical question but one I find intriguing:  A 400mm f/2.8 lens attached to a 1.4 teleconverter becomes, in effect a 560mm f/4 lens.  If it were possible, using an APSC sized sensor, to add a `speed booster' would the resulting combination become, in effect, a 560mm f/2.8 set up?  And, if it can be, or has been done, what would the I.Q. be like?   

[fdisilvestro]

You will end up with a 400mm f2.8 which on APS-C will have the equivalent field of view of a 560mm on full frame.

Basically the speed booster cancels the effect of the teleconverter and you gain nothing, actually you can get lower quality

[BernardLanguillier]

Don't think I qualify as a guru, but will share my views anyway.

You are right in your expectation in terms of getting a 560 f2.8 with an APS-C coverage (I believe that Fransisco is not correct here). In fact I own all the piece needed to do a test some day:
- Nikon 400mm f2.8
- teleconverter
- speedbooster F -> E
- sony a5100

Will test when I find the time.

Cheers,
Bernard