Why does the signal detector D0 in the delayed choice experiment always show a blur? As explained in many answers such as https://physics.stackexchange.com/a/516512/64051, the delayed choice quantum eraser cannot be used to send information back in time because the detector D0 always show the same blur pattern regardless of whether the measurement of idler photons are detected or erased, and it is not possible to know whether there was measurement or erasure without comparing the detector results through classical transmission of information.
However, I am confused as to why D0 always show the same pattern. My understanding is that what D0 observes will be equivalent to that of a double slit experiment, and double slit experiment actually shows interference pattern on the detector without needing to compare anything.
 A: In the DCE the blur is only there we when look at all coincidences together. When we look at the collection of signals when D0 and D1 are at the same time we see the pattern.  Also when we look at D0 with D2 we get an opposite pattern.
The DCE has been disproven to have anything to do with looking back (or forward) in time.
Added explanation to question/comment below:
So let's just some things out of the way .... interference is a very historical word and is somewhat inaccurate when taken literally ... the word interference is still taught in high school/1st/2nd year university etc for its simplicity.  But in truth 2 photons never cancel each other (that is a violation of conservation of energy), Dirac, Feynman and many other scientists know that each photon takes its own probable path (Dirac used to say "the photon is interfering with itself" but the modern view is that the photon determines its path).
So let us start with an excited atom/electron (which can be in this state for a while), we will eventually get photon emission BUT we can never really have emission if we don't have absorption i.e a photon is really nothing until it is absorbed/observed ..... all photons are eventually absorbed by an atom/electron!  And that defines the path taken.
The excited electron/atom are capable of exerting forces thru what are known as (or conjectured as) virtual photons (only force no energy). The EM field is very dynamic, it hosts both the real photon (EM wave) and as well virtual photons (which carry real momentum).
Thus it is very possible that the photon determines its path via these virtual forces.
IN the DCE, just like in the DSE, the excited source electron/photon will choose one of a few probable paths .... Feynman path integral is a way to calculate these and the result is similar to the classic "interference" calculation. BY ADDING the beam splitter after the down converter a few more VIABLE path choices are offered to the electron/photon ... and one of the probable paths is for the photon to go directly thru one slit .... thru the crystal... thru the beam splitter directly to a detector ... it is this path that does not "perceive" or is not affected by the double slit and hence is really going thru one slit only ... THUS the lack of the pattern.
The last part in italics is my interpretation ... there may be additional factors that allow the "photon wavefunction" (probable path taken) to only perceive one slit. An experiment could be done with the DSE with just adding a beam splitter afterwards (we don't need the DCE stuff) ... but it has not been done as far as I know.  Important matters in physics today are quantum computing, nuclear fusion, electric cars ... the DSE is kind of an older less practicle experiment these days.
