Nitpicking about the active/passive usage of "correlated" I hesitate whether to ask this here in the stats StackExchange or in the linguistics/English one, but I reckon there might be more language-nitpicking users on here than stats-savvy users in the other forum ;)
I often read reports that mention correlation as a verb in the active voice, as in "We then correlated A with B and found...". To me, this verb only makes sense in the passive voice, as when saying for instance that "We found that A and B were significantly correlated". I might be wrong that this really constitutes active vs passive voice grammatically, but what I describe is the difference between doing something to A and B such that they each end up changed, versus computing a third variable (e.g. an R coeff) from them.
One can, of course, actively DEcorrelate two variables, but it seems to me that to "correlate" them, rather than referring to something active, is simply used as a shorthand for checking whether a significant such correlation exists!
Am I wrong? Does it make any other sense statistically to say that you [actively] correlated A with B?
 A: "Correlate" is a back formation of "correlation", which comes from "co" (with) and "relation". Which I suppose is a bit redundant, as a relation is always with something else. It would be acceptable to say "We related X to Y", so I think that from a "lay" perspective, it makes sense to say "We correlated X to Y". One could argue that in a math context, "correlate" has a specific meaning that precludes this use, but that raises questions such as "What is that meaning?" "How was it established?", and "In what circumstances is it reasonable to call for math specific usage?". For instance, there was a Jeopardy! clue along the lines of "It's the set of points within a fixed distance of a central point." The "correct" response was "What is sphere", but mathematically the correct response was "What is ball?" Even though they were discussing math, this is a program directed at the general populace, so making the distinction was reasonable. 
So I would say that it is reasonable to make the distinction yourself, and even reasonable to expect someone speaking to a math audience to make the distinction, but it's acceptable in more lay contexts to not do so.

I might be wrong that this really constitutes active vs passive voice grammatically

I think you are. Generally speaking, if something is in the passive voice, then you can add a "by ..." at the end, e.g. "The passive voice is frequently used [by writers]".

but what I describe is the difference between doing something to A and B such that they each end up changed

I don't think that's an accurate description. If someone were to say "We compared A and B", would they be implying that A and B were changed? Just because something is grammatically the object of a verb, doesn't mean that anything was actually done to it.
A: I don't think this is nitpicking at all. 
The first time I heard someone say "We correlated A with B", the speaker had the ability to influence A. I took their saying to mean "A and B were first uncorrelated, but we then altered A so as to have it strongly correlated with B". I spent a lot of time trying to figure out why they had done this. Eventually I realized that they meant "we found a correlation between A and B", and their motivation became much more clear at that point. 
A: I see where you are coming with this – if you say something like "we correlated A with B", you might risk giving the impression that you introduced correlation between A and B where perhaps none existed before.
In my view, there are better ways to say this, such as: "we investigated whether A and B were correlated" or "we studied the (linear?) association/relationship between A and B".
Can you get away with using "we correlated A and B" from a grammatical and/or statistical viewpoint? The answer is yes. Is that the best way you can get your point across? My own answer to this last question would be No.
A: I don't think this is a grammatical issue, just a question of how words are used, or should be best used, in practice.
A meta-lesson I have learned over several years is that a claim that something is ungrammatical is fragile. There is always another grammarian who can be found who will dispute the assertion. (I am of a generation firmly told never to split infinitives because, supposedly, the practice is totally ungrammatical; that was rebutted as bogus logic (a misconceived analogy with Latin) long before I was taught this in the 1960s; my teachers were, I guess now, just passing on what they had been told in their youth, and so forth. Nevertheless I still can't split an infinitive willingly.)
I would understand "we correlated $X$ and $Y$" easily as "we calculated the correlation between $X$ and $Y$". It's fairly common usage, I think. Even if it isn't common usage, I don't see what is ungrammatical about it. There is an associated question of how far the correlation exists as an inevitable consequence of the data, as a mathematical or even real fact, before its value is calculated, or indeed regardless of whether that is done. I can't say I have ever worried about that.
But I wouldn't want to write that in a paper or catch myself saying it in a presentation. That is mostly a question of personal style, and as always agreement and disagreement about style are both to be expected.
I can't imagine saying "We plotted $Y$ against $X$", because I would just say "Here is a plot..." or "Figure 2 is a plot ...". Similarly, at most, I would just say "The correlation is ...".
It's worth remembering that Francis Galton hijacked correlation, which was a fairly unusual but long-existing word, for the present statistical purpose. Now I guess that the statistical sense of correlation (or a more diluted or generalised sense of it) is primary usage.
Notes:

*

*You want nit-pickers to comment, so in that vein I will say that $A$ and $B$ are not congenial notation for variables, even in complete abstraction.


*Never heard of "decorrelated"!
A: Correlate is now commonly used as a verb. You pointed to the use of this word as transitive vs. intransitive, and stated that the latter is right and the former is, perhaps, wrong.
Note, unlike you, I'm not framing this as the difference between active and passive forms, because that distinction is just a red herring in this case. Consider this, the form that you find more comfortable to use is "A is correlated to B" is passive. However, it's not the fact that it's passive that makes it more natural to you. It's that it's intransitive, as in active form "A correlates to B," as opposed to its transitive form "we correlate A to B," that makes it sound right to you.
I must agree that the intransitive form sounds more natural, both in passive and active forms. Moreover, when Galton first introduced the term, he used it only as an intransitive verb, in passive form, e.g. "the length of the arm is said to be correlated with that of the leg." According to Pearson, it was Galton who first defined the term as a statistical concept in "Co-relations and their Measurement, chiefly from Anthropometric Data” in 1888. Although the word itself was used before in other contexts. Pearson's paper "Notes on the History of Correlation" is here.
Now, I have to break a bad news: both forms have been in use for quite some time. Here's an example from The Standard American Encyclopedia of Arts... published in 1898!



-- verb intransitive – correlate, correlating. To have reciprocal relation, to be
reciprocally relates, as father and son. -- verb transitive. To place in
reciprocal relation: to determine the relations between, as between
several objects or phenomena which bear a resemblance to one another

As you can see both intransitive and transitive forms are described, i.e. "A correlates to B" and "we correlate A to B" are both fine. See also this discussion.
The verb "correlate" was created by back-formation from the noun. For instance, apparently, a verb "translate" was created similarly from a noun "translation".
@kjetilbhalvorsen brought up an example "to google", but it's a different mechanism of word formation called verbing, and a special case of it too. Normally, verbing is making verbs from nouns like "medal" $to$ "to medal." In this case we take an eponym "Google" and make a verb "to google." It's similar to "Xerox" $\to$ "to xerox", and even an older example of a guy named Charles Boycott $\to$ "to boycott."
What's even more interesting about Google case, is that it's made from a recently made up word "googol."
A: This usage of the verb correlate may be uncommon but it is grammatically correct since it can be used as a transitive verb.

correlate: to present or set forth so as to show relationship. "He correlates the findings of the scientists, the psychologists, and the mystics."

See this Definition for reference.
