Abstract art, especially painting, is art with no subject -- or rather, the subject is the art itself. By removing the "picture" from the picture, the artist is free to concentrate their effort on

Commonly called the plastic elements of art. (This list is illustrative not definitive)

It is these elements that determine the quality of a painting -- not the subject. The subject of an Abstract Painting is the painting itself.

In response to Gamaliel -- I disagree with your analysis. Picasso painted representationally. His cubist paintings, often mistaken for abstract art, are simply another way to look at real things. The abstract paintings of Ad Reinhart or Robert Motherwell are a different kettle of fish entirely. Piet Mondrian was a minimalist.
Further response -- I am glad you brought in a different perspective -- my responses are in no way to be construed as attacks.

As a painter, it makes perfect sense to use the term in such a narrow fashion. Using the term Abstract as a catch all for any painting not strictly representational weakens it, removes its power and accuracy. It causes the lumping together of cubism, futurism, dada, abstract expressionism, minimalism, surrealism, etc, under one umbrella. These are distinct schools of art with different motivations, different points to make.

The artists you list are all theoretically quite different -- it's similar to grouping a Harley, a VW Beetle, a Cadillac, and a bicycle together and saying 20th Century transportation is enough to describe them all.

I would like to call your attention to Gamaliel's dissent below -- after careful reading, I believe we both have a valid point, and in the end, the decision, like all art decisions, will be made
in the eye of the beholder.