You didn't say which part of Italy you visited, but when I was in Tuscany last summer for three months, I bought a book called Le Insalate di Campo (Salad of the Fields) about edible plants in Tuscany, and it lists one Amaranth plant. The one it includes is Amaranthus retroflexus, which looks pretty much like the plant in your picture.

ThePlants for a Future database says that all plants in the Amaranth genus are edible (doesn't mean they all taste good, though). It mentions that young leaves of this one are mild in flavor and can be eaten cooked or raw, and that this plant is a good one to use as a grain source. The PFAF page gives common names for the plant, including Redroot Amaranth. I've seen it referred to as Redroot Pigweed on other sites.

When I was out walking the Tuscan countryside near the town we stayed in (for my husband's job), I loved the diversity of flowers but also that I could identify so many of them -- because they've spread to the U.S.! I have started a Tuscany Wildflower page on my website, to help me remember the walks. So far, it only has about a dozen wildflowers on it, but I have a lot more pictures to work with.