Hi,
I was lucky enough to have very heavy clay in my garden. Some 20-30cm below the surface it was yellow/orange, good for making pottery and not for gardening. Now I have beautiful clay-loam, loose, airy and very fertile. Some 50-60 cm depth of it last time I checked. Its probably around 80-90 cm down today.
Lets start with pointing what not to do. Sadly i can see lots of very bad advise. Damaging really.
Never add sand or gravel to clay !
It will form sort of concrete. Pretty much unworkable.
Clay particles surround grains of sand/gravel and texture wont change. The sand/gravel will just make it more difficult to put nything down into the soil.
Do an experiment to see yourself the mechanics. Take a glass of flour - as you can see not much air in between the particles. Then take a glass of peas - you can see lots of air. Now take a glass of flour and handful of peas to it - result, peas will mix, resulting in flour filling all the voids between the peas and still no air pockets!
Same goes with clay and sand/gravel with added hindrance of sand/gravel making more difficult to put anything down into the sail as mentioned above.
I do not know if ground chalk will loosen the clay up or not. Meant to do an experiment long time ago, but haven't come around to do that yet.
As per opinion above about the "chalk boulders being unusable" - it is very soft rock and it doesn't take much to beak it down or powderise. Strong, preferably metal, bucket, some metal bar, preferably wide and few minutes of pondering in the bucket will turn the chalk into a powder.
Best way, to break it up - three ingredients, organic matter and charcoal (once "charged with biology it is often called biochar), plants.
Organic matter should be dug in, the finer the better. Saw dust is great, The more the better.
At the same time dug in lots of ground charcoal.
The charcoal should be powderized. Again, bucket and metal/hard wood bar. Make sure that the charcoal is moist so you don't generate cloud of dust which is not very good for health when breath in.
You can dug in bigger chunks, but the smaller the charcoal particles the more effective it is at breaking up clay.
Organic matter will break down over time and has to be replenished, but no dig is needed. Actually any digging is damaging to the soil.
Charcoal will stay in soil for years (dozens?hundreds?) doing its magic.
I could write a book on benefits of charcoal in the soil, here enough to say that it is single best soil amendment known to man kind.
Once you dig in the stuff above you can plant your veggies.
At the harvest remember to never pull out the roots. Of course if the root is not your crop - cucumbers, tomato, lettuce etc. Leave the roots in the soil, so they can rot providing organic matter in the soil. This organic matter is not fertilizer. It is food for soil microbes which fertilize your plants. Any spare space in between your veggies use to sow some stuff to chop later on. Plants like wheat, rye, rape, mustard. Point is that these plants grow and spreading roots down in the soil. Once chopped at surface level they will not regrow from roots. Instead the roots till rot adding organic matter to the soil. The top dump into the compost.
And that is it. No more needs to be done and you will enjoy beautiful, fertile clay-loam.
Never dig the patch again. Just add few cm of compost on top every now and then when planting/sawing new veggies.
Never fertilize, never use pesticides.
Healthy population of microbes in the soil will provide all the nutrients your plants need. Digging up, fertilizing, using pesticides kills off the good guys in the soil and do more harm to you plants than help. Once good guys are killed off, the bad guys have space to thrive.
Very beneficial is to inoculating the patch with fungi - like parasol mushroom, portobello mushroom etc. Not only they will assist your plants with gathering nutrients, break down organic matter in the soil, but as a bonus you can enjoy very tasty mushroom every now and then
The only exception when you need to dig up the patch is when you want to grow early succession plants like brassicas.
The above is just a top an iceberg. If you want to know more search online for "no dig method" or "soil food web".
Enjoy