To amplify my thoughts somewhat. I advocate each passenger being allowed one piece of carryon, limited in size as has been mentioned. I advocate tsa or equivalent paper sealing same and placing it in the overheads where it should be safe.
It should be 'relatively' easy and inexpensive to install temporary locks like a large bicycle cable lock which runs the length of the cabin before more elaborate electronic locks are installed at the next major overhaul.
I would submit the cost is a good deal less than losing a third of your flights out of Heathrow. It would keep people in their seats longer upon docking as well. At the same time, I think airlines should be encouraged (or forced) to do a far better job with the hold baggage than has been their history.