Each tag sends out a radio message when a reader lights it up with radio frequency waves (the reader’s signal also provides the power for the tag). The message is a large number, large enough that it is astronomically unlikely you’ll find two with the same number in the same shop, burned into each tag when it was manufactured.
When the shop gets items with such tags (or sticks them onto stuff), the number of the tag gets entered into the shop’s database. When the cashier pulls the item across their reader, the number gets removed. If a reader in the security gates detects a tag with a number still in the inventory, you get an alarm.
The detectors at the doors send power the tag as it passes through them, when powered the tag emits a signal back to the detectors.
This is called wireless electricity, Nikolas Tesla came up with it, and many others have tried it on small scales. Think about how you can send and receive signals with with a small, low power, wifi emitter. Boost up the power of the emitter and you can power a low power device.
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