HIGH is normally used for digital inputs/outputs, where there are only two possible states: HIGH/LOW.
It is not normally used to refer to analogue inputs, which can have 1024 different values between 0 and 1023.
Somewhere HIGH is defined as '1', so only 1 value out of the 1024 possible analogue values is regarded as HIGH.
An ADC reading of 0 isn't regarded as HIGH, neither are ADC counts of 2, 3, 4, 5, 6, 7, . . . . . . . 1021, 1022, and 1023.
You only need approximately 5mV of noise to go from a reading of '0' to 'HIGH'.
So it is unlikely that you are always going to get a reading of '0', when the voltage is supposed to be 0V.