If two states $A$ and $ B$ are connected by a reversible path, they can never be connected by an irreversible path during an adiabatic process because:

$\underbrace{\Delta U}_{\text{state function}} = \underbrace{\Delta W}_{\text{path function}}$

And $W_{\text{reversible}}> W_{\text{irreversible}}$

Now, Atkins' Physical Chemistry (Section $3.2$) states that:

To calculate the difference in entropy between any two states of a system, we find a reversible path between them, and integrate the energy supplied as heat at each stage of the path divided by the temperature at which heating occurs.

But the problem is that we cannot define a reversible path for an adiabatic irreversible process. If that's the case, how can we calculate the entropy change for an adiabatic irreversible process?

If two states $A$ and $ B$ are connected by a reversible path, they can never be connected by an irreversible path during an adiabatic process because:

$\underbrace{\Delta U}_{\text{state function}} = \underbrace{\Delta W}_{\text{path function}}$

And $W_{\text{reversible}}\neq W_{\text{irreversible}}$

Now, Atkins' Physical Chemistry (Section $3.2$) states that:

To calculate the difference in entropy between any two states of a system, we find a reversible path between them, and integrate the energy supplied as heat at each stage of the path divided by the temperature at which heating occurs.

But the problem is that we cannot define a reversible path for an adiabatic irreversible process. If that's the case, how can we calculate the entropy change for an adiabatic irreversible process?