# Are Q values estimated from a DQN different from a duelling DQN with the same number of layers and filters?

I am confused about the Q values of a duelling deep Q network (DQN). As far as I know, duelling DQNs have 2 outputs

1. Advantage: how good it is to be in a particular state $$s$$

2. Value: the advantage of choosing a particular action $$a$$

We can make these two outputs into Q values (reward for choosing particular action $$a$$ when in state $$s$$) by adding them together.

However, in a DQN, we get Q values from the single output layer of the network.

Now, suppose that I use the same DQN model with the very same weights in my input and hidden layers and changing the output layer which gives us Q values to advantage and value outputs. Then, during training, if I add them together, will it give me the same Q value for a particular state, supposing all the parameters of both my algorithms are the same except for the output layers?

On the other hand, since we only have the target Q-value, separating the Q-value into state value and advantage result in the identifiability issue. That is the network might simply learn $$V(s)=0$$, $$A(s,a)=Q(s,a)$$ for every state.
$$Q(s,a;\theta,\alpha,\beta)=V(s;\theta,\beta)+(A(s,a;\theta,\alpha)-\frac{1}{|A|}\sum\limits_{a'}A(s,a';\theta,\alpha))$$