(Modified comparative advantage model)Can trade with a country that only has one industry be mutually beneficial? Suppose there are two countries, China and Japan. Suppose there are also only two goods, namely food and bronze mirrors. China produces both food and bronze mirrors, while Japan produces only food. In order to produce 1 unit of food China has to give up 0.3 units of bronze mirrors, while Japan needs to give up 0 units of bronze mirrors due to absence of bronze mirrors production.
In order to produce 1 bronze mirror China has to give up 3.33 units of food, while Japan can't sacrifice anything for 1 bronze mirror because it doesn't produce bronze mirrors in the first place.

Given this situation, can it make sense for China to start specializing in producing and trading bronze mirrors with Japan?

Normally we would compare opportunity costs for both industries in both countries in order to conclude who has comparative advantage in what industry and then make countries specialize and trade accordingly, but in this example this will work only with one country. In this example Japan has comparative advantage in food production because it's opportunity cost of producing food is zero. So Japan should specialize in producing food. BUT, it's unclear how can we justify trade where China specializes in producing bronze mirrors. After all, it's incorrect to say that China has COMPARATIVE advantage in producing bronze mirrors (how can there be any comparison if there is nothing to compare with? Besides, from more mathy point of view, attempt of such comparison will fail due to division by zero).
I did not encounter this kind of example before but I think comparative advantage would still apply. In Ricardian model opportunity cost can be expressed as a ratio of outputs $a$ for food $f$ and bronze mirrors $b$ as $a_f/a_b$. If Japan can create A units of food but 0 bronze mirrors the opportunity cost would be given by $A/0$ now this is undefined in mathematics, because limit of the expression would not give you the same value from both sides, but since we are working only with positive quantities (presumably output cant be negative) we could evaluate the limit only from positive side which will give you $\infty$ meaning that no matter how much you sacrifice food you will get no mirrors.

The rest of the problem could be solved in this way, the equilibrium price that should be $a_{f}^{'}/ a_b^{'} &lt; P_f /P_b &lt; a_f/a_b$  would just have the upper bound equal to $\infty$. The exact quantities demanded would be given by the assumptions on individual demands in countries etc. Even in case this would not be mathematically rigorous I think you could carry out virtually the same analysis with just assuming some small infinitesimal production possibility for Japan and examining how the solution behaves as it goes to 0.