The question is harder to answer than you might think. Of course, it helps to know that, in philosophy, a zombie is something quite a bit different from the brain-eating monster we find in movies.
In this context “zombie” refers to someone who has all the physical traits of a person, including behavior patterns, but no consciousness. If you tell a zombie that she isn’t a thinking person, her face might flush, and she might scream at you, but it would all just be a mechanical reaction, there is actually no consciousness driving that behavior.
Consciousness is not something we can test for physically. We simply infer it from behavior patterns. But, if materialism is true, there is no reason to think that consciousness is really needed in order to control behavior. The complex patterns of electrons, as they bounce around the brain, is enough.
If that is the case, there’s no logical reason (again, if one is a materialist) to assume that other people have consciousness. Of course, (most) materialists do believe that other people are conscious. In general, they tend to say that consciousness just is the pattern of electrons–or that it arises naturally from those patterns.
But, unless one is willing to posit some kind of psychic property to electrons specifically, then this is the position that anything which had such a pattern would be conscious: rocks tied to springs, a system of gears and levers, or the nation of China sending each other text messages.
It seems absurd to say that such things would be conscious in the way that a human is, but that is precisely what one means if one is going to say that consciousness is nothing more than a pattern of electrons.
And there remains the fact that, unless we are willing to propose something outside the purview of science, there is no more reason to think that the people we see every day are conscious than a very complex system of rocks whacking one another.
One assumes that others are conscious because one experiences consciousness personally. We all know such a thing exists; we don’t need science to tell us that. As such, we find it very plausible to believe that creatures like us are conscious. This is particularly true given that the alternative is demanding scientific evidence for something that science can’t find in one’s self.
But, if one is willing to accept that other people have consciousness, then one has accepted that there are ways of knowing other than our physical senses. That being the case, it becomes very hard to understand why “show me physical evidence” is a reason to reject any of our non-physical perceptions (such as our moral sense) as valid paths to knowledge.
And this, of course, opens the door to several arguments for theism.