Let me start by answering the question honestly. I don't know, so I am not advocating a zero-carb diet as a therapy for cancer. However, I think it is an interesting hypothesis that deserves to be studied.
As background, consumed carbohydrates are readily converted into glucose in the bloodstream, which provides nourishment to all cells in the body. In high fat and high protein diets, where carbohydrates are absent, ketones are produced, which are an alternative fuel. When the percentage of carbohydrates are reduced below some critical level, which varies between individuals, a state of nutritional ketosis results (not to be confused with the dangerous condition in diabetics).
As an example, I am particularly sensitive to carbohydrates so to maintain ketosis, I try to keep my carbohydrate intake as low as possible, which is around 5% to 10% of the total calories I consume. Protein also can interfere with ketosis, so in the case of my diet, I average about 70% fat, 22% protein and 8% carbohydrates. How did I determine these ratios? I used urine strips and a blood ketosis device to monitor my ketone levels at differing proportions of these three macro nutrients. Incidentally, the Atkins diet is incorrectly called a high protein diet but is in fact a high fat diet. Eating large amounts of protein can be dangerous, so when decreasing carbohydrates, fat consumption must increase.
The cultural preoccupation with fat as being bad is undoubtedly the reason why the Atkins diet stressed protein. Since the purpose of this post is not to discus this complex issue, I will not give an analysis here but will merely point out that high fat intake is dangerous when combined with high carbohydrate consumption, not when carbohydrates are eliminated.
It takes the body about two to three weeks to acclimate to a ketogenic diet. After this period, the body readily converts consumed fat into ketones and efficiently uses ketones as an energy source. The body then also more readily converts fat in cells to ketones, an explanation of why ketogenic diets lead to weight loss. Ketogenic diets are starting to be used by high-endurance athletes for energy and reducing bonking, where the brain shuts down due to a short supply of glucose. In contrast, fat cells in an athlete in nutritional ketosis more readily provide ketones that powers the brain and other organs.
Since cancer cells ravenously consume glucose to grow and spread, reducing this fuel may weaken them. If cancer cells have a preference for glucose over ketones, and since the body seems to function efficiently on ketones in acclimated individuals, a ketogenic diet may add an extra punch to traditional chemotherapy or radiation therapy.
A Phase I study is in progress to test whether or not a Ketogenic Diet with concurrent chemoradiation for non-small cell lung cancer improves longevity.
On a related topic, it is well known that many animals live substantially longer on a very low calorie diet. Controlled studies of such diets in humans are not possible; how many people would volunteer to be at the edge of starvation for a lifetime, and how could cheating be monitored? These are issues that make any long-term observational studied of populations flawed and impractical.
A hypothesis for this observation is that glucose - the mitochondrial fuel source, is responsible for shortening life spans. If this were so, replacing glucose with ketones as an alternative fuel might do the trick.
Cynthia Kenyon of UC San Fransisco has been studying the effects of diets in tiny worms. She found that worms on a diet devoid of sugar can outlive their sweet-tooth brethren by a factor of two, a dramatic effect that has a sound genetic basis; sugar is part of a biochemical pathway that interferes with cell repair. These results may not carry over to humans, but it is an interesting hypothesis that also deserves a test.
After discovering the affect of sugar on worm longevity, Kenyon herself went on a low glycemic index diet, totally avoiding sugar accept for those in dark chocolates. As a scientist, she does not advocate such diets.
Given that a zero-carb diet maximizes ketosis, which might starve cancer cells and turn on the cell-repair mechanism, I would have a serious discussion with my physician about ketongenic diets if I or someone close to me were struggling with cancer.