Here's Nobel prize winning economist Amartya Sen to explain how communism has indeed "worked", using the example of China's impressive development (from Hunger and Public Action: Studies in Development Economics):

"Because of its radical commitment to the elimination of poverty and to improving living conditions — a commitment in which Maoist as well as Marxist ideas and ideals played an important part — China did achieve many things that the Indian leadership failed to press for and pursue with any vigor. The elimination of widespread hunger, illiteracy, and ill health falls solidly in this category. When state action operates in the right direction, the results can be quite remarkable, as is illustrated by the social achievements of the pre-reform period." —Amartya Sen, [30]

Another important comment summarizing the findings of the study is as follows:

"We argue, in particular, that the accomplishments relating to education, healthcare, land reforms, and social change in the pre-reform period made significantly positive contributions to the achievements of the post-reform period. This is so not only in terms of their role in sustained high life expectancy and related achievements, but also in providing firm support for economic expansion based on market reforms."
—Amartya Sen, [30]

Professor Sen states here that the prereform period saw enormous increases in quality of life for the Chinese people, as well as important economic developments, without which the economic expansion following the 1979 market reforms most likely could not have taken place. He notes that during this period, a ‘remarkable reduction in chronic undernourishment took place,’ attributing this to the policies implemented by the people’s republic:

"The casual processes through which the reduction of undernourishment was achieved involved extensive state action including redistributive policies, nutritional support, and of course health care (since undernourishment is frequently caused by parasitic diseases and other illnesses)."
—Amartya Sen, [30]

Professor Sen focuses more attention on the remarkable advances in healthcare during this period:

"China’s achievements in the field of health during the pre-reform period include a dramatic reduction of infant and child mortality and a remarkable expansion of longevity."
—Amartya Sen, [30]

It is also noted that the PRC’s life expectancy approximately doubled from approximately 35 years in 1949 to 68 years in 1981 (when the market reforms began to take effect).[31] Another source on public health in the short twentieth century PRC comes from the journal Population Studies, in a study conducted by researchers from Stanford University and the National Bureau for Economic Research. One important comment is as follows:

"China’s growth in life expectancy at birth from 35–40 years in 1949 to 65.5 years in 1980 is among the most rapid sustained increases in documented global history."
—Babiarz KS, Eggleston K, Miller G, Zhang Q, [32]

This alone goes to show the massive benefits that the revolutionary socialist movement attained. More important information is provided in the study, dealing with hospital and medical resources:

"Physician and hospital supply grew dramatically under Mao due to a variety of factors (including increases in government financing, the introduction of social insurance for urban public employees, and the launch of China’s Rural Cooperative Medical System in the mid-1950s). Rural Cooperative Medical Schemes (CMS) were vigorously promoted and became widespread in the late 1960s as part of the Cultural Revolution."
—Babiarz KS, Eggleston K, Miller G, Zhang Q, [32]

They also quote other research which found that the rapid gains in Chinese healthcare can be attributed to the specific policies that the communists implemented:

"China’s mortality decline between 1953 and 1957, which resembles that of the US between 1900 and 1930, was “primarily due to the unique social organisation of Chinese public health practices.”
—Babiarz KS, Eggleston K, Miller G, Zhang Q, [32]

Note that China achieved in four years what Imperial America took three decades to accomplish, due to their differing systems (i.e. planned economics vs. market economics). The study also confirms the immense success of the PRC’s vaccination programs:

"Systematic efforts to vaccinate the population against polio, measles, diphtheria, whooping cough, scarlet fever, and cholera were rapid and reputedly successful (China nearly eradicated smallpox within the span of only three years, with the last documented cases occurring in Tibet and Yunnan in 1960)."
—Babiarz KS, Eggleston K, Miller G, Zhang Q, [32]

Additional citations for the claims in the above quotes are provided in the original study.

"India seems to manage to fill its cupboard with more skeletons every eight years than China put there in its years of shame." - Amartya Sen, Hunger and Public Action: Studies in Development Economics