Some of the most intriguing problems in solid-state physics arise when the motion of one electron dramatically affects the motion of surrounding electrons. Traditionally, such highly correlated electron systems have been studied mainly in materials with complex transition metal chemistry1, 2. Over the past decade, researchers have learned to confine one or a few electrons within a nanometre-scale semiconductor 'artificial atom', and to understand and control this simple system in detail3. Here we combine artificial atoms to create a highly correlated electron system within a nano-engineered semiconductor structure3. We tune the system in situ through a quantum phase transition between two distinct states, each a version of the Kondo state4, in which a bound electron interacts with surrounding mobile electrons. The boundary between these competing Kondo states is a quantum critical point—namely, the exotic and previously elusive two-channel Kondo state5, 6, in which electrons in two reservoirs are entangled through their interaction with a single localized spin.