We consider the ground state of simple quantum systems coupled to an environment. In general the system is entangled with its environment. As a consequence, even at zero temperature, the energy of the system is not sharp: a projective measurement can find the system in an excited state. We show that energy fluctuation measurements at zero temperature provide entanglement information. For two-state systems which exhibit a persistent current in the ground state, energy fluctuations and persistent current fluctuations are closely related. The harmonic oscillator serves to illustrate energy fluctuations in a system with an infinite number of states. In addition to the energy distribution we discuss the energy-energy time-correlation function in the zero-temperature limit.