Recent quantum thermodynamic resource theories and derivations of the second law in the quantum regime are predicated upon the unitary operation as a basic building block. The unitary itself is usually described by an external observer that manipulates an interaction. Including this control into a fully quantum description, a so-called “quantum clock”, is thus a critical step to placing quantum protocols on a firm footing, especially since due to information gain-disturbance principles, it is impossible to perform these operations perfectly. Here we present a quantum clock that performs a general energy-preserving unitary autonomously with an error that is exponentially small in both the dimension and the energy of the clock. The full quantum setup —system to be controlled plus clock— is described by a time independent Hamiltonian. This is crucial if one desires to understand the full quantum limitations to control, since a time dependent Hamiltonian would require external control, not explicitly accounted for. The main result is to show that this setup with a clock initially in a Gaussian superposition state can implement to any desired precision, any energy preserving unitary on the system during an arbitrarily small time interval with a backreaction on the clock which is exponentially small in both energy and clock dimension. How fast as a function of energy and dimension the error in the back-reaction approaches zero is of paramount importance for understanding resource theories, since if the decay in error is too slow, one would have to invest a lot of work to correcting the error, representing an unaccounted for cost to quantum thermodynamic resource theories. Previous to this work, it was only known that unitaries can be implemented perfectly in the infinite dimensional limit, from which it is impossible to estimate the thermodynamic cost of control. The model we present for the quantum clock is based on a model introduced by Eugene Wigner in General Relativity and later investigated in the non-relativistic regime by Asher Peres. Crucially, we consider a quantum superposition of so-called “clock states”, in contrast to Asher’s study. This is a crucial difference, which due to quantum constructive/destructive interference, leads to a much more accurate clock which can achieve the exponentially small error. In conclusion, our work has implications for the validity of resource theories, and is both a benchmark for future implementations, as well as a conjecture on the fundamental limitations of clocks and control.

Preprint arXiv:1607.04591