Weak values and multivariate traces: Structure and efficient estimation

Weak values of quantum observables are a powerful tool for investigating quantum phenomena. Some methods for measuring weak values in the laboratory require weak interactions and postselection, while others are deterministic, but require statistics over a number of experiments that grows linearly with the dimension of the measured system in the worst case over all possible observables. Here we propose a deterministic dimension-independent scheme for estimating weak values of arbitrary observables. The scheme is based on controlled SWAP operations, and associates states and observables in the mathematical expression of the weak value to preparations devices and measurements devices in the experimental setup, respectively. Thanks to this feature, it provides insights into the relation between states of two identical quantum systems at a single moment of time and states of a single quantum system at two moments of time, also known as two-time states. Specifically, our scheme provides an alternative expression for two-time states, and establishes a link between two-time states accessible through the controlled-SWAP scheme and bipartite quantum states with positive partial transpose. Furthermore, we discuss its generalizations applicable to a situation where N systems are given and unknown, and classical information on M systems (M < N) is available, allowing estimation of multivariate traces of order N+M. The use of classical information on some of the states enables circuits on fewer qubits and with fewer gates, decreasing the experimental requirements for their estimation.