title: Direct measurement of large-scale quantum states via expectation values of non-Hermitian matrices.
abstract: In quantum mechanics, predictions are made by way of calculating expectation values of observables, which take the form of Hermitian operators. Non-Hermitian operators, however, are not necessarily devoid of physical significance, and they can play a crucial role in the characterization of quantum states. Here we show that the expectation values of a particular set of non-Hermitian matrices, which we call column operators, directly yield the complex coefficients of a quantum state vector. We provide a definition of the state vector in terms of measurable quantities by decomposing these column operators into observables. The technique we propose renders very-large-scale quantum states significantly more accessible in the laboratory, as we demonstrate by experimentally characterizing a 100,000-dimensional entangled state. This represents an improvement of two orders of magnitude with respect to previous phase-and-amplitude characterizations of discrete entangled states.
authors.affiliation: Institute of Photonics and Quantum Sciences, School of Engineering &Physical Sciences, Heriot-Watt University, David Brewster Building, Edinburgh EH14 4AS, UK.
authors.forename: Eliot
authors.initials: E
authors.lastname: Bolduc
authors.affiliation: Institute of Photonics and Quantum Sciences, School of Engineering &Physical Sciences, Heriot-Watt University, David Brewster Building, Edinburgh EH14 4AS, UK.
authors.forename: Genevieve
authors.initials: G
authors.lastname: Gariepy
authors.affiliation: Institute of Photonics and Quantum Sciences, School of Engineering &Physical Sciences, Heriot-Watt University, David Brewster Building, Edinburgh EH14 4AS, UK.
authors.forename: Jonathan
authors.initials: J
authors.lastname: Leach