  In quantum mechanics, predictions are made by way of calculating expectation
values of observables, which take the form of Hermitian operators. It is far
less common to exploit non-Hermitian operators to perform measurements. 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 the 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 characterising a 100 000-dimensional entangled state. This
represents an improvement of two orders of magnitude with respect to previous
characterisations of discrete entangled states. Furthermore, in numerical
studies, we consider mixed quantum states and show that for purities greater
that 0.81, we can reliably extract the most significant eigenvector of the
density matrix with a probability greater than 99%. We anticipate that our
method will prove to be a useful asset in the quest for understanding and
manipulating large-scale quantum systems.
