Reconstructing the quantum state of oscillator networks with a single qubit

We introduce a scheme to reconstruct arbitrary states of networks composed of quantum oscillators--e.g., the motional state of trapped ions or the radiation state of coupled cavities. The scheme uses minimal resources, in the sense that it i) requires only the interaction between one-qubit probe and one constituent of the network; ii) provides the reconstructed state directly from the data, avoiding any tomographic transformation; iii) involves the tuning of only one coupling parameter. In addition, we show that a number of quantum properties can be extracted without full reconstruction of the state. The scheme can be used for probing quantum simulations of anharmonic many-body systems and quantum computations with continuous variables. Experimental implementation with trapped ions is also discussed and shown to be within reach of current technology.