Efficient and feasible state tomography of quantum many-body systems

We present a novel method to perform quantum state tomography for many-particle systems which are particularly suitable for estimating states in lattice systems such as of ultra-cold atoms in optical lattices. We show that the need for measuring a tomographically complete set of observables can be overcome by letting the state evolve under some suitably chosen random circuits followed by the measurement of a single observable. We generalize known results about the approximation of unitary 2-designs, i.e., certain classes of random unitary matrices, by random quantum circuits and connect our findings to the theory of quantum compressed sensing. We show that for ultra-cold atoms in optical lattices established techniques like optical super-lattices, laser speckles, and time-of-flight measurements are sufficient to perform fully certified, assumption-free tomography. Combining our approach with tensor network methods - in particular the theory of matrix-product states - we identify situations where the effort of reconstruction is even constant in the number of lattice sites, allowing in principle to perform tomography on large-scale systems readily available in present experiments.Comment: 10 pages, 3 figures, minor corrections, discussion added, emphasizing that no single-site addressing is needed at any stage of the scheme when implemented in optical lattice system

A Generalization of Quantum Stein's Lemma

We present a generalization of quantum Stein's Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d.. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein's Lemma, in terms of the quantum relative entropy. Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition.Comment: 30 pages. (see posting by M. Piani arXiv:0904.2705 for a different proof of the strict positiveness of the regularized relative entropy of entanglement on every entangled state). published version