Some Ulam's reconstruction problems for quantum states

Provided a complete set of putative $k$-body reductions of a multipartite quantum state, can one determine if a joint state exists? We derive necessary conditions for this to be true. In contrast to what is known as the quantum marginal problem, we consider a setting where the labeling of the subsystems is unknown. The problem can be seen in analogy to Ulam's reconstruction conjecture in graph theory. The conjecture - still unsolved - claims that every graph on at least three vertices can uniquely be reconstructed from the set of its vertex-deleted subgraphs. When considering quantum states, we demonstrate that the non-existence of joint states can, in some cases, already be inferred from a set of marginals having the size of just more than half of the parties. We apply these methods to graph states, where many constraints can be evaluated by knowing the number of stabilizer elements of certain weights that appear in the reductions. This perspective links with constraints that were derived in the context of quantum error-correcting codes and polynomial invariants. Some of these constraints can be interpreted as monogamy-like relations that limit the correlations arising from quantum states. Lastly, we provide an answer to Ulam's reconstruction problem for generic quantum states.Comment: 22 pages, 3 figures, v2: significantly revised final versio

Positive maps and trace polynomials from the symmetric group

With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials $X_{\alpha_1} \cdots X_{\alpha_r}$ and their traces $\operatorname{tr}(X_{\alpha_1} \cdots X_{\alpha_r})$. Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.Comment: 28 pages, 3 figures, 2 tables. Extensively rewritten: asymmetric maps, proof for Motzkin matrix polynomial, and connections to QIT added. Comments welcome

FPGA-based operational concept and payload data processing for the Flying Laptop satellite

Flying Laptop is the first small satellite developed by the Institute of Space Systems at the Universität Stuttgart. It is a test bed for an on-board computer with a reconfigurable, redundant and self-controlling high computational ability based on the field pro- grammable gate arrays (FPGAs). This Technical Note presents the operational concept and the on-board payload data processing of the satellite. The designed operational concept of Flying Laptop enables the achievement of mission goals such as technical demonstration, scientific Earth observation, and the payload data processing methods. All these capabilities expand its scientific usage and enable new possibilities for real-time applications. Its hierarchical architecture of the operational modes of subsys- tems and modules are developed in a state-machine diagram and tested by means of MathWorks Simulink-/Stateflow Toolbox. Furthermore, the concept of the on-board payload data processing and its implementation and possible applications are described

Entanglement detection with trace polynomials

We provide a systematic method for nonlinear entanglement detection based on trace polynomial inequalities. In particular, this allows to employ multi-partite witnesses for the detection of bipartite states, and vice versa. We identify witnesses for which linear detection of an entangled state fails, but for which nonlinear detection succeeds. With the trace polynomial formulation a great variety of witnesses arise from immamant inequalities, which can be implemented in the laboratory through randomized measurements