TY  - JOUR
TI  - Some Ulam's reconstruction problems for quantum states
UR  - https://doi.org/10.1088/1751-8121/aadd1e
PB  - IOP PUBLISHING LTD
A1  - Huber, F
A1  - Severini, S
KW  - Science & Technology
KW  -  Physical Sciences
KW  -  Physics
KW  -  Multidisciplinary
KW  -  Physics
KW  -  Mathematical
KW  -  Physics
KW  -  quantum marginal problem
KW  -  graph states
KW  -  Ulam reconstruction problem
KW  -  legitimate deck problem
KW  -  CODES
KW  -  GRAPH
KW  -  INVARIANTS
KW  -  CONJECTURE
IS  - 43
EP  - 22
SN  - 1751-8121
AV  - public
N1  - This version is the author accepted manuscript. For information on re-use, please refer to the publisher?s terms and conditions.
JF  - Journal of Physics A: Mathematical and Theoretical
Y1  - 2018/10/26/
N2  - 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.
VL  - 51
ID  - discovery10068305
ER  -