Felix HuberPhD student
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Felix Huber and Simone Severini
Some Ulam's reconstruction problems for quantum states
Provided by 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 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.
Felix Huber, Christopher Eltschka, Jens Siewert and Otfried Gühne
Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity
J. Phys. A: Math. Theor. 51, 175301 (2018), arXiv:1708.06298
A pure multipartite quantum state is called absolutely maximally entangled (AME), if all reductions obtained by tracing out at least half of its parties are maximally mixed. Maximal entanglement is then present across every bipartition. The existence of such states is in many cases unclear. With the help of the weight enumerator machinery known from quantum error correction and the generalized shadow inequalities, we obtain new bounds on the existence of AME states in dimensions larger than two. To complete the treatment on the weight enumerator machinery, the quantum MacWilliams identity is derived in the Bloch representation. Finally, we consider AME states whose subsystems have different local dimensions, and present an example for a $2 \times3 \times 3 \times 3$ system that shows maximal entanglement across every bipartition.
The set of correlations between particles in multipartite quantum systems is larger than those in classical systems. Nevertheless, it is subject to restrictions by the underlying quantum theory. In order to better understand the structure of this set, a possible strategy is to divide all correlations into two components, depending on the question of whether they involve an odd or an even number of particles. For pure multi-qubit states we prove that these two components are inextricably interwoven and often one type of correlations completely determines the other. As an application, we prove that all pure qubit states with an odd number of qubits are uniquely determined among all mixed states by the odd component of the correlations. In addition, our approach leads to invariants under the time evolution with Hamiltonians containing only odd correlations and can simplify entanglement detection.
Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a seven-qubit state whose three-body marginals are all maximally mixed, or equivalently, a pure $((7,1,4))_2$ quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves the seven-particle problem as the last open case concerning maximally entangled states of qubits.
We show that generic pure states (states drawn according to the Haar measure) of four particles of equal internal dimension are uniquely determined among all other pure states by their two-body marginals. In fact, certain subsets of three of the two-body marginals suffice for the characterization. We also discuss generalizations of the statement to pure states of more particles, showing that these are almost always determined among pure states by three of their $(n-2)$-body marginals. Finally, we present special families of symmetric pure four-particle states that share the same two-body marginals and are therefore undetermined. These are four-qubit Dicke states in superposition with generalized GHZ states.
The question whether a given quantum state is a ground or thermal state of a few-body Hamiltonian can be used to characterize the complexity of the state and is important for possible experimental implementations. We provide methods to characterize the states generated by two- and, more generally, k-body Hamiltonians as well as the convex hull of these sets. This leads to new insights into the question which states are uniquely determined by their marginals and to a generalization of the concept of entanglement. Finally, certification methods for quantum simulation can be derived.