Multipartite quantum states that cannot be uniquely determined by their reduced states of all proper subsets of the parties exhibit some inherit `high-order' correlation. This paper elaborates this issue by giving necessary and sufficient conditions for a pure multipartite state to be locally undetermined, and moreover, characterizing precisely all the pure states sharing the same set of reduced states with it. Interestingly, local determinability of pure states is closely related to a generalized notion of Schmidt decomposition. Furthermore, we find that locally undetermined states have some applications to the well-known consensus problem in distributed computing. To be specific, given some physically separated agents, when communication between them, either classical or quantum, is unreliable, then there exists a totally correct and completely fault-tolerant protocol for them to reach a consensus if and only if they share a priori a locally undetermined quantum state.
Multipartite quantum states that cannot be uniquely determined by their reduced states of all proper subsets of the parties exhibit some inherit ‘high-order ’ correlation. This paper elaborates this issue by giving necessary and sufficient conditions for a pure multipartite state to be locally undetermined, and moreover, characterizing precisely all the pure states sharing the same set of reduced states with it. Interestingly, we find that locally undetermined states have some applications to the well-known consensus problem in distributed computation. To be specific, given some physically separated agents, when communication between them, either classical or quantum, is unreliable and they are not allowed to use local ancillary quantum systems, then there exists a totally correct and completely fault-tolerant protocol for them to reach a consensus if and only if they share a priori a locally undetermined quantum state. 1
Multipartite quantum states that cannot be uniquely determined by their reduced states of all proper subsets of the parties exhibit some inherit `high-order' correlation. This paper elaborates this issue by giving necessary and sufficient conditions for a pure multipartite state to be locally undetermined, and moreover, characterizing precisely all the pure states sharing the same set of reduced states with it. Interestingly, local determinability of pure states is closely related to a generalized notion of Schmidt decomposition. Furthermore, we find that locally undetermined states have some applications to the well-known consensus problem in distributed computation. To be specific, given some physically separated agents, when communication between them, either classical or quantum, is unreliable and they are not allowed to use local ancillary quantum systems, then there exists a totally correct and completely fault-tolerant protocol for them to reach a consensus if and only if they share a priori a locally undetermined quantum state.
Graph states form a rich class of entangled states that exhibit important aspects of multi-partite entanglement. At the same time, they can be described by a number of parameters that grows only moderately with the system size. They have a variety of applications in quantum information theory, most prominently as algorithmic resources in the context of the one-way quantum computer, but also in other fields such as quantum error correction and multi-partite quantum communication, as well as in the study of foundational issues such as non-locality and decoherence. In this review, we give a tutorial introduction into the theory of graph states. We introduce various equivalent ways how to define graph states, and discuss the basic notions and properties of these states. The focus of this review is on their entanglement properties. These include aspects of non-locality, bi-partite and multi-partite entanglement and its classification in terms of the Schmidt measure, the distillability properties of mixed entangled states close to a pure graph state, as well as the robustness of their entanglement under decoherence. We review some of the known applications of graph states, as well as proposals for their experimental implementation.
Graph states form a rich class of entangled states that exhibit important aspects of multi-partite entanglement. At the same time, they can be described by a number of parameters that grows only moderately with the system size. They have a variety of applications in quantum information theory, most prominently as algorithmic resources in the context of the one-way quantum computer, but also in other fields such as quantum error correction and multi-partite quantum communication, as well as in the study of foundational issues such as non-locality and decoherence. In this review, we give a tutorial introduction into the theory of graph states. We introduce various equivalent ways how to define graph states, and discuss the basic notions and properties of these states. The focus of this review is on their entanglement properties. These include aspects of non-locality, bi-partite and multi-partite entanglement and its classification in terms of the Schmidt measure, the distillability properties of mixed entangled states close to a pure graph state, as well as the robustness of their entanglement under decoherence. We review some of the known applications of graph states, as well as proposals for their experimental implementation.
At the heart of the curious phenomenon of quantum entanglement lies the relation between the whole and its parts. In my thesis, I explore different aspects of this theme in the multipartite setting by drawing connections to concepts from statistics, graph theory, and quantum error-correcting codes: first, I address the case when joint quantum states are determined by their few-body parts and by Jaynes’ maximum entropy principle. This can be seen as an extension of the notion of entanglement, with less complex states already being determined by their few-body marginals. Second, I address the conditions for certain highly entangled multipartite states to exist. In particular, I present the solution of a long-standing open problem concerning the existence of an absolutely maximally entangled state on seven qubits. This sheds light on the algebraic properties of pure quantum states, and on the conditions that constrain the sharing of entanglement amongst multiple particles. Third, I investigate Ulam’s graph reconstruction problems in the quantum setting, and obtain legitimacy conditions of a set of states to be the reductions of a joint graph state. Lastly, I apply and extend the weight enumerator machinery from quantum error correction to investigate the existence of codes and highly entangled states in higher dimensions. This clarifies the physical interpretation of the weight enumerators and of the quantum MacWilliams identity, leading to novel applications in multipartite entanglement.