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Register a new userDegrees of entanglement for multipartite systems
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We propose a unified mathematical scheme, based on a classical tensor isomorphism, for characterizing entanglement that works for pure states of multipartite systems of any number of particles. The degree of entanglement is indicated by a set of absolute values of the determinants for each subspace of the multipartite systems. Unlike other schemes, our scheme provides indication of the degrees of entanglement when the qubits are measured or lost successively, and leads naturally to the necessary and sufficient conditions for multipartite pure states to be separable. For systems with a large number of particles, a rougher indication of the degree of entanglement is provided by the set of mean values of the determinantal values for each subspace of the multipartite systems.
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Quantum entanglement between an arbitrary number of remote qubits is examined analytically. We show that there is a non-probabilistic way to address in one context the management of entanglement of an arbitrary number of mixed-state qubits by engaging quantitative measures of entanglement and a specific external control mechanism. Both all-party entanglement and weak inseparability are considered. We show that for $Nge4$, the death of all-party entanglement is permanent after an initial collapse. In contrast, weak inseparability can be deterministically managed for an arbitrarily large number of qubits almost indefinitely. Our result suggests a picture of the path that the system traverses in the Hilbert space.
The standard definition of genuine multipartite entanglement stems from the need to assess the quantum control over an ever-growing number of quantum systems. We argue that this notion is easy to hack: in fact, a source capable of distributing bipartite entanglement can, by itself, generate genuine $k$-partite entangled states for any $k$. We propose an alternative definition for genuine multipartite entanglement, whereby a quantum state is genuinely network $k$-entangled if it cannot be produced by applying local trace-preserving maps over several $k$-partite states distributed among the parties, even with the aid of global shared randomness. We provide analytic and numerical witnesses of genuine network entanglement, and we reinterpret many past quantum experiments as demonstrations of this feature.
In recent papers, the theory of representations of finite groups has been proposed to analyzing the violation of Bell inequalities. In this paper, we apply this method to more complicated cases. For two partite system, Alice and Bob each make one of $d$ possible measurements, each measurement has $n$ outcomes. The Bell inequalities based on the choice of two orbits are derived. The classical bound is only dependent on the number of measurements $d$, but the quantum bound is dependent both on $n$ and $d$. Even so, when $d$ is large enough, the quantum bound is only dependent on $d$. The subset of probabilities for four parties based on the choice of six orbits under group action is derived and its violation is described. Restricting the six orbits to three parties by forgetting the last party, and guaranteeing the classical bound invariant, the Bell inequality based on the choice of four orbits is derived. Moreover, all the corresponding nonlocal games are analyzed.
Beyond the simplest case of bipartite qubits, the composite Hilbert space of multipartite systems is largely unexplored. In order to explore such systems, it is important to derive analytic expressions for parameters which characterize the systems state space. Two such parameters are the degree of genuine multipartite entanglement and the degree of mixedness of the systems state. We explore these two parameters for an N-qubit system whose density matrix has an X form. We derive the class of states that has the maximum amount of genuine multipartite entanglement for a given amount of mixedness. We compare our results with the existing results for the N=2 case. The critical amount of mixedness above which no N-qubit X-state possesses genuine multipartite entanglement is derived. It is found that as N increases, states with higher mixedness can still be entangled.
Distribution and distillation of entanglement over quantum networks is a basic task for Quantum Internet applications. A fundamental question is then to determine the ultimate performance of entanglement distribution over a given network. Although this question has been extensively explored for bipartite entanglement-distribution scenarios, less is known about multipartite entanglement distribution. Here we establish the fundamental limit of distributing multipartite entanglement, in the form of GHZ states, over a quantum network. In particular, we determine the multipartite entanglement distribution capacity of a quantum network, in which the nodes are connected through lossy bosonic quantum channels. This setting corresponds to a practical quantum network consisting of optical links. The result is also applicable to the distribution of multipartite secret key, known as common key, for both a fully quantum network and trusted-node based quantum key distribution network. Our results set a general benchmark for designing a network topology and network quantum repeaters (or key relay in trusted nodes) to realize efficient GHZ state/common key distribution in both fully quantum and trusted-node-based networks. We show an example of how to overcome this limit by introducing a network quantum repeater. Our result follows from an upper bound on distillable GHZ entanglement introduced here, called the recursive-cut-and-merge bound, which constitutes major progress on a longstanding fundamental problem in multipartite entanglement theory. This bound allows for determining the distillable GHZ entanglement for a class of states consisting of products of bipartite pure states.
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