Quantifying experimentally created entanglement could in principle be accomplished by measuring the entire density matrix and calculating an entanglement measure of choice thereafter. Due to the tensor structure of the Hilbert space, this approach becomes unfeasible even for medium-sized systems. Here we present methods for quantifying the entanglement of arbitrarily large two-colorable graph states from simple measurements. The presented methods provide non-trivial bounds on the entanglement for any state as long as there is sufficient fidelity with such a graph state. The measurement data considered here is merely given by stabilizer measurements, thus leading to an exponential reduction in the number of measurements required. We provide analytical results for the robustness of entanglement and the relative entropy of entanglement.
Quantifying experimentally created entanglement could in principle be accomplished by measuring the entire density matrix and calculating an entanglement measure of choice thereafter. Due to the tensor structure of the Hilbert space, this approach becomes unfeasible even for medium-sized systems. Here we present methods for quantifying the entanglement of arbitrarily large two-colorable graph states from simple measurements. The presented methods provide non-trivial bounds on the entanglement for any state as long as there is sufficient fidelity with such a graph state. The measurement data considered here is merely given by stabilizer measurements, thus leading to an exponential reduction in the number of measurements required. We provide analytical results for the robustness of entanglement and the relative entropy of entanglement.
We introduce a method for extracting meaningful entanglement measures of tensor network states in general dimensions.Current methods require the explicit reconstruction of the density matrix, which is highly demanding, or the contraction of replicas, which requires an effort exponential in the number of replicas and which is costly in terms of memory.In contrast, our method requires the stochastic sampling of matrix elements of the classically represented reduced states with respect to random states drawn from simple product probability measures constituting frames.Even though not corresponding to physical operations, such matrix elements are straightforward to calculate for tensor network states, and their moments provide the Rényi entropies and negativities as well as their symmetry-resolved components.We test our method on the one-dimensional critical XX chain and the two-dimensional toric code in a checkerboard geometry.Although the cost is exponential in the subsystem size, it is sufficiently moderate so that-in contrast with other approaches-accurate results can be obtained on a personal computer for relatively large subsystem sizes.
Recently, there are tremendous developments on the number of controllable qubits in several quantum computing systems. For these implementations, it is crucial to determine the entanglement structure of the prepared multipartite quantum state as a basis for further information processing tasks. In reality, evaluation of a multipartite state is in general a very challenging task owing to the exponential increase of the Hilbert space with respect to the number of system components. In this work, we propose a systematic method using very few local measurements to detect multipartite entanglement structures based on the graph state --- one of the most important classes of quantum states for quantum information processing. Thanks to the close connection between the Schmidt coefficient and quantum entropy in graph states, we develop a family of efficient witness operators to detect the entanglement between subsystems under any partitions and hence the entanglement intactness. We show that the number of local measurements equals to the chromatic number of the underlying graph, which is a constant number, independent of the number of qubits. In reality, the optimization problem involved in the witnesses can be challenging with large system size. For several widely-used graph states, such as 1-D and 2-D cluster states and the Greenberger-Horne-Zeilinger state, by taking advantage of the area law of entanglement entropy, we derive analytical solutions for the witnesses, which only employ two local measurements. Our method offers a standard tool for entanglement structure detection to benchmark multipartite quantum systems.
Abstract Topological quantum error correcting codes have emerged as leading candidates towards the goal of achieving large-scale fault-tolerant quantum computers. However, quantifying entanglement in these systems of large size in the presence of noise is a challenging task. In this paper, we provide two different prescriptions to characterize noisy stabilizer states, including the surface and the color codes, in terms of localizable entanglement over a subset of qubits. In one approach, we exploit appropriately constructed entanglement witness operators to estimate a witness-based lower bound of localizable entanglement, which is directly accessible in experiments. In the other recipe, we use graph states that are local unitary equivalent to the stabilizer state to determine a computable measurement-based lower bound of localizable entanglement. If used experimentally, this translates to a lower bound of localizable entanglement obtained from single-qubit measurements in specific bases to be performed on the qubits outside the subsystem of interest. Towards computing these lower bounds, we discuss in detail the methodology of obtaining a local unitary equivalent graph state from a stabilizer state, which includes a new and scalable geometric recipe as well as an algebraic method that applies to general stabilizer states of arbitrary size. Moreover, as a crucial step of the latter recipe, we develop a scalable graph-transformation algorithm that creates a link between two specific nodes in a graph using a sequence of local complementation operations. We develop open-source Python packages for these transformations, and illustrate the methodology by applying it to a noisy topological color code, and study how the witness and measurement-based lower bounds of localizable entanglement varies with the distance between the chosen qubits.
Abstract Recently, there are tremendous developments on the number of controllable qubits in several quantum computing systems. For these implementations, it is crucial to determine the entanglement structure of the prepared multipartite quantum state as a basis for further information processing tasks. In reality, evaluation of a multipartite state is in general a very challenging task owing to the exponential increase of the Hilbert space with respect to the number of system components. In this work, we propose a systematic method using very few local measurements to detect multipartite entanglement structures based on the graph state—one of the most important classes of quantum states for quantum information processing. Thanks to the close connection between the Schmidt coefficient and quantum entropy in graph states, we develop a family of efficient witness operators to detect the entanglement between subsystems under any partitions and hence the entanglement intactness. We show that the number of local measurements equals to the chromatic number of the underlying graph, which is a constant number, independent of the number of qubits. In reality, the optimization problem involved in the witnesses can be challenging with large system size. For several widely used graph states, such as 1-D and 2-D cluster states and the Greenberger–Horne–Zeilinger state, by taking advantage of the area law of entanglement entropy, we derive analytical solutions for the witnesses, which only employ two local measurements. Our method offers a standard tool for entanglement-structure detection to benchmark multipartite quantum systems.
Many experiments in quantum information aim at creating graph states. Quantifying the purity of an experimentally achieved graph state could in principle be accomplished using full-state tomography. This method requires a number of measurement settings growing exponentially with the number of constituents involved. Thus, full-state tomography becomes experimentally infeasible even for a moderate number of qubits. In this paper, we present a method to estimate the purity of experimentally achieved graph states with simple measurements. The observables we consider are the stabilizers of the underlying graph. Then, we formulate the problem as: "What is the state with the least purity that is compatible with the measurement data?" We solve this problem analytically and compare the obtained bounds with results from full-state tomography for simulated data.
Many experiments in quantum information aim at creating graph states. Quantifying the purity of an experimentally achieved graph state could in principle be accomplished using full-state tomography. This method requires a number of measurement settings growing exponentially with the number of constituents involved. Thus, full-state tomography becomes experimentally infeasible even for a moderate number of qubits. In this paper we present a method to estimate the purity of experimentally achieved graph states with simple measurements. The observables we consider are the stabilizers of the underlying graph. Then, we formulate the problem as: "What is the state with the least purity that is compatible with the measurement data?" We solve this problem analytically and compare the obtained bounds with results from full-state tomography for simulated data.