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arXiv:2211.01064 · Physical Review A

Localizing genuine multipartite entanglement in noisy stabilizer states

K J Harikrishnan, Amit Kumar Pal

Published Sep 5, 2023• 4 citations
arXivHTMLPDF

Abstract

Characterizing large noisy multiparty quantum states using genuine multipartite entanglement is a challenging task. In this paper, we calculate lower bounds of genuine multipartite entanglement localized over a chosen multiparty subsystem of multiqubit stabilizer states in the noiseless and noisy scenarios. In the absence of noise, adopting a graph-based technique, we perform the calculation for arbitrary graph states as representatives of the stabilizer states and show that the graph operations required for the calculation have a polynomial scaling with the system size. As demonstrations, we compute the localized genuine multipartite entanglement over subsystems of large graphs having linear, ladder, and square structures. We also extend the calculation for graph states subjected to single-qubit Markovian or non-Markovian Pauli noise on all qubits and demonstrate, for a specific lower bound of the localizable genuine multipartite entanglement corresponding to a specific Pauli measurement setup, the existence of a critical noise strength beyond which all of the postmeasured states are biseparable. The calculation is also useful for arbitrary large stabilizer states under noise due to the local unitary connection between stabilizer states and graph states. We demonstrate this by considering a toric code defined on a square lattice and computing a lower bound of localizable genuine multipartite entanglement over a nontrivial loop of the code. Similar to the graph states, we show the existence of the critical noise strength in this case also and discuss its interesting features.

DOI
10.1103/physreva.108.032404

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Characterizing large noisy multiparty quantum states using genuine multipartite entanglement is a challenging task. In this paper, we calculate lower bounds of genuine multipartite entanglement localized over a chosen multiparty subsystem of multiqubit stabilizer states in the noiseless and noisy scenarios. In the absence of noise, adopting a graph-based technique, we perform the calculation for arbitrary graph states as representatives of the stabilizer states and show that the graph operations required for the calculation have a polynomial scaling with the system size. As demonstrations, we compute the localized genuine multipartite entanglement over subsystems of large graphs having linear, ladder, and square structures. We also extend the calculation for graph states subjected to single-qubit Markovian or non-Markovian Pauli noise on all qubits and demonstrate, for a specific lower bound of the localizable genuine multipartite entanglement corresponding to a specific Pauli measurement setup, the existence of a critical noise strength beyond which all of the postmeasured states are biseparable. The calculation is also useful for arbitrary large stabilizer states under noise due to the local unitary connection between stabilizer states and graph states. We demonstrate this by considering a toric code defined on a square lattice and computing a lower bound of localizable genuine multipartite entanglement over a nontrivial loop of the code. Similar to the graph states, we show the existence of the critical noise strength in this case also and discuss its interesting features.
We study the task of localizing multipartite entanglement in pure quantum states onto a subsystem by measuring the remaining systems. To this end, we fix a multipartite entanglement measure and consider two quantities: the multipartite entanglement of assistance (MEA), defined as the entanglement measure averaged over the post-measurement states and maximized over arbitrary measurements; and the localizable multipartite entanglement (LME), defined in the same way but restricted to only local single-system measurements. Both quantities generalize previously considered bipartite entanglement localization measures. In our work we choose the n-tangle, the genuine multipartite entanglement concurrence and the concentratable entanglement (CE) as the underlying seed measure, and discuss the resulting MEA and LME quantities. First, we prove easily computable upper and lower bounds on MEA and LME and establish Lipschitz-continuity for the n-tangle and CE-based LME and MEA. Using these bounds we investigate the typical behavior of entanglement localization by deriving concentration inequalities for the MEA evaluated on Haar-random states and performing numerical studies for small tractable system sizes. We then turn our attention to protocols that transform graph states. We give a simple criterion based on a matrix equation to decided whether states with a specified n-tangle value can be obtained from a given graph state, providing no-go theorems for a broad class of such graph state transformations beyond the usual “local Clifford plus local Pauli measurement” framework. This analysis is generalized to weighted graph states, which provide a realistic error model in current experiments preparing graph state. Our entanglement localization framework certifies the near-optimality of recently discussed local-measurement protocols to transform uniformly weighted line graph states into GHZ states, even when considering arbitrary entangled measurements. Finally, we demonstrate how our MEA and LME quantities can be used to detect critical phenomena such as phase transitions in transversal field Ising models. Since entanglement localization is operationally relevant throughout quantum networking and measurement-based quantum computation, our framework of results based on the MEA and LME has the potential for broad applications in these fields.
Abstract Detecting genuine multipartite entanglement (GME) is a state-characterization task that benchmarks coherence and experimental control in quantum systems. Existing GME tests often require joint measurements on many qubits, posing challenges for systems like time-bin encoded qubits and microwave photons from superconducting circuits, where qubit connectivity is limited or measurement noise grows with the number of jointly measured qubits. Here we introduce versatile GME and k -inseparability criteria applicable to any state, which only require measuring O ( n 2 ) out of 2 n (at most) m -body stabilizers of n -qubit target graph states, with m upper-bounded by twice the graph’s maximum degree. For cluster or ring-graph states, only constant-weight stabilizers are needed. Using semidefinite programming (and sometimes graph-local complementations), we can reduce the number or weight of required stabilizers. Analytical and numerical results show that our criteria are noise-robust and may infer state infidelity from certified k -inseparability in microwave photonic graph states generated under realistic conditions.
Detecting genuine multipartite entanglement (GME) is a state-characterization task that benchmarks coherence and experimental control in quantum systems. Existing GME tests often require joint measurements on many qubits, posing challenges for systems like time-bin encoded qubits and microwave photons from superconducting circuits, where qubit connectivity is limited or measurement noise grows with the number of jointly measured qubits. Here we introduce versatile GME and k-inseparability criteria applicable to any state, which only require measuring O(n2) out of 2n (at most) m-body stabilizers of n-qubit target graph states, with m upper-bounded by twice the graph’s maximum degree. For cluster or ring-graph states, only constant-weight stabilizers are needed. Using semidefinite programming (and sometimes graph-local complementations), we can reduce the number or weight of required stabilizers. Analytical and numerical results show that our criteria are noise-robust and may infer state infidelity from certified k-inseparability in microwave photonic graph states generated under realistic conditions.
Abstract Genuine multipartite entanglement (GME) is arguably the most valuable form of entanglement in the multipartite case with application for instance in quantum metrology. In order to detect that form of entanglement in multipartite quantum states, one typically uses entanglement witnesses. The aim of this paper is to generalize the results of Tóth and Gühne (2005 Phys. Rev. A 72 022340) in order to provide a construction of witnesses of GME tailored to entangled subspaces originating from the multi-qudit stabilizer formalism—a framework well known for its role in quantum error correction, which also provides a very convenient description of a broad class of entangled multipartite states (both pure and mixed). Our construction includes graph states of arbitrary local dimension. We then show that in certain situations, the obtained witnesses detecting GME in quantum systems of higher local dimension are superior in terms of noise robustness to those derived for multiqubit states.
Complete characterization of a noisy multipartite quantum state in terms of entanglement requires full knowledge of how the entanglement content in the state is affected by the spatial distribution of noise in the state. Specifically, we find that if the measurement basis in the protocol of computing localizable entanglement and the basis of the Kraus operator representing the local noisy channel do not commute, the information regarding the noise is retained in the system even after the qubit is traced out after measurement. Using this result and the basic properties of entanglement under noise, we present a set of hierarchies that localizable entanglement over a specific subsystem in a multiqubit state can obey when local noise acts on the subparts or on all the qubits of the whole system. In particular, we propose two types of hierarchies---one tailored according to the number of noisy unmeasured qubits, and the other one that depends additionally on the cardinality of the set of noisy measured qubits, leading to the classification of quantum states. We report the percentage of states satisfying the proposed hierarchies in the case of random three- and four-qubit systems and show, using both analytical methods and numerical simulations, that in almost all the cases, anticipated hierarchies tend to hold with the variation of the strength of noise.
Despite the exponential overhead to describe general multiqubit quantum states and processes, efficient methods for certain state families and operations have been developed and utilized. The stabilizer formalism and the Gottesman-Knill theorem, where pure stabilizer or graph states are manipulated by Clifford operations and Pauli measurements, are prominent examples, and these states play a major role in many applications in quantum technologies. Here we develop a noisy stabilizer formalism, i.e., a method that allows one to efficiently describe and follow not only pure states under Clifford operations and Pauli measurements but also Pauli noise processes acting on such stabilizer states, including uncorrelated and correlated dephasing and single- or multiple-qubit depolarizing noise. The method scales linearly in the number of qubits of the initial state, but exponentially in the size of the target state. Thus, whenever a noisy stabilizer state is manipulated by means of local Pauli measurements such that a multipartite entangled state of a few qubits is generated, one can efficiently describe the resulting state.