In this work, we study the asymptotic behavior of protocols that localize entanglement in large multi-qubit states onto a subset of qubits by measuring the remaining qubits. We use the maximal average n-tangle that can be generated on a fixed subsystem by measuring its complement -- either with local or global measurements -- as our key figure of merit. These quantities are known respectively as the localizable entanglement (LE) and the entanglement of assistance (EA). We build upon the work of [arXiv:2411.04080] that proposed a polynomial-time test, based on the EA, for whether it is possible to transform certain graph states into others using local measurements. We show, using properties of the EA, that this test is effective and useful in large systems for a wide range of sizes of the measured subsystem. In particular, we use this test to demonstrate the surprising result that general local unitaries and global measurements will typically not provide an advantage over the more experimentally feasible local Clifford unitaries and local Pauli measurements in transforming large linear cluster states into GHZ states. Finally, we derive concentration inequalities for the LE and EA over Haar-random states which indicate that the localized entanglement structure has a striking dependence on the locality of the measurement. In deriving these concentration inequalities, we develop several technical tools that may be of independent interest.
In this work, we study the asymptotic behavior of protocols that localize entanglement in large multi-qubit states onto a subset of qubits by measuring the remaining qubits. We use the maximal average n-tangle that can be generated on a fixed subsystem by measuring its complement -- either with local or global measurements -- as our key figure of merit. These quantities are known respectively as the localizable entanglement (LE) and the entanglement of assistance (EA). We build upon the work of [arXiv:2411.04080] that proposed a polynomial-time test, based on the EA, for whether it is possible to transform certain graph states into others using local measurements. We show, using properties of the EA, that this test is effective and useful in large systems for a wide range of sizes of the measured subsystem. In particular, we use this test to demonstrate the surprising result that general local unitaries and global measurements will typically not provide an advantage over the more experimentally feasible local Clifford unitaries and local Pauli measurements in transforming large linear cluster states into GHZ states. Finally, we derive concentration inequalities for the LE and EA over Haar-random states which indicate that the localized entanglement structure has a striking dependence on the locality of the measurement. In deriving these concentration inequalities, we develop several technical tools that may be of independent interest.
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.
Computing localizable entanglement for noisy many-particle quantum states is difficult due to the optimization over all possible sets of local projection measurements. Therefore, it is crucial to develop lower bounds, which can provide useful information about the behaviour of localizable entanglement, and which can be determined by measuring a limited number of operators, or by performing the least number of measurements on the state, preferably without performing a full state tomography. In this paper, we adopt two different yet related approaches to obtain a witness-based, and a measurement-based lower bounds for localizable entanglement. The former is determined by the minimal amount of entanglement that can be present in a subsystem of the multipartite quantum state, which is consistent with the expectation value of an entanglement witness. Determining this bound does not require any information about the state beyond the expectation value of the witness operator, which renders this approach highly practical in experiments. The latter bound of localizable entanglement is computed by restricting the local projection measurements over the qubits outside the subsystem of interest to a suitably chosen basis. We discuss the behaviour of both lower bounds against local physical noise on the qubits, and discuss their dependence on noise strength and system size. We also analytically determine the measurement-based lower bound in the case of graph states under local uncorrelated Pauli noise.
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.
Characterizing increasingly complex quantum systems is a central task in quantum information science, yet experimental costs often scale prohibitively with system size. Certifying key properties using simple local measurements is highly desirable but challenging. In this work, we introduce a highly general certification framework based on a physical phenomenon that we call localizable quantumness: for generic many-body states, essential quantum properties are robustly preserved within the projected ensembles on small subsystems after performing local projective measurements on the rest of the system. Leveraging this insight, we develop protocols to certify global properties—including multipartite entanglement, circuit complexity, and quantum magic—by witnessing them on a small, accessible subsystem. Remarkably, randomizing the local measurement bases extends this capability to certify state fidelity. Relying solely on local Pauli measurements, these protocols achieve constant sample complexity and robustness for almost all quantum states, including a wide range of physically relevant states. For certifying the fidelity of n -qubit states, this O ( 1 ) scaling dramatically improves upon state-of-the-art protocols requiring O ( n 4 ) samples. Our unified framework provides both a practical toolkit for large-scale quantum certification and a novel lens into complex many-body systems.
Quantum entanglement is the ability of joint quantum systems to possess global properties (correlation among systems) even when subsystems have no definite individual property. Whilst the 2-dimensional (qubit) case is well-understood, currently, tools to characterise entanglement in high dimensions are limited. We experimentally demonstrate a new procedure for entanglement certification that is suitable for large systems, based entirely on information-theoretics. It scales more efficiently than Bell's inequality and entanglement witness. The method we developed works for arbitrarily large system dimension d and employs only two local measurements of complementary properties. This procedure can also certify whether the system is maximally entangled. We illustrate the protocol for families of bipartite states of qudits with dimension up to 32 composed of polarisation-entangled photon pairs.
Localizability of entanglement in fully inseparable states is a key ingredient of assisted quantum information protocols as well as measurement-based models of quantum computing. We investigate the existence of fully inseparable states with nonlocalizable entanglement, that is, with entanglement which cannot be localized between any pair of subsystems by any measurement on the remaining part of the system. It is shown, that the nonlocalizable entanglement occurs already in suitable mixtures of a three-qubit GHZ state and white noise. Further, we generalize this set of states to a two-parametric family of fully inseparable three-qubit states with nonlocalizable entanglement. Finally, we demonstrate experimentally the existence of nonlocalizable entanglement by preparing and characterizing one state from the family using correlated single photons and linear optical circuit.