Recently, a technique known as quantum symmetry test has gained increasing attention for detecting bipartite entanglement in pure quantum states. In this work we show that, beyond qualitative detection, a family of well-defined measures of bipartite and multipartite entanglement can be obtained with symmetry tests. We propose and benchmark several efficient methods to estimate these measures, and derive near-optimal sampling strategies for each. Despite the nonlinearity of the methods, we demonstrate that the sampling error scales no worse than $O(N_{\mathrm{tot}}^{-1/2})$ with the total number of copies $N_{\mathrm{tot}}$, which suggests experimental feasibility. By exploiting symmetries we compute our measures for large number of copies, and derive the asymptotic decay exponents for relevant states in many-body systems. Using these results we identify tradeoffs between estimation complexity and sensitivity of the presented entanglement measures, oriented to practical implementations.
Recently, a technique known as quantum symmetry test has gained increasing attention for detecting bipartite entanglement in pure quantum states. In this work we show that, beyond qualitative detection, a family of well-defined measures of bipartite and multipartite entanglement can be obtained with symmetry tests. We propose and benchmark several efficient methods to estimate these measures, and derive near-optimal sampling strategies for each. Despite the nonlinearity of the methods, we demonstrate that the sampling error scales no worse than O(Ntot−1/2) with the total number of copies Ntot, which suggests experimental feasibility. By exploiting symmetries we compute our measures for large number of copies, and derive the asymptotic decay exponents for relevant states in many-body systems. Using these results we identify tradeoffs between estimation complexity and sensitivity of the presented entanglement measures, oriented to practical implementations.
We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, m-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.
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.
Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
The focal point of this thesis is strongly based on quantum entanglement within Quantum Information Science. We provide an indepth analysis of entanglement quantification in both bipartite and multipartite quantum systems, with an emphasis on many-body entanglement. We examine the definitions of entanglement, introduce several entanglement measures, and examine the challenges in expanding these measures to multipartite scenarios. For bipartite systems, we detail entropy-based measures such as von Neumann and Rényi entropies, the entanglement spectrum and important algebraic metrics such as negativity and concurrence. We then elaborate on multipartite entanglement, introducing modern measures and detection techniques including geometric and global entanglement measurements, generalised multipartite concurrence, multipartite negativity, fidelity-based entanglement benchmarks, and entanglement witnesses. Some examples include Greenberger-Horne-Zeilinger (GHZ), and the W states to illustrate the distinction between biseparable and truly multi-partite-entangled states, as well as the SLOCC classification of families of entangled states. We also briefly introduce practical ways of measuring entanglement, such as the SWAP test, quantum state tomography for experimental estimation of entanglement and consequently randomised measurements. We apply these measures to specific quantum states (GHZ, W , graph states, and random states), analysing how entanglement evolves under operations and how different metrics capture distinct features of multi-qubit entanglement. Finally, we conclude with a summary of findings, a discussion of current limitations such as scalability of measures. The thesis is organised into five chapters: Introduction, Entanglement, Multipartite Entanglement Measures, Negativity vs. Three-Tangle, and Conclusion.
We show how to estimate a broad class of multipartite entanglement measures from Bell-basis measurement data. In addition to lowering the experimental requirements relative to previously known methods of estimating these measures, our proposed scheme also enables a simpler analysis of the number of measurement repetitions required to achieve an $\ensuremath{\epsilon}$-close approximation of the measures, which we provide for each. We focus our analysis on the recently introduced concentratable entanglements [Beckey et al., Phys. Rev. Lett. 127, 140501 (2021)] because many other well-known multipartite entanglement measures are recovered as special cases of this family of measures. We extend the definition of the concentratable entanglements to mixed states and show how to construct lower bounds on the mixed state concentratable entanglements that can also be estimated using only Bell-basis measurement data. Finally, we demonstrate the feasibility of our methods by classically simulating their implementation on a noisy Rydberg atom quantum computer.
Genuine multipartite entanglement (GME) is a valuable resource for quantum technologies, but quantifying it in many-body systems can be demanding since measures defined through optimizations over bipartitions are often assumed to require large joint measurements and many measurement settings. We introduce the concepts of GME evaluation length and GME evaluation number, which capture, respectively, the largest subsystem that must be jointly measured and the number of distinct subsystems that need to be measured to evaluate multipartite entanglement measures defined via bipartition optimizations. Taking the genuine multipartite concurrence (GMC) as a canonical example, we show that, for N-qubit Haar-random pure states, the GMC can be evaluated using only N single-qubit purity measurements. Thus, for typical many-body states, evaluating the GMC requires considering only the N single-site bipartitions, rather than the exponentially many distinct bipartitions. We further corroborate this behavior numerically for physically accessible states generated by local random quantum circuits. Our results show that the practical cost of quantifying GME can be dramatically lower than suggested by its formal definition, establishing measurement complexity as a complementary operational lens for multipartite entanglement.