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arXiv:2311.00805

A Witness of GHZ Entanglement Using Only Collective Spin Measurements

Lin Htoo Zaw, Khoi-Nguyen Huynh-Vu, Valerio Scarani

Published Nov 1, 2023• 0 citations
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Abstract

Of existing entanglement witnesses that utilize only collective measurements of a spin ensemble, not all can detect genuine multipartite entanglement (GME), and none can detect Greenberger-Horne-Zeilinger (GHZ) states beyond the tripartite case. We fill this gap by introducing an entanglement witness that detects GME of spin ensembles, whose total spin is half-integer, using only collective spin measurements. Our witness is based on a nonclassicality test introduced by Tsirelson, and solely requires the measurement of total angular momentum along different directions. States detected by our witness are close to a family of GHZ-like states, which includes GHZ states of an odd number of spin-half particles. We also study the robustness of our witness under depolarizing noise, and derive exact noise bounds for detecting noisy GHZ states.

DOI
10.48550/arxiv.2311.00805

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Of existing entanglement witnesses that utilize only collective measurements of a spin ensemble, not all can detect genuine multipartite entanglement (GME), and none can detect Greenberger-Horne-Zeilinger (GHZ) states beyond the tripartite case. We fill this gap by introducing an entanglement witness that detects GME of spin ensembles, whose total spin is half-integer, using only collective spin measurements. Our witness is based on a nonclassicality test introduced by Tsirelson, and solely requires the measurement of total angular momentum along different directions. States detected by our witness are close to a family of GHZ-like states, which includes GHZ states of an odd number of spin-half particles. We also study the robustness of our witness under depolarizing noise, and derive exact noise bounds for detecting noisy GHZ states.
We introduce entanglement witnesses for spin ensembles which detect genuine multipartite entanglement using only measurements of the total angular momentum. States that are missed by most other angular-momentum-based witnesses for spin ensembles, which include Greenberger-Horne-Zeilinger states and certain superpositions of Dicke states, can be effectively detected by our witness. The protocol involves estimating the probability that the total angular momentum is positive along equally spaced directions on a plane. Alternatively, one could measure along a single direction at different times, under the assumption that the total spins undergo a uniform precession. Genuine multipartite entanglement is detected when the observed score exceeds a separable bound. Exact analytical expressions for the separable bound are obtained for spin ensembles ${j}_{1}\ensuremath{\bigotimes}{j}_{2}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{j}_{N}$ such that the total spin is a half-integer and numerical results are reported for the other cases. Finally, we conjecture an expression for the separable bound when the total spin is not known, which is well supported by the numerical results.
We construct an entanglement witness for many-qubit systems, based on symmetric two-body correlations with two measurement settings. This witness is able to detect the entanglement of some Dicke states for any number of particles and such detection exhibits some robustness against white noise and thermal noise under the Lipkin-Meshkov-Glick Hamiltonian. In addition, it detects the entanglement of spin-squeezed states, with a detection strength that approaches the maximal value for sufficiently large numbers of particles. As spin-squeezed states can be experimentally generated, the properties of the witness with respect to these states may be amenable to experimental investigation. We show that while the witness is unable to detect Greenberger-Horne-Zeilinger (GHZ) states, it is instead able to detect superpositions of Dicke states with GHZ states.
Abstract This chapter takes a closer look at entanglement in multipartite systems. An examination of pure tripartite systems serves a starting point from which we discover the equivalence classes of Greenberger-Horne-Zeilinger (GHZ) states and W states. We then continue with the mixed-state case and discuss the notions of full separability, partition-separability and biseparability versus genuine tripartite entanglement, before stating the GHZ theorem in the formulation following Mermin. For systems of three or more parties we then formulate the definitions for k-separability and genuine multipartite entanglement (GME), as well as for k-producibility and entanglement depth, followed by prominent examples for GME states such as the generalized GHZ states and Dicke states. We give an overview of the problem of detecting GME using various linear and non-linear GME witnesses, lifted witnesses, PPT mixers, before discussing challenges for the characterization and quantification of multipartite entanglement. We close by analysing the phenomenon of entangled entanglement.
Characterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.
Entanglement is a key resource for quantum information processing. A widely used tool for detecting entanglement is the entanglement witness, where the measurement of the witness operator is guaranteed to be positive for all separable states and can be negative for certain entangled states. In reality, due to exponentially increasing the Hilbert-space dimension with respect to the system size, it is very challenging to construct an efficient entanglement witness for general multipartite entangled states. For NNN-partite Greenberger-Horne-Zeilinger (GHZ)-like states, the most robust witness scheme requires N+1N+1N+1 local measurement settings and can tolerate up to 1/21/21/2 white noise. As a comparison, the most efficient witness for GHZ-like states only needs two local measurement settings and can tolerate up to 1/31/31/3 white noise. There is a tradeoff between the realization efficiency, the number of measurement settings, and the detection robustness, the maximally tolerable white noise. In this paper, we study this tradeoff by proposing a family of entanglement witnesses with kkk ($2\ensuremath{\le}k\ensuremath{\le}N+1$) local measurement settings. Considering symmetric local measurements, we calculate the maximal tolerable noise for any given number of measurement settings. Consequently, we design the optimal witness with a minimal number of settings for any given level of white noise. Our theoretical analysis can be applied to other multipartite entangled states with a strong symmetry. Our witnesses can be easily implemented in experiment and applied in practical multipartite entanglement detection under different noise conditions.
We introduce an even-parity precession protocol that can detect the nonclassicality of some quantum states using only measurements of a uniformly precessing variable at different points in time. Depending on the system under study, the protocol may detect the Wigner negativity of a single quantum harmonic oscillator or of a single spin $j\ensuremath{\ge}2$; the non-Gaussian entanglement of two harmonic oscillators; or genuine multipartite entanglement of a spin ensemble, whose total spin is an integer. Unlike in other nonclassicality tests, simultaneous or sequential measurements are not required. Our protocol can also detect states that commute with the parity operator, which were missed by similar protocols built from Tsirelson's original precession protocol. This work also closes a long-standing gap by showing the possibility of detecting the Greenberger-Horne-Zeilinger entanglement of an even number of qubits using only collective spin measurements.