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arXiv:2307.09685 · Physical Review A

Entanglement statistics of randomly interacting spins

Paulo Freitas Gomes, Marcel Novaes, Fernando Parisio

Published Sep 13, 2023• 0 citations
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Abstract

We investigate the entanglement in the ground state of systems comprising two and three qubits with random interactions. Since the Hamiltonians also contain deterministic one-body terms, by varying the interaction strength, one can continuously interpolate between deterministic separable eigenstates and fully random entangled eigenstates, with nontrivial intermediate behavior. Entanglement strongly depends on the underlying topology of the interaction among the qubits. Since W states correspond to a zero-measure set as compared to the set of Greenberger-Horne-Zeilinger (GHZ) states, in all investigated cases the ground states are of the latter type. However, for a certain class of interactions (nonseparable collective potential) high GHZ entanglement is produced, while for fully separable pairwise interactions the marginal GHZ ground states concentrate in the vicinity of W states.

DOI
10.1103/physreva.108.032208

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We investigate the entanglement in the ground state of systems comprising two and three qubits with random interactions. Since the Hamiltonians also contain deterministic one-body terms, by varying the interaction strength, one can continuously interpolate between deterministic separable eigenstates and fully random entangled eigenstates, with nontrivial intermediate behavior. Entanglement strongly depends on the underlying topology of the interaction among the qubits. Since W states correspond to a zero-measure set as compared to the set of Greenberger-Horne-Zeilinger (GHZ) states, in all investigated cases the ground states are of the latter type. However, for a certain class of interactions (nonseparable collective potential) high GHZ entanglement is produced, while for fully separable pairwise interactions the marginal GHZ ground states concentrate in the vicinity of W states.
Entanglement refers to the inability of many-body quantum mechanical systems to be separated into independent subsystems. This has been well investigated in systems consisting of two entangled subsystems. In larger systems, more complex types of entanglement exist. In a system consisting of three subsystems (e.g., qubits), it is possible that all three subsystems are entangled with each other in a way that cannot be reduced to bipartite entanglement, and it is known that two different, inequivalent forms of tripartite entanglement exist such as the GHZ and WWW states (GHZ denotes ``Greenberger-Horne-Zeilinger''). Here, we investigate a particularly interesting system with competing one-, two-, and three-body interactions. Its ground state can be a product state, a GHZ state, or a WWW state, depending on the type and strength of the spin-spin couplings. By varying an external control parameter, the system can be made to undergo quantum transitions between the various ground-state-entanglement phases. We implement the system in an NMR quantum simulator and use adiabatic evolution of the effective Hamiltonian to drive the system through the quantum transitions. In the experimental and numerical simulations, we check the suitability of different observables for making the quantum transitions visible and for characterizing the different phases.
The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely, the Greenberger-Horne-Zeilinger (GHZ) type and the WWW type. A state belonging to one of these classes can be stochastically transformed only into a state within the same class by local operations and classical communications. We provide local quantum operations, consisting of the most general two-outcome measurement operators, for the deterministic transformations of three-qubit pure states in which the initial and the target states are in the same class. We explore these transformations, originally having standard GHZ and standard WWW states, under the local measurement operations carried out by a single party and ppp (p=2,3p=2,3p=2,3) parties (successively). We find a notable result that the standard GHZ state cannot be deterministically transformed to a GHZ-type state in which all its bipartite entanglements are nonzero, i.e., a transformation can be achieved with unit probability when the target state has at least one vanishing bipartite concurrence.
Some properties of three-qubit systems interacting with a noisy environment are discussed. The amount of the survival entanglement is quantified for the Greenberger–Horne–Zeilinger (GHZ) and W states. It is shown that the entanglement decay depends on the noise type (correlated or non-correlated), the number of interacting qubits with the environment, and the initial state that passes through this noisy environment. In general, the GHZ state is more fragile than the W state. The phenomenon of entanglement sudden death appears in the GHZ state only for non-correlated noise.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
Abstract In most scenarios, particularly when quantum systems are measured to extract information, W states demonstrate greater robustness compared to Greenberger–Horne–Zeilinger (GHZ) states. This study investigates the evolution of entanglement properties in multi-qubit W and GHZ states under continuous measurement. Our findings reveal that, under specific conditions, GHZ states can exhibit superior entanglement properties relative to W states when qubits are measured using a measurement operator. For the three-qubit case, the GHZ state demonstrates stronger entanglement when all three qubits are measured simultaneously. In the four-qubit scenario, the post-measurement entanglement of the GHZ state surpasses that of the W state when the two central qubits are measured simultaneously. Similarly, for the five-qubit system, the GHZ state achieves higher entanglement than the W state when the three central qubits are measured. Furthermore, as the number of qubits increases, the entanglement between qubits in both GHZ and W states diminishes after measurement.
We investigate the possibility to obtain higly multipartite-entangled states as non-degenerate eigenstates of Hamiltonians that involve only short-range and few-body interactions. We study small-size systems (with a number of qubits ranging from three to five) and search for Hamiltonians with a maximally multipartite entangled state (MMES) as a non-degenerate eigenstate. We then find conditions, including bounds on the number of coupled qubits, to build a Hamiltonian with a Greenberger–Horne–Zeilinger (GHZ) state as a non-degenerate eigenstate. We finally comment on possible applications.