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Mermin Devices for Generalized Dicke States

Roman V. Buniy, Thomas W. Kephart

TL;DR

The paper investigates nonlocal correlations in multi-qubit entangled states by bounding Bell-Mermin operators $M_n$ using Mermin devices. It generalizes the original GHZ construction to four qubits and to generalized Dicke states, deriving complete eigenoperator structures and exact bounds on $M_n$-expectations for key states, including the three-qubit W state and its Dicke generalizations. It shows that instructional-set (local-hidden-variable) descriptions cannot account for all measurement outcomes in several cases (notably GHZ and some Dicke states), while in other cases some Dicke states permit such descriptions under restricted operator choices. These results quantify nonlocality and robustness of entanglement in a broad class of multi-qubit states and lay groundwork for extending the analysis to larger systems (BK2026).

Abstract

We present here several new exact results for a number of entangled states: the W-state of three qubits and its generalization -- Dicke states for more than three qubits. We derive these results by bounding the expected values of the Bell-Mermin operators. We review the three qubit GHZ Mermin device, make its generalization to four qubits, and then construct analogous Mermin devices for the generalized Dicke states of three and four qubits. As a result of studying if their operations can be fully explained by Mermin's instructional sets, we show that the GHZ and Dicke states of three qubits and the GHZ state of four qubits do not allow such a description. However, among the two generalized Dicke states of four qubits, one does allow and the other does not allow such a description.

Mermin Devices for Generalized Dicke States

TL;DR

The paper investigates nonlocal correlations in multi-qubit entangled states by bounding Bell-Mermin operators using Mermin devices. It generalizes the original GHZ construction to four qubits and to generalized Dicke states, deriving complete eigenoperator structures and exact bounds on -expectations for key states, including the three-qubit W state and its Dicke generalizations. It shows that instructional-set (local-hidden-variable) descriptions cannot account for all measurement outcomes in several cases (notably GHZ and some Dicke states), while in other cases some Dicke states permit such descriptions under restricted operator choices. These results quantify nonlocality and robustness of entanglement in a broad class of multi-qubit states and lay groundwork for extending the analysis to larger systems (BK2026).

Abstract

We present here several new exact results for a number of entangled states: the W-state of three qubits and its generalization -- Dicke states for more than three qubits. We derive these results by bounding the expected values of the Bell-Mermin operators. We review the three qubit GHZ Mermin device, make its generalization to four qubits, and then construct analogous Mermin devices for the generalized Dicke states of three and four qubits. As a result of studying if their operations can be fully explained by Mermin's instructional sets, we show that the GHZ and Dicke states of three qubits and the GHZ state of four qubits do not allow such a description. However, among the two generalized Dicke states of four qubits, one does allow and the other does not allow such a description.
Paper Structure (8 sections, 24 equations, 1 figure, 1 table)

This paper contains 8 sections, 24 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. States
  3. Eigenoperators
  4. Mermin devices
  5. GHZ states
  6. Generalized Dicke states
  7. Bounds
  8. Conclusions

Figures (1)

  • Figure 1: Contour plots of $\mu_{3,+}(v_{3,1})$, $\mu_{4,+}(v_{4,1})$ and $\mu_{4,+}(v_{4,2})$ as functions of $x_{3}$ and $y_{3}$. Extremal values in \ref{['m_3_v_3_1']}, \ref{['m_4_v_4_1']} and \ref{['m_4_v_4_2']} are at the locations in these figures that are given by the coordinates $(x_{3},y_{3})$ in \ref{['x_3_y_3_v_3_1']}, \ref{['x_3_y_3_v_4_1']} and \ref{['x_3_y_3_v_4_2']}.