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Laser Physics Letters

Logarithmic measure of quantum entanglement of multiqubit systems

Anna Boeva

Published Dec 1, 2025• 0 citations
Publisher

Abstract

Abstract The importance of quantum entanglement in the development of quantum cryptography, quantum communications, and other areas of science related to modern quantum technologies cannot be overestimated. While the measures of entanglement for a two-level system in a pure state are well-known, the mathematical description of quantum entanglement for multi-level systems is still an open question. The paper proposes a formula that allows us to calculate the numerical value of the quantum entanglement of systems in pure states based on their state vector. The value of quantum entanglement lies in the range [0, 1], and for Greenberg–Horne–Zeilinger states, this value is equal to one.

DOI
10.1088/1612-202x/ae1e6c

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Abstract The importance of quantum entanglement in the development of quantum cryptography, quantum communications, and other areas of science related to modern quantum technologies cannot be overestimated. While the measures of entanglement for a two-level system in a pure state are well-known, the mathematical description of quantum entanglement for multi-level systems is still an open question. The paper proposes a formula that allows us to calculate the numerical value of the quantum entanglement of systems in pure states based on their state vector. The value of quantum entanglement lies in the range [0, 1], and for Greenberg–Horne–Zeilinger states, this value is equal to one.
We propose a measure of entanglement between two subsystems of arbitrary (but finite) number of levels. This measure is invariant under unitary transformations of the subsystems and varies between 0 (product states) and 1 (maximum entangled states). The case of two two-level systems is worked out explicitly, where the known complementarity between one- and two-particle interference results in a very natural way from a sum rule. Generalizations to more than two subsystems are discussed. Here we show that for three particles there cannot exist a pure state that is completely characterized by three-particle entanglement alone. For the example of the Greenberger-Horne-Zeilinger (GHZ) state, an experimental setup is proposed, in which its corresponding two-particle entanglement would show up.
We investigate the relation between the amount of entanglement localized on a chosen subsystem of a multiqubit system via local measurements on the rest of the system, and the bipartite entanglement that is lost during this measurement process. We study a number of paradigmatic pure states, including the generalized Greenberger--Horne--Zeilinger (GHZ), the generalized WWW (gW), Dicke, and the generalized Dicke states. For the generalized GHZ and WWW states, we analytically derive bounds on localizable entanglement in terms of the entanglement present in the system prior to the measurement. Also, for the Dicke and the generalized Dicke states, we demonstrate that with increasing system size, localizable entanglement tends to be equal to the bipartite entanglement present in the system over a specific partition before measurement. We extend the investigation numerically in the case of arbitrary multiqubit pure states. We also analytically determine the modification of these results, including the proposed bounds, in situations where these pure states are subjected to single-qubit phase-flip noise on all qubits. Additionally, we study one-dimensional paradigmatic quantum spin models, namely, the transverse-field XY model and the XXZ model in an external field, and numerically demonstrate a cubic dependence of the localized entanglement on the lost entanglement. We show that this relation is robust even in the presence of disorder in the strength of the external field.
We present calculations of the normally ordered characteristic function (NOCF) of Bose operators for the Greenberger–Horne–Zeilinger states and the Dicke ones. The NOCF is shown to contain information about two types of photon correlations: pseudoclassical and quantum ones. Quantum correlations are connected with interference terms of the NOCF, whereas pseudoclassical photon correlations and their cumulants satisfy the relations for classical random variables. We introduce a measure of the quantumness for entangled multimode states not related to the entropy approach. The maximum of the measure of quantumness is demonstrated to explain an intuitive choice of numeric values for the coefficients in a multiqubit state vector.