arXiv:2409.13457 · Physical review. B./Physical review. B

Multipartite entanglement and quantum sensing in a spin- 5 2 Heisenberg molecular iron(III) triangle

Hamid Arian Zad, Jozef Strečka, Winfried Plass

Published Jan 22, 2025• 3 citations

Abstract

This study provides insights into the static and dynamic quantum properties of the trinuclear high-spin iron(III) molecular complex $[{\mathrm{Fe}}_{3}{\mathrm{Cl}}_{3}({\mathrm{saltag}}^{\mathrm{Br}}){(\mathrm{py})}_{6}]{\mathrm{ClO}}_{4}$, abbreviated as ${\mathrm{Fe}}_{3}$. Using exact diagonalization of a spin-$\frac{5}{2}$ Heisenberg triangle in a magnetic field, we model the corresponding quantum behavior of the molecular compound ${\mathrm{Fe}}_{3}$. Our rigorous analysis employs various key metrics to explore the rich quantum behavior of this molecular compound. At sufficiently low temperatures, the bipartite negativity reveals that the pairwise entanglement between any pair of iron(III) magnetic ions of the molecular complex ${\mathrm{Fe}}_{3}$ can be significantly enhanced by a small magnetic field. This enhancement is followed by unconventional steplike changes characterized by a sequence of plateaus and sudden downturns as the magnetic field further increases. A qualitatively similar behavior is also observed in the genuine tripartite entanglement among all three iron(III) magnetic ions in the trinuclear complex ${\mathrm{Fe}}_{3}$. Notably, the bipartite and tripartite entanglements persist in the molecular complex ${\mathrm{Fe}}_{3}$ up to moderate temperatures of approximately 30 and 70 K, respectively. Additionally, we demonstrate the achievement of quantum-enhanced sensitivity by initializing the molecular complex ${\mathrm{Fe}}_{3}$ in Dicke states. Finally, we investigate a quantum sensing protocol by applying a local magnetic field specifically to one iron(III) magnetic ion of the molecular compound ${\mathrm{Fe}}_{3}$ and performing readout sequentially on one of the two remaining iron(III) magnetic ions.

DOI: 10.1103/physrevb.111.024420

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Room-Temperature Entanglement of the Nickel-Radical Molecular Complex (Et3NH)[Ni(hfac)2L] Reinforced by the Magnetic Field

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Magnetic-Field-Orientation Dependent Thermal Entanglement of a Spin-1 Heisenberg Dimer: The Case Study of Dinuclear Nickel Complex with an Uniaxial Single-Ion Anisotropy

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Tripartite entanglement of {Cu3} single molecular magnet with magnetic field in thermal equilibrium

Yidan Zheng, Bin Zhou

Acta Physica SinicaJan 1, 20161 citation

Effect of Single-Ion Anisotropy on Stability of Quantum and Thermal Entanglement in a Mixed-Spin Heisenberg Trimer

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arXiv:2509.01451IEEE Transactions on MagneticsJan 12, 20260 citations

arXiv:2409.13457 · Physical review. B./Physical review. B

Multipartite entanglement and quantum sensing in a spin- 5 2 Heisenberg molecular iron(III) triangle

Hamid Arian Zad, Jozef Strečka, Winfried Plass

Published Jan 22, 2025• 3 citations

arXiv HTML PDF

Abstract

This study provides insights into the static and dynamic quantum properties of the trinuclear high-spin iron(III) molecular complex

[{\mathrm{Fe}}_{3}{\mathrm{Cl}}_{3}({\mathrm{saltag}}^{\mathrm{Br}}){(\mathrm{py})}_{6}]{\mathrm{ClO}}_{4}

, abbreviated as

{\mathrm{Fe}}_{3}

. Using exact diagonalization of a spin-

\frac{5}{2}

Heisenberg triangle in a magnetic field, we model the corresponding quantum behavior of the molecular compound

{\mathrm{Fe}}_{3}

. Our rigorous analysis employs various key metrics to explore the rich quantum behavior of this molecular compound. At sufficiently low temperatures, the bipartite negativity reveals that the pairwise entanglement between any pair of iron(III) magnetic ions of the molecular complex

{\mathrm{Fe}}_{3}

can be significantly enhanced by a small magnetic field. This enhancement is followed by unconventional steplike changes characterized by a sequence of plateaus and sudden downturns as the magnetic field further increases. A qualitatively similar behavior is also observed in the genuine tripartite entanglement among all three iron(III) magnetic ions in the trinuclear complex

{\mathrm{Fe}}_{3}

. Notably, the bipartite and tripartite entanglements persist in the molecular complex

{\mathrm{Fe}}_{3}

up to moderate temperatures of approximately 30 and 70 K, respectively. Additionally, we demonstrate the achievement of quantum-enhanced sensitivity by initializing the molecular complex

{\mathrm{Fe}}_{3}

in Dicke states. Finally, we investigate a quantum sensing protocol by applying a local magnetic field specifically to one iron(III) magnetic ion of the molecular compound

{\mathrm{Fe}}_{3}

and performing readout sequentially on one of the two remaining iron(III) magnetic ions.

DOI: 10.1103/physrevb.111.024420

Related papers

Molecular nanomagnet Cu 2 + Ni 2 + Cu 2 + as a resource for bipartite and tripartite quantum entanglement and coherence

Azadeh Ghannadan, Hamid Arian Zad, Saeed Haddadi +3 more

We investigate key quantum characteristics of the mixed spin-(1/2,1,1/2) Heisenberg trimer under the influence of an external magnetic field. Specifically, we analyze the distributions of bipartite and tripartite entanglement quantified through the respective negativities and the ${l}_{1}$ -norm of coherence with the help of rigorous analytical and numerical methods. Our findings suggest that the heterotrinuclear molecular nanomagnet $[{{{\text{Cu}}^{2+}\text{L}}}_{2}{\text{Ni}}^{2+}{({\text{H}}_{2}\text{O})}_{2}]{({\text{ClO}}_{4})}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}3{\text{H}}_{2}\text{O}$, which represents an experimental realization of the mixed spin-(1/2,1,1/2) Heisenberg trimer, exhibits a significant bipartite entanglement between ${\text{Cu}}^{2+}$ and ${\text{Ni}}^{2+}$ magnetic ions along with robust tripartite entanglement among all three constituent ${\text{Cu}}^{2+}{\text{Ni}}^{2+}{\text{Cu}}^{2+}$ magnetic ions. The significant bipartite and tripartite entanglement persists even at relatively high temperatures up to $37\phantom{\rule{0.16em}{0ex}}\text{K}$ and magnetic fields up to $46\phantom{\rule{0.16em}{0ex}}\text{T}$ , whereby the coherence is maintained even at elevated temperatures. It is evidenced that the aforementioned molecular complex with the magnetic core ${\text{Cu}}^{2+}{\text{Ni}}^{2+}{\text{Cu}}^{2+}$ provides an intriguing quantum resource, which exhibits a star-shaped state within the singlet eigenstate at low magnetic fields and W-like state within the triplet eigenstate at moderate magnetic fields.We investigate key quantum characteristics of the mixed spin-(1/2,1,1/2) Heisenberg trimer under the influence of an external magnetic field. Specifically, we analyze the distributions of bipartite and tripartite entanglement quantified through the respective negativities and the ${l}_{1}$-norm of coherence with the help of rigorous analytical and numerical methods. Our findings suggest that the heterotrinuclear molecular nanomagnet $[{{{\text{Cu}}^{2+}\text{L}}}_{2}{\text{Ni}}^{2+}{({\text{H}}_{2}\text{O})}_{2}]{({\text{ClO}}_{4})}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}3{\text{H}}_{2}\text{O}$, which represents an experimental realization of the mixed spin-(1/2,1,1/2) Heisenberg trimer, exhibits a significant bipartite entanglement between ${\text{Cu}}^{2+}$ and ${\text{Ni}}^{2+}$ magnetic ions along with robust tripartite entanglement among all three constituent ${\text{Cu}}^{2+}{\text{Ni}}^{2+}{\text{Cu}}^{2+}$ magnetic ions. The significant bipartite and tripartite entanglement persists even at relatively high temperatures up to $37\phantom{\rule{0.16em}{0ex}}\text{K}$ and magnetic fields up to $46\phantom{\rule{0.16em}{0ex}}\text{T}$, whereby the coherence is maintained even at elevated temperatures. It is evidenced that the aforementioned molecular complex with the magnetic core ${\text{Cu}}^{2+}{\text{Ni}}^{2+}{\text{Cu}}^{2+}$ provides an intriguing quantum resource, which exhibits a star-shaped state within the singlet eigenstate at low magnetic fields and W-like state within the triplet eigenstate at moderate magnetic fields.

arXiv:2407.07037Physical Review AFeb 7, 20258 citations

Quantum entanglement generation in heterometallic Ni 4 2 + Gd 4 3 + complexes

Hamid Arian Zad, M. Jaščur, Azam Zoshki +2 more

We investigate various types of quantum entanglement in the octanuclear heterometallic $3d/4f$ complexes denoted as ${\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}$ under an external magnetic field, using the exact diagonalization approach. These molecular magnets, which can be effectively described by Heisenberg spin models, consist of two identical { ${\mathrm{Ni}}_{2}^{2+}{\mathrm{Gd}}_{2}^{3+}$ } cubane subunits bridged by acetate and hydroxide ligands. Our analysis reveals that their magnetization exhibits intermediate plateaus at low temperatures, indicating distinct ground states characteristic of Ni-containing compounds. Using negativity as a measure of quantum entanglement, we examine the influence of single-ion anisotropy and magnetic field on tetrapartite, bipartite, 1--3 tangle, and 2--2 tangle entanglements in two families of ${\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}$ complexes: (1) without anisotropy and (2) with anisotropy. Complex (1) exhibits strong bipartite entanglement between Ni ions, which persists up to $T\ensuremath{\approx}3.0\phantom{\rule{0.28em}{0ex}}\text{K}$ and $B\ensuremath{\approx}4.0\phantom{\rule{0.28em}{0ex}}\text{T}$, but shows significantly weaker tetrapartite entanglement and vanishing bipartite entanglement between $\mathrm{Gd}\ensuremath{\cdots}\mathrm{Gd}$ and $\mathrm{Ni}\ensuremath{\cdots}\mathrm{Gd}$ pairs. In contrast, complex (2) displays nonzero and sizable values for all types of entanglement considered. These findings emphasis the crucial role of single-ion anisotropy in generating and shaping the entanglement landscape of heterometallic ${\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}$ complexes. Notably, we find that the 1--3 tangle entanglement between a Ni ion and the remaining sites in a cubane unit serves as a reliable indicator of ground-state phase transitions, exhibiting distinct changes across phase boundaries irrespective of the presence of single-ion anisotropy.We investigate various types of quantum entanglement in the octanuclear heterometallic $3d/4f$ complexes denoted as ${\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}$ under an external magnetic field, using the exact diagonalization approach. These molecular magnets, which can be effectively described by Heisenberg spin models, consist of two identical {${\mathrm{Ni}}_{2}^{2+}{\mathrm{Gd}}_{2}^{3+}$} cubane subunits bridged by acetate and hydroxide ligands. Our analysis reveals that their magnetization exhibits intermediate plateaus at low temperatures, indicating distinct ground states characteristic of Ni-containing compounds. Using negativity as a measure of quantum entanglement, we examine the influence of single-ion anisotropy and magnetic field on tetrapartite, bipartite, 1--3 tangle, and 2--2 tangle entanglements in two families of ${\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}$ complexes: (1) without anisotropy and (2) with anisotropy. Complex (1) exhibits strong bipartite entanglement between Ni ions, which persists up to $T\ensuremath{\approx}3.0\phantom{\rule{0.28em}{0ex}}\text{K}$ and $B\ensuremath{\approx}4.0\phantom{\rule{0.28em}{0ex}}\text{T}$, but shows significantly weaker tetrapartite entanglement and vanishing bipartite entanglement between $\mathrm{Gd}\ensuremath{\cdots}\mathrm{Gd}$ and $\mathrm{Ni}\ensuremath{\cdots}\mathrm{Gd}$ pairs. In contrast, complex (2) displays nonzero and sizable values for all types of entanglement considered. These findings emphasis the crucial role of single-ion anisotropy in generating and shaping the entanglement landscape of heterometallic ${\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}$ complexes. Notably, we find that the 1--3 tangle entanglement between a Ni ion and the remaining sites in a cubane unit serves as a reliable indicator of ground-state phase transitions, exhibiting distinct changes across phase boundaries irrespective of the presence of single-ion anisotropy.

arXiv:2503.20311Physical review. B./Physical review. BAug 26, 20250 citations

Room-Temperature Entanglement of the Nickel-Radical Molecular Complex (Et3NH)[Ni(hfac)2L] Reinforced by the Magnetic Field

Jozef Strečka, Elham Shahhosseini Shahrabadi

Bipartite entanglement is comprehensively investigated in the mononuclear molecular complex (Et3NH)[Ni(hfac)2L], where HL denotes 2-(2-hydroxy-3-methoxy-5-nitrophenyl)-4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazol-3-oxide-1-oxyl and hfacH stands for hexafluoroacetylacetone. From the magnetic point of view, the molecular compound (Et3NH)[Ni(hfac)2L] consists of an exchange-coupled spin-1 Ni2+ magnetic ion and a spin-12 nitronyl-nitroxide radical substituted nitrophenol. The nickel-radical molecular complex affords an experimental realization of a mixed spin-(12, 1) Heisenberg dimer with a strong antiferromagnetic exchange coupling, J/kB = 505 K, and two distinct g-factors, gRad = 2.005 and gNi = 2.275. By adopting this set of magnetic parameters, we demonstrate that the Zeeman splitting of a quantum ferrimagnetic ground-state doublet due to a weak magnetic field may substantially reinforce the strength of bipartite entanglement at low temperatures. The molecular compound (Et3NH)[Ni(hfac)2L] maintains sufficiently strong thermal entanglement, even at room temperature, vanishing only above 546 K. Specifically, the thermal entanglement in the nickel-radical molecular complex retains approximately 40% of the maximum value, corresponding to perfectly entangled Bell states at room temperature, which implies that this magnetic compound provides a suitable platform of a molecular qubit with potential implications for room-temperature quantum computation and quantum information processing.Bipartite entanglement is comprehensively investigated in the mononuclear molecular complex (Et3NH)[Ni(hfac)2L], where HL denotes 2-(2-hydroxy-3-methoxy-5-nitrophenyl)-4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazol-3-oxide-1-oxyl and hfacH stands for hexafluoroacetylacetone. From the magnetic point of view, the molecular compound (Et3NH)[Ni(hfac)2L] consists of an exchange-coupled spin-1 Ni2+ magnetic ion and a spin-12 nitronyl-nitroxide radical substituted nitrophenol. The nickel-radical molecular complex affords an experimental realization of a mixed spin-(12, 1) Heisenberg dimer with a strong antiferromagnetic exchange coupling, J/kB = 505 K, and two distinct g-factors, gRad = 2.005 and gNi = 2.275. By adopting this set of magnetic parameters, we demonstrate that the Zeeman splitting of a quantum ferrimagnetic ground-state doublet due to a weak magnetic field may substantially reinforce the strength of bipartite entanglement at low temperatures. The molecular compound (Et3NH)[Ni(hfac)2L] maintains sufficiently strong thermal entanglement, even at room temperature, vanishing only above 546 K. Specifically, the thermal entanglement in the nickel-radical molecular complex retains approximately 40% of the maximum value, corresponding to perfectly entangled Bell states at room temperature, which implies that this magnetic compound provides a suitable platform of a molecular qubit with potential implications for room-temperature quantum computation and quantum information processing.

arXiv:2406.02144InorganicsMar 31, 20245 citations

Magnetic-Field-Orientation Dependent Thermal Entanglement of a Spin-1 Heisenberg Dimer: The Case Study of Dinuclear Nickel Complex with an Uniaxial Single-Ion Anisotropy

Azadeh Ghannadan, Jozef Strečka

The bipartite entanglement in pure and mixed states of a quantum spin-1 Heisenberg dimer with exchange and uniaxial single-ion anisotropies is quantified through the negativity in a presence of the external magnetic field. At zero temperature the negativity shows a marked stepwise dependence on a magnetic field with two abrupt jumps and plateaus, which can be attributed to the quantum antiferromagnetic and quantum ferrimagnetic ground states. The magnetic-field-driven phase transition between the quantum antiferromagnetic and quantum ferrimagnetic ground states manifests itself at nonzero temperatures by a local minimum of the negativity, which is followed by a peculiar field-induced rise of the negativity observable in a range of moderately strong magnetic fields. The rising temperature generally smears out abrupt jumps and plateaus of the negativity, which cannot be distinguished in the relevant dependencies above a certain temperature. It is shown that the thermal entanglement is most persistent against rising temperature at the magnetic field, for which an energy gap between a ground state and a first excited state is highest. Besides, temperature variations of the negativity of the spin-1 Heisenberg dimer with an easy-axis single-ion anisotropy may exhibit a singular point-kink, at which the negativity has discontinuity in its first derivative. The homodinuclear nickel complex [Ni2(Medpt)2(μ-ox)(H2O)2](ClO4)2·2H2O provides a suitable experimental platform of the antiferromagnetic spin-1 Heisenberg dimer, which allowed us to estimate a strength of the bipartite entanglement between two exchange-coupled Ni2+ magnetic ions on the grounds of the interaction constants reported previously from the fitting procedure of the magnetization data. It is verified that the negativity of this dinuclear compound is highly magnetic-field-orientation dependent due to presence of a relatively strong uniaxial single-ion anisotropy.The bipartite entanglement in pure and mixed states of a quantum spin-1 Heisenberg dimer with exchange and uniaxial single-ion anisotropies is quantified through the negativity in a presence of the external magnetic field. At zero temperature the negativity shows a marked stepwise dependence on a magnetic field with two abrupt jumps and plateaus, which can be attributed to the quantum antiferromagnetic and quantum ferrimagnetic ground states. The magnetic-field-driven phase transition between the quantum antiferromagnetic and quantum ferrimagnetic ground states manifests itself at nonzero temperatures by a local minimum of the negativity, which is followed by a peculiar field-induced rise of the negativity observable in a range of moderately strong magnetic fields. The rising temperature generally smears out abrupt jumps and plateaus of the negativity, which cannot be distinguished in the relevant dependencies above a certain temperature. It is shown that the thermal entanglement is most persistent against rising temperature at the magnetic field, for which an energy gap between a ground state and a first excited state is highest. Besides, temperature variations of the negativity of the spin-1 Heisenberg dimer with an easy-axis single-ion anisotropy may exhibit a singular point-kink, at which the negativity has discontinuity in its first derivative. The homodinuclear nickel complex [Ni2(Medpt)2(μ-ox)(H2O)2](ClO4)2·2H2O provides a suitable experimental platform of the antiferromagnetic spin-1 Heisenberg dimer, which allowed us to estimate a strength of the bipartite entanglement between two exchange-coupled Ni2+ magnetic ions on the grounds of the interaction constants reported previously from the fitting procedure of the magnetization data. It is verified that the negativity of this dinuclear compound is highly magnetic-field-orientation dependent due to presence of a relatively strong uniaxial single-ion anisotropy.

MoleculesJun 5, 202120 citations

Tripartite entanglement of {Cu3} single molecular magnet with magnetic field in thermal equilibrium

Yidan Zheng, Bin Zhou

Quantum entanglement is one of the most fundamental properties of quantum mechanics. Because of the nonlocality, quantum entanglement is widely used in quantum computation and quantum information. Considering the fact that thermal fluctuation suppresses quantum effects, the concept of thermal entanglement is introduced to refer to the idea that the effect of temperature should be viewed as external control in the preparation of entangled state. It has been found that nanoscale single molecular magnet has a novel quantum effect at low temperature. Furthermore, single-molecular magnet is viewed as a promising candidate for realizing encoding and manipulation of quantum information. Na9[Cu3Na3(H2O)9(-AsW9O33)2]26H2O (denoted as {Cu3} for convenience) is one of the typical representatives of nanoscale single molecular magnets. In this paper, we will theoretically analyze the properties of tripartite entanglement in {Cu3} with an external magnetic field in thermal equilibrium. The tripartite negativity is used to characterize the tripartite entanglement. The tripartite negativity of {Cu3} single molecular magnet is calculated numerically by using the equivalent spin model and experimental fitting parameters. We consider the magnetic fields along the vertical and the parallel directions of triangular spin ring, respectively, and the case with a tilted magnetic field is also discussed in this paper. It is shown that the magnitude and direction of magnetic field, and temperature have importance effects on the tripartite negativity of the system. It is found that the larger extra strong magnetic field will inhibit the generation of the quantum state of tripartite entanglement at higher temperature. In addition, compared with the magnetic field along the parallel direction of triangular spin ring and the tilted magnetic field, the magnetic field along the vertical direction of triangular spin ring obtains larger values of tripartite negativity under the same temperature and magnetic field. We also plot the variations of the critical temperature with the magnetic field along different directions, and from the critical temperature-magnetic field phase diagrams one can obtain the range of parameters in which the tripartite entanglement of the system exists. We also find that entanglement revival behaviors may occur in the specific range of parameters. Therefore, the properties of the tripartite entanglement in the {Cu3} triangular spin ring can be controlled and enhanced by choosing appropriate magnitude and direction of the magnetic field and temperature.Quantum entanglement is one of the most fundamental properties of quantum mechanics. Because of the nonlocality, quantum entanglement is widely used in quantum computation and quantum information. Considering the fact that thermal fluctuation suppresses quantum effects, the concept of thermal entanglement is introduced to refer to the idea that the effect of temperature should be viewed as external control in the preparation of entangled state. It has been found that nanoscale single molecular magnet has a novel quantum effect at low temperature. Furthermore, single-molecular magnet is viewed as a promising candidate for realizing encoding and manipulation of quantum information. Na9[Cu3Na3(H2O)9(-AsW9O33)2]26H2O (denoted as {Cu3} for convenience) is one of the typical representatives of nanoscale single molecular magnets. In this paper, we will theoretically analyze the properties of tripartite entanglement in {Cu3} with an external magnetic field in thermal equilibrium. The tripartite negativity is used to characterize the tripartite entanglement. The tripartite negativity of {Cu3} single molecular magnet is calculated numerically by using the equivalent spin model and experimental fitting parameters. We consider the magnetic fields along the vertical and the parallel directions of triangular spin ring, respectively, and the case with a tilted magnetic field is also discussed in this paper. It is shown that the magnitude and direction of magnetic field, and temperature have importance effects on the tripartite negativity of the system. It is found that the larger extra strong magnetic field will inhibit the generation of the quantum state of tripartite entanglement at higher temperature. In addition, compared with the magnetic field along the parallel direction of triangular spin ring and the tilted magnetic field, the magnetic field along the vertical direction of triangular spin ring obtains larger values of tripartite negativity under the same temperature and magnetic field. We also plot the variations of the critical temperature with the magnetic field along different directions, and from the critical temperature-magnetic field phase diagrams one can obtain the range of parameters in which the tripartite entanglement of the system exists. We also find that entanglement revival behaviors may occur in the specific range of parameters. Therefore, the properties of the tripartite entanglement in the {Cu3} triangular spin ring can be controlled and enhanced by choosing appropriate magnitude and direction of the magnetic field and temperature.

Acta Physica SinicaJan 1, 20161 citation

Effect of Single-Ion Anisotropy on Stability of Quantum and Thermal Entanglement in a Mixed-Spin Heisenberg Trimer

Hana Vargová, Jozef Strečka

The effect of uniaxial single-ion anisotropy on quantum entanglement is rigorously quantified using negativity in a mixed spin-(1,1/2,1) Heisenberg trimer, accounting for different exchange coupling constants between identical and distinct spins. Bipartite negativities between the single-spin entity and the remaining spin dimer are analyzed alongside the global tripartite negativity (gTN) of the whole trimer under the effect of an external magnetic field and both easy-axis and easy-plane types of single-ion anisotropy. Interestingly, the single-ion anisotropy significantly influences the degree of entanglement by altering the stability regions of energetically preferred ground states and it may also introduce additional phases in the overall ground-state phase diagram. Moreover, even an arbitrarily small single-ion anisotropy is sufficient to generate global tripartite entanglement in a state that remains biseparable in the isotropic limit. Finally, the thermal stability of entanglement is discussed in detail, including the emergence of a peculiar local minimum at finite temperatures. The obtained theoretical results may offer deeper insights into bipartite and tripartite entanglement in trimetallic Ni2+-Cu2+-Ni2+molecular compounds.The effect of uniaxial single-ion anisotropy on quantum entanglement is rigorously quantified using negativity in a mixed spin-(1,1/2,1) Heisenberg trimer, accounting for different exchange coupling constants between identical and distinct spins. Bipartite negativities between the single-spin entity and the remaining spin dimer are analyzed alongside the global tripartite negativity (gTN) of the whole trimer under the effect of an external magnetic field and both easy-axis and easy-plane types of single-ion anisotropy. Interestingly, the single-ion anisotropy significantly influences the degree of entanglement by altering the stability regions of energetically preferred ground states and it may also introduce additional phases in the overall ground-state phase diagram. Moreover, even an arbitrarily small single-ion anisotropy is sufficient to generate global tripartite entanglement in a state that remains biseparable in the isotropic limit. Finally, the thermal stability of entanglement is discussed in detail, including the emergence of a peculiar local minimum at finite temperatures. The obtained theoretical results may offer deeper insights into bipartite and tripartite entanglement in trimetallic Ni2+-Cu2+-Ni2+molecular compounds.

arXiv:2509.01451IEEE Transactions on MagneticsJan 12, 20260 citations

This study provides insights into the static and dynamic quantum properties of the trinuclear high-spin iron(III) molecular complex

[{\mathrm{Fe}}_{3}{\mathrm{Cl}}_{3}({\mathrm{saltag}}^{\mathrm{Br}}){(\mathrm{py})}_{6}]{\mathrm{ClO}}_{4}

, abbreviated as

{\mathrm{Fe}}_{3}

. Using exact diagonalization of a spin-

\frac{5}{2}

Heisenberg triangle in a magnetic field, we model the corresponding quantum behavior of the molecular compound

{\mathrm{Fe}}_{3}

{\mathrm{Fe}}_{3}

{\mathrm{Fe}}_{3}

. Notably, the bipartite and tripartite entanglements persist in the molecular complex

{\mathrm{Fe}}_{3}

up to moderate temperatures of approximately 30 and 70 K, respectively. Additionally, we demonstrate the achievement of quantum-enhanced sensitivity by initializing the molecular complex

{\mathrm{Fe}}_{3}

in Dicke states. Finally, we investigate a quantum sensing protocol by applying a local magnetic field specifically to one iron(III) magnetic ion of the molecular compound

{\mathrm{Fe}}_{3}

and performing readout sequentially on one of the two remaining iron(III) magnetic ions.

{l}_{1}

-norm of coherence with the help of rigorous analytical and numerical methods. Our findings suggest that the heterotrinuclear molecular nanomagnet $[{{{\text{Cu}}^{2+}\text{L}}}_{2}{\text{Ni}}^{2+}{({\text{H}}_{2}\text{O})}_{2}]{({\text{ClO}}_{4})}_{2}\ifmmode\cdot\else\textperiodcentered\fi{}3{\text{H}}_{2}\text{O}$, which represents an experimental realization of the mixed spin-(1/2,1,1/2) Heisenberg trimer, exhibits a significant bipartite entanglement between

{\text{Cu}}^{2+}

and

{\text{Ni}}^{2+}

magnetic ions along with robust tripartite entanglement among all three constituent

{\text{Cu}}^{2+}{\text{Ni}}^{2+}{\text{Cu}}^{2+}

magnetic ions. The significant bipartite and tripartite entanglement persists even at relatively high temperatures up to

37\phantom{\rule{0.16em}{0ex}}\text{K}

and magnetic fields up to

46\phantom{\rule{0.16em}{0ex}}\text{T}

, whereby the coherence is maintained even at elevated temperatures. It is evidenced that the aforementioned molecular complex with the magnetic core

{\text{Cu}}^{2+}{\text{Ni}}^{2+}{\text{Cu}}^{2+}

provides an intriguing quantum resource, which exhibits a star-shaped state within the singlet eigenstate at low magnetic fields and W-like state within the triplet eigenstate at moderate magnetic fields.

We investigate various types of quantum entanglement in the octanuclear heterometallic

3d/4f

complexes denoted as

{\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}

under an external magnetic field, using the exact diagonalization approach. These molecular magnets, which can be effectively described by Heisenberg spin models, consist of two identical {

{\mathrm{Ni}}_{2}^{2+}{\mathrm{Gd}}_{2}^{3+}

} cubane subunits bridged by acetate and hydroxide ligands. Our analysis reveals that their magnetization exhibits intermediate plateaus at low temperatures, indicating distinct ground states characteristic of Ni-containing compounds. Using negativity as a measure of quantum entanglement, we examine the influence of single-ion anisotropy and magnetic field on tetrapartite, bipartite, 1--3 tangle, and 2--2 tangle entanglements in two families of

{\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}

complexes: (1) without anisotropy and (2) with anisotropy. Complex (1) exhibits strong bipartite entanglement between Ni ions, which persists up to $T\ensuremath{\approx}3.0\phantom{\rule{0.28em}{0ex}}\text{K}$ and $B\ensuremath{\approx}4.0\phantom{\rule{0.28em}{0ex}}\text{T}$, but shows significantly weaker tetrapartite entanglement and vanishing bipartite entanglement between $\mathrm{Gd}\ensuremath{\cdots}\mathrm{Gd}$ and $\mathrm{Ni}\ensuremath{\cdots}\mathrm{Gd}$ pairs. In contrast, complex (2) displays nonzero and sizable values for all types of entanglement considered. These findings emphasis the crucial role of single-ion anisotropy in generating and shaping the entanglement landscape of heterometallic

{\mathrm{Ni}}_{4}^{2+}{\mathrm{Gd}}_{4}^{3+}

complexes. Notably, we find that the 1--3 tangle entanglement between a Ni ion and the remaining sites in a cubane unit serves as a reliable indicator of ground-state phase transitions, exhibiting distinct changes across phase boundaries irrespective of the presence of single-ion anisotropy.