arXiv:2407.07037 · Physical Review A

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, Jozef Strečka, Zhirayr Adamyan, Vadim Ohanyan

Published Feb 7, 2025• 8 citations

Abstract

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.

DOI: 10.1103/physreva.111.022605

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

Hana Vargová, Jozef Strečka

arXiv:2509.01451IEEE Transactions on MagneticsJan 12, 20260 citations

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

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

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

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

Hamid Arian Zad, Jozef Strečka, Winfried Plass

arXiv:2409.13457Physical review. B./Physical review. BJan 22, 20253 citations

Genuine tripartite entanglement in a mixed spin-(1/2,1) Heisenberg tetramer

Hana Vargová, Jozef Strečka

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Unconventional strengthening of the bipartite entanglement of a mixed spin-(1/2,1) Heisenberg dimer achieved through Zeeman splitting

Hana Čenčariková, Jozef Strečka

arXiv:2008.12555Physical review. B./Physical review. BNov 16, 202037 citations

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

Jozef Strečka, Elham Shahhosseini Shahrabadi

arXiv:2406.02144InorganicsMar 31, 20245 citations

arXiv:2407.07037 · Physical Review A

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, Jozef Strečka, Zhirayr Adamyan, Vadim Ohanyan

Published Feb 7, 2025• 8 citations

arXiv HTML PDF

Abstract

{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.

DOI: 10.1103/physreva.111.022605

Related papers

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

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

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

Hamid Arian Zad, Jozef Strečka, Winfried Plass

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.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.

arXiv:2409.13457Physical review. B./Physical review. BJan 22, 20253 citations

Genuine tripartite entanglement in a mixed spin-(1/2,1) Heisenberg tetramer

Hana Vargová, Jozef Strečka

A genuine tripartite entanglement of a mixed spin-(1/2,1) Heisenberg tetramer is rigorously analyzed in a presence of external magnetic field. The couple of mixed spin-(1/2,1) dimers is arranged in a perfect rectangular square plaquette involving two nonequivalent Heisenberg exchange couplings $J$ and $J_1$ . The degree of a genuine tripartite entanglement is evaluated according to the genuine tripartite negativity ${\cal N}_{ABC}$ defined as a geometric mean of all possible bipartite negativities corresponding to a decomposition into a single spin and the remaining spin dimer ${\cal N}_{A|BC}$ , ${\cal N}_{B|AC}$ and ${\cal N}_{C|AB}$ after degrees of freedom of the last fourth spin $D$ are traced out. Due to the symmetry of a mixed spin-(1/2,1) Heisenberg tetramer two different genuine tripartite negativities for the trimeric system $1/2\!-\!1\!-\!1$ and $1/2\!-\!1/2\!-\!1$ were identified. It was found that the genuine tripartite negativity for the interaction ratio $J_1/J\!<\!1$ becomes nonzero solely in the tripartite system $1/2\!-\!1\!-\!1$ at low-enough magnetic fields. The opposite interaction limit $J_1/J\!>\!1$ gives rise to the nonzero genuine tripartite negativity in both tripartite systems in a presence of external magnetic field until the classical ferromagnetic state is achieved. It was shown, that the genuine tripartite negativity of a mixed spin-(1/2,1) Heisenberg tetramer can be detected also at nonzero temperatures. An enhancement of the thermal genuine tripartite negativity through the enlargement of the total spin number of a tripartite system is evidenced. The correlation between the bipartite negativity of two spins and the genuine tripartite negativity is discussed in detail.A genuine tripartite entanglement of a mixed spin-(1/2,1) Heisenberg tetramer is rigorously analyzed in a presence of external magnetic field. The couple of mixed spin-(1/2,1) dimers is arranged in a perfect rectangular square plaquette involving two nonequivalent Heisenberg exchange couplings $J$ and $J_1$. The degree of a genuine tripartite entanglement is evaluated according to the genuine tripartite negativity ${\cal N}_{ABC}$ defined as a geometric mean of all possible bipartite negativities corresponding to a decomposition into a single spin and the remaining spin dimer ${\cal N}_{A|BC}$, ${\cal N}_{B|AC}$ and ${\cal N}_{C|AB}$ after degrees of freedom of the last fourth spin $D$ are traced out. Due to the symmetry of a mixed spin-(1/2,1) Heisenberg tetramer two different genuine tripartite negativities for the trimeric system $1/2\!-\!1\!-\!1$ and $1/2\!-\!1/2\!-\!1$ were identified. It was found that the genuine tripartite negativity for the interaction ratio $J_1/J\!<\!1$ becomes nonzero solely in the tripartite system $1/2\!-\!1\!-\!1$ at low-enough magnetic fields. The opposite interaction limit $J_1/J\!>\!1$ gives rise to the nonzero genuine tripartite negativity in both tripartite systems in a presence of external magnetic field until the classical ferromagnetic state is achieved. It was shown, that the genuine tripartite negativity of a mixed spin-(1/2,1) Heisenberg tetramer can be detected also at nonzero temperatures. An enhancement of the thermal genuine tripartite negativity through the enlargement of the total spin number of a tripartite system is evidenced. The correlation between the bipartite negativity of two spins and the genuine tripartite negativity is discussed in detail.

arXiv:2311.17444Nov 29, 20230 citations

Unconventional strengthening of the bipartite entanglement of a mixed spin-(1/2,1) Heisenberg dimer achieved through Zeeman splitting

Hana Čenčariková, Jozef Strečka

Bipartite quantum and thermal entanglement is quantified within pure and mixed states of a mixed spin-( $1/2,1$ ) Heisenberg dimer with the help of negativity. It is shown that the negativity, which may serve as a measure of the bipartite entanglement at zero as well as nonzero temperatures, strongly depends on intrinsic parameters---for instance, exchange and uniaxial single-ion anisotropy---in addition to extrinsic parameters such as temperature and magnetic field. It turns out that a rising magnetic field unexpectedly reinforces the bipartite entanglement due to the Zeeman splitting of energy levels, which lifts the twofold degeneracy of the quantum ferrimagnetic ground state. The maximal bipartite entanglement is thus reached within a quantum ferrimagnetic phase at sufficiently low but nonzero magnetic fields under the assumption that the gyromagnetic $g$ factors of the spin-1/2 and spin-1 magnetic ions are equal and the uniaxial single-ion anisotropy is half of the exchange constant. It is suggested that the heterodinuclear complex $[\mathrm{Ni}(\mathrm{dpt})({\mathrm{H}}_{2}\mathrm{O})\mathrm{Cu}(\mathrm{pba})]\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$ [pba = 1,3-propylenebis(oxamato) and dpt = bis-(3-aminopropyl)amine], which affords an experimental realization of the mixed spin-( $1/2,1$ ) Heisenberg dimer, remains strongly entangled up to relatively high temperatures (about 115 K) and magnetic fields (about 140 T) that are comparable with the relevant exchange constant.Bipartite quantum and thermal entanglement is quantified within pure and mixed states of a mixed spin-($1/2,1$) Heisenberg dimer with the help of negativity. It is shown that the negativity, which may serve as a measure of the bipartite entanglement at zero as well as nonzero temperatures, strongly depends on intrinsic parameters---for instance, exchange and uniaxial single-ion anisotropy---in addition to extrinsic parameters such as temperature and magnetic field. It turns out that a rising magnetic field unexpectedly reinforces the bipartite entanglement due to the Zeeman splitting of energy levels, which lifts the twofold degeneracy of the quantum ferrimagnetic ground state. The maximal bipartite entanglement is thus reached within a quantum ferrimagnetic phase at sufficiently low but nonzero magnetic fields under the assumption that the gyromagnetic $g$ factors of the spin-1/2 and spin-1 magnetic ions are equal and the uniaxial single-ion anisotropy is half of the exchange constant. It is suggested that the heterodinuclear complex $[\mathrm{Ni}(\mathrm{dpt})({\mathrm{H}}_{2}\mathrm{O})\mathrm{Cu}(\mathrm{pba})]\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$ [pba = 1,3-propylenebis(oxamato) and dpt = bis-(3-aminopropyl)amine], which affords an experimental realization of the mixed spin-($1/2,1$) Heisenberg dimer, remains strongly entangled up to relatively high temperatures (about 115 K) and magnetic fields (about 140 T) that are comparable with the relevant exchange constant.

arXiv:2008.12555Physical review. B./Physical review. BNov 16, 202037 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

{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+}

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.

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.

J

and

J_1

. The degree of a genuine tripartite entanglement is evaluated according to the genuine tripartite negativity

{\cal N}_{ABC}

defined as a geometric mean of all possible bipartite negativities corresponding to a decomposition into a single spin and the remaining spin dimer

{\cal N}_{A|BC}

{\cal N}_{B|AC}

and

{\cal N}_{C|AB}

after degrees of freedom of the last fourth spin

D

are traced out. Due to the symmetry of a mixed spin-(1/2,1) Heisenberg tetramer two different genuine tripartite negativities for the trimeric system

1/2\!-\!1\!-\!1

and

1/2\!-\!1/2\!-\!1

were identified. It was found that the genuine tripartite negativity for the interaction ratio

J_1/J\!<\!1

becomes nonzero solely in the tripartite system

1/2\!-\!1\!-\!1

at low-enough magnetic fields. The opposite interaction limit

J_1/J\!>\!1

gives rise to the nonzero genuine tripartite negativity in both tripartite systems in a presence of external magnetic field until the classical ferromagnetic state is achieved. It was shown, that the genuine tripartite negativity of a mixed spin-(1/2,1) Heisenberg tetramer can be detected also at nonzero temperatures. An enhancement of the thermal genuine tripartite negativity through the enlargement of the total spin number of a tripartite system is evidenced. The correlation between the bipartite negativity of two spins and the genuine tripartite negativity is discussed in detail.

Bipartite quantum and thermal entanglement is quantified within pure and mixed states of a mixed spin-(

1/2,1

) Heisenberg dimer with the help of negativity. It is shown that the negativity, which may serve as a measure of the bipartite entanglement at zero as well as nonzero temperatures, strongly depends on intrinsic parameters---for instance, exchange and uniaxial single-ion anisotropy---in addition to extrinsic parameters such as temperature and magnetic field. It turns out that a rising magnetic field unexpectedly reinforces the bipartite entanglement due to the Zeeman splitting of energy levels, which lifts the twofold degeneracy of the quantum ferrimagnetic ground state. The maximal bipartite entanglement is thus reached within a quantum ferrimagnetic phase at sufficiently low but nonzero magnetic fields under the assumption that the gyromagnetic

g

factors of the spin-1/2 and spin-1 magnetic ions are equal and the uniaxial single-ion anisotropy is half of the exchange constant. It is suggested that the heterodinuclear complex $[\mathrm{Ni}(\mathrm{dpt})({\mathrm{H}}_{2}\mathrm{O})\mathrm{Cu}(\mathrm{pba})]\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$ [pba = 1,3-propylenebis(oxamato) and dpt = bis-(3-aminopropyl)amine], which affords an experimental realization of the mixed spin-(

1/2,1

) Heisenberg dimer, remains strongly entangled up to relatively high temperatures (about 115 K) and magnetic fields (about 140 T) that are comparable with the relevant exchange constant.