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

Evidence of a significant full tripartite entanglement in a mixed spin-(1/2,1) Heisenberg tetramer

Hana Vargová, Jozef Strečka

We rigorously analyze the full tripartite entanglement of a mixed spin-(1/2,1) Heisenberg tetramer in the presence of an external magnetic field. The tetramer consists of two mixed spin-(1/2,1) dimers arranged to form a square plaquette, with distinct exchange couplings J and J1 between spins from the same and different dimers, respectively. The full tripartite entanglement is classified according to the tripartite negativity NABC defined as a geometric mean of all possible bipartite negativities NA|BC, NB|AC, and NC|AB, when the degrees of freedom of the fourth spin D of the tetrapartite system ABCD are traced out. We demonstrate that for J1/J>1, both trimeric subsystems 1/2-1-1 and 1/2-1/2-1 exhibit tripartite entanglement mediated through the central spin. This entanglement becomes more robust upon the application of an external magnetic field. The opposite limit J1/J<1 leads to the spontaneous emergence of full tripartite entanglement solely within the subsystem 1/2-1-1. However, the increasing magnetic field swiftly disrupts the three-particle correlation across the relevant ground-state phase boundary. Additionally, we identify that in the isotropic limit J1/J=1, the ground-state degeneracy significantly reduces the strength of full tripartite entanglement and favors nontrivial subtypes of full nonseparable states with none or just one spin-dimer entanglement. Finally, we discuss in detail the thermal stability of all fully tripartite nonseparable states.We rigorously analyze the full tripartite entanglement of a mixed spin-(1/2,1) Heisenberg tetramer in the presence of an external magnetic field. The tetramer consists of two mixed spin-(1/2,1) dimers arranged to form a square plaquette, with distinct exchange couplings J and J1 between spins from the same and different dimers, respectively. The full tripartite entanglement is classified according to the tripartite negativity NABC defined as a geometric mean of all possible bipartite negativities NA|BC, NB|AC, and NC|AB, when the degrees of freedom of the fourth spin D of the tetrapartite system ABCD are traced out. We demonstrate that for J1/J>1, both trimeric subsystems 1/2-1-1 and 1/2-1/2-1 exhibit tripartite entanglement mediated through the central spin. This entanglement becomes more robust upon the application of an external magnetic field. The opposite limit J1/J<1 leads to the spontaneous emergence of full tripartite entanglement solely within the subsystem 1/2-1-1. However, the increasing magnetic field swiftly disrupts the three-particle correlation across the relevant ground-state phase boundary. Additionally, we identify that in the isotropic limit J1/J=1, the ground-state degeneracy significantly reduces the strength of full tripartite entanglement and favors nontrivial subtypes of full nonseparable states with none or just one spin-dimer entanglement. Finally, we discuss in detail the thermal stability of all fully tripartite nonseparable states.

Results in PhysicsMay 5, 20245 citations

Thermal entanglement properties of small spin clusters

Indrani Bose, Amit Tribedi

Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude $\frac{1}{2}$ , in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength ${J}_{1}$ , ${J}_{2}$ , and $K$ , respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a $S=\frac{1}{2}$ antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with ${J}_{2}=0$ , $K=0$ , in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility $\ensuremath{\chi}$ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature ${T}_{c}$ below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron $({J}_{1}={J}_{2},K=0)$ ---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature ${T}_{E}$ below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude $\frac{1}{2}$, in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength ${J}_{1}$, ${J}_{2}$, and $K$, respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a $S=\frac{1}{2}$ antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with ${J}_{2}=0$, $K=0$, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility $\ensuremath{\chi}$ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature ${T}_{c}$ below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron $({J}_{1}={J}_{2},K=0)$---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature ${T}_{E}$ below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.

arXiv:cond-mat/0503170Physical Review AAug 15, 200557 citations

Distribution of Bipartite Entanglement of a Random Pure State

Satya N. Majumdar

Jan 1, 20120 citations

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.

Exchange interactions in spin systems can give rise to quantum entanglement in the ground and thermal states of the systems. We consider a spin tetramer, with spins of magnitude

\frac{1}{2}

, in which the spins interact via nearest-neighbor, diagonal, and four-spin interactions of strength

{J}_{1}

{J}_{2}

, and

K

, respectively. The ground- and thermal-state entanglement properties of the tetramer are calculated analytically in the various limiting cases. Both bipartite and multipartite entanglements are considered and a signature of the quantum phase transition (QPT), in terms of the entanglement ratio, is identified. The first-order QPT is accompanied by discontinuities in the nearest-neighbor and diagonal concurrences. The magnetic properties of a

S=\frac{1}{2}

antiferromagnetic polyoxovanadate compound V12 are well explained by tetramers, with

{J}_{2}=0

K=0

, in which the spins interact via the isotropic Heisenberg exchange interaction Hamiltonian. Treating the magnetic susceptibility $\ensuremath{\chi}$ as an entanglement witness (EW), an estimate of the lower bound of the critical entanglement temperature

{T}_{c}

below which entanglement is present in the experimental compound, is determined. Two other cases considered include the symmetric tetramer---i.e., tetrahedron

({J}_{1}={J}_{2},K=0)

---and the symmetric trimer. In both the cases, there is no entanglement between a pair of spins in the thermal state but multipartite entanglement is present. A second EW based on energy provides an estimate of the entanglement temperature

{T}_{E}

below which the thermal state is definitely entangled. This EW detects bipartite entanglement in the case of the tetramer describing a square of spins (the case of V12 ) and multipartite entanglement in the cases of the tetrahedron and symmetric trimer.