We present detailed analytical calculations for an 1D Ising ring of arbitrary number of spin-1/2 particles, in order to reveal entanglement properties of the stationary states. We show that the ground state and specific eigenstates of the Ising Hamiltonian posses remarkable entanglement properties that can reveal new insight into quantum correlations present in the Ising model. This correlations might be exploited in quantum information processing. We propose an intuitive picture of the behaviour of multipartite entanglement and discuss a relation of our results to some aspects of phase transitions in the Ising model.We present detailed analytical calculations for an 1D Ising ring of arbitrary number of spin-1/2 particles, in order to reveal entanglement properties of the stationary states. We show that the ground state and specific eigenstates of the Ising Hamiltonian posses remarkable entanglement properties that can reveal new insight into quantum correlations present in the Ising model. This correlations might be exploited in quantum information processing. We propose an intuitive picture of the behaviour of multipartite entanglement and discuss a relation of our results to some aspects of phase transitions in the Ising model.

arXiv:quant-ph/0312154Dec 18, 20031 citation

On the ground state of Ising chains with finite range interactions

M. Bundaru, N. Angelescu, G. Nenciu

Physics Letters AFeb 1, 197313 citations

The phase-diagram of the Blume-Capel-Haldane-Ising spin chain

Christian Boudreault, S. A. Owerre, Manu B. Paranjape

We consider the one-dimensional spin chain for arbitrary spin $s$ on a periodic chain with $N$ sites, the generalization of the chain that was studied by Blume and Capel \cite{bc}: $H=\sum_{i=1}^N \left(a (S^z_i)^2+ b S^z_iS^z_{i+1}\right).$ The Hamiltonian only involves the $z$ component of the spin thus it is essentially an Ising \cite{Ising} model. The Hamiltonian also figures exactly as the anisotropic term in the famous model studied by Haldane \cite{haldane} of the large spin Heisenberg spin chain \cite{bethe}. Therefore we call the model the Blume-Capel-Haldane-Ising model. Although the Hamiltonian is trivially diagonal, it is actually not always obvious which eigenstate is the ground state. In this paper we establish which state is the ground state for all regions of the parameter space and thus determine the phase diagram of the model. We observe the existence of solitons-like excitations and we show that the size of the solitons depends only on the ratio $a/b$ and not on the number of sites $N$ . Therefore the size of the soliton is an intrinsic property of the soliton not determined by boundary conditions.We consider the one-dimensional spin chain for arbitrary spin $s$ on a periodic chain with $N$ sites, the generalization of the chain that was studied by Blume and Capel \cite{bc}: $$H=\sum_{i=1}^N \left(a (S^z_i)^2+ b S^z_iS^z_{i+1}\right).$$ The Hamiltonian only involves the $z$ component of the spin thus it is essentially an Ising \cite{Ising} model. The Hamiltonian also figures exactly as the anisotropic term in the famous model studied by Haldane \cite{haldane} of the large spin Heisenberg spin chain \cite{bethe}. Therefore we call the model the Blume-Capel-Haldane-Ising model. Although the Hamiltonian is trivially diagonal, it is actually not always obvious which eigenstate is the ground state. In this paper we establish which state is the ground state for all regions of the parameter space and thus determine the phase diagram of the model. We observe the existence of solitons-like excitations and we show that the size of the solitons depends only on the ratio $a/b$ and not on the number of sites $N$. Therefore the size of the soliton is an intrinsic property of the soliton not determined by boundary conditions.

arXiv:1601.06195Jan 22, 20160 citations

Ground- and excited-state energies extraction via Trotterization on IBM quantum computers

Fernando Espinoza-Ortiz, Chungwei Lin, Chih-Chun Chien

We implement the Hadamard test with Trotterized time-evolution operators on IBM quantum computers to simultaneously extract ground- and excited-state energies of the transverse field Ising model (TFIM) and transverse longitudinal field Ising model (TLFIM). The Trotterization circuits for the TFIM admit constant-depth circuits (CDCs) for arbitrary time, allowing us to locate a large number of eigen-energies above the background noise for up to six spins. Via circuit synthesis we show that the three-spin TLFIM has constant-depth structure although it does not meet the known CDC criteria. The CDCs enable the extraction of the ground and first-excited state energies of the three-site TLFIM via its dynamics. We also address complications from the noisy background and discrete Fourier transform to enhance the reliability of the extraction process and compare the results from different generations of IBM hardware to highlight the improvement.We implement the Hadamard test with Trotterized time-evolution operators on IBM quantum computers to simultaneously extract ground- and excited-state energies of the transverse field Ising model (TFIM) and transverse longitudinal field Ising model (TLFIM). The Trotterization circuits for the TFIM admit constant-depth circuits (CDCs) for arbitrary time, allowing us to locate a large number of eigen-energies above the background noise for up to six spins. Via circuit synthesis we show that the three-spin TLFIM has constant-depth structure although it does not meet the known CDC criteria. The CDCs enable the extraction of the ground and first-excited state energies of the three-site TLFIM via its dynamics. We also address complications from the noisy background and discrete Fourier transform to enhance the reliability of the extraction process and compare the results from different generations of IBM hardware to highlight the improvement.

quant-pharXiv:2606.18534Jun 18, 20260 citations

Quantum-information approach to the Ising model: Entanglement in chains of qubits

Peter Štelmachovič, Vladimír Bužek

Simple physical interactions between spin- $1∕2$ particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite systems. In particular, we present a detailed investigation of the well-known Ising model of a chain (ring) of spin- $1∕2$ particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin- $1∕2$ particles in the chain in the standard (computer) basis, and we investigate quantum entanglement between individual qubits. We analyze bipartite as well as multipartite entanglement in the ground state of the model. In particular, we show that bipartite entanglement between pairs of qubits of the Ising chain (measured in terms of a concurrence) as a function of the parameter $\ensuremath{\lambda}$ has a maximum around the point $\ensuremath{\lambda}=1$, and it monotonically decreases for large values of $\ensuremath{\lambda}$. We prove that in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ this state is locally unitary equivalent to an $N$ -partite Greenberger-Horn-Zeilinger state. We also analyze a very specific eigenstate of the Ising Hamiltonian with a zero eigenenergy (we denote this eigenstate as the $X$ -state). This $X$ -state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset $A$ of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of $N=2n+1$ qubits is maximally entangled with the remaining qubits (set $B$ ) in the chain. In addition, we prove that by performing a local operation just on the subset $B$ , one can transform the $X$ -state into a direct product of $k$ singlets shared by the parties $A$ and $B$ . This property of the $X$ -state can be utilized for new secure multipartite communication protocols.Simple physical interactions between spin-$1∕2$ particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite systems. In particular, we present a detailed investigation of the well-known Ising model of a chain (ring) of spin-$1∕2$ particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin-$1∕2$ particles in the chain in the standard (computer) basis, and we investigate quantum entanglement between individual qubits. We analyze bipartite as well as multipartite entanglement in the ground state of the model. In particular, we show that bipartite entanglement between pairs of qubits of the Ising chain (measured in terms of a concurrence) as a function of the parameter $\ensuremath{\lambda}$ has a maximum around the point $\ensuremath{\lambda}=1$, and it monotonically decreases for large values of $\ensuremath{\lambda}$. We prove that in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ this state is locally unitary equivalent to an $N$-partite Greenberger-Horn-Zeilinger state. We also analyze a very specific eigenstate of the Ising Hamiltonian with a zero eigenenergy (we denote this eigenstate as the $X$-state). This $X$-state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset $A$ of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of $N=2n+1$ qubits is maximally entangled with the remaining qubits (set $B$) in the chain. In addition, we prove that by performing a local operation just on the subset $B$, one can transform the $X$-state into a direct product of $k$ singlets shared by the parties $A$ and $B$. This property of the $X$-state can be utilized for new secure multipartite communication protocols.

Physical Review ASep 17, 200485 citations

We consider the one-dimensional spin chain for arbitrary spin

s

on a periodic chain with

N

sites, the generalization of the chain that was studied by Blume and Capel \cite{bc}:

H=\sum_{i=1}^N \left(a (S^z_i)^2+ b S^z_iS^z_{i+1}\right).

The Hamiltonian only involves the

z

component of the spin thus it is essentially an Ising \cite{Ising} model. The Hamiltonian also figures exactly as the anisotropic term in the famous model studied by Haldane \cite{haldane} of the large spin Heisenberg spin chain \cite{bethe}. Therefore we call the model the Blume-Capel-Haldane-Ising model. Although the Hamiltonian is trivially diagonal, it is actually not always obvious which eigenstate is the ground state. In this paper we establish which state is the ground state for all regions of the parameter space and thus determine the phase diagram of the model. We observe the existence of solitons-like excitations and we show that the size of the solitons depends only on the ratio

a/b

and not on the number of sites

N

. Therefore the size of the soliton is an intrinsic property of the soliton not determined by boundary conditions.

Simple physical interactions between spin-

1∕2

particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite systems. In particular, we present a detailed investigation of the well-known Ising model of a chain (ring) of spin-

1∕2

particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin-

1∕2

particles in the chain in the standard (computer) basis, and we investigate quantum entanglement between individual qubits. We analyze bipartite as well as multipartite entanglement in the ground state of the model. In particular, we show that bipartite entanglement between pairs of qubits of the Ising chain (measured in terms of a concurrence) as a function of the parameter $\ensuremath{\lambda}$ has a maximum around the point $\ensuremath{\lambda}=1$, and it monotonically decreases for large values of $\ensuremath{\lambda}$. We prove that in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ this state is locally unitary equivalent to an

N

-partite Greenberger-Horn-Zeilinger state. We also analyze a very specific eigenstate of the Ising Hamiltonian with a zero eigenenergy (we denote this eigenstate as the

X

-state). This

X

-state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset

A

of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of

N=2n+1

qubits is maximally entangled with the remaining qubits (set

B

) in the chain. In addition, we prove that by performing a local operation just on the subset

B

, one can transform the

X

-state into a direct product of

k

singlets shared by the parties

A

and

B

. This property of the

X

-state can be utilized for new secure multipartite communication protocols.