We investigate the entanglement of the ground state in the quantum networks that their nodes are considered as quantum harmonic oscillators. To this aim, the Schmidt numbers and entanglement entropy between two arbitrary partitions of a network, are calculated. In partitioning an arbitrary graph into two parts there are some nodes in each parts which are not connected to the nodes of the other part. So these nodes of each part, can be in distinct subsets. Therefore the graph separates into four subsets. The nodes of the first and last subsets are those which are not connected to the nodes of other part. In theorem I, by using generalized Schur complement method in these four subsets, we prove that all graphs which their connections between all two alternative subsets are complete, have the same entropy. A large number of graphs satisfy this theorem. Then the entanglement entropy in the limit of large coupling and large size of system, is investigated in these graphs. One of important quantities about partitioning, is conductance of graph. The con- ductance of graph is considered in some various graphs. In these graphs we compare the conductance of graph and the entanglement entropy.We investigate the entanglement of the ground state in the quantum networks that their nodes are considered as quantum harmonic oscillators. To this aim, the Schmidt numbers and entanglement entropy between two arbitrary partitions of a network, are calculated. In partitioning an arbitrary graph into two parts there are some nodes in each parts which are not connected to the nodes of the other part. So these nodes of each part, can be in distinct subsets. Therefore the graph separates into four subsets. The nodes of the first and last subsets are those which are not connected to the nodes of other part. In theorem I, by using generalized Schur complement method in these four subsets, we prove that all graphs which their connections between all two alternative subsets are complete, have the same entropy. A large number of graphs satisfy this theorem. Then the entanglement entropy in the limit of large coupling and large size of system, is investigated in these graphs. One of important quantities about partitioning, is conductance of graph. The con- ductance of graph is considered in some various graphs. In these graphs we compare the conductance of graph and the entanglement entropy.

arXiv:1407.4044Jul 15, 20140 citations

Entanglement entropy in the Hamming networks

M. A. Jafarizadeh, S. Nami, Eghbalifam, F.

We investigate the Hamming networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information each part of a network shares with the rest of the system via quantum fluctuations. Therefore, the Schmidt numbers and entanglement entropy between two special parts of Hamming network, can be calculated. To this aim, first we use the stratification method to rewrite the adjacency matrix of the network in the stratification basis. Then the entanglement entropy and Schmidt number for special partitions are calculated analytically by using the generalized Schur complement method. Also, we calculate the entanglement entropy between two arbitrary subsets (two equal subsets have the same number of vertices) in H(2; 3) and H(2; 4) numerically, and we give the minimum and maximum values of entanglement entropy in these two Hamming network.We investigate the Hamming networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information each part of a network shares with the rest of the system via quantum fluctuations. Therefore, the Schmidt numbers and entanglement entropy between two special parts of Hamming network, can be calculated. To this aim, first we use the stratification method to rewrite the adjacency matrix of the network in the stratification basis. Then the entanglement entropy and Schmidt number for special partitions are calculated analytically by using the generalized Schur complement method. Also, we calculate the entanglement entropy between two arbitrary subsets (two equal subsets have the same number of vertices) in H(2; 3) and H(2; 4) numerically, and we give the minimum and maximum values of entanglement entropy in these two Hamming network.

arXiv:1503.04986Mar 17, 20153 citations

Entanglement entropy in the Hypercube networks

M. A. Jafarizadeh, F. Eghbalifam, S. Nami

We investigate the hypercube networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information each part of a network shares with the rest of the system via quantum fluctuations. Therefore, the Schmidt numbers and entanglement entropy between two special parts of Hypercube network, can be calculated. To this aim, first we use the stratification method to rewrite the adjacency matrix of the network in the stratification basis which is the matrix representation of the angular momentum. Then the entanglement entropy and Schmidt number for special partitions are calculated an- alytically by using the generalized Schur complement method. Also, we calculate the entanglement entropy between two arbitrary equal subsets (two equal subsets have the same number of vertices) in H(3, 2) and H(4, 2) numerically, and we give the minimum and maximum values of entanglement entropy in these two Hypercube network. Then we can conjecture the minimum and maximum values of entanglement entropy for equal subsets in H(d, 2).We investigate the hypercube networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information each part of a network shares with the rest of the system via quantum fluctuations. Therefore, the Schmidt numbers and entanglement entropy between two special parts of Hypercube network, can be calculated. To this aim, first we use the stratification method to rewrite the adjacency matrix of the network in the stratification basis which is the matrix representation of the angular momentum. Then the entanglement entropy and Schmidt number for special partitions are calculated an- alytically by using the generalized Schur complement method. Also, we calculate the entanglement entropy between two arbitrary equal subsets (two equal subsets have the same number of vertices) in H(3, 2) and H(4, 2) numerically, and we give the minimum and maximum values of entanglement entropy in these two Hypercube network. Then we can conjecture the minimum and maximum values of entanglement entropy for equal subsets in H(d, 2).

arXiv:1411.0310Nov 2, 20142 citations

Entanglement of Coupled Harmonic Oscillators

H. Snijder

The entropy of a black hole scales with its area. This is a puzzling feature in the context of statistical mechanics, where entropy scales with volume. It is argued that entanglement entropy underlays such an area law. This quantity is hard to calculate because a precise understanding of the underlying quantum state is needed. Therefore it is interesting to study a Schwarzschild analogue in a photon condensate which is an exactly known quantum system in the ground-state mimicking the behaviour of an astronomic black hole. A harmonic chain of size $N$ with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out $N-l$ oscillators. This separation of two regions can be interpret as separating the inner and outer regions of a black hole. The resulting entanglement entropy is considered for the discrete chain as well for $N\\rightarrow\\infty$ , which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large $N$ , the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by $S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421$ in the discrete case and $S=\\frac{0.3744}{3} \\ln{l} + 0.6421$ in the infinite case.The entropy of a black hole scales with its area. This is a puzzling feature in the context of statistical mechanics, where entropy scales with volume. It is argued that entanglement entropy underlays such an area law. This quantity is hard to calculate because a precise understanding of the underlying quantum state is needed. Therefore it is interesting to study a Schwarzschild analogue in a photon condensate which is an exactly known quantum system in the ground-state mimicking the behaviour of an astronomic black hole. A harmonic chain of size $N$ with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out $N-l$ oscillators. This separation of two regions can be interpret as separating the inner and outer regions of a black hole. The resulting entanglement entropy is considered for the discrete chain as well for $N\\rightarrow\\infty$, which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large $N$, the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by $S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421$ in the discrete case and $S=\\frac{0.3744}{3} \\ln{l} + 0.6421$ in the infinite case.

Jan 1, 20190 citations

Dynamics of quantum entanglement in two time-dependent coupled harmonic oscillators

Ayoub Ghaba, Radouan Hab-arrih, Elhoussine Atmani +1 more

We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy $S_L=1-\mathcal{P}$ for arbitrary excitation numbers $(n,m)$ , allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters $θ$ and $\vartheta_2$ , the frequency parameter $β_0$ and the coupling strength $ε$ : the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case $ω_1=ω_2=1$ and strong couplings $ε\approx 0.99$ , the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy $S_L=1-\mathcal{P}$ for arbitrary excitation numbers $(n,m)$, allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters $θ$ and $\vartheta_2$, the frequency parameter $β_0$ and the coupling strength $ε$: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case $ω_1=ω_2=1$ and strong couplings $ε\approx 0.99$, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.

quant-pharXiv:2606.29617Jun 30, 20260 citations

Information sharing in quantum complex networks

Alessio Cardillo, Fernando Galve, David Zueco +1 more

We introduce the use of entanglement entropy as a tool for studying the amount of information shared between the nodes of quantum complex networks. By considering the ground state of a network of coupled quantum harmonic oscillators, we compute the information that each node has on the rest of the system. We show that the nodes storing the largest amount of information are not the ones with the highest connectivity, but those with intermediate connectivity, thus breaking down the usual hierarchical picture of classical networks. We show both numerically and analytically that the mutual information characterizes the network topology. As a by-product, our results point out that the amount of information available for an external node connecting to a quantum network allows one to determine the network topology.We introduce the use of entanglement entropy as a tool for studying the amount of information shared between the nodes of quantum complex networks. By considering the ground state of a network of coupled quantum harmonic oscillators, we compute the information that each node has on the rest of the system. We show that the nodes storing the largest amount of information are not the ones with the highest connectivity, but those with intermediate connectivity, thus breaking down the usual hierarchical picture of classical networks. We show both numerically and analytically that the mutual information characterizes the network topology. As a by-product, our results point out that the amount of information available for an external node connecting to a quantum network allows one to determine the network topology.

arXiv:1211.2580Physical Review AMay 15, 201317 citations

N

with a constant spring constant is studied as a toy model, of which the entanglement entropy is calculated by tracing out

N-l

oscillators. This separation of two regions can be interpret as separating the inner and outer regions of a black hole. The resulting entanglement entropy is considered for the discrete chain as well for

N\\rightarrow\\infty

, which represents the Klein-Gordon field on a flat background and with a UV cut-off. For large

N

, the entropy converge to the limit case. The performed numerical simulations hints at an area law with logarithmic corrections, given by

S=\\frac{0.2776}{3} \\ln{\\bigg{(} \\frac{N}{\\pi }\\sin{\\frac{\\pi l}{N}}\\bigg{)}} + 0.6421

in the discrete case and

S=\\frac{0.3744}{3} \\ln{l} + 0.6421

in the infinite case.

S_L=1-\mathcal{P}

for arbitrary excitation numbers

(n,m)

, allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters

θ

and

\vartheta_2

, the frequency parameter

β_0

and the coupling strength

ε

: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case

ω_1=ω_2=1

and strong couplings

ε\approx 0.99

, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.