We present entanglement measures between spatially separated regions and between two distinguishable or indistinguishable particles in one-dimensional two-particle continuous-time quantum walks governed by the Hubbard Hamiltonian. The left-right entanglement checks the entropy of coarse-grained states counting the numbers of particles on the left and right halves of the lattice while the particle-particle entanglement is based on the entropy of the singular values of the time-evolved Fock state. With separable, entangled, and doubly occupied initial states, we examine initial entanglement and the following growth in different entanglement measures. While the entanglement measures of the indistinguishable cases resemble those of the distinguishable cases when the initial states are comparable, the long-time limits of the entanglement measures are typically non-monotonic as the onsite repulsion increases. We also discuss possible implications for future research of entanglement in multi-particle quantum dynamics.We present entanglement measures between spatially separated regions and between two distinguishable or indistinguishable particles in one-dimensional two-particle continuous-time quantum walks governed by the Hubbard Hamiltonian. The left-right entanglement checks the entropy of coarse-grained states counting the numbers of particles on the left and right halves of the lattice while the particle-particle entanglement is based on the entropy of the singular values of the time-evolved Fock state. With separable, entangled, and doubly occupied initial states, we examine initial entanglement and the following growth in different entanglement measures. While the entanglement measures of the indistinguishable cases resemble those of the distinguishable cases when the initial states are comparable, the long-time limits of the entanglement measures are typically non-monotonic as the onsite repulsion increases. We also discuss possible implications for future research of entanglement in multi-particle quantum dynamics.

quant-pharXiv:2606.02505Jun 2, 20260 citations

Measuring Many-Body Entanglement

Anonymous

PhysicsJul 12, 20120 citations

Introduction Of Quantum Entanglement Measure Based On The Expectation Values Of Pauli Operators

Mahmood Zeheiry

In this paper, firstly considering that in separable states, the measurement of one particle has no effect on the measurement of the second particle, we show that Alice and Bob can find directions in which the results of their measurements on the spin of the particle are always maximized. In other words, the state of the particle is an eigenstate for the operator that is applied in that direction, so the sum of the spins of two particles can have a maximum value. We will argue that in entangled states, due to the effect of particle measurement results on each other, Alice and Bob cannot find the desired operators. Therefore, in such measurements, the total spin of the particles will always be less than the mentioned maximum. But we ask them to try and measure in directions that will get the most value. Because this value is maximum for separable states and minimum for fully entangled states, and for the rest of the states, it will be proportional to the degree of entanglement between the two maximum and minimum values, we set this parameter as We are calling it the "separability index". Then, based on this index, the measure of entanglement was introduced and extended to states with higher dimensions. In the end, examples of di-qubit states di-qutrit states, and di-qudit states were investigated and the efficiency of the measure was confirmed by the results of the examples. Considering that in this measure, the values of entanglement are calculated based on the expectation values and we can measure the expectation values in the experiment, we hope to be one step closer to the testability of the entanglement value.In this paper, firstly considering that in separable states, the measurement of one particle has no effect on the measurement of the second particle, we show that Alice and Bob can find directions in which the results of their measurements on the spin of the particle are always maximized. In other words, the state of the particle is an eigenstate for the operator that is applied in that direction, so the sum of the spins of two particles can have a maximum value. We will argue that in entangled states, due to the effect of particle measurement results on each other, Alice and Bob cannot find the desired operators. Therefore, in such measurements, the total spin of the particles will always be less than the mentioned maximum. But we ask them to try and measure in directions that will get the most value. Because this value is maximum for separable states and minimum for fully entangled states, and for the rest of the states, it will be proportional to the degree of entanglement between the two maximum and minimum values, we set this parameter as We are calling it the "separability index". Then, based on this index, the measure of entanglement was introduced and extended to states with higher dimensions. In the end, examples of di-qubit states di-qutrit states, and di-qudit states were investigated and the efficiency of the measure was confirmed by the results of the examples. Considering that in this measure, the values of entanglement are calculated based on the expectation values and we can measure the expectation values in the experiment, we hope to be one step closer to the testability of the entanglement value.

arXiv:2211.08981Nov 14, 20220 citations

Entanglement of Indistinguishable Particles and its Quantification

Florian Sokoli, Burkhard Kümmerer

We introduce geometric measures of entanglement for indistinguishable particles, which apply to mixed states, multipartite systems, and arbitrary dimensions. They are based on generalized (i.e., not necessarily finite) norms on the set of quantum states and lead to the first necessary and sufficient computational separability criterion in this general setting. The coherent approach developed in the paper allows us to compare, in particular, entanglement for fermionic and distinguishable particles: The entanglement measure for fermionic particles coincides with the corresponding entanglement measure for distinguishable particles up to a factor of $k!$ where $k$ is the number of particles involved. By this result the amount of entanglement emerging from fermi statistics alone is clearly separated from the overall amount of entanglement. Finally, our techniques are applied to entanglement related to Schmidt and Slater numbers.We introduce geometric measures of entanglement for indistinguishable particles, which apply to mixed states, multipartite systems, and arbitrary dimensions. They are based on generalized (i.e., not necessarily finite) norms on the set of quantum states and lead to the first necessary and sufficient computational separability criterion in this general setting. The coherent approach developed in the paper allows us to compare, in particular, entanglement for fermionic and distinguishable particles: The entanglement measure for fermionic particles coincides with the corresponding entanglement measure for distinguishable particles up to a factor of $k!$ where $k$ is the number of particles involved. By this result the amount of entanglement emerging from fermi statistics alone is clearly separated from the overall amount of entanglement. Finally, our techniques are applied to entanglement related to Schmidt and Slater numbers.

arXiv:1507.04615Jul 16, 20150 citations

Measures of Quantum Entanglement

P. K. Aravind

Springer eBooksJan 1, 19962 citations

k!

where

k

is the number of particles involved. By this result the amount of entanglement emerging from fermi statistics alone is clearly separated from the overall amount of entanglement. Finally, our techniques are applied to entanglement related to Schmidt and Slater numbers.