We investigate the entanglement patterns of photon-added and photon-subtracted four-mode squeezed vacuum states. Entanglements in different scenarios are analyzed by varying the number of photons added or subtracted in certain modes, which are referred to as the ``player'' modes, the others being ``spectators.'' We find that the photon-subtracted state can give us higher entanglement than the photon-added state which is in contrast to the two-mode situation. We also study the logarithmic negativity of the two-mode reduced density matrix obtained from the four-mode state which again shows that the state after photon subtraction can possess higher entanglement than that of the photon-added state, and we then compare it to that of the two-mode squeezed vacuum state. Moreover, we examine the non-Gaussianity of the photon-added and photon-subtracted states to find that the rich features provided by entanglement cannot be captured by the measure of nonclassicality.
We investigate the entanglement patterns of photon-added and photon-subtracted four-mode squeezed vacuum states. Entanglements in different scenarios are analyzed by varying the number of photons added or subtracted in certain modes, which are referred to as the player'' modes, the others being spectators.'' We find that the photon-subtracted state can give us higher entanglement than the photon-added state which is in contrast to the two-mode situation. We also study the logarithmic negativity of the two-mode reduced density matrix obtained from the four-mode state which again shows that the state after photon subtraction can possess higher entanglement than that of the photon-added state, and we then compare it to that of the two-mode squeezed vacuum state. Moreover, we examine the non-Gaussianity of the photon-added and photon-subtracted states to find that the rich features provided by entanglement cannot be captured by the measure of nonclassicality.
The non-Gaussian operations effected by adding or subtracting a photon on entangled optical beams emerging from a parametric down-conversion process have been suggested to enhance entanglement. Heralded photon addition or subtraction is, as a matter of fact, at the heart of continuous-variable entanglement distillation. The use of such processes has recently been experimentally demonstrated in the context of the generation of optical coherent-state superpositions or the verification of canonical commutation relations. Here, we carry out a systematic study of the effect of local photon additions and subtractions on a two-mode squeezed vacuum state, showing that the entanglement generally increases with the number of such operations. This is analytically proven when additions or subtractions are restricted to one mode only, while we observe that the highest entanglement is achieved when these operations are equally shared between the two modes. We also note that adding photons typically provides a stronger entanglement enhancement than subtracting photons, while photon subtraction performs better in terms of energy efficiency. Furthermore, we analyze the interplay between entanglement and non-Gaussianity, showing that it is more subtle than previously expected.
The entanglement properties of non-Gaussian states are investigated, which are obtained by performing the photon addition, photon subtraction, photon-addition-then-subtraction, and photon-subtraction-then-addition operations on the two-mode squeezed vacuum state. We show that the partial von Neumann entropy of all the resulting states is greater than that of the original squeezed state, but only the photon-subtracted states and the photon-added-then-subtracted states have the stronger Einstein-Podolsky-Rosen correlation than the original squeezed state. Quantum teleportation of Braunstein and Kimble protocol is studied for coherent states, squeezed states, and mixed Gaussian states with the non-Gaussian entangled resources. For all the states to be teleported, the fidelity with the photon-subtracted and the photon-added-then-subtracted entangled resources is higher than that with the two-mode squeezed vacuum resource. Based on Bures fidelity, we find that quantum teleportation for mixed and classical single-mode Gaussian states is more faithful than for single-mode Gaussian states with high purity and nonclassicality.
We theoretically analyze the Einstein-Podolsky-Rosen (EPR) correlation, the quadrature squeezing, and the continuous-variable quantum teleportation when considering non-Gaussian entangled states generated by applying multiple-photon subtraction and multiple-photon addition to a two-mode squeezed vacuum state (TMSVs). Our results indicate that in the case of the multiple-photon-subtracted TMSVs with symmetric operations, the corresponding EPR correlation, the two-mode squeezing degree, the sum squeezing, and the fidelity of teleporting a coherent state or a squeezed vacuum state can be enhanced for any squeezing parameter r and these enhancements increase with the number of subtracted photons in the low-squeezing regime, while asymmetric multiple-photon subtractions will generally reduce these quantities. For the multiple-photon-added TMSVs, although it holds stronger entanglement, its EPR correlation, two-mode squeezing, sum squeezing, and the fidelity of a coherent state are always smaller than that of the TMSVs. Only when considering the case of teleporting a squeezed vacuum state does the symmetric photon addition make somewhat of an improvement in the fidelity for large-squeezing parameters. In addition, we analytically prove that a one-mode multiple-photon-subtracted TMSVs is equivalent to that of the one-mode multiple-photon-added one. And one-mode multiple-photon operations will diminish the above four quantities for any squeezing parameter r.
We investigate the entanglement swapping of non-Gaussian states, including squeezed number states, two-mode photon-added states, and two-mode photon-subtracted squeezed states and analyze the entanglement of the swapped states by adopting logarithmic negativity as the measure of entanglement. Furthermore, we examine the fidelity of teleportation protocols for different input states where the swapped states serve as the quantum channel. All these results are compared with that obtained in the case of Gaussian two-mode squeezed vacuum.
We investigate the entanglement enhancement of the two-mode squeezed vacuum state using various combinations of photon additions and photon subtractions. We suppose that a sequence of N operations are performed on each mode and on a single mode of the two-mode squeezed vacuum state, each operation being photon addition or photon subtraction; then there are 2N possible combinations of operations in total. We calculate the entanglement for all the photon-added/subtracted states generated by all possible combinations for N=2, N=3, and N=4, respectively, and deduce that the combination of operations alternating between photon addition and photon subtraction enhances the entanglement most effectively, while adding or subtracting N photons repeatedly enhances the entanglement least effectively.
Genuine multimode entanglement in continuous variable systems can be quantified by exploring the geometry of the state space, namely, via the generalized geometric measure (GGM). It is defined by the shortest distance of a given multimode state from a nongenuinely multimode entangled state. For the multimode Gaussian states, we derive a closed-form expression of GGM in terms of the symplectic eigenvalues of the reduced states. Following that prescription, the characteristics of GGM for typical three- and four-mode Gaussian states are investigated. In the non-Gaussian paradigm, we compute GGM for photon-added as well as -subtracted states having three and four modes and find that both addition and subtraction of photons enhance the genuine multimode entanglement of the state compared to its Gaussian counterpart. Our analysis reveals that, when an initial three-mode vacuum state is evolved according to an interacting Hamiltonian, photon addition is more beneficial in increasing GGM compared to photon subtraction, while the scenario reverses when one considers the four mode non-Gaussian states. Specifically, subtracting photons from four-mode squeezed vacuum states almost always results in higher multimode entanglement content than that of photon addition to both single as well as multimode and constrained as well as unconstrained operations. Furthermore, we observe that GGM freezes under subtraction of photons involving multiple modes, in some specific cases. This feature does not appear while adding photons. Finally, we relate the enhancements of GGM with the distance-based non-Gaussianity measure.