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Journal of the Optical Society of America B

Enhancing entanglement of the two-mode squeezed vacuum state by arbitrary combinations of photon additions and subtractions

Yang Yang

Published Jan 2, 2019• 10 citations
Publisher

Abstract

We investigate the problem of enhancing quantum entanglement of the two-mode squeezed vacuum state by asymmetric combinations of photon additions and subtractions. Suppose a combination of N operations is performed on the two-mode squeezed vacuum state, in which each operation may be either photon addition or photon subtraction and may be performed on either mode A or mode B of the state; then there are (N+1)2N possible combinations in all. We calculate in detail the entanglement enhancement by all possible combinations for each given N∈{1,2,…,5,6}, respectively, and find that when the squeezing parameter is small, combinations such as aa†aa†…, a†aa†a…, and their equivalent ones always achieve the greatest enhancement of entanglement, while combinations such as aaaa…, a†a†a†a†…, and their equivalent ones always achieve the lowest enhancement of entanglement; when the squeezing parameter is great enough, entanglement enhancement by each combination is nearly the same. We also compare the maximal and minimal entanglement achieved by such combinations for N=1 to 6, and find that when the squeezing parameter is small, a combination of more operations may generate even less entanglement than a combination of fewer operations.

DOI
10.1364/josab.36.000178

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We investigate the problem of enhancing quantum entanglement of the two-mode squeezed vacuum state by asymmetric combinations of photon additions and subtractions. Suppose a combination of N operations is performed on the two-mode squeezed vacuum state, in which each operation may be either photon addition or photon subtraction and may be performed on either mode A or mode B of the state; then there are (N+1)2N possible combinations in all. We calculate in detail the entanglement enhancement by all possible combinations for each given N∈{1,2,…,5,6}, respectively, and find that when the squeezing parameter is small, combinations such as aa†aa†…, a†aa†a…, and their equivalent ones always achieve the greatest enhancement of entanglement, while combinations such as aaaa…, a†a†a†a†…, and their equivalent ones always achieve the lowest enhancement of entanglement; when the squeezing parameter is great enough, entanglement enhancement by each combination is nearly the same. We also compare the maximal and minimal entanglement achieved by such combinations for N=1 to 6, and find that when the squeezing parameter is small, a combination of more operations may generate even less entanglement than a combination of fewer operations.
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.
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.
Single-photon-level non-Gaussian operations—photon addition, photon subtraction, and their coherent superposition—are powerful tools with which to increase entanglement in continuous-variable optical states. Although such operations are typically not deterministic, there may be other advantages such as noiseless manipulation. Therefore, to fully account for the efficacy of a particular non-Gaussian operation in a practical scenario, we develop figures of merit which trade off the advantages of such protocols against their success probability. Specifically, we define 'entanglement enhancement rate' as the increase in entanglement per trial of a generic non-Gaussian operation on a two-mode squeezed vacuum (TMSV) state. We consider states generated by photon-subtraction, photon-addition and a coherent superposition of subtraction and addition. We compare each strategy when applied to one or both modes of a TMSV state, and also in the presence of channel losses prior to the operation. In many cases, additional properties are analytically calculable, including excess noise arising from the operation and the fidelity of the resulting states to particular Gaussian and non-Gaussian states. Finally, by incorporating loss, we derive optimal interaction parameters for each non-Gaussian operation which maximize the effectiveness of the particular protocol under investigation.
Subtracting photons from a two-mode squeezed state is a well-known method to increase entanglement. We analyze different strategies of local photon subtraction from a two-mode squeezed state in terms of entanglement gain and success probability. We develop a general framework that incorporates imperfections and losses in all stages of the process: before, during, and after subtraction. By combining all three effects into a single efficiency parameter, we provide analytical and numerical results for subtraction strategies using photon-number-resolving and threshold detectors. We compare the entanglement gain afforded by symmetric and asymmetric subtraction scenarios across the two modes. For a given amount of loss, we identify an optimized set of parameters, such as initial squeezing and subtraction beam splitter transmissivity, that maximize the entanglement gain rate. We identify regimes for which asymmetric subtraction of different Fock states on the two modes outperforms symmetric strategies. In the lossless limit, subtracting a single photon from one mode always produces the highest entanglement gain rate. In the lossy case, the optimal strategy depends strongly on the losses on each mode individually, such that there is no general optimal strategy. Rather, taking losses on each mode as the only input parameters, we can identify the optimal subtraction strategy and required beam splitter transmissivities and initial squeezing parameter. Finally, we discuss the implications of our results for the distillation of continuous-variable quantum entanglement.