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arXiv:2312.02066 · Physical Review A

Majorization theoretical approach to entanglement enhancement via local filtration

Zacharie Van Herstraeten, Nicolas J. Cerf, Saikat Guha, Christos N. Gagatsos

Published Oct 28, 2024• 0 citations

Abstract

From the perspective of majorization theory, we study how to enhance the entanglement of a two-mode squeezed vacuum (TMSV) state by using local filtration operations. We present several schemes achieving entanglement enhancement with photon addition and subtraction, and then consider filtration as a general probabilistic procedure consisting in acting with local (nonunitary) operators on each mode. From this, we identify a sufficient set of two conditions for these filtration operators to successfully enhance the entanglement of a TMSV state, namely, the operators must be Fock orthogonal (i.e., preserving the orthogonality of Fock states) and Fock amplifying (i.e., giving larger amplitudes to larger Fock states). Our results notably prove that ideal photon addition, subtraction, and any concatenation thereof always enhance the entanglement of a TMSV state in the sense of majorization theory. We further investigate the case of realistic photon addition (subtraction) and are able to upper bound the distance between a realistic photon-added (-subtracted) TMSV state and a nearby state that is provably more entangled than the TMSV, thus extending entanglement enhancement to practical schemes via the use of a notion of approximate majorization. Finally, we consider the state resulting from $k$-photon addition (on each of the two modes) on a TMSV state. We prove analytically that the state corresponding to $k=1$ majorizes any state corresponding to $2\ensuremath{\le}k\ensuremath{\le}8$ and we conjecture the validity of the statement for all $k\ensuremath{\ge}9$.

DOI: 10.1103/physreva.110.042430

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From the perspective of majorization theory, we study how to enhance the entanglement of a two-mode squeezed vacuum (TMSV) state by using local filtration operations. We present several schemes achieving entanglement enhancement with photon addition and subtraction, and then consider filtration as a general probabilistic procedure consisting in acting with local (nonunitary) operators on each mode. From this, we identify a sufficient set of two conditions for these filtration operators to successfully enhance the entanglement of a TMSV state, namely, the operators must be Fock orthogonal (i.e., preserving the orthogonality of Fock states) and Fock amplifying (i.e., giving larger amplitudes to larger Fock states). Our results notably prove that ideal photon addition, subtraction, and any concatenation thereof always enhance the entanglement of a TMSV state in the sense of majorization theory. We further investigate the case of realistic photon addition (subtraction) and are able to upper bound the distance between a realistic photon-added (-subtracted) TMSV state and a nearby state that is provably more entangled than the TMSV, thus extending entanglement enhancement to practical schemes via the use of a notion of approximate majorization. Finally, we consider the state resulting from

k

-photon addition (on each of the two modes) on a TMSV state. We prove analytically that the state corresponding to

k=1

majorizes any state corresponding to

2\ensuremath{\le}k\ensuremath{\le}8

and we conjecture the validity of the statement for all

k\ensuremath{\ge}9

.

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.

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.

For the first time, to the best of our knowledge, based on a two-mode squeezed vacuum (TMSV), we introduce photon-added orthogonal TMSVs and photon-subtracted orthogonal TMSVs via multiphoton addition a † k b † l and multiphoton subtraction a k b l ( k , l are arbitrary integers). As a comparison, we also consider photon-added TMSVs and photon-subtracted TMSVs. We theoretically analyze Einstein–Podolsky–Rosen (EPR) correlation and teleportation fidelity for all these two-mode states. Our results indicate that orthogonalization cannot enhance EPR correlation and teleportation fidelity, at least so far.