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

Yang Yang

doi:10.1364/JOSAB.36.000178

Journal of the Optical Society of America B
Vol. 36,
Issue 2,
pp. 178-186
(2019)
•https://doi.org/10.1364/JOSAB.36.000178

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

Yang Yang

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Yang Yang, "Enhancing entanglement of the two-mode squeezed vacuum state by arbitrary combinations of photon additions and subtractions," J. Opt. Soc. Am. B 36, 178-186 (2019)

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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 $(N + 1) 2^{N}$ possible combinations in all. We calculate in detail the entanglement enhancement by all possible combinations for each given $N \in {1, 2, \dots, 5, 6}$ , respectively, and find that when the squeezing parameter is small, combinations such as $a a^{†} a a^{†} \dots$ , $a^{†} a a^{†} a \dots$ , and their equivalent ones always achieve the greatest enhancement of entanglement, while combinations such as $a a a a \dots$ , $a^{†} a^{†} a^{†} a^{†} \dots$ , 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.

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Equations (15)

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CP	(1, 0)	(0, 1)
$f_{1}^{(1)} (n) = n + 1$	$a, a^{†}$	$b, b^{†}$

CPs	(2, 0)	(1, 1)	(0, 2)
$f_{1}^{(2)} (n) = {(n + 1)}^{2}$	$a a^{†}, a^{†} a$	$a b, a^{†} b^{†}$	$b b^{†}, b^{†} b$
$f_{2}^{(2)} (n) = (n + 1) (n + 2)$	$a^{2}, a^{† 2}$	$a b^{†}, a^{†} b$	$b^{2}, b^{† 2}$

CPs	(3, 0)	(2, 1)
$f_{1}^{(3)} (n) = {(n + 1)}^{3}$	$a a^{†} a, a^{†} a a^{†}$	$a^{†} a b, a a^{†} b^{†}$
$f_{2}^{(3)} (n) = {(n + 1)}^{2} (n + 2)$	$a^{† 2} a, a^{†} a^{2}$	$a^{†} a b^{†}, a^{† 2} b^{†}$
$f_{3}^{(3)} (n) = (n + 1) {(n + 2)}^{2}$	$a a^{† 2}, a^{2} a^{†}$	$a^{2} b, a a^{†} b$
$f_{4}^{(3)} (n) = (n + 1) (n + 2) (n + 3)$	$a^{† 3}, a^{3}$	$a^{† 2} b, a^{2} b^{†}$
	$(1, 2)$	$(0, 3)$
	$a b^{†} b, a^{†} b b^{†}$	$b b^{†} b, b^{†} b b^{†}$
	$a^{†} b^{† 2}, a^{†} b^{†} b$	$b^{† 2} b, b^{†} b^{2}$
	$a b^{2}, a b b^{†}$	$b b^{† 2}, b^{2} b^{†}$
	$a b^{† 2}, a^{†} b^{2}$	$b^{† 3}, b^{3}$

CPs	(4, 0)
$f_{1}^{(4)} (n) = {(n + 1)}^{4}$	$a a^{†} a a^{†}, a^{†} a a^{†} a$
$f_{2}^{(4)} (n) = {(n + 1)}^{3} (n + 2)$	$a^{† 2} a a^{†}, a a^{†} a^{2}$
$f_{3}^{(4)} (n) = {(n + 1)}^{2} {(n + 2)}^{2}$	$a^{† 2} a^{2}, a^{2} a^{† 2}, a^{†} a^{2} a^{†}, a a^{† 2} a$
$f_{4}^{(4)} (n) = {(n + 1)}^{2} (n + 2) (n + 3)$	$a^{† 3} a, a^{†} a^{3}$
$f_{5}^{(4)} (n) = (n + 1) {(n + 2)}^{3}$	$a^{†} a a^{† 2}, a^{2} a^{†} a$
$f_{6}^{(4)} (n) = (n + 1) (n + 2) {(n + 3)}^{2}$	$a a^{† 3}, a^{3} a^{†}$
$f_{7}^{(4)} (n) = (n + 1) (n + 2) (n + 3) (n + 4)$	$a^{4}, a^{† 4}$

CPs	(5, 0)
$f_{1}^{(5)} (n) = {(n + 1)}^{5}$	$a a^{†} a a^{†} a, a^{†} a a^{†} a a^{†}$
$f_{2}^{(5)} (n) = {(n + 1)}^{4} (n + 2)$	$a^{†} a a^{†} a a, a^{†} a^{†} a a^{†} a$
$f_{3}^{(5)} (n) = {(n + 1)}^{3} {(n + 2)}^{2}$	$a a a^{†} a^{†} a, a a^{†} a a a^{†}, a a^{†} a^{†} a a^{†}, a^{†} a a a^{†} a^{†}$
$f_{4}^{(5)} (n) = {(n + 1)}^{3} (n + 2) (n + 3)$	$a a^{†} a a a, a^{†} a^{†} a^{†} a a^{†}$
$f_{5}^{(5)} (n) = {(n + 1)}^{2} {(n + 2)}^{3}$	$a a^{†} a^{†} a a, a^{†} a a a^{†} a, a^{†} a a^{†} a^{†} a, a^{†} a^{†} a a a^{†}$
$f_{6}^{(5)} (n) = {(n + 1)}^{2} {(n + 2)}^{2} (n + 3)$	$a^{†} a^{†} a a a, a^{†} a^{†} a^{†} a a$
$f_{7}^{(5)} (n) = (n + 1) {(n + 2)}^{4}$	$a a a^{†} a a^{†}, a a^{†} a a^{†} a^{†}$
$f_{8}^{(5)} (n) = {(n + 1)}^{2} (n + 2) {(n + 3)}^{2}$	$a a^{†} a^{†} a^{†} a, a^{†} a a a a^{†}$
$f_{9}^{(5)} (n) = (n + 1) {(n + 2)}^{3} (n + 3)$	$a a a^{†} a a, a^{†} a^{†} a a^{†} a^{†}$
$f_{10}^{(5)} (n) = {(n + 1)}^{2} (n + 2) (n + 3) (n + 4)$	$a^{†} a a a a, a^{†} a^{†} a^{†} a^{†} a$
$f_{11}^{(5)} (n) = (n + 1) {(n + 2)}^{2} {(n + 3)}^{2}$	$a a a a^{†} a^{†}, a a a^{†} a^{†} a^{†}$
$f_{12}^{(5)} (n) = (n + 1) (n + 2) {(n + 3)}^{3}$	$a a a a^{†} a, a^{†} a a^{†} a^{†} a^{†}$
$f_{13}^{(5)} (n) = (n + 1) (n + 2) (n + 3) {(n + 4)}^{2}$	$a a a a a^{†}, a a^{†} a^{†} a^{†} a^{†}$
$f_{14}^{(5)} (n) = (n + 1) (n + 2) (n + 3) (n + 4) (n + 5)$	$a a a a a, a^{†} a^{†} a^{†} a^{†} a^{†}$

CPs	(6, 0)
$f_{1}^{(6)} (n) = {(n + 1)}^{6}$	$a a^{†} a a^{†} a a^{†}, a^{†} a a^{†} a a^{†} a$
$f_{2}^{(6)} (n) = {(n + 1)}^{5} (n + 2)$	$a a^{†} a a^{†} a a, a^{†} a^{†} a a^{†} a a^{†}$
$f_{3}^{(6)} (n) = {(n + 1)}^{4} {(n + 2)}^{2}$	$a a a^{†} a^{†} a a^{†}, a a^{†} a a a^{†} a^{†}, a a^{†} a^{†} a a^{†} a, a^{†} a a a^{†} a^{†} a, a^{†} a a^{†} a a a^{†}, a^{†} a^{†} a a^{†} a a$
$f_{4}^{(6)} (n) = {(n + 1)}^{4} (n + 2) (n + 3)$	$a^{†} a a^{†} a a a, a^{†} a^{†} a^{†} a a^{†} a$
$f_{5}^{(6)} (n) = {(n + 1)}^{3} {(n + 2)}^{3}$	$a a a^{†} a^{†} a a, a a^{†} a a a^{†} a, a^{†} a a^{†} a^{†} a a^{†}, a^{†} a^{†} a a a^{†} a^{†}$
$f_{6}^{(6)} (n) = {(n + 1)}^{2} {(n + 2)}^{4}$	$a a a^{†} a a^{†} a^{†}, a a^{†} a a^{†} a^{†} a, a a^{†} a^{†} a a a^{†}, a^{†} a a a^{†} a a^{†}, a^{†} a a^{†} a^{†} a a, a^{†} a^{†} a a a^{†} a$
$f_{7}^{(6)} (n) = {(n + 1)}^{3} (n + 2) {(n + 3)}^{2}$	$a a^{†} a a a a^{†}, a a^{†} a^{†} a^{†} a a^{†}$
$f_{8}^{(6)} (n) = {(n + 1)}^{2} {(n + 2)}^{3} (n + 3)$	$a a^{†} a^{†} a a a, a^{†} a a a^{†} a a, a^{†} a^{†} a a^{†} a^{†} a, a^{†} a^{†} a^{†} a a a^{†}$
$f_{9}^{(6)} (n) = {(n + 1)}^{3} (n + 2) (n + 3) (n + 4)$	$a a^{†} a a a a, a^{†} a^{†} a^{†} a^{†} a a^{†}$
$f_{10}^{(6)} (n) = (n + 1) {(n + 2)}^{5}$	$a a a^{†} a a^{†} a, a^{†} a a^{†} a a^{†} a^{†}$
$f_{11}^{(6)} (n) = {(n + 1)}^{2} {(n + 2)}^{2} {(n + 3)}^{2}$	$a a a a^{†} a^{†} a^{†}, a a a^{†} a^{†} a^{†} a, a a^{†} a^{†} a^{†} a a, a^{†} a a a a^{†} a^{†}, a^{†} a^{†} a a a a^{†}, a^{†} a^{†} a^{†} a a a$
$f_{12}^{(6)} (n) = {(n + 1)}^{2} {(n + 2)}^{2} (n + 3) (n + 4)$	$a^{†} a^{†} a a a a, a^{†} a^{†} a^{†} a^{†} a a$
$f_{13}^{(6)} (n) = {(n + 1)}^{2} (n + 2) {(n + 3)}^{3}$	$a^{†} a a a a^{†} a, a^{†} a a^{†} a^{†} a^{†} a$
$f_{14}^{(6)} (n) = (n + 1) {(n + 2)}^{3} {(n + 3)}^{2}$	$a a a a^{†} a^{†} a, a a a^{†} a a a^{†}, a a^{†} a^{†} a a^{†} a^{†}, a^{†} a a a^{†} a^{†} a^{†}$
$f_{15}^{(6)} (n) = (n + 1) {(n + 2)}^{3} (n + 3) (n + 4)$	$a a a^{†} a a a, a^{†} a^{†} a^{†} a a^{†} a^{†}$
$f_{16}^{(6)} (n) = {(n + 1)}^{2} (n + 2) (n + 3) {(n + 4)}^{2}$	$a a^{†} a^{†} a^{†} a^{†} a, a^{†} a a a a a^{†}$
$f_{17}^{(6)} (n) = {(n + 1)}^{2} (n + 2) (n + 3) (n + 4) (n + 5)$	$a^{†} a a a a a, a^{†} a^{†} a^{†} a^{†} a^{†} a$
$f_{18}^{(6)} (n) = (n + 1) (n + 2) {(n + 3)}^{4}$	$a a a a^{†} a a^{†}, a a^{†} a a^{†} a^{†} a^{†}$
$f_{19}^{(6)} (n) = (n + 1) (n + 2) {(n + 3)}^{3} (n + 4)$	$a a a a^{†} a a, a^{†} a^{†} a a^{†} a^{†} a^{†}$
$f_{20}^{(6)} (n) = (n + 1) (n + 2) {(n + 3)}^{2} {(n + 4)}^{2}$	$a a a a a^{†} a^{†}, a a a^{†} a^{†} a^{†} a^{†}$
$f_{21}^{(6)} (n) = (n + 1) (n + 2) (n + 3) {(n + 4)}^{3}$	$a a a a a^{†} a, a^{†} a a^{†} a^{†} a^{†} a^{†}$
$f_{22}^{(6)} (n) = (n + 1) (n + 2) (n + 3) (n + 4) {(n + 5)}^{2}$	$a a a a a a^{†}, a a^{†} a^{†} a^{†} a^{†} a^{†}$
$f_{23}^{(6)} (n) = (n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (n + 6)$	$a a a a a a, a^{†} a^{†} a^{†} a^{†} a^{†} a^{†}$

Abstract

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Equations (15)

Journal of the Optical Society of America B