Majorization theoretical approach to entanglement enhancement via local filtration

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 (non-unitary) operators on each mode. From this, we identify a sufficient set of two conditions on filtration operators for successfully enhancing 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 $2\leq k \leq 8$ and we conjecture the validity of the statement for all $k\geq 9$.