Majorization theoretical approach to entanglement enhancement via local filtration

TL;DR

The paper develops a majorization-theoretic framework to enhance entanglement of a two-mode squeezed vacuum (TMSV) via local filtration. By introducing Fock-orthogonal and Fock-amplifying operators, it proves that ideal photon addition/subtraction, and any concatenation of such operations, always yields a more entangled output in the majorization sense, relative to the original TMSV, with a constructive mapping to a column-stochastic matrix. It further extends the analysis to realistic photon-addition/subtraction schemes, providing approximate majorization bounds based on a total-variation distance to a majorized state, and demonstrates, for k-photon addition, a partial majorization ladder with k = 1 majorizing k = 2…8 (with conjecture for all k ≥ 9). Collectively, these results establish a principled, probabilistic route for non-Gaussian entanglement enhancement in continuous-variable systems and offer practical guidance for implementing non-unitary local filters in CV quantum networks.

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 (non-unitary) 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 $2\leq k \leq 8$ and we conjecture the validity of the statement for all $k\geq 9$.

Figures (7)

• Figure 1: A pumped crystal produces a TMSV via spontaneous parametric down conversion. Then, we act locally on both modes with some operators possibly including measurements. We call that process filtration. We will study the conditions for filtration to produce an output state that is more entangled than the TMSV, in the majorization sense.
• Figure 2: Illustration of the 3 different schemes considered in Section \ref{['sec:schemes']}. Each (orange) round-cornered box is an ideal photon addition/subtraction ($k,l\in\mathbb{Z}$), or a more general single-mode quantum operator ($\hat{F},\hat{G}$). Scheme (a) corresponds to dual-mode single addition/subtraction (see also Ref. navarrete2012enhancing) and yields the pure state $\vert\Phi_\lambda^{k,l}\rangle$. Scheme (b) corresponds to single-mode multi addition/subtraction and yields the pure state $\vert\Phi_\lambda^{\mathbf{k}}\rangle$. Finally, scheme (c) corresponds to the filtration scheme and yields the pure state ${\vert\Phi^{\hat{F},\hat{G}}_\lambda\rangle}$. Note that, for each scheme, we consider that the output state $\ket{\Phi}$ is normalized. Scheme (c) is a generalization of scheme (b), which itsef generalizes scheme (a).
• Figure 3: Realistic schemes of photon addition and subtraction. On the left (resp. right), a two-mode squeezer with gain $g$ (resp. beam-splitter with transmittance $\eta$) acts on an input $\hat{\rho}$ and a vacuum environment $\ket{0}$; then, conditionally on the measurement of $k$ photons at the environment output, the (non-normalized) resulting state is $\hat{A}_k\hat{\rho}\hat{A}^\dagger_k$ (resp. $\hat{B}_k\hat{\rho}\hat{B}^\dagger_k$). The expression of the Kraus operators $\hat{A}_k$ and $\hat{B}_k$ is given in Eq. \ref{['eq:krauss_op']}.
• Figure 4: Each point on this graph corresponds to a couple $(\eta,\lambda)$. The parameter $\lambda$ defines a thermal state $\hat{\tau}_\lambda$, the parameter $\eta$ defines a set of realistic photon subtraction Kraus operators $\lbrace\hat{B}_k\rbrace$ (see Eq. \ref{['eq:krauss_op']}). We then compare $\hat{\tau}_\lambda$ to $\hat{\sigma}_k=\hat{B}_k\hat{\tau}_\lambda\hat{B}^\dagger_k/\mathrm{Tr}[\hat{B}_k\hat{\tau}_\lambda\hat{B}^\dagger_k]$ for $k\in\lbrace 1,...,6\rbrace$. The color of the point $(\eta,\lambda)$ is related to the values of $k$ above which entanglement enhancement is successful ($\hat{\sigma}_k\prec\hat{\tau}_\lambda$). Note that we get identical results for realistic photon addition, since it produces equivalent output as realistic photon subtraction when $g=1/\eta$. For ideal photon addition ($g=1$) or ideal photon subtraction ($\eta=1$), the majorization relation always holds, as expected from Theorem \ref{['theorem:single_mode']}. Note that the squeezing expressed in dB is related to $\lambda$ as: $\lambda[\mathrm{dB}]=(20/\ln10)\tanh^{-1}(\sqrt{\lambda})$.
• Figure 5: Each couple $(\eta,\lambda)$ defines a thermal state $\hat{\tau}$ and a state $\hat{\sigma}=\hat{A}_1\hat{\tau}\hat{A}^\dagger_1/\mathrm{Tr}[\hat{A}_1\hat{\tau}\hat{A}^\dagger_1]$, where $\hat{A}_1$ is the realistic single photon addition operator (see Eq. \ref{['eq:krauss_op']} with $k=1$). This logarithmic plot associates each point $(\eta,\lambda)$ to a value $d$, such that we can ensure that the state $\hat{\sigma}$ is at a distance of at most $10^{-d}$ (in TVD) from another state that is majorized by $\hat{\tau}$, see Eq. \ref{['eq:bound_major_dist']}. Note that no distinction is made among values above $7$, and among values below $0$. This figure should be compared with Fig. \ref{['fig:num_addsub']} for $k=1$.

Theorems & Definitions (15)

• Theorem 1: Nielsen Nielsen1999-ee
• Definition 1: Fock-orthogonal operator
• Definition 2: Fock-amplifying operator
• Theorem 2
• proof
• Definition 3: Jointly Fock-amplifying operator pair
• Theorem 3
• proof
• Definition 4: Fock-preserving operator
• Theorem 4: Concatenability