Majorization relations and entanglement generation in a beam splitter

We prove that a beam splitter, one of the most common optical components, fulfills several classes of majorization relations, which govern the amount of quantum entanglement that it can generate. First, we show that the state resulting from $k$ photons impinging on a beam splitter majorizes the corresponding state with any larger photon number $k^{\prime }>k$, implying that the entanglement monotonically grows with $k$. Then we examine parametric infinitesimal majorization relations as a function of the beam-splitter transmittance and find that there exists a parameter region where majorization is again fulfilled, implying a monotonic increase of entanglement by moving towards a balanced beam splitter. We also identify regions with a majorization default, where the output states become incomparable. In this latter situation, we find examples where catalysis may nevertheless be used in order to recover majorization. The catalyst states can be as simple as a path-entangled single-photon state or a two-mode squeezed vacuum state.

Figure 1

Majorization relations with respect to the input photon number. In the upper scheme, the input state is $|k,0\rangle$, while in the lower scheme it is $|k+1,0\rangle$. The output state $|{\Psi }^{\left(k\right)}\left(\theta \right)\rangle$ majorizes $|{\Psi }^{(k+1)}\left(\theta \right)\rangle$, hence the generated entanglement $\mu \left({\Psi }^{\left(k\right)}\left(\theta \right)\right)$ monotonically grows with $k$ for all $\theta$.

Figure 2

Majorization relations with respect to the coupling parameter $\theta$ (or transmittance $\tau ={\cos }^2\theta$). In both schemes the input state is $|k,0\rangle$ but the angles differ by $\varepsilon$. In a specific parameter region, the corresponding output states $|{\Psi }^{\left(k\right)}\left(\theta \right)\rangle$ and $|{\Psi }^{\left(k\right)}(\theta +\varepsilon )\rangle$ are proven to satisfy a majorization relation.

Figure 3

Same situation as in Fig. 2, viewed as a sequence of two beam splitters with angles $\theta$ and $\varepsilon$. In a specific parameter region, $|{\Psi }^{\left(k\right)}\left(\theta \right)\rangle$ majorizes $|{\Psi }^{\left(k\right)}(\theta +\varepsilon )\rangle$ for all $k$.

Figure 4

Entanglement Rényi entropies resulting from a two-photon state impinging on a beam splitter as a function of the coupling parameter $\theta$ (related to the transmittance $\tau ={\cos }^2\theta$). The vertical dashed line denotes the boundary between parameter regions $r=1$ and $r=2$. The von Neumann entropy ($\alpha \to 1$) keeps increasing in region $r=2$, while higher-order Rényi entropies have a different behavior and start decreasing somewhere in region $r=2$. The min-entropy ($\alpha \to \infty$) exhibits a nondifferentiable point right at the crossover point and decreases throughout the entire region $r=2$, reflecting the default of majorization.

Figure 5

Entanglement Rényi entropies resulting from a three-photon state impinging on a beam splitter as a function of $\theta$. The two vertical dashed lines at ${\theta }_1$ and ${\theta }_2$ separate the three regions $r=1,2,3$, while the dotted line corresponds to the local minimum of the min-entropy. Majorization is violated in the region $r=2$ from the left boundary of this region at ${\theta }_1$ up to the local minimum of the min-entropy. The majorization violation throughout the entire region $r=3$ is not manifested by the behavior of the Rényi entropies.

Figure 6

Schematic of the catalyzed conversion between the incomparable states resulting from a three-photon state impinging on a beam splitter with angles $\theta =0.62$ and $\theta +\omega =0.72$. The catalyst is the entangled state obtained from a beam splitter with angle $\theta =0.7$ and one single-photon input state.