Determination of the Schmidt number

Optimized, necessary, and sufficient conditions for the identification of the Schmidt number will be derived in terms of general Hermitian operators. These conditions apply to arbitrary mixed quantum states. The optimization procedure delivers equations similar to the eigenvalue problem of an operator. The properties of the solution of these equations will be studied. We solve these equations for classes of operators. The solutions will be applied to phase randomized two-mode squeezed-vacuum states in continuous variable systems.

Figure 1

The experimental realization for the considered setup is the following. In both beam-splitter inputs, we have squeezed-vacuum states. The squeezing is considered in orthogonal quadratures. The 50:50 beam splitter delivers an entangled two-mode squeezed-vacuum state. The detectors ${\mathrm{D}}_A$, ${\mathrm{D}}_B$ resemble the measurement of a, so far unknown, test operator $L$. The fluctuation of the distance between the beam-splitter output and the detector ${\mathrm{D}}_A$ yield the phase randomization $\pm \delta \phi$. The distance between the beam-splitter output and the detector ${\mathrm{D}}_B$ is fixed.

Figure 2

Here, the detected entanglement for a two-mode squeezed-vacuum state is given. We choose a squeezing of $3\mathrm{dB}$. In addition, the state is manipulated by an equally distributed phase randomization with the interval of the length $2\delta \phi$. The functions are normalized to the maximal value. The entanglement can be verified until $\delta \phi ={79}^{°}$ (blue solid function). We also identified a greater Schmidt rank than 2 for phase diffusion of $\delta \phi ={25}^{°}$ (red dashed function).

Figure 3

Here, the detected entanglement is given for higher squeezing, $10\mathrm{dB}$, and the test operator in Eq. (79). In the numerical calculation, the entanglement can be verified for a phase randomization up to $\delta \phi ={178}^{°}$ (blue solid function). We also identified a greater SN than 2 for phase diffusion up to $\delta \phi ={102}^{°}$ (red dashed function).