Operational Gaussian Schmidt-number witnesses

The general class of Gaussian Schmidt-number witness operators for bipartite systems is studied. It is shown that any member of this class is reducible to a convex combination of two types of Gaussian operators using local operations and classical communications. This gives rise to a simple operational method, which is solely based on measurable covariance matrices of quantum states. Our method bridges the gap between theory and experiment of entanglement quantification. In particular, we certify lower bounds of the Schmidt number of squeezed thermal and phase-randomized squeezed vacuum states as examples of Gaussian and non-Gaussian quantum states, respectively.

Figure 1

Logarithmic sketches of the normalized expectation value of the test operator ${\widehat{\mathcal{L}}}_n$ for squeezed thermal states ($\overline{m}=0$) versus the mean global thermal noise (blue, full lines). The dashed lines represent the minimal SN-$r$ eigenvalues $g_r$. From top to bottom, the value of $r=1,2,\cdots$ is increasing; the lowest (red, dotted) curves represent $g_{\infty }$. The squeezing parameters are (a) $\gamma =0.7$ and (b) $\gamma =0.98$.

Figure 2

Logarithmic sketches of the normalized expectation value of the test operator ${\widehat{\mathcal{L}}}_n$ for squeezed thermal states (${\overline{n}}_1={\overline{n}}_2=0$) versus the mean thermal occupation of the locally added noise (blue, full lines). The dashed lines represent the minimal SN-$r$ eigenvalues $g_r$. From top to bottom, the value of $r=1,2,\cdots$ is increasing, the lowest (red, dotted) curves represent $g_{\infty }$. The squeezing parameters are (a) $\gamma =0.7$ and (b) $\gamma =0.98$.

Figure 3

The normalized SN levels and expectation values of ${\widehat{\mathcal{L}}}_n$ for three squeezed thermal states versus squeezing parameter (full lines). The dashed lines represent the minimal SN-$r$ eigenvalues $g_r$. From top to bottom, the value of $r=1,2,\cdots$ is increasing, the lowest (red, dotted) curve represents $g_{\infty }$. Apparently, the entanglement is getting stronger with decreasing thermal noise.

Figure 4

Logarithmic sketches of the normalized expectation value of the test operator ${\widehat{\mathcal{L}}}_n$ for a phase randomized, bipartite squeezed-state ${\widehat{\varrho }}_{\sigma }$ versus phase-randomization parameter $\sigma$ (blue, full lines). The dashed lines represent the minimal SN-$r$ eigenvalues $g_r$. From top to bottom, the value of $r=1,2,...$ is increasing, the lowest (red, dotted) curves represent $g_{\infty }$. For the squeezing parameter of (a) $\gamma =0.7$ and (b) $\gamma =0.98$ entanglement can be detected up to a phase randomization of $\sigma \cong 0.775$ and $\sigma \cong 0.665$, respectively.