Unified framework to determine Gaussian states in continuous-variable systems

Gaussian states are the backbone of quantum information protocols with continuous-variable systems whose power relies fundamentally on the entanglement between the different modes. In the case of global pure states, knowledge of the reduced states in a given bipartition of a multipartite quantum system bears information on the entanglement in such bipartition. For Gaussian states, the reduced states are also Gaussian, so their determination requires essentially the experimental determination of their covariance matrix. Here we develop strategies to determine the covariance matrix of an arbitrary $n$-mode bosonic Gaussian state through measurement of the total phase acquired when appropriate metaplectic evolutions, associated with quadratic Hamiltonians, are applied. Simply one-mode metaplectic evolutions, such rotations, squeezing, and shear transformations, in addition to a single two-mode rotation, allows us to determine all the covariance matrix elements of an $n$-mode bosonic system. All the single-mode metaplectic evolutions are applied conditionally to a state in which an ancilla qubit is entangled with the $n$-mode system. The ancillary system provides, after measurement, the value of the total phase of each evolution. The proposed method is experimentally suited to implement in the most currently used continuous-variable systems.

Figure 1

Experimental setup to implement the conditional squeezing transformation on the transverse spatial degree of freedom of single photons, for example, generated with the SPDC process. Here the conditional squeezing transformation is implemented on the idler photon. On the signal photon it has to be implemented in an optical image system in both transverse degrees of freedom (not shown) before the coincidence measurement of the twin photons. The dashed rectangles correspond to cylindrical lenses of focal length $f$, acting in the transverse $y$ degree of freedom. The acronym HWP denotes the half waveplate and SLM the spatial light modulator. See the text for details.

Figure 2

Experimental setup to implement the conditional position shear transformation on the transverse spatial degree of freedom of single photons, for example, generated with the SPDC process. Here the conditional position shear transformation is implemented on the idler photon. On the signal photon it has to be implemented in an optical image system in both transverse degrees of freedom (not shown) before the coincidence measurement of the twin photons. The shaded ovals correspond to spherical lenses with focal length $f$. See the text for details.

Figure 3

Experimental setup of the measurement protocol of the total phase of a rotation of the reduced one-mode Gaussian state ${\widehat{\varrho }}^{\left(j\right)}$ corresponding to the $j\text{th}$ mode of a multimode Gaussian state. The initial qubit-ancilla state corresponds to the mode-entangled state of one photon (of frequency $\omega$) in the interferometer after the first beam splitter. The $j\text{th}$ mode of frequency ${\omega }^{\left(j\right)}$ is injected and extracted from one of the arms of the interferometer through suitable dichroic mirrors (DM). The rotation over ${\widehat{\varrho }}^{\left(j\right)}$ is implemented by the Kerr medium conditioned to the one-photon occupation of the upper arm or the interferometer. Finally, the rotation of the qubit ancilla is performed by the second beam splitter and the measurement of the number of photons at the output modes determines, through Eq. (76), the total phase ${\varphi }_{{\mathsf{R}}^{\left(\mathsf{j}\right)}}$ once we choose $\phi =0$ and $\phi =\pi /2$.

Figure 4

Same experimental setup as in Fig. 3, but now before the conditional rotation in the Kerr medium a squeezing transformation over the state ${\widehat{\varrho }}^{\left(j\right)}$ is performed. This is done with a stimulated degenerate parametric down-conversion in a type-I nonlinear crystal, characterized by a second-order electric susceptibility ${\chi }^{\left(2\right)}$. Thus, the down-conversion process pumped by the field of frequency $2{\omega }^{\left(j\right)}$ is stimulated by the mode of frequency ${\omega }^{\left(j\right)}$ in the quantum state ${\widehat{\varrho }}^{\left(j\right)}$. The converted fields are in the same mode ${\omega }^{\left(j\right)}$ producing a squeezing effect on ${\widehat{\varrho }}^{\left(j\right)}$.