Wigner tomography measurements. (b) and (c) The measured joint Wigner function $W_{12}$ of the Bell states ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$ on the Re-Re and Im-Re planes, respectively. (d) and (e) Real parts of the density matrices of the states ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$ measured with the decoding and state tomography sequence as shown in Fig. \[fig:fig3\](b), respectively. Solid black outlines are for the ideal density matrices. Measured imaginary parts for both states are smaller than 0.04 and not shown. The fidelities for ${\ensuremath{\left|\Phi_{\pm}\right\rangle}}$ are 0.957 and 0.930, respectively.[]{data-label="fig:fig4"}](Figure4_final.pdf)

We finally show that our conditional dynamics can be used to deterministically create high-fidelity single-photon Bell states ${\ensuremath{\left|\Phi_{\pm}\right\rangle}} = \left({\ensuremath{\left|01\right\rangle}}\pm{\ensuremath{\left|10\right\rangle}}\right)/\sqrt{2}$. The approach is an extension of the previously reported SNAP operation for universal control of one cavity [@Krastanov2015; @Heeres2015] to two cavities. When combined with the single-cavity SNAP gates, our method can be used to realize arbitrary universal multi-cavity control. The experimental sequence is shown in Fig. \[fig:fig4\](a), where a conditional $2\pi$ rotation on qubit $Q_3$ is sandwiched in between two pairs of phase-space displacements of the cavities. With help of the two ancillary qubits, joint Wigner tomography of the two cavities is performed and two slice cuts of the measured two-mode Wigner function are shown in Figs. \[fig:fig4\](b-c). The density matrices of ${\ensuremath{\left|\Phi_{+}\right\rangle}}$ and ${\ensuremath{\left|\Phi_{-}\right\rangle}}$, reconstructed by mapping the state of the two cavities to qubits $Q_1$ and $Q_2$ and then jointly measuring the state of these qubits \[as in the tomography measurement of Fig. \[fig:fig3\](b)\], a