Experimental mutual coherence from separable coherent qubits

Quantum coherence is a fundamental resource in modern quantum physics with important applications in quantum technologies. In composite quantum systems, various forms of coherence emerge and play an important role, such as the global coherence, coherence of local subsystems, and the recently introduced mutual coherence. We investigate states that maximize the mutual coherence in various subspaces of the overall two-qubit Hilbert space and discover a nontrivial asymmetric optimal state in the three-dimensional subspace. We experimentally generate this optimal state from two factorized photonic qubits by a strictly incoherent probabilistic quantum operation that projects the input state onto the desired three-dimensional subspace. For comparison, we also experimentally test the preparation of states with maximal mutual coherence by unitary transformations of input product states. These proof-of-principle tests demonstrate the initial steps of control of mutual quantum coherence in qubit systems.

Figure 1

Mutual coherence generation by a strictly incoherent quantum filter from input product state with $C_M=0$. Shown are the conceptual scheme of (a) the protocol and (b) the experimental setup. HWP: half-wave plate; QWP: quarter-wave plate; GT: Glan-Taylor polarizer; PPBS: partially polarizing beam splitter; BD: beam-displacing crystal; FC: fiber coupler; and SPAD: single-photon avalanche detector [27].

Figure 2

Generation of mutual coherence from a product two-qubit state. As an example, we plot the real parts of the density matrices of (a) the theoretical input product state with $p=p_{\mathrm{th}}$, (b) the corresponding theoretical output state $|{\psi }_3\rangle$ obtained after application of the filter ${\widehat{M}}_{-}$, and (c) the actual experimental output state. The imaginary parts of the theoretical density matrices vanish. We also display the dependence of the mutual coherence of the output state on (d) the input state excitation probability $p$ and (e) the residual population of the unwanted level $|00\rangle$. Blue dots represent experimental data, and the orange crosses in (d) are experimental data after numerical elimination of the level $|00\rangle$ by filter ${\widehat{M}}_0=\widehat{I}-|00\rangle \langle 00|$. Solid lines indicate predictions of a theoretical model of the setup, and the dashed line is the ideal theoretical dependence for a perfect setup. The employed quantum filters are specified in the main text.

Figure 3

Generation of the two-qubit states (8) by unitary operations. As an example, we plot the real parts of the density matrices of (a) a theoretical asymmetric input product state for $c=0.277$, (b) the corresponding theoretical output state obtained by application of CZ gate ${\widehat{U}}_{\mathrm{CZ}}$ and local unitary operation ${\widehat{V}}_B$, and (c) the actual output experimental state prepared with nominal $c=0.264$. The imaginary parts of the theoretical density matrices vanish. Also shown is (d) the dependence of the mutual coherence of the output state on ${\left|c\right|}^2$ and (e) the residual population of level $|00\rangle$ in the experimentally prepared states. Blue dots represent experimental data, the solid lines indicate predictions of a theoretical model of the setup, and the dashed line shows the ideal theoretical dependence.

Figure 4

Generation of the optimal state $|{\psi }_4\rangle$ with quantum CZ gate. Shown are the real parts of the density matrices of (a) the theoretical input symmetric product state for $p=0.5$, (b) the theoretical maximally entangled output state $|{\psi }_4\rangle$, and (c) the experimentally prepared state. The imaginary parts of the theoretical density matrices vanish, and the imaginary parts of the matrix elements of the experimental state in (c) are smaller than 0.011.