Two-Mode Bosonic State Tomography with Single-Shot Joint-Parity Measurement of a Trapped Ion

The full characterization of a continuous-variable quantum system is a challenging problem. For the trapped-ion system, a number of methods of measuring the quantum states have been developed, including the measurement of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>Q</a:mi> </a:math> quasiprobability function and the density-matrix elements in the Fock basis, but these approaches are often slow and difficult to scale to multimode states. Here, we demonstrate a novel and powerful scheme for measuring a continuous-variable quantum state that uses the direct single-shot measurement of the joint parity of the phonon states of a trapped ion. We drive a spin-dependent bichromatic beam-splitter interaction that coherently exchanges phonons between different harmonic oscillator modes of the ion. This interaction encodes the joint-parity information into the relative phase between the two spin states, enabling measurement of the combined phonon-number parity across multiple modes in a single shot. Leveraging this capability, we directly measure multimode Wigner quasiprobability distributions to perform quantum state tomography of an entangled coherent state, and calculate various quantum informational quantities with a model-based estimation of the density matrix. We further show that the single-shot joint-parity measurement can be used to detect parity-flip errors in real time. By postselecting the parity-measurement outcomes, we experimentally demonstrate the partial recovery of coherence, effectively implementing an error-mitigation technique. Lastly, we identify the various sources of error affecting the fidelity of the spin-dependent beam-splitter operation and study the feasibility of high-fidelity operations. The interaction studied in this work can be extended to more than two modes, and is highly relevant to continuous-variable quantum computing and quantum metrology.