One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements, since it requires a global reconstruction. Here we experimentally demonstrate a scheme that can be used to directly measure individual density matrix elements of general quantum states. The scheme relies on measuring a sequence of three observables, each complementary to the last. The first two measurements are made weak to minimize the disturbance they cause to the state, while the final measurement is strong. We perform this joint measurement on polarized photons in pure and mixed states to directly measure their density matrix. The weak measurements are achieved using two walk-off crystals, each inducing a polarization-dependent spatial shift that couples the spatial and polarization degrees of freedom of the photons. This direct measurement method provides an operational meaning to the density matrix and promises to be especially useful for large dimensional states.Shortly after the inception of the quantum state, Pauli questioned its measurability, and in particular, whether or not a wave function can be obtained from position and momentum measurements [1]. This question, now referred to as the Pauli problem, draws on concepts such as complementarity and measurement in an attempt to demystify the physical significance of the quantum state. Indeed, the task of determining a quantum state is a central issue in quantum physics due to both its foundational and practical implications. For instance, a method to verify the production of complicated states is desirable in quantum information and quantum metrology applications. Moreover, since a state fully characterizes a system, any possible measurement outcome can be predicted once the state is determined.A wave function describes a quantum system that can be isolated from its environment, meaning the two are non-interacting and the system is in a pure state. More generally, open quantum systems can interact with their environment and the two can become entangled. In such cases, or even in the presence of classical noise, the system is in a statistical mixture of states (i.e. mixed state), and one requires a density matrix to fully describe the quantum system. In fact, some regard the density matrix as more fundamental than the wave function because of its generality and its relationship to classical measurement theory [2].The standard way of measuring the density matrix is by using quantum state tomography (QST). In QST, one performs an often overcomplete set of measurements in incompatible bases on identically prepared copies of the state. Then, one fits a candidate state to the measurement results with the help of a reconstruction algorithm [3]. Many efforts have been made to optimize QST [4][5][6][7], but the scalability of the experimental apparatus and the complexity of the reconstruction algorithm renders the task increasingly difficult for large dimensional systems. In addition, since QST requires a global reconstruction, it ...

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Decoherence and noise from optical absorption in superconducting circuits hinder development of microwave to optical quantum transducers. Addressing these issues, we fabricate niobium-based resonators and qubits, and study them under laser illumination at milliKelvin temperatures.

Coherent transduction of quantum states from the microwave to the optical domain can play a key role in quantum networking and distributed quantum computing. We present the design of a piezo-optomechanical device formed in a hybrid lithium niobate on silicon platform, that is suitable for microwave-to-optical quantum transduction. Our design is based on acoustic hybridization of an ultra-low mode volume piezoacoustic cavity with an optomechanical crystal cavity. The strong piezoelectric nature of lithium niobate allows us to mediate transduction via an acoustic mode which only minimally interacts with the lithium niobate, and is predominantly silicon-like, with very low electrical and acoustic loss. We estimate that this transducer can realize an intrinsic conversion efficiency of up to 35% with <0.5 added noise quanta when resonantly coupled to a superconducting transmon qubit and operated in pulsed mode at 10 kHz repetition rate. The performance improvement gained in such hybrid lithium niobate-silicon transducers make them suitable for heralded entanglement of qubits between superconducting quantum processors connected by optical fiber links.

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