We discuss how quantum state tomography of entangled photon pairs can be used in the lab as a natural follow-up to popular Bell inequalities experiments for graduate students and to introduce them to more advanced quantum concepts. In addition, we discuss how the measured density matrix can be used to quantitatively estimate the quantum state quality and optimize the Bell measurement to reach its maximum expected value for an arbitrary two-photon entangled state.
We discuss how quantum state tomography of entangled photon pairs can be used in the lab as a natural follow-up to popular Bell inequalities experiments for graduate students and to introduce them to more advanced quantum concepts. In addition, we discuss how the measured density matrix can be used to quantitatively estimate the quantum state quality and optimize the Bell measurement to reach its maximum expected value for an arbitrary two-photon entangled state.
The growth of quantum technologies is attracting the interest of many students eager to learn concepts such as quantum entanglement or quantum superposition. However, the non-intuitive nature of these concepts poses a challenge to understanding them. Here, we present an entangled photon system which can perform a Bell test, i.e. the CHSH inequality, and can obtain the complete tomography of the two-photon state. The proposed setup is versatile, cost-effective and allows for multiple classroom operating modes. We present two variants, both facilitating the measurement of Bell inequalities and quantum state tomography. Experimental results showcase successful manipulation of the quantum state of the photons, achieving high-fidelity entangled states and significant violations of Bell's inequalities. Our setup's simplicity and affordability enhances accessibility for less specialized laboratories, allowing students to familiarize themselves with quantum physics concepts.
Bell’s inequality plays an important role in quantum mechanics. Many experiments have been made to test the Bell’s inequality by using various bipartite entangled sources. However, the maximal violation of the Bell inequality (i.e., the Cirel’son’s limit) has not been reached yet, due to various practically-existing imperfections (e.g., entangled source is not the pure maximally-entangled state, and the measurement settings used are not the best ones). In this paper, we investigate experimentally an approach to enhance the violation degree of Bell’s inequality with the commercialized entangled photons by optimizing the measurement settings for the experimentally-reconstructed density matrix of the entangled photons. It is shown that the measurement setting related to the pure entangled state can not get maximally violation generally. However, upon the density matrix reconstructed by tomographic technique, optimal measurement settings can be found. With these optimal settings, we consequently found that the violation degree of the Bell’s inequality can be increased. This suggests a possible approach to optimize the test of Bell’s inequality.
Two-photon entangled quantum states are a fundamental tool for quantum information and quantum cryptography. A complete description of a generic quantum state is provided by its density matrix: the technique allowing experimental reconstruction of the density matrix is called quantum state tomography. Entangled states density matrix reconstruction requires a large number of measurements on many identical copies of the quantum state. An alternative way of certifying the amount of entanglement in two-photon states is represented by the estimation of specific parameters, e.g., negativity and concurrence. If we have a priori partial knowledge of our state, it's possible to develop several estimators for these parameters that require lower amount of measurements with respect to full density matrix reconstruction. The aim of this work is to introduce and test different estimators for negativity and concurrence for a specific class of two-photon states.
The reconstruction of density matrices from measurement data (quantum state tomography) is the most comprehensive method for assessing the accuracy and performance of quantum devices. Existing methods to reconstruct two-photon density matrices require the detection of both photons unless a priori information is available. Based on the concept of quantum-induced coherence by path identity, we present an approach to quantum state tomography that circumvents the requirement of detecting both photons. We show that an arbitrary two-qubit density matrix, which can contain up to fifteen free parameters, can be fully reconstructed from single-photon measurement data without any postselection and a priori information. In addition to advancing an alternative approach to quantum state measurement problems, our results also have notable practical implications. A practical challenge in quantum state measurement arises from the fact that effective single-photon detectors are not readily accessible for a wide spectral range. Our method, which eliminates the need for coincidence measurements, enables quantum state tomography in the case where one of the two photons is challenging or impossible to detect. It therefore opens the door to measuring quantum states hitherto inaccessible.
Abstract High-quality quantum state generation is essential for advanced quantum information processing, including quantum communication, quantum sensing, and quantum computing. In realistic experiments, however, the generated quantum state is degraded by multiple experimental error sources whose individual contributions are often difficult to quantify and calibrate. Quantum state tomography (QST) provides experimental access to the resulting density matrix, but it does not by itself attribute observed deviations to specific error sources. To address this problem, we propose an automated method for quantifying error sources by combining simulation and parameter optimization, in which a parametrized simulator of state preparation and measurement is optimized so that the simulated density matrix best fits the experimentally reconstructed density matrix obtained via QST. We focus on the experimental generation of time-bin entangled photon pairs, for which we model the relevant error sources and simulate the density matrix with adjustable model parameters, thereby optimizing the parameters and minimizing the trace distance to the experimental data. Optimization of the parameters reduced the trace distance from 0.177 to 0.024, indicating that our modeled error sources explain 86% of the errors. Reducing the predicted error sources improves the state quality, consistent with our predictions and thus validating the proposed method. The proposed framework provides a practical route to automated, quantitative error attribution and calibration directly from experimental QST data.
Tomography allows one to estimate the density matrix describing the state an ensemble of quantum systems are prepared in (for example, polarization tomography determines the polarization state of a beam of identically prepared photons). In general, it is not possible to uniquely decompose the density matrix into its pure state components. Agarwal et al. proposed a protocol which, for a mixture composed of any two pure states of a qubit (with arbitrary probabilities), allows an observer to infer not only the density matrix but the identity of those specific pure states and their weights - the additional requirement being that the qubits arrive in pairs, where both qubits in each pair are in the same state. We experimentally demonstrate this learning-from-pairs concept using photons in the polarization degree of freedom. We use tomography to measure a sequence of single photons and make use of their time-of-arrival information to 'pair up' the photons after the measurement. From here we are able to infer the photons' polarization states and their respective probabilities, and we demonstrate this for various different choices of polarization states and ratios. Finally, we investigate our ability to discriminate between two equal mixtures of distinct pairs of orthogonal polarization states. We find that on the order of approx. 10e4 photons is typically enough to achieve tomography fidelities of approximately 0.9999. This is sufficient to discriminate between two different preparations of the same mixed state, differing by angles of less than 5 degrees between the pure states used in the two preparations.