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Journal of Physics Conference Series

Optimal estimation of parameters of an entangled quantum state

Salvatore Virzì, Alessio Avella, F Piacentini, Marco Gramegna, G. Brida, Ivo Pietro Degiovanni, Marco Genovese

Published May 1, 2017• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/841/1/012033

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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.
Entanglement not only plays a crucial role in quantum technologies, but is key to our understanding of quantum correlations in many-body systems. However, in an experiment, the only way of measuring entanglement in a generic mixed state is through reconstructive quantum tomography, requiring an exponential number of measurements in the system size. Here, we propose an operational scheme to measure the entanglement --- as given by the negativity --- between arbitrary subsystems of size NAN_ANA​ and NBN_BNB​, with O(NA+NB)\mathcal{O}(N_A + N_B)O(NA​+NB​) measurements, and without any prior knowledge of the state. We propose how to experimentally measure the partially transposed moments of a density matrix, and using just the first few of these, extract the negativity via Chebyshev approximation or machine learning techniques. Our procedure will allow entanglement measurements in a wide variety of systems, including strongly interacting many body systems in both equilibrium and non-equilibrium regimes.
We have performed quantum-state tomography on several different single-photon entangled states, that is, states in which a single photon is shared between two possible paths. We begin by doing a tomographic reconstruction of density matrices in the subspace where a single photon is shared by two spatial modes. In this subspace we are able to create high-fidelity, path-entangled states. Also within this subspace we use the Akaike information criterion and Monte Carlo simulations to help us estimate the amount and source of state drift in our system. We find that the primary state drift is due to phase drifts and fluctuations on the order of $\ensuremath{\pi}/400$ in our experimental apparatus. We then use the single-photon subspace density matrices and further measurements to estimate density matrices in the larger space consisting of two modes containing up to one photon each. In this larger space we find that the concurrence of the density matrices is $C\ensuremath{\cong}0.08$ and is greater than 0 by at least 45 standard deviations, indicating that our states are indeed entangled.
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.
Quantum computation depends on quantum entanglement, a correlation between subsystems that cannot occur classically. A variety of theoretical measures exist for quantifying the degree entanglement in such schemes, all of which are functions of the system density matrix. How can the entanglement be measured experimentally? Using quantum tomography techniques developed for two photon entangled states, the density matrix can be reconstructed from the appropriate experimental data. In this case the state tomography gives the complete characterization of the physical system (for the relevant degree of freedom, such as spin or polarization). It gives information on both the degree of nonclassical correlation, that is entanglement, as well as the amount of decoherence in the system. In this proceedings we discuss the general state tomography procedure required to characterize a few qubit quantum computer, for any architecture.
It is shown that, despite strong nonlinearity, entanglement of formation of a two-qubit state can be measured without prior state reconstruction. Collective measurements on a small number of copies are provided that allow one to determine quantum concurrence via estimation of only four parameters. It is also pointed out that another entanglement measure based on so-called "negativity" can also be measured in a similar way. The result is related to the general problem: What kind of information can be extracted efficiently from an unknown quantum state?
We address the experimental determination of entanglement for systems made of a pair of polarization qubits. We exploit quantum estimation theory to derive optimal estimators, which are then implemented to achieve ultimate bound to precision. In particular, we present a set of experiments aimed at measuring the amount of entanglement for states belonging to different families of pure and mixed two-qubit two-photon states. Our scheme is based on visibility measurements of quantum correlations and achieves the ultimate precision allowed by quantum mechanics in the limit of Poissonian distribution of coincidence counts. Although optimal estimation of entanglement does not require the full tomography of the states we have also performed state reconstruction using two different sets of tomographic projectors and explicitly shown that they provide a less precise determination of entanglement. The use of optimal estimators also allows us to compare and statistically assess the different noise models used to describe decoherence effects occurring in the generation of entanglement.