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Physical Review A

Tomographic reconstruction of time-bin-entangled qudits

Samantha J. Nowierski, Neal N. Oza, Prem Kumar, Gregory S. Kanter

Published Oct 19, 2016• 18 citations
Publisher

Abstract

We describe an experimental implementation to generate and measure high-dimensional time-bin-entangled qudits. Two-photon time-bin entanglement is generated via spontaneous four-wave mixing in single-mode fiber. Unbalanced Mach-Zehnder interferometers transform selected time bins to polarization entanglement, allowing standard polarization-projective measurements to be used for complete quantum state tomographic reconstruction. Here we generate maximally entangled qubits $(d=2)$, qutrits $(d=3)$, and ququarts $(d=4)$, as well as other phase-modulated nonmaximally entangled qubits and qutrits. We reconstruct and verify all generated states using maximum-likelihood estimation tomography.

DOI
10.1103/physreva.94.042328

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We describe an experimental implementation to generate and measure high-dimensional time-bin-entangled qudits. Two-photon time-bin entanglement is generated via spontaneous four-wave mixing in single-mode fiber. Unbalanced Mach-Zehnder interferometers transform selected time bins to polarization entanglement, allowing standard polarization-projective measurements to be used for complete quantum state tomographic reconstruction. Here we generate maximally entangled qubits (d=2)(d=2)(d=2), qutrits (d=3)(d=3)(d=3), and ququarts (d=4)(d=4)(d=4), as well as other phase-modulated nonmaximally entangled qubits and qutrits. We reconstruct and verify all generated states using maximum-likelihood estimation tomography.
Quantum state tomography (QST) is an important method for evaluating the quality of entangled photon pairs, and has been widely used to measure polarization entanglement. However, QST has not been applied to time-bin entanglement, which is a type of entanglement suitable for fiber transmission. In this paper, we clarify the way to implement QST on time-bin entangled photon pairs using a 1-bit delayed interferometer. We also provide experimental results for a demonstration of QST for time-bin entangled photon pairs generated using spontaneous four-wave mixing in a dispersion shifted fiber.
We generate time-bin entangled photons and measure the resulting quantum state with a tomography system that uses asymmetric interferometers having $3\ifmmode\times\else\texttimes\fi{}3$ output couplers. This configuration allows for measurements to be made simultaneously in all bases that are required for tomographic reconstruction. By eliminating the burden of tuning interferometer phases and by measuring all the spatial and temporal modes, substantial improvements in measurement speed are observed. Raw fidelities of 84% with respect to an ideal entangled state are measured in about 2 s using a 12-MHz entangled state generator, with corresponding accidental count subtracted fidelities exceeding 90%.
We present a comprehensive framework for quantum state tomography of time-bin qudits sent through a fiber. Starting from basic assumptions, we define a positive-operator valued measure which is then applied to the quantum state reconstruction problem. A realistic scenario is considered where the time uncertainty of the detector is treated as a source of experimental noise. The performance of the quantum tomography framework is examined through a series of numerical simulations conducted for different parameters describing the apparatus. The quality of state recovery, quantified by the notion of minimum fidelity, is depicted on graphs for a range of fiber lengths. Special attention is paid to relative phase reconstruction for qubits and qutrits. The results present relevant interdependence between the fiber length and the detector jitter.
Quantum state tomography (QST) is an essential tool for characterizing an unknown quantum state. Recently, QST has been performed for entangled qudits based on orbital angular momentum, time-energy uncertainty, and frequency bins. Here, we propose a QST for time-bin qudits, with which the number of measurement settings scales linearly with dimension d . Using the proposed scheme, we performed QST for a four-dimensional time-bin maximally entangled state with 16 measurement settings. We successfully reconstructed the density matrix of the entangled qudits, with which the average fidelity of the state was calculated to be 0.950.