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Quantum Information and Measurement (QIM)

Quantum State Tomography in a Single Setup, with a Single Observable

Dikla Oren, Maor Mutzafi, Yonina C. Eldar, Mordechai Segev

Published Jan 1, 2017• 0 citations
Publisher

Abstract

We propose a scheme for recovering quantum states in a single experimental setup, using a single observable, by adding a known ancilla state, introducing mixing between degrees of freedom, and utilizing structure in the states.
DOI
10.1364/qim.2017.qt6a.58

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The field of quantum information relies on the crucial issue of characterizing quantum states from measurements. This is performed through a process called quantum state tomography (QST). However, QST requires a large number of measurements, each derived from a different physical observable corresponding to a different experimental setup. Changing the setup results in unwanted changes to the data, prolongs the measurement, and impairs assumptions made about noise. Here, we propose to overcome these drawbacks by performing QST with a single observable. A single observable can often be realized by a single setup, thereby considerably reducing the experimental effort. However, the information contained in a single observable is insufficient for full QST. To overcome the lack of sufficient measurements in a single observable, we increase the system dimension by adding an ancilla that couples to the information in the system and exploit the fact that the sought state is often close to a pure state. We demonstrate our approach on multiphoton states by recovering structured quantum states from a single observable in a single experimental setup.
Quantum information has been drawing a wealth of research in recent years, shedding light on questions at the heart of quantum mechanics, as well as advancing fields such as complexity theory, cryptography, key distribution, and chemistry. These fundamental and applied aspects of quantum information rely on a crucial issue: the ability to characterize a quantum state from measurements, through a process called Quantum State Tomography (QST). However, QST requires a large number of measurements, each derived from a different physical observable corresponding to a different experimental setup. Unfortunately, changing the setup results in unwanted changes to the data, prolongs the measurement and impairs the assumptions that are always made about the stationarity of the noise. Here, we propose to overcome these drawbacks by performing QST with a single observable. A single observable can often be realized by a single setup, thus considerably reducing the experimental effort. In general, measurements of a single observable do not hold enough information to recover the quantum state. We overcome this lack of information by relying on concepts inspired by Compressed Sensing (CS), exploiting the fact that the sought state - in many applications of quantum information - is close to a pure state (and thus has low rank). Additionally, we increase the system dimension by adding an ancilla that couples to information evolving in the system, thereby providing more measurements, enabling the recovery of the original quantum state from a single-observable measurements. We demonstrate our approach on multi-photon states by recovering structured quantum states from a single observable, in a single experimental setup. We further show how this approach can be used to recover quantum states without number-resolving detectors.
Photonic quantum technologies utilize various degrees of freedom (DOFs) of light, such as polarization, frequency, and spatial modes, to encode quantum information. In the effort of further improving channel capacity of quantum communication, and for increasing the complexity of available quantum operations, high-dimensional and hyperentangled states are now gaining interest. However, efficiently measuring these high dimensional states is challenging due to the large number of measurements required for reconstructing the full density matrix via quantum state tomography (QST), and the fact that each measurement requires some modification in the experimental setup. Here, we propose a framework for reconstructing the density matrix of a single-photon hyperentangled across multiple DOFs using a single intensity-measurement obtainable from traditional cameras, and discuss extensions for multiphoton hyperentangled states. Our method hinges on the spatial DOF of the photon and uses it to encode the quantum information from the other DOFs. We numerically demonstrate this method for single-photon OAM-spin and OAM-frequency entangled states using an ideal coupler and a multimode fiber, to perform the information mixing and transfer the encoding to spatial information, where it is detected using a simple camera. This technique simplifies the experimental setup and reduces acquisition time compared to traditional QST-based methods. Moreover, it allows the recovery of DOFs that conventional cameras cannot detect, such as polarization, thus eliminating the need for projection measurements.
Wheeler's delayed-choice experiment delays the decision to observe either the wave or particle behavior of a photon until after it has entered the interferometer, and the quantum delayed-choice experiment provides the possibility of observing the wave and particle behaviors with a single experimental setup by introducing quantum control. We here propose a modified quantum delayed-choice experiment without quantum control or entanglement assistance, in which a photon can be prepared in a wave-particle superposition state and the morphing behavior of wave-to-particle transition can be observed easily. It is demonstrated that the presented scheme can allow us to rule out classical hidden-variable models in a device-independent manner via violating dimension witness. We also extend the scheme to the situation of two degrees of freedom, first constructing a hybrid quantum delayed-choice experiment which enables a photon respectively exhibits wave and particle behaviors in different degrees of freedom, and then proposing a scheme to prepare the single-photon wave-particle entanglement. This study is not only meaningful to explore the wave-particle complementarity of photons, but also provides potential for the research of the single-particle nonlocality from the perspective of the wave-particle degree of freedom.