Unitary transformations are the fundamental building blocks of gates and operations in quantum information processing, allowing the complete manipulation of quantum systems in a coherent manner. In the case of photons, optical elements that can perform unitary transformations are readily available only for some degrees of freedom, e.g., wave plates for polarization. However, for high-dimensional states encoded in the transverse spatial modes of light, performing arbitrary unitary transformations remains a challenging task for both theoretical proposals and actual implementations. Following the idea of multi-plane light conversion, we show that it is possible to perform a broad variety of unitary operations at high quality by using only a few phase modulation planes. More importantly, we experimentally implement several high-dimensional quantum gates for up to five-dimensional states encoded in the full-field mode structure of photons. In particular, we realize cyclic and quantum Fourier transformations, known as Pauli X ^ -gates and Hadamard H ^ -gates, respectively, with an average visibility of more than 90%. In addition, we demonstrate near-perfect “unitarity” by means of quantum process tomography, unveiling a process purity of 99%. Last, we demonstrate the benefit of the two independent spatial degrees of freedom, i.e., azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Thus, our demonstrations open up new paths to implement high-dimensional quantum operations, which can be applied to various tasks in quantum communication, computation, and sensing schemes.
Unitary transformations are the fundamental building blocks of gates and operations in quantum information processing, allowing the complete manipulation of quantum systems in a coherent manner. In the case of photons, optical elements that can perform unitary transformations are readily available only for some degrees of freedom, e.g., wave plates for polarization. However, for high-dimensional states encoded in the transverse spatial modes of light, performing arbitrary unitary transformations remains a challenging task for both theoretical proposals and actual implementations. Following the idea of multi-plane light conversion, we show that it is possible to perform a broad variety of unitary operations at high quality by using only a few phase modulation planes. More importantly, we experimentally implement several high-dimensional quantum gates for up to five-dimensional states encoded in the full-field mode structure of photons. In particular, we realize cyclic and quantum Fourier transformations, known as Pauli X ^ -gates and Hadamard H ^ -gates, respectively, with an average visibility of more than 90%. In addition, we demonstrate near-perfect “unitarity” by means of quantum process tomography, unveiling a process purity of 99%. Last, we demonstrate the benefit of the two independent spatial degrees of freedom, i.e., azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Thus, our demonstrations open up new paths to implement high-dimensional quantum operations, which can be applied to various tasks in quantum communication, computation, and sensing schemes.
Structure photons invoke interesting fundamental properties and enable novel applications, e.g. their use as high-dimensional quantum states, which are known to be beneficial in various quantum information tasks. However, to use their full potential the ability to perform any unitary operations is indispensable. Here, we present a scheme to perform arbitrary unitary operation for all transverse spatial modes, i.e. a so-called multiport for high-dimensional quantum states. We realize it by using multiple consecutive phase modulations, which are designed by wave front matching techniques. At first, we perform near-perfect measurements of full-field modes, i.e. modes with an azimuthal and radial structure [1]. Our method only requires three consecutive phase modulations followed by a single mode fiber and is, in principle, error-free and lossless. We achieve an average error of 4.2% for a set of 9 different full-field Laguerre-Gauss and Hermite-Gauss modes with an efficiency of up to 70%. We further demon-strate its potential in a quantum cryptography protocol and in high-dimensional quantum state tomography. We then use the multiport to implement high-dimensional quantum gates [2], such as cyclic and quantum Fourier transformations, known as Pauli X-gates and Hadamard H-gates, respectively. We achieve an aver-age visibility of more than 90% for dimensions up to 5 and obtain a process purity of 99% by means of a full quantum process tomography. We also demonstrate the benefit of the two independent spatial degrees of freedom, i.e. azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Using more efficient phase modulations will pave the way to novel multi-particle high-dimensional quantum information experiments. [1] M. Hiekkamäki, S. Prabhakar, R. Fickler, Near-perfect measuring of full-field transverse-spatial modes of light, arXiv:1909.01685 [2] F. Brandt, M. Hiekkamäki, F. Bouchard, M. Huber, R. Fickler, High-dimensional quantum gates using full-field spatial modes of photons, arXiv:1907.13002
Spatial modes have attracted a lot of attention in the quantum optics community because of their possible application in high-dimensional quantum bits, i.e. qudits. One crucial task in quantum communication and computation applications, is the ability to perform unitary transformations on these qudits. However, arbitrary unitary transformations between full-field spatial modes have not been realized so far. Thus, in order to bring spatial modes closer to communication and computation applications, methods for performing these transformations need to be devised. \n \nIn this thesis, we introduce a method for generating these unitary transformations. The method is based on multi-plane light conversion (MPLC), where the spatial structure of light is transformed through multiple consecutive transverse-phase modulations. The method we use for generating these task specific phase-modulations is called wavefront matching (WFM). \n \nThis thesis consists of three main sections. First, we will introduce some necessary theory behind high-dimensional quantum-states and single photons. We will also explore some theory behind the generation of single photons and introduce light's spatial modes in more detail. Second, we will introduce WFM along with other methods we needed for testing the unitary transformations experimentally. Finally, we will experimentally test the mode transformations in a spatial mode filter and in multi-mode transformations. We apply the mode filter in quantum key distribution (QKD) and quantum state tomography (QST) measurements, and we implement different high-dimensional quantum gates using the multi-mode transformations. Additionally, we introduce a method of potentially observing two-photon interference with these unitary transformations. \n \nWith the mode filter, we achieved error rates below five percent and in the QKD application we measured a theoretical data transmission rate of 1.98 bits per measured photon. With the quantum gates, we achieved accuracies of up to 98\\% in all mutually unbiased bases (MUBs). The results shown in this thesis demonstrate the efficiency and unitarity of WFM in a broad set of different tasks. All the shown tasks are important in high-dimensional quantum communication and computation, and hence we believe that WFM will become an important tool in high-dimensional quantum information processing. We also give an outlook on how WFM could potentially be improved, and list some additional tasks, in which WFM could be useful. We believe that the broad applicability of WFM will enable some unexplored quantum information or photonics applications.
High-dimensional degrees of freedom of photons can encode more quantum information than their two-dimensional counterparts. While the increased information capacity has advantages in quantum applications (such as quantum communication), controlling and manipulating these systems has been challenging. Here we show a method to perform deterministic arbitrary high-dimensional Pauli X gates for single photons carrying orbital angular momentum. The X gate consists of a cyclic permutation of qudit basis vectors and, together with the Z gate, forms the basis for performing arbitrary transformations. The proposed experimental setups only use two basic optical elements such as mode sorters and mode shifters and thus could be implemented in any system where these experimental tools are available. Furthermore the number of involved interferometers scales logarithmically with the dimension, which is important for practical implementation.
The information-carrying capacity of a single photon can be vastly expanded by exploiting its multiple degrees of freedom: spatial, temporal, and polarization. Although multiple qubits can be encoded per photon, to date only two-qubit single-photon quantum operations have been realized. Here, we report an experimental demonstration of three-qubit single-photon, linear, deterministic quantum gates that exploit photon polarization and the two-dimensional spatial-parity-symmetry of the transverse single-photon field. These gates are implemented using a polarization-sensitive spatial light modulator that provides a robust, non-interferometric, versatile platform for implementing controlled unitary gates. Polarization here represents the control qubit for either separable or entangling unitary operations on the two spatial-parity target qubits. Such gates help generate maximally entangled three-qubit Greenberger-Horne-Zeilinger and W states, which is confirmed by tomographical reconstruction of single-photon density matrices. This strategy provides access to a wide range of three-qubit states and operations for use in few-qubit quantum information processing protocols.Photons are essential for quantum information processing, but to date only two-qubit single-photon operations have been realized. Here the authors demonstrate experimentally a three-qubit single-photon linear deterministic quantum gate by exploiting polarization along with spatial-parity symmetry.
The field of high-dimensional quantum photonics involves the use of multimode photonic degrees-of-freedom such as the spatial, temporal, or spectral structure of light to encode multi-level quantum states. Recent years have seen rapid progress in the development of methods to generate, manipulate, and distribute such quantum states of light and their use in a range of quantum technology applications that offer practical advantages over conventional qubit-based approaches. High-dimensional quantum states of light encoded in photonic time-bins, frequency-bins, transverse-spatial modes, waveguide paths, and temporal modes have enabled noise-robust fundamental tests of quantum mechanics, error-resilient and high-capacity quantum communication protocols, andas well as efficient approaches for quantum information processing, to name just a few examples. However, research in this field has progressed fairly independently, with little exchange across different photonic degrees-of-freedom or between experiment and theory and no comprehensive comparison between degrees-of-freedom. This roadmap aims to bridge this gap by surveying progress in each area and identifying shared challenges and opportunities that cut across two or more photonic degrees-of-freedoms. We review early work and state-of-the-art experimental techniques under development for high-dimensional quantum states encoded in single and entangled photons, as well as theoretical tools for their measurement and certification. We outline the main outstanding challenges for theory and each experimental degree-of-freedom, identifying promising future directions of research that may enable these to be overcome. We end by discussing interconnections and shared challenges centered around their distribution, measurement, and manipulation, with a view towards their integration into next-generation quantum technology platforms and applications.
The controlled quantum gate is an essential element for the realization of quantum computer and has been implemented in various types of two-level systems. However, high-dimensional controlled quantum gates are rarely investigated both theoretically and experimentally although high-dimensional quantum systems exhibit remarkable advantages over two-level systems for some of quantum information and computing tasks. Here we experimentally demonstrate the four-dimensional controlled X gate and its unique higher orders based on orbital-angular-momentum and polarization degrees of freedom of a single photon. The controlled gates have been examined with various input states, and all the conversion efficiency of the unitary operations are above 93% with 100% utility of photons. This work is an important step towards the goal of achieving arbitrary d-dimensional quantum circuit and paves a way for the implementation of quantum communication, computation in high-dimensional quantum systems.