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Quantum Operations on Integrated Photonic Circuits: State Tomography and Coherence Measurements

Wilder R. Cardoso

Published Jul 9, 2020• 0 citations
OpenAlex

Abstract

This work proposes to carry out a study on integrated photonic circuits. Our goal is to develop a methodology for obtaining the parameters of a photonic circuit that allows us to perform a set of quantum operations necessary for the realization of a given task or quantum processing. More specifically, the quantum tasks involve the implementation of a POVM (Positive Operator Value Measure) through integrated photonic circuits. Thus, we seek to obtain optimized configurations for these devices so that the reduction of their structural complexity simplifies their manufacturing process. In addition, it is also our goal that such objects are robust against errors and losses and that the experimental applications that involve them present a minimization of the time necessary for their realization. We studied the symmetries of the proposed POVMs to, from them, configure the circuit in an optimized way. Initially, we studied the case of quantum state tomography for qubits and qutrits. We obtained planar circuits that had a smaller number of optical elements and an optical depth than those obtained by methods already known in the literature. In addition, these circuits implement all POVM elements at once, making it possible to obtain all their associated probabilities at once, minimizing the time spent on the experiment compared to experiments on an optical table or with reconfigurable circuits, where it is necessary to make changes to the configuration of the experimental apparatus from one POVM element to another. Looking for a way to generalize our protocol to larger systems, we developed a method to obtain three-dimensional photonic circuits that can be used in the experimental realization of quantum state tomography in qudits of any finite dimension. Our protocol generates circuits whose number of beam splitters and optical depth scale with functions less than in other protocols already known, showing a drastic reduction in their complexity. In addition, these circuits were highly robust in relation to the occurrence of losses. The same protocol used to obtain three-dimensional circuits for performing quantum tomography can be used to obtain circuits to implement quantum coherence measurements. The circuits obtained here also have the same advantages as the tomographic case.

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•Jan 1, 2012•2 citations
This work proposes to carry out a study on integrated photonic circuits. Our goal is to develop a methodology for obtaining the parameters of a photonic circuit that allows us to perform a set of quantum operations necessary for the realization of a given task or quantum processing. More specifically, the quantum tasks involve the implementation of a POVM (Positive Operator Value Measure) through integrated photonic circuits. Thus, we seek to obtain optimized configurations for these devices so that the reduction of their structural complexity simplifies their manufacturing process. In addition, it is also our goal that such objects are robust against errors and losses and that the experimental applications that involve them present a minimization of the time necessary for their realization. We studied the symmetries of the proposed POVMs to, from them, configure the circuit in an optimized way. Initially, we studied the case of quantum state tomography for qubits and qutrits. We obtained planar circuits that had a smaller number of optical elements and an optical depth than those obtained by methods already known in the literature. In addition, these circuits implement all POVM elements at once, making it possible to obtain all their associated probabilities at once, minimizing the time spent on the experiment compared to experiments on an optical table or with reconfigurable circuits, where it is necessary to make changes to the configuration of the experimental apparatus from one POVM element to another. Looking for a way to generalize our protocol to larger systems, we developed a method to obtain three-dimensional photonic circuits that can be used in the experimental realization of quantum state tomography in qudits of any finite dimension. Our protocol generates circuits whose number of beam splitters and optical depth scale with functions less than in other protocols already known, showing a drastic reduction in their complexity. In addition, these circuits were highly robust in relation to the occurrence of losses. The same protocol used to obtain three-dimensional circuits for performing quantum tomography can be used to obtain circuits to implement quantum coherence measurements. The circuits obtained here also have the same advantages as the tomographic case.
Manipulation of qudits in optical tables is a difficult and nonscalable task. The use of integrated optical circuits opens new possibilities for the generation, manipulation, and characterization of high dimensional states besides the ease of transmission of these states through an optical fiber. In this work we propose photonic circuits to perform minimum quantum state tomography of path qudits and show how to determine all the constituents parameters of these circuits (beam splitters and phase shifters). Our strategies were based on the symmetries of the involved POVMs (positive operator-valued measures) suggested for minimum tomography and allowed us to obtain interferometers smaller than those obtained by other already known methods. The calculations of the transmittances and reflectivities of the beam splitters were made using the definition of probability operators in extended Hilbert spaces and the application of Naimark's theorem. The employment of equidistant states for the definition of the POVM elements allowed us to develop a recipe applicable to the tomography of qudits of any dimension, generalizing our scheme.
Knowing that quantum coherence plays a central role in quantum phenomena, we propose, in this work, three-dimensional photonic circuits with compact designs that allow measurement of the quantum coherence of NNN-dimensional quantum systems from the probabilities of detecting the photons that emerge from their outputs. Here, we present two proposals: one based on the positive operator-valued measure (POVM) proposed by Carmeli et al. [New J. Phys. 20, 063038 (2018)] and the other based on the sum of the input-state partial visibilities, obtained by interfering pairs of base states. Based on the form of the POVM that we define for this task, the proposed interferometers are given by a sequence of different unitary operations. These operations act independently in each vertical sector, or layer, of the circuit, which simplifies its determination. We have thus obtained circuits whose number of beam splitters obeys a quadratic polynomial function and has a linear optical depth, which confirms its compactness and its reduced complexity.
In this work, we propose three-dimensional photonic circuit designs that guarantee a considerable reduction in the complexity of circuits for the purpose of performing quantum state tomography of N-dimensional path qudits. The POVM (Positive Operator-Valued Measure) chosen in this work ensures that, for odd dimensions, such process is minimal. Our proposal consists of organizing the waveguides that form the circuit as a square array formed by N vertical sectors composed of N waveguides each, arranged in the vertical direction. Based on the symmetry of the chosen POVM, the interferometer acting on the initial quantum system can be divided into a sequence of three different unitary operations. These operations act independently on each vertical sector, or layer, of the circuit, which simplifies their determination. We have thus obtained circuits such that the number of beam splitters obeys a polynomial function of degree 3 with the quantum system dimension, whereas in current proposals this quantity grows with a polynomial function of degree 4. Besides that, the optical depth is reduced from a quadratic to a linear function of the quantum system dimension in our scheme. These results confirm the remarkable reduction of the complexity of the photonic circuits in our proposal.
Abstract We propose compact three-dimensional photonic circuit designs that guarantee a remarkable reduction in the complexity of circuits to perform quantum state tomography of N -dimensional path qudits. We obtain solutions such that the number of beam splitters obeys a polynomial function of degree three with the quantum system dimension, whereas, in current proposals, this quantity grows with a polynomial function of degree four. In addition, the optical depth reduces from quadratic to linear over the dimension of the Hilbert space.
In this paper, we present a protocol to obtain photonic circuits that can be used in the implementation of contextuality tests on qutrit systems. The use of photonic integrated circuits offers several advantages for performing this type of task. These include scalability, accuracy, robustness, high-speed and efficient quantum measurements, precise control over the phase properties of photons by using electrically driven heaters to induce a thermo-optic phase shift and resistance to noise. We relate the average values that appear in the inequalities with the probability of photon counting in the circuit outputs and present a realizable configuration for the desired device, taking into account state-dependent and state-independent contextuality tests.
Photons have proven to be an effective test-bed for the fundamental concepts and elements of quantum-enhanced technologies. As systems become increasingly complex, however, practical considerations make the traditional approach of bulk optics and free-space propagation progressively more difficult. The major obstacles are the physical space necessary to realise and operate such a complex system, its stability, and maintaining low losses. In order to address these issues, quantum optical technologies can take a cue from their classical counterparts and look towards an integrated architecture to provide miniaturisation, greatly enhanced stability, less alignment, and low loss interfaces between different system components. In this thesis the feasibility of chip-based waveguides as a platform for metrology and information processing will be explored. In Part I, the necessary criteria for a metrology system to out-perform its classical counterpart will be investigated. It will be found that loss is a major barrier to this aim and, critically, that it is unlikely to have been achieved to date by any experiment which consumes resources of a fixed photon number. The issue of loss will be addressed by developing a scalable heralded source of a class of entangled photonic states which are both robust to losses and practically feasible to prepare. A novel tomographic technique will be developed to characterize these states and it will be explicitly demonstrated how it is possible to beat some bounds on classical performance without being able to out-perform a comparable classical system. Finally, a proof of principle demonstration of a waveguide-based interferometer with an integrated phase-shifter will be undertaken. It will be shown that the device preserves quantum interference, making it suitable for use in quantum-enhanced metrology applications. In Part II, integrated optics in the context of information processing will be discussed. First, a novel characterization technique will be developed which enables the behaviour of complex circuits to be predicted. The technique is independent of loss in the device being characterized. A method of simulating these circuits will be outlined that takes advantage of the computational speed-up available from parallelisation and sparse matrix operations. A key increase in complexity for integrated photonic systems will be demonstrated by showing quantum interference of three photons from two separate sources in eight spatial modes. The resulting interference has a visibility which beats all possible classical interference visibilities for similar circuits. Finally, a fully integrated waveguide-coupled photon-number-resolving detector will be developed and demonstrated. This proof of concept demonstration will show good resolution of different photon number events. The device will be modelled and routes to high efficiency operation will be explored.