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arXiv:1512.07605 · Physical Review A

Quantum error correction against photon loss using NOON states

Marcel Bergmann, Peter van Loock

Published Jul 7, 2016• 82 citations
arXivHTMLPDF

Abstract

The so-called NOON states are quantum optical resources known to be useful especially for quantum lithography and metrology. At the same time, they are known to be very sensitive to photon losses and rather hard to produce experimentally. Concerning the former, here we present a scheme where NOON states are the elementary resources for building quantum error-correction codes against photon losses, thus demonstrating that such resources can also be useful to suppress the effect of loss. Our NOON code is an exact code that can be systematically extended from one-photon to higher-number losses. Its loss scaling depending on the codeword photon number is the same as for some existing, exact loss codes such as bosonic and quantum parity codes, but its codeword mode number is intermediate between that of the other codes. Another generalization of the NOON code is given for arbitrary logical qudits instead of logical qubits. While, in general, the final codewords are always obtainable from multimode NOON states through application of beam splitters, both codewords for the one-photon-loss qubit NOON code can be simply created from single-photon states with beam splitters. We give various examples and also discuss a potential application of our qudit code for quantum communication.

DOI
10.1103/physreva.94.012311

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The so-called NOON states are quantum optical resources known to be useful especially for quantum lithography and metrology. At the same time, they are known to be very sensitive to photon losses and rather hard to produce experimentally. Concerning the former, here we present a scheme where NOON states are the elementary resources for building quantum error-correction codes against photon losses, thus demonstrating that such resources can also be useful to suppress the effect of loss. Our NOON code is an exact code that can be systematically extended from one-photon to higher-number losses. Its loss scaling depending on the codeword photon number is the same as for some existing, exact loss codes such as bosonic and quantum parity codes, but its codeword mode number is intermediate between that of the other codes. Another generalization of the NOON code is given for arbitrary logical qudits instead of logical qubits. While, in general, the final codewords are always obtainable from multimode NOON states through application of beam splitters, both codewords for the one-photon-loss qubit NOON code can be simply created from single-photon states with beam splitters. We give various examples and also discuss a potential application of our qudit code for quantum communication.
We analyze a generalized quantum error-correction code against photon loss where a logical qubit is encoded into a subspace of a single oscillator mode that is spanned by distinct multicomponent cat states (coherent-state superpositions). We present a systematic code construction that includes the extension of an existing one-photon-loss code to higher numbers of losses. When subject to a photon loss (amplitude damping) channel, the encoded qubits are shown to exhibit a cyclic behavior where the code and error spaces each correspond to certain multiples of losses, half of which can be corrected. As another generalization we also discuss how to protect logical qudits against photon losses, and as an application we consider a one-way quantum communication scheme in which the encoded qubits are periodically recovered while the coherent-state amplitudes are restored as well at regular intervals.
Abstract Multi-mode NOON states can quantum-enhance multiple-phase estimation in the absence of photon loss. However, a multi-mode NOON state is known to be vulnerable to photon loss, and its quantum-enhancement can be dissipated by lossy environment. In this work, we demonstrate that a quantum advantage in estimate precision can still be achieved in the presence of photon loss. This is accomplished by optimizing the weights of the multi-mode NOON states according to photon loss rates in the multiple modes, including the reference mode which defines the other phases. For practical relevance, we also show that photon-number counting via a multi-mode beam-splitter achieves the useful, albeit sub-optimal, quantum advantage. We expect this work to provide valuable guidance for developing quantum-enhanced multiple-phase estimation techniques in lossy environments.
Quantum entanglement is a fascinating physical resource that is central to the field of quantum information processing with important applications in such areas as quantum computing, quantum communication, quantum metrology, and quantum imaging. \nIt describes the physical phenomenon in which the quantum state of a multipartite system cannot be written as consisting of constituents that are independent of each other. In this case, the state of the system is called an entangled state. In this \ndissertation, the entanglement of photons--the elementary excitations of the quantized electromagnetic field--is studied. A focus is made on a particular type of entangled states of light called the d-mode N-photon NOON state, which consists of N photons with the N photons as an ensemble appearing at d orthogonal modes simultaneously and are inseparable. \nIn quantum metrology, a field that investigates the ultimate precision of measurements of unknown physical parameters, the two-mode NOON state has been studied extensively with respect to its ability to achieve super-resolution and super-sensitivity in the estimation of a single parameter. A lot of theoretical research has been conducted on the generation of two-mode NOON states, and two-mode NOON states with up to 4 photons have been experimentally and efficiently produced. On the other hand, recently there has been increasing interest in the study of the simultaneous estimation of multiple parameters. Accordingly, the multi-mode NOON state has been attracting more and more attention in view of its potential for enhanced efficiency as compared to using multiple two-mode NOON states separately to estimate the parameters. Nevertheless, no known generation method of multi-mode NOON states with more than 2 photons or 2 modes exists so far. \nIn this dissertation, several scalable generation methods of multi-mode NOON states are proposed. These methods take advantage of multi-photon quantum interference and they can theoretically be applied to produce NOON states with an \narbitrary number of modes and an arbitrary number of photons. The intrinsic generation probability for each method is calculated, and the methods are compared in regards to their feasibility and efficiency. The advantages of using multi-mode NOON \nstates in quantum metrology are also analyzed and discussed in detail.
Distributed quantum information processing is based on the transmission of quantum data over lossy channels between quantum processing nodes. These nodes may be separated by a few microns or on planetary scale distances, but transmission losses due to absorption and/or scattering in the channel are the major source of error for most distributed quantum information tasks. Of course, quantum error correction (QEC) and detection techniques can be used to mitigate such effects, but error detection approaches have severe performance limitations due to the signaling constraints between nodes, and so error correction approaches are preferable-assuming one has sufficient high quality local operations. Typically, performance comparisons between loss-mitigating codes assume one encoded qubit per photon. However, single photons can carry more than one qubit of information and so our focus in this Letter is to explore whether loss-based QEC codes utilizing quantum multiplexed photons are viable and advantageous, especially as photon loss results in more than one qubit of information being lost. We show that quantum multiplexing enables significant resource reduction, in terms of the number of single-photon sources, while at the same time maintaining (or even lowering) the number of 2-qubit gates required. Further, our multiplexing approach requires only conventional optical gates already necessary for the implementation of these codes.
We propose a protocol for the generation of NOON states of resonator modes. The physical model is composed of two Kerr-nonlinear resonators and a four-level qudit. Using the off-resonant couplings between the resonators and the qudit, qudit-level-dependent frequency shifts on the two resonators are induced. The frequency shifts allow us to drive different resonators to the N-photon state when the qudit is in different intermediate levels. Consequently, the generation of NOON states with arbitrary photon number N can be completed in only three steps, i.e., driving the qudit to a superposition state of the two intermediate levels, driving one of the resonators to its N-photon state, and driving the qudit back to its ground level. Numerical simulations show that, in the regime of strong Kerr nonlinearity and coupling strengths, the protocol can produce the NOON state with high fidelity in the cases of different photon numbers. In addition, it is possible for the protocol to produce acceptable fidelity in the presence of systematic errors and decoherence factors. Therefore, the protocol may provide some useful perspectives for effective generation of photonic NOON states.
We consider quantum error-correction codes for multimode bosonic systems, such as optical fields, that are affected by amplitude damping. We demonstrate that the most accessible method of transforming optical systems with the help of passive linear networks has limited usefulness in preparing and manipulating such codes. These limitations stem directly from the recoverability condition for one-photon loss. We introduce a three-photon code protecting against the first order of amplitude damping, i.e., a single photon loss, and discuss its preparation using linear optics with single-photon sources and conditional detection. Quantum state and process tomography in the code subspace can be implemented using passive linear optics and photon counting. An experimental proof-of-principle demonstration of elements of the proposed quantum error correction scheme for a one-photon erasure lies well within present technological capabilities.