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arXiv:quant-ph/0702075 · Physical Review A

Protecting an optical qubit against photon loss

Wojciech Wasilewski, Konrad Banaszek

Published Apr 16, 2007• 55 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.75.042316

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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.
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.
Traditional quantum error correction involves the redundant encoding of kkk quantum bits using nnn quantum bits to allow the detection and correction of any ttt bit error. The smallest general t=1t=1t=1 code requires n=5n=5n=5 for k=1k=1k=1. However, the dominant error process in a physical system is often well known, thus inviting the following question: Given a specific error model, can more efficient codes be devised? We demonstrate alternative codes that correct just amplitude damping errors that allow, for example, a t=1t=1t=1, k=1k=1k=1 code using effectively n=4.6n=4.6n=4.6. Our scheme is based on using bosonic states of photons in a finite number of optical modes. We present necessary and sufficient conditions for the codes and describe construction algorithms, physical implementation, and performance bounds.
The cat code is a promising encoding scheme for bosonic quantum error correction as it allows for correction against losses---the dominant error mechanism in most bosonic systems. However, it has remained unclear how the required syndrome measurement and recovery can be implemented in the optical regime. Here, we introduce a teleportation-based error-correction scheme for the cat code, using elements suitable for an optical setting. The scheme detects and corrects single-photon losses while restoring the amplitude of the cat states, thereby greatly suppressing the accumulation of errors in lossy channels.
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 propose and investigate a method of error detection and noise correction for bosonic linear networks using a method of unitary averaging. The proposed error averaging does not rely on ancillary photons or control and feedforward correction circuits, remaining entirely passive in its operation. We construct a general mathematical framework for this technique and then give a series of proof of principle examples including numerical analysis. Two methods for the construction of averaging are then compared to determine the most effective manner of implementation and probe the related error thresholds. Finally we discuss some of the potential uses of this scheme.