We propose a linear-optical implementation of a hyperentanglement-assisted quantum error-correcting code. The code is hyperentanglement-assisted because the shared entanglement resource is a photonic state hyperentangled in polarization and orbital angular momentum. It is possible to encode, decode, and diagnose channel errors using linear-optical techniques. The code corrects for polarization "flip" errors and is thus suitable only for a proof-of-principle experiment. The encoding and decoding circuits use a Knill-Laflamme-Milburn-like scheme for transforming polarization and orbital angular momentum photonic qubits. A numerical optimization algorithm finds a unit-fidelity encoding circuit that requires only three ancilla modes and has success probability equal to 0.0097.
We propose a simple scheme for complete Bell-state measurement of photons using hyperentangled states - entangled in multiple degrees of freedom. In addition to hyperentanglement, our scheme requires only linear optics and single photon detectors, and is realizable with current technology. At the cost of additional classical communication, our Bell-state measurement can be implemented non-locally. We discuss the possible application of these results to quantum dense coding and quantum teleportation.
Hyperentanglement has attracted much attention for its fascinating applications in quantum communication. However, it is impossible to purify a pair of photon systems in a mixed hyperentangled state with errors in two degrees of freedom using linear optical elements only, far different from all the existing entanglement purification protocols in a degree of freedom (DOF) for quantum systems. Here, we investigate the possibility of purifying a spatial-polarization mixed hyperentangled Bell state with the errors in both the spatial mode and polarization DOFs, resorting to the nonlinear optics of…
We present two efficient schemes for the deterministic generation and the complete nondestructive analysis of hyperentangled Bell states in both the polarization and spatial-mode degrees of freedom (DOFs) of two-photon systems, assisted by the nitrogen-vacancy (NV) centers in diamonds coupled to microtoroidal resonators as a result of cavity quantum electrodynamics (QED). With the input-output process of photons, two-photon polarization-spatial hyperentangled Bell states can be generated in a deterministic way and their complete nondestructive analysis can be achieved. These schemes can be gen…
We use hyperentangled photons to experimentally implement an entanglement-assisted quantum process tomography technique known as Direct Characterization of Quantum Dynamics. Specifically, hyperentanglement-assisted Bell-state analysis enabled us to characterize a variety of single-qubit quantum processes using far fewer experimental configurations than are required by Standard Quantum Process Tomography (SQPT). Furthermore, we demonstrate how known errors in Bell-state measurement may be compensated for in the data analysis. Using these techniques, we have obtained single-qubit process fidelit…
Photon-efficient quantum repeater chains via hyperentanglement-assisted purification
University of Pisa（比萨大学）；Institute for Informatics and Telematics (IIT), National Research Council (CNR)（意大利国家研究委员会信息与遥测研究所）；Department of Physics, University of Naples Federico II（那不勒斯费德里科二世大学物理系）；National Institute of Optics (INO), National Research Council (CNR)（意大利国家研究委员会光学国家研究所）
We report an optical one-way quantum computing experiment with stationary quantum memory involved. First we create a hybrid four-qubit cluster state with two qubits propagating as photons and the other two stationary and stored in a laser-cooled atomic-ensemble quantum memory, and characterize it with entanglement witness and quantum state tomography. Then, by making use of this cluster state and fast operations of electro-optic modulators, we realize memory-assisted feedforward operations and demonstrate deterministic single-qubit rotation as an example.
Recently, two-photon six-qubit hyperentangled states were produced in experiment and they can improve the channel capacity of quantum communication largely. Here we present a scheme for the hyperentanglement purification of nonlocal two-photon systems in three degrees of freedom (DOFs), including the polarization, the first-longitudinal-momentum, and the second longitudinal momentum DOFs. Our hyperentanglement purification protocol (hyper-EPP) is constructed with two steps resorting to parity-check quantum nondemolition measurement on the three DOFs and SWAP gates, respectively. With these two…
Bell-state analysis (BSA) is essential in quantum communication, but it is impossible to distinguish unambiguously the four Bell states in the polarization degree of freedom (DOF) of two-photon systems with only linear optical elements, except for the case in which the BSA is assisted with hyperentangled states, the simultaneous entanglement in more than one DOF. Here, we propose a scheme to distinguish completely the 16 hyperentangled Bell states in both the polarization and the spatial-mode DOFs of two-photon systems, by using the giant nonlinear optics in quantum dot-cavity systems. This sc…
A complete and deterministic Bell state measurement was realized by a simple linear optics experimental scheme which adopts 2-photon polarization-momentum hyperentanglement. The scheme, which is based on the discrimination among the single photon Bell states of the hyperentangled state, requires the adoption of standard single photon detectors. The four polarization Bell states have been measured with average fidelity $F=0.889\pm0.010$ by using the linear momentum degree of freedom as the ancilla. The feasibility of the scheme has been characterized as a function of the purity of momentum enta…
Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of pre-existing entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard QECCs which can only be constructed from dual-containing codes. Operator quantum error-correcting codes (OQECCs) allow certain errors to be corrected (or prevented) passively, reducing the complexity of the correction procedure. We combine these two extensions of standard quantum error correction into a unified entanglement-assisted quantum err…
We construct an error-detected block, assisted by the quantum-dot spins in double-sided optical microcavities. With this block, we propose three error-detected schemes for the deterministic generation, the complete analysis, and the complete nondestructive analysis of hyperentangled Bell states in both the polarization and spatial-mode degrees of freedom of two-photon systems. In these schemes, the errors can be detected, which can improve their fidelities largely, far different from other previous schemes assisted by the interaction between the photon and the quantum-dot-cavity system. Our sc…
Hyperentanglement, the entanglement in several degrees of freedom (DOFs) of a quantum system, has attracted much attention as it can be used to increase both the channel capacity of quantum communication and its security largely. Here, we present the first scheme to completely distinguish the hyperentangled Bell states of two-photon systems in three DOFs with the help of cross-Kerr nonlinearity without destruction, including two longitudinal momentum DOFs and the polarization DOF. We use cross-Kerr nonlinearity to construct quantum nondemolition detectors which can be used to make a parity-che…
The scheme of entanglement-assisted quantum error-correcting (EAQEC) codes assumes that the ebits of the receiver are error-free. In practical situations, errors on these ebits are unavoidable, which diminishes the error-correcting ability of these codes. We consider two different versions of this problem. We first show that any (nondegenerate) standard stabilizer code can be transformed into an EAQEC code that can correct errors on the qubits of both sender and receiver. These EAQEC codes are equivalent to standard stabilizer codes, and hence the decoding techniques of standard stabilizer cod…
Contrary to the assumption that most quantum error-correcting codes (QECC) make, it is expected that phase errors are much more likely than bit errors in physical devices. By employing the entanglement-assisted stabilizer formalism, we develop a new kind of error-correcting protocol which can flexibly trade error correction abilities between the two types of errors, such that high error correction performance is achieved both in symmetric and in asymmetric situations. The characteristics of the QECCs can be optimized in an adaptive manner during information transmission. The proposed entanglem…
Quantum error correcting codes (QECCs) in quantum communi- cation systems has been known to exhibit improved performance with the use of error-free entanglement bits (ebits). In practical situations, ebits inevitably suffer from errors, and as a result, the error-correcting capability of the code is diminished. Prior studies have proposed two different schemes as a solu- tion. One uses only one QECC to correct errors on the receiver's side (i.e., Bob) and on the sender's side (i.e., Alice). The other uses different QECCs on each side. In this paper, we present a method to correct errors on bot…
We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary for standard quantum error correcting codes, thus allowing us to ``quantize'' all of classical linear coding theory. In particular, efficient modern classical codes that attain the Shannon capacity can be made into entanglement-assisted quantum codes attaining the hashing bound (closely related to the quantum capacity). For systems without large amounts of shared entanglement,…
In this paper, we introduce a unified framework to construct entanglement-assisted quantum error-correcting codes, including additive and nonadditive codes, based on the codeword stabilized framework on subsystems. The codeword stabilized (CWS) framework is a scheme to construct quantum error-correcting codes (QECCs) including both additive and nonadditive codes, and gives a method to construct a QECC from a classical error-correcting code in standard form. Entangled pairs of qubits (ebits) can be used to improve capacity of quantum error correction. In addition, it gives a method to overcome …
Entanglement-assisted quantum error-correcting codes (EAQECCs) to desired rate, error-correcting capability and maximum shared entanglement are constructed. Thus for a required rate $R$, required error-correcting capability to correct $t$ errors, mds (maximum distance separable) EAQECCs of the form $[[n,r,d;c]]$ with $R=\frac{r}{n}, d\geq (2t+1), c = (n-r), d= (n-r+1)$ are constructed. Series of such codes may be constructed where the rate and the relative distance approach non-zero constants as $n$ approaches infinity. The codes may also be constructed over prime order fields in which modular…
We prove that the known formulae for computing the optimal number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs) over the binary field hold for codes over arbitrary finite fields as well. We also give a Gilbert-Varshamov bound for EAQECCs and constructions of EAQECCs coming from punctured self-orthogonal linear codes which are valid for any finite field.
If entanglement is available, the error-correcting ability of quantum codes can be increased. We show how to optimize the minimum distance of an entanglement-assisted quantum error-correcting (EAQEC) code, obtained by adding ebits to a standard quantum error-correcting code, over different encoding operators. By this encoding optimization procedure, we found several new EAQEC codes, including a family of [[n, 1, n; n-1]] EAQEC codes for n odd and code parameters [[7, 1, 5; 2]], [[7, 1, 5; 3]], [[9, 1, 7; 4]], [[9, 1, 7; 5]], which saturate the quantum singleton bound for EAQEC codes. A random …
We present a general formalism for quantum error-correcting codes that encode both classical and quantum information (the EACQ formalism). This formalism unifies the entanglement-assisted formalism and classical error correction, and includes encoding, error correction, and decoding steps such that the encoded quantum and classical information can be correctly recovered by the receiver. We formally define this kind of quantum code using both stabilizer and symplectic language, and derive the appropriate error-correcting conditions. We give several examples to demonstrate the construction of su…
We provide a systematic way of constructing entanglement-assisted quantum error-correcting codes via graph states in the scenario of preexisting perfectly protected qubits. It turns out that the preexisting entanglement can help beat the quantum Hamming bound and can enhance (not only behave as an assistance) the performance of the quantum error correction. Furthermore we generalize the error models to the case of not-so-perfectly-protected qubits and introduce the quantity infidelity as a figure of merit and show that our code outperforms also the ordinary quantum error-correcting codes.
The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is…
We show how extra entanglement shared between sender and receiver reduces the memory requirements for a general entanglement-assisted quantum convolutional code. We construct quantum convolutional codes with good error-correcting properties by exploiting the error-correcting properties of an arbitrary basic set of Pauli generators. The main benefit of this particular construction is that there is no need to increase the frame size of the code when extra shared entanglement is available. Then there is no need to increase the memory requirements or circuit complexity of the code because the fram…
Having protected quantum information is essential to perform quantum computations. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so called qudit quantum computers. In this paper we show that codes for these systems can be derived from already known codes, and in particular this procedure removes the need for shared entanglement in entanglement-assisted quantum error-correcting codes, which is a result which could prove to be useful for fault-tolerant qudit, and even qubit, quantum computers.
In this paper, we provide a framework for constructing entanglement-assisted quantum error-correcting codes (EAQECCs) from classical additive codes over a finite commutative local Frobenius ring $\mathcal{R}$. At the heart of the framework, and this is one of the main technical contributions of our paper, is a procedure to construct, for an additive code $\mathcal{C}$ over $\mathcal{R}$, a generating set for $\mathcal{C}$ that is in standard form, meaning that it consists purely of isotropic generators and hyperbolic pairs. Moreover, when $\mathcal{R}$ is a Galois ring, we give an exact expres…
Entangled qubit can increase the capacity of quantum error correcting codes based on stabilizer codes. In addition, by using entanglement quantum stabilizer codes can be construct from classical linear codes that do not satisfy the dual-containing constraint. We show that it is possible to construct both additive and non-additive quantum codes using the codeword stabilized quantum code framework. Nonadditive codes may offer improved performance over the more common sta- bilizer codes. Like other entanglement-assisted codes, the encoding procedure acts only the qubits on Alice's side, and only …
The entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum…
Quantum error correction is an important building block for reliable quantum information processing. A challenging hurdle in the theory of quantum error correction is that it is significantly more difficult to design error-correcting codes with desirable properties for quantum information processing than for traditional digital communications and computation. A typical obstacle to constructing a variety of strong quantum error-correcting codes is the complicated restrictions imposed on the structure of a code. Recently, promising solutions to this problem have been proposed in quantum informat…
Though the entanglement-assisted formalism provides a universal connection between a classical linear code and an entanglement-assisted quantum error-correcting code (EAQECC), the issue of maintaining large amount of pure maximally entangled states in constructing EAQECCs is a practical obstacle to its use. It is also conjectured that the power of entanglement-assisted formalism to convert those good classical codes comes from massive consumption of maximally entangled states. We show that the above conjecture is wrong by providing families of EAQECCs with an entanglement consumption rate that…
Known quantum error correction schemes are typically able to take advantage of only a limited class of classical error-correcting codes. Entanglement-assisted quantum error correction is a partial solution which made it possible to exploit any classical linear codes over the binary or quaternary finite field. However, the known entanglement-assisted scheme requires noiseless qubits that help correct quantum errors on noisy qubits, which can be too severe an assumption. We prove that a more relaxed and realistic assumption is sufficient by presenting encoding and decoding operations assisted by…
Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction
Xanadu；Department of Mathematics & Statistics, University of Guelph（圭尔夫大学数学与统计系）；Perimeter Institute for Theoretical Physics（理论物理珀米特研究所）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）
Brun, Devetak, and Hsieh [Science 314, 436 (2006)] demonstrated that pre-shared entanglement between sender and receiver enables quantum communication protocols that have better parameters than schemes without the assistance of entanglement. Subsequently, the same authors derived a version of the so-called quantum Singleton bound that relates the parameters of the entanglement-assisted quantum-error correcting codes proposed by them. We present a new entanglement-assisted quantum communication scheme with parameters violating this bound in certain ranges.
Quantum error-correcting codes will be the ultimate enabler of a future quantum computing or quantum communication device. This theory forms the cornerstone of practical quantum information theory. We provide several contributions to the theory of quantum error correction--mainly to the theory of "entanglement-assisted" quantum error correction where the sender and receiver share entanglement in the form of entangled bits (ebits) before quantum communication begins. Our first contribution is an algorithm for encoding and decoding an entanglement-assisted quantum block code. We then give severa…
The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is defined from the orthogonal group of a simplified stabilizer group. From the Poisson summation formula, this duality leads to the MacWilliams identities and linear programming bounds for EAQEC codes. We establish a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits.
We show how to protect a stream of quantum information from decoherence induced by a noisy quantum communication channel. We exploit preshared entanglement and a convolutional coding structure to develop a theory of entanglement-assisted quantum convolutional coding. Our construction produces a Calderbank-Shor-Steane (CSS) entanglement-assisted quantum convolutional code from two arbitrary classical binary convolutional codes. The rate and error-correcting properties of the classical convolutional codes directly determine the corresponding properties of the resulting entanglement-assisted quan…
In the previous research by Grassl, Huber and Winter, they proved a theorem which can make entanglement-assisted quantum error-correcting codes (EAQECC) from general quantum error-correcting codes (QECC). In this paper, we prove that the shortened EAQECC is a stabilizer code if the original EAQECC is a stabilizer code.
Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum (QUENTA) codes. In this paper, we use algebraic geometry codes to construct several families of QUENTA codes via the Euclidean and the Hermitian construction. Two of the families crea…
Entanglement-assisted quantum (QUENTA) codes are a subclass of quantum error-correcting codes which use entanglement as a resource. These codes can provide error correction capability higher than the codes derived from the traditional stabilizer formalism. In this paper, it is shown a general method to construct QUENTA codes from cyclic codes. Afterwards, the method is applied to Reed-Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of QUENTA codes. Two families of QUENTA codes are maximal distance separable (MDS), and one is almost MDS or alm…
We develop the theory of entanglement-assisted quantum error correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to pre-shared entanglement. Conventional stabilizer codes are equivalent to dual-containing symplectic codes. In contrast, EAQEC codes do not require the dual-containing condition, which greatly simplifies their construction. We show how any quaternary classical code can be made into a EAQEC code. In particular, efficient modern codes, like LDPC codes, which attain the Shannon capacity, can be made into E…
We report on the feedback stabilization of the zero-phonon emission frequency of a single InAs quantum dot. The spectral separation of the phonon-assisted component of the resonance fluorescence provides a probe of the detuning between the zero-phonon transition and the resonant driving laser. Using this probe in combination with active feedback, we stabilize the zero-phonon transition frequency against environmental fluctuations. This protocol reduces the zero-phonon fluorescence intensity noise by a factor of 22 by correcting for environmental noise with a bandwidth of 191 Hz, limited by the…
An Entanglement-Assisted Stabilizer Framework for Distributed Sensing of Local Phases
School of Electrical Engineering, Korea University（韩国大学电气工程学院）；Quantum Network Research Center, Korea Institute of Science and Technology Information（韩国科学技术信息研究院量子网络研究中心）
We propose a teleportation-based scheme to implement a universal set of quantum gates with a four-component cat code, assisted by appropriate entangled resource states and photon number resolving detection. The four-component cat code features the ability to recover from single photon loss. Here, we propose a concrete procedure to correct the single photon loss, including detecting the single photon loss event and recovering the initial states. By concatenating with standard qubit error correcting codes, we estimate the loss threshold for fault-tolerant quantum computation and obtain a signifi…
