Distillation Protocol Research Articles

Fault-tolerant preparation of stabilizer states for quantum Calderbank-Shor-Steane codes by classical error-correcting codes

Stabilizer states are extensively studied in quantum information theory for their structures based on the Pauli group. Calderbank-Shor-Steane (CSS) stabilizer states are of particular importance in their application to fault-tolerant quantum computation (FTQC). However, how to fault-tolerantly prepare arbitrary CSS stabilizer states for general CSS stabilizer codes is still unknown, and their preparation can be highly costly in computational resources. In this paper, we show how to prepare a large class of CSS stabilizer states useful for FTQC. We propose distillation protocols using syndrome encoding by classical codes or quantum CSS codes. Along the same lines, we show that classical coding techniques can reduce the ancilla consumption in Steane syndrome extraction by using additional transversal controlled-NOT gates and classical computing power. In the scenario of a fixed ancilla consumption rate, we can increase the frequency of quantum error correction and effectively lower the error rate.

Distilling perfect GHZ states from two copies of non-GHZ-diagonal mixed states

It has been shown that a nearly pure Greenberger–Horne–Zeilinger (GHZ) state could be distilled from a large (even infinite) number of GHZ-diagonal states that can be obtained by depolarizing general multipartite mixed states (non-GHZ-diagonal states) through sequences of (probabilistic) local operations and classical communications. We here demonstrate that perfect GHZ states can be extracted, with certain probabilities, from two copies of non-GHZ-diagonal mixed states when some conditions are satisfied. This result implies that it is not necessary to depolarize these entangled mixed states to the GHZ-diagonal type, and that they are better than GHZ-diagonal states for distillation of pure GHZ states. We find a wide class of multipartite entangled mixed states that fulfill the requirements. Moreover, we display that the obtained result can be applied to practical noisy environments, e.g., amplitude-damping channels. Our findings provide an important complementarity to conventional GHZ-state distillation protocols (designed for GHZ-diagonal states) in theory, as well as having practical applications.

Nonlocality Distillation and Trivial Communication Complexity for High-Dimensional Systems**Supported by the National Basic Research Program of China under Grant No 2012CB921900, and the National Natural Science Foundation of China under Grant Nos 11175089 and 11475089.

A nonlocality distillation protocol for arbitrary high-dimensional systems is proposed. We study the nonlocality distillation in the 2-input d-output bi-partite case. Firstly, we give the one-parameter nonlocal boxes and their correlated distilling protocol. Then, we generalize the one-parameter nonlocality distillation protocol to the two-parameter case. Furthermore, we introduce a contracting protocol testifying that the 2-input d-output nonlocal boxes make communication complexity trivial.

Efficient entanglement distillation without quantum memory.

Entanglement distribution between distant parties is an essential component to most quantum communication protocols. Unfortunately, decoherence effects such as phase noise in optical fibres are known to demolish entanglement. Iterative (multistep) entanglement distillation protocols have long been proposed to overcome decoherence, but their probabilistic nature makes them inefficient since the success probability decays exponentially with the number of steps. Quantum memories have been contemplated to make entanglement distillation practical, but suitable quantum memories are not realised to date. Here, we present the theory for an efficient iterative entanglement distillation protocol without quantum memories and provide a proof-of-principle experimental demonstration. The scheme is applied to phase-diffused two-mode-squeezed states and proven to distil entanglement for up to three iteration steps. The data are indistinguishable from those that an efficient scheme using quantum memories would produce. Since our protocol includes the final measurement it is particularly promising for enhancing continuous-variable quantum key distribution.

Effect of weak measurement on entanglement distribution over noisy channels.

Being able to implement effective entanglement distribution in noisy environments is a key step towards practical quantum communication, and long-term efforts have been made on the development of it. Recently, it has been found that the null-result weak measurement (NRWM) can be used to enhance probabilistically the entanglement of a single copy of amplitude-damped entangled state. This paper investigates remote distributions of bipartite and multipartite entangled states in the amplitudedamping environment by combining NRWMs and entanglement distillation protocols (EDPs). We show that the NRWM has no positive effect on the distribution of bipartite maximally entangled states and multipartite Greenberger-Horne-Zeilinger states, although it is able to increase the amount of entanglement of each source state (noisy entangled state) of EDPs with a certain probability. However, we find that the NRWM would contribute to remote distributions of multipartite W states. We demonstrate that the NRWM can not only reduce the fidelity thresholds for distillability of decohered W states, but also raise the distillation efficiencies of W states. Our results suggest a new idea for quantifying the ability of a local filtering operation in protecting entanglement from decoherence.

Entanglement distillation using the exchange interaction

A key ingredient of quantum repeaters is entanglement distillation, i.e., the generation of high-fidelity entangled qubits from a larger set of pairs with lower fidelity. Here, we present entanglement distillation protocols based on qubit couplings that originate from exchange interaction. First, we make use of asymmetric bilateral two-qubit operations generated from anisotropic exchange interaction and show how to distill entanglement using two input pairs. We furthermore consider the case of three input pairs coupled through isotropic exchange. Here, we characterize a set of protocols which are optimizing the tradeoff between the fidelity increase and the probability of a successful run.

High-efficient entanglement distillation from photon loss and decoherence

We illustrate an entanglement distillation protocol (EDP) for a mixed photon-ensemble which composed of four kinds of entangled states and vacuum states. Exploiting the linear optics and local entanglement resource (four-qubit entangled GHZ state), we design the nondemolition parity-checking and qubit amplifying (PCQA) setup for photonic polarization degree of freedom which are the key device of our scheme. With the PCQA setup, a high-fidelity entangled photon-pair system can be achieved against the transmission losses and the decoherence in noisy channels. And in the available purification range for our EDP, the fidelity of this ensemble can be improved to the maximal value through iterated operations. Compared to the conventional entanglement purification schemes, our scheme largely reduces the initialization requirement of the distilled mixed quantum system, and overcomes the difficulties posed by inherent channel losses during photon transmission. All these advantages make this scheme more useful in the practical applications of long-distance quantum communication.

An FPGA-Based 4 Mbps Secret Key Distillation Engine for Quantum Key Distribution Systems

Quantum key distribution (QKD) enables provably secure communication between two parties over an optical fiber that arguably withstands any form of attack. Besides the need for a suitable physical signalling scheme and the corresponding devices, QKD also requires a secret key distillation protocol. This protocol and the involved signal processing handle the reliable key agreement process over the fragile quantum channel, as well as the necessary post-processing of key bits to avoid leakage of secret key information to an eavesdropper. In this paper we present in detail an implementation of a key distillation engine for a QKD system based on the coherent one-way (COW) protocol. The processing of key bits by the key distillation engine includes agreement on quantum bit detections (sifting), information reconciliation with forward error correction coding, parameter estimation, and privacy amplification over an authenticated channel. We detail the system architecture combining all these processing steps, and discuss the design trade-offs for each individual system module. We also assess the performance and efficiency of our key distillation implementation in terms of throughput, error correction capabilities, and resource utilization. On a single-FPGA (Xilinx Virtex-6 LX240T) platform, the system supports distilled key rates of up to 4 Mbps.

Axiomatic approach to contextuality and nonlocality

We present a unified axiomatic approach to contextuality and nonlocality based on the fact that both are resource theories. In those theories, the main objects are consistent boxes, which can be transformed by certain operations to achieve certain tasks. The amount of resource is quantified by appropriate measures of the resource. Following a recent paper [J. I. de Vicente, J. Phys. A: Math. Theor. 47, 424017 (2014)], and recent development of abstract approach to resource theories, such as entanglement theory, we propose axioms and welcome properties for operations and measures of resources. As one of the axioms of the measure we propose the asymptotic continuity: the measure should not differ on boxes that are close to each other by more than the distance with a factor depending logarithmically on the dimension of the boxes. We prove that relative entropy of contextuality is asymptotically continuous. Considering another concept from entanglement theory (the convex roof of a measure), we prove that for some nonlocal and contextual polytopes, the relative entropy of a resource is upper bounded up to a constant factor by the cost of the resource. Finally, we prove that providing a measure $X$ of resource does not increase under allowed class of operations such as, e.g., wirings, the maximal distillable resource which can be obtained by these operations is bounded from above by the value of $X$ up to a constant factor. We show explicitly which axioms are used in the proofs of presented results, so that analogous results may remain true in other resource theories with analogous axioms. We also make use of the known distillation protocol of bipartite nonlocality to show how contextual resources can be distilled.

Quantum macroscopicity versus distillation of macroscopic superpositions

We suggest a way to quantify a type of macroscopic entanglement via distillation of Greenberger-Horne-Zeilinger states by local operations and classical communication. We analyze how this relates to an existing measure of quantum macroscopicity based on the quantum Fisher information in several examples. Both cluster states and Kitaev surface code states are found to not be macroscopically quantum but can be distilled into macroscopic superpositions. We look at these distillation protocols in more detail and ask whether they are robust to perturbations. One key result is that one-dimensional cluster states are not distilled robustly but higher-dimensional cluster states are.

On the Power of PPT-Preserving and Non-Signalling Codes

We derive one-shot upper bounds for quantum noisy channel codes. We do so by regarding a channel code as a bipartite operation with an encoder belonging to the sender and a decoder belonging to the receiver, and imposing constraints on the bipartite operation. We investigate the power of codes whose bipartite operation is non-signalling from Alice to Bob, positive-partial transpose (PPT) preserving, or both, and derive a simple semidefinite program for the achievable entanglement fidelity. Using the semidefinite program, we show that the non-signalling assisted quantum capacity for memoryless channels is equal to the entanglement-assisted capacity. We also relate our PPT-preserving codes and the PPT-preserving entanglement distillation protocols studied by Rains. Applying these results to a concrete example, the 3-dimensional Werner-Holevo channel, we find that codes that are non-signalling and PPT-preserving can be strictly less powerful than codes satisfying either one of the constraints, and therefore provide a tighter bound for unassisted codes. Furthermore, PPT-preserving non-signalling codes can send one qubit perfectly over two uses of the channel, which has no quantum capacity. We discuss whether this can be interpreted as a form of superactivation of quantum capacity.

Exploring ethanol: Engaging first‐year undergraduates in the biochemistry of ethyl alcohol

We sought to expose incoming Skidmore College students to principles and applications of biochemistry through a series of experiment‐based activities in an interdisciplinary first‐year seminar course, “Under the Influence: Alcohol in Science and Society.” To provide the diverse student population with first‐hand experience of ethanol's biochemical origins, we developed simple fermentation and distillation protocols using grocery store supplies and standard lab glassware. In subsequent weeks, students used agar plates, spirit lamps, baking supplies, and fruit flies to characterize ethanol as an antiseptic, fuel, solvent, and intoxicant. Activities were deliberately open‐ended to engage students in experimental design as well as laboratory materials and skills. All protocols were developed and validated during a summer faculty/student collaborative research project, supporting pedagogical as well as career development goals. Sample data and outcomes indicated appreciable student engagement in the process as well as content of the activities, and supported new models for accessible biochemistry active‐learning opportunities for a broad range of undergraduates.

Nonlocality distillation for high-dimensional correlated boxes

Most of the existing distillation protocols only work for binary-input binary-output nonlocal boxes (two-dimensional boxes), and they cannot be generalized to the binary-input multi-output nonlocal boxes (high-dimensional boxes) in a trivial way. We will design some comparator-based protocols to distill high-dimensional nonlocal boxes. Our protocols are more powerful and universal than the previous ones in the sense that they can distill the arbitrary-dimensional boxes rather than the limited two-dimensional ones. The initial nonlocalities and the wiring manners between the boxes are two main factors deciding the distillation efficiency.

Deterministic entanglement distillation for secure double-server blind quantum computation.

Blind quantum computation (BQC) provides an efficient method for the client who does not have enough sophisticated technology and knowledge to perform universal quantum computation. The single-server BQC protocol requires the client to have some minimum quantum ability, while the double-server BQC protocol makes the client's device completely classical, resorting to the pure and clean Bell state shared by two servers. Here, we provide a deterministic entanglement distillation protocol in a practical noisy environment for the double-server BQC protocol. This protocol can get the pure maximally entangled Bell state. The success probability can reach 100% in principle. The distilled maximally entangled states can be remaind to perform the BQC protocol subsequently. The parties who perform the distillation protocol do not need to exchange the classical information and they learn nothing from the client. It makes this protocol unconditionally secure and suitable for the future BQC protocol.

Towards quantum networks of single spins: analysis of a quantum memory with an optical interface in diamond.

Single defect centers in diamond have emerged as a powerful platform for quantum optics experiments and quantum information processing tasks. Connecting spatially separated nodes via optical photons into a quantum network will enable distributed quantum computing and long-range quantum communication. Initial experiments on trapped atoms and ions as well as defects in diamond have demonstrated entanglement between two nodes over several meters. To realize multi-node networks, additional quantum bit systems that store quantum states while new entanglement links are established are highly desirable. Such memories allow for entanglement distillation, purification and quantum repeater protocols that extend the size, speed and distance of the network. However, to be effective, the memory must be robust against the entanglement generation protocol, which typically must be repeated many times. Here we evaluate the prospects of using carbon nuclear spins in diamond as quantum memories that are compatible with quantum networks based on single nitrogen vacancy (NV) defects in diamond. We present a theoretical framework to describe the dephasing of the nuclear spins under repeated generation of NV spin-photon entanglement and show that quantum states can be stored during hundreds of repetitions using typical experimental coupling parameters. This result demonstrates that nuclear spins with weak hyperfine couplings are promising quantum memories for quantum networks.

Exponential Decay of Correlations Implies Area Law

We prove that a finite correlation length, i.e., exponential decay of correlations, implies an area law for the entanglement entropy of quantum states defined on a line. The entropy bound is exponential in the correlation length of the state, thus reproducing as a particular case Hastings’s proof of an area law for groundstates of 1D gapped Hamiltonians. As a consequence, we show that 1D quantum states with exponential decay of correlations have an efficient classical approximate description as a matrix product state of polynomial bond dimension, thus giving an equivalence between injective matrix product states and states with a finite correlation length. The result can be seen as a rigorous justification, in one dimension, of the intuition that states with exponential decay of correlations, usually associated with non-critical phases of matter, are simple to describe. It also has implications for quantum computing: it shows that unless a pure state quantum computation involves states with long-range correlations, decaying at most algebraically with the distance, it can be efficiently simulated classically. The proof relies on several previous tools from quantum information theory—including entanglement distillation protocols achieving the hashing bound, properties of single-shot smooth entropies, and the quantum substate theorem—and also on some newly developed ones. In particular we derive a new bound on correlations established by local random measurements, and we give a generalization to the max-entropy of a result of Hastings concerning the saturation of mutual information in multiparticle systems. The proof can also be interpreted as providing a limitation on the phenomenon of data hiding in quantum states.

Entanglement distillation for quantum communication network with atomic-ensemble memories.

Atomic ensembles are effective memory nodes for quantum communication network due to the long coherence time and the collective enhancement effect for the nonlinear interaction between an ensemble and a photon. Here we investigate the possibility of achieving the entanglement distillation for nonlocal atomic ensembles by the input-output process of a single photon as a result of cavity quantum electrodynamics. We give an optimal entanglement concentration protocol (ECP) for two-atomic-ensemble systems in a partially entangled pure state with known parameters and an efficient ECP for the systems in an unknown partially entangled pure state with a nondestructive parity-check detector (PCD). For the systems in a mixed entangled state, we introduce an entanglement purification protocol with PCDs. These entanglement distillation protocols have high fidelity and efficiency with current experimental techniques, and they are useful for quantum communication network with atomic-ensemble memories.

Resource cost results for one-way entanglement distillation and state merging of compound and arbitrarily varying quantum sources

We consider one-way quantum state merging and entanglement distillation under compound and arbitrarily varying source models. Regarding quantum compound sources, where the source is memoryless, but the source state an unknown member of a certain set of density matrices, we continue investigations begun in the work of Bjelaković et al. [“Universal quantum state merging,” J. Math. Phys. 54, 032204 (2013)] and determine the classical as well as entanglement cost of state merging. We further investigate quantum state merging and entanglement distillation protocols for arbitrarily varying quantum sources (AVQS). In the AVQS model, the source state is assumed to vary in an arbitrary manner for each source output due to environmental fluctuations or adversarial manipulation. We determine the one-way entanglement distillation capacity for AVQS, where we invoke the famous robustification and elimination techniques introduced by Ahlswede. Regarding quantum state merging for AVQS we show by example that the robustification and elimination based approach generally leads to suboptimal entanglement as well as classical communication rates.

Quantum error correction and detection: Quantitative analysis of a coherent-state amplitude-damping code

We re-examine a non-Gaussian quantum error correction code designed to protect optical coherent-state qubits against errors due to an amplitude damping channel. We improve on a previous result [Phys. Rev. A 81, 062344 (2010)] by providing a tighter upper bound on the performance attained when considering realistic assumptions which constrain the operation of the gates employed in the scheme. The quantitative characterization is performed through measures of fidelity and concurrence, the latter obtained by employing the code as an entanglement distillation protocol. We find that, when running the code in fully-deterministic error correction mode, direct transmission can only be beaten for certain combinations of channel and input state parameters, whereas in error detection mode, the usage of higher repetition encodings remains beneficial throughout.

Distillation protocols for Fourier states in quantum computing

Fourier states are multi-qubit registers that facilitate phase rotations in fault-tolerant quantum computing. We propose distillation protocols for constructing the fundamental, $n$-qubit Fourier state with error $O(2^{-n})$ at a cost of $O(n \log n)$ Toffoli gates and Clifford gates, or any arbitrary Fourier state using $O(n^2)$ gates. We analyze these protocols with methods from digital signal processing. These results suggest that phase kickback, which uses Fourier states, could be the current lowest-overhead method for generating arbitrary phase rotations.