Optimal Entanglement Swapping in Quantum Repeaters.

We formulate the problem of finding the optimal entanglement swapping scheme in a quantum repeater chain as a Markov decision process and present its solution for different repeaters' sizes. Based on this, we are able to demonstrate that the commonly used "doubling" scheme for performing probabilistic entanglement swapping of probabilistically distributed entangled qubit pairs in quantum repeaters does not always produce the best possible raw rate. Focusing on this figure of merit, without considering additional probabilistic elements for error suppression such as entanglement distillation on higher "nesting levels," our approach reveals that a power-of-two number of segments has no privileged position in quantum repeater theory; the best scheme can be constructed for any number of segments. Moreover, classical communication can be included into our scheme, and we show how this influences the raw waiting time for different numbers of segments, confirming again the optimality of "nondoubling" in some relevant parameter regimes. Thus, our approach provides the minimal possible waiting time of quantum repeaters in a fairly general physical setting.

Efficient Backcasting Search for Optical Quantum State Synthesis.

Non-Gaussian states are essential for many optical quantum technologies. The so-called optical quantum state synthesizer (OQSS), consisting of Gaussian input states, linear optics, and photon-number resolving detectors, is a promising method for non-Gaussian state preparation. However, an inevitable and crucial problem is the complexity of the numerical simulation of the state preparation on a classical computer. This problem makes it very challenging to generate important non-Gaussian states required for advanced quantum information processing. Thus, an efficient method to design OQSS circuits is highly desirable. To circumvent the problem, we offer a scheme employing a backcasting approach, where the circuit of OQSS is divided into some sublayers, and we simulate the OQSS backwards from final to first layers. Moreover, our results show that the detected photon number by each detector is at most 2, which can significantly reduce the requirements for the photon-number resolving detector. By virtue of the potential for the preparation of a wide variety of non-Gaussian states, the proposed OQSS can be a key ingredient in general optical quantum information processing.

Phase Locking between Two All-Optical Quantum Memories.

Optical approaches to quantum computation require the creation of multimode photonic quantum states in a controlled fashion. Here we experimentally demonstrate phase locking of two all-optical quantum memories, based on a concatenated cavity system with phase reference beams, for the time-controlled release of two-mode entangled single-photon states. The release time for each mode can be independently determined. The generated states are characterized by two-mode optical homodyne tomography. Entanglement and nonclassicality are preserved for release-time differences up to 400 ns, confirmed by logarithmic negativities and Wigner-function negativities, respectively.

All-Optical Storage of Phase-Sensitive Quantum States of Light.

We experimentally demonstrate storage and on-demand release of phase-sensitive, photon-number superposition states of the form α|0⟩+ße^{iθ}|1⟩ for an optical quantized oscillator mode. For this purpose, we newly developed a phase-probing mechanism compatible with a storage system composed of two concatenated optical cavities, which was previously employed for storage of phase-insensitive single-photon states [Phys. Rev. X 3, 041028 (2013)PRXHAE2160-330810.1103/PhysRevX.3.041028]. This is the first demonstration of all-optically storing highly nonclassical and phase-sensitive quantum states of light. The strong nonclassicality of the states after storage becomes manifest as a negative region in the corresponding Wigner function shifted away from the origin in phase space. This negativity is otherwise, without the phase information of the memory system, unobtainable. While our scheme includes the possibility of optical storage, on-demand release and synchronization of arbitrary single-rail qubit states, it is not limited to such states. In fact, our technique is extendible to more general phase-sensitive states such as multiphoton superposition or entangled states, and thus it represents a significant step toward advanced optical quantum information processing, where highly nonclassical states are utilized as resources.

Deterministic quantum teleportation of photonic quantum bits by a hybrid technique.

Quantum teleportation allows for the transfer of arbitrary unknown quantum states from a sender to a spatially distant receiver, provided that the two parties share an entangled state and can communicate classically. It is the essence of many sophisticated protocols for quantum communication and computation. Photons are an optimal choice for carrying information in the form of 'flying qubits', but the teleportation of photonic quantum bits (qubits) has been limited by experimental inefficiencies and restrictions. Main disadvantages include the fundamentally probabilistic nature of linear-optics Bell measurements, as well as the need either to destroy the teleported qubit or attenuate the input qubit when the detectors do not resolve photon numbers. Here we experimentally realize fully deterministic quantum teleportation of photonic qubits without post-selection. The key step is to make use of a hybrid technique involving continuous-variable teleportation of a discrete-variable, photonic qubit. When the receiver's feedforward gain is optimally tuned, the continuous-variable teleporter acts as a pure loss channel, and the input dual-rail-encoded qubit, based on a single photon, represents a quantum error detection code against photon loss and hence remains completely intact for most teleportation events. This allows for a faithful qubit transfer even with imperfect continuous-variable entangled states: for four qubits the overall transfer fidelities range from 0.79 to 0.82 and all of them exceed the classical limit of teleportation. Furthermore, even for a relatively low level of the entanglement, qubits are teleported much more efficiently than in previous experiments, albeit post-selectively (taking into account only the qubit subspaces), and with a fidelity comparable to the previously reported values.

Ultrafast Long-Distance Quantum Communication with Static Linear Optics.

We propose a projection measurement onto encoded Bell states with a static network of linear optical elements. By increasing the size of the quantum error correction code, both Bell measurement efficiency and photon-loss tolerance can be made arbitrarily high at the same time. As a main application, we show that all-optical quantum communication over large distances with communication rates similar to those of classical communication is possible solely based on local state teleportations using optical sources of encoded Bell states, fixed arrays of beam splitters, and photon detectors. As another application, generalizing state teleportation to gate teleportation for quantum computation, we find that in order to achieve universality the intrinsic loss tolerance must be sacrificed and a minimal amount of feedforward has to be added.

Entanglement swapping between discrete and continuous variables.

We experimentally realize "hybrid" entanglement swapping between discrete-variable (DV) and continuous-variable (CV) optical systems. DV two-mode entanglement as obtainable from a single photon split at a beam splitter is robustly transferred by means of efficient CV entanglement and operations, using sources of squeezed light and homodyne detections. The DV entanglement after the swapping is verified without postselection by the logarithmic negativity of up to 0.28±0.01. Furthermore, our analysis shows that the optimally transferred state can be postselected into a highly entangled state that violates a Clauser-Horne-Shimony-Holt inequality by more than 4 standard deviations, and thus it may serve as a resource for quantum teleportation and quantum cryptography.

3/4-Efficient Bell measurement with passive linear optics and unentangled ancillae.

It is well known that an unambiguous discrimination of the four optically encoded Bell states is possible with a probability of 50% at best, when using static, passive linear optics and arbitrarily many vacuum-mode ancillae. By adding unentangled single-photon ancillae, we are able to surpass this limit and reach a success probability of at least 75%. We discuss the error robustness of the proposed scheme and a generalization to reach a success probability arbitrarily close to 100%.

Noiseless Conditional Teleportation of a Single Photon.

We experimentally demonstrate the noiseless teleportation of a single photon by conditioning on quadrature Bell measurement results near the origin in phase space and thereby circumventing the photon loss that otherwise occurs even in optimal gain-tuned continuous-variable quantum teleportation. In general, thanks to this loss suppression, the noiseless conditional teleportation can preserve the negativity of the Wigner function for an arbitrary pure input state and an arbitrary pure entangled resource state. In our experiment, the positive value of the Wigner function at the origin for the unconditional output state, W(0,0)=0.015±0.001, becomes clearly negative after conditioning, W(0,0)=-0.025±0.005, illustrating the advantage of noiseless conditional teleportation.

Exploring a new regime for processing optical qubits: squeezing and unsqueezing single photons.

We implement the squeezing operation as a genuine quantum gate, deterministically and reversibly acting "online" upon an input state no longer restricted to the set of Gaussian states. More specifically, by applying an efficient and robust squeezing operation for the first time to non-Gaussian states, we demonstrate a two-way conversion between a particlelike single-photon state and a wavelike superposition of coherent states. Our squeezing gate is reliable enough to preserve the negativities of the corresponding Wigner functions. This demonstration represents an important and necessary step towards hybridizing discrete and continuous quantum protocols.

Logical states for fault-tolerant quantum computation with propagating light.

To harness the potential of a quantum computer, quantum information must be protected against error by encoding it into a logical state that is suitable for quantum error correction. The Gottesman-Kitaev-Preskill (GKP) qubit is a promising candidate because the required multiqubit operations are readily available at optical frequency. To date, however, GKP qubits have been demonstrated only at mechanical and microwave frequencies. We realized a GKP state in propagating light at telecommunication wavelength and verified it through homodyne measurements without loss corrections. The generation is based on interference of cat states, followed by homodyne measurements. Our final states exhibit nonclassicality and non-Gaussianity, including the trident shape of faint instances of GKP states. Improvements toward brighter, multipeaked GKP qubits will be the basis for quantum computation with light.

Beating the one-half limit of ancilla-free linear optics Bell measurements.

We show that optically encoded two-qubit Bell states can be unambiguously discriminated with a success probability of more than 50% in both single-rail and dual-rail encodings by using active linear-optical resources that include Gaussian squeezing operations. These results are in contrast to the well-known upper bound of 50% for unambiguous discrimination of dual-rail Bell states using passive, static linear optics and arbitrarily many vacuum modes. We present experimentally feasible schemes that improve the success probability to 64.3% in dual-rail and to 62.5% in single-rail for a uniform random distribution of Bell states. Conceptually, this demonstrates that neither interactions that induce nonlinear mode transformations (such as Kerr interactions) nor auxiliary entangled photons are required to go beyond the one-half limit. We discuss the optimality of our single-rail scheme and talk about an application of our dual-rail scheme in quantum communication.

Bell-state measurement exceeding 50% success probability with linear optics.

Bell-state projections serve as a fundamental basis for most quantum communication and computing protocols today. However, with current Bell-state measurement schemes based on linear optics, only two of four Bell states can be identified, which means that the maximum success probability of this vital step cannot exceed 50%. Here, we experimentally demonstrate a scheme that amends the original measurement with additional modes in the form of ancillary photons, which leads to a more complex measurement pattern, and ultimately a higher success probability of 62.5%. Experimentally, we achieve a success probability of (57.9 ± 1.4)%, a substantial improvement over the conventional scheme. With the possibility of extending the protocol to a larger number of ancillary photons, our work paves the way toward more efficient realizations of quantum technologies based on Bell-state measurements.

How to decompose arbitrary continuous-variable quantum operations.

We present a general, systematic, and efficient method for decomposing any given exponential operator of bosonic mode operators, describing an arbitrary multimode Hamiltonian evolution, into a set of universal unitary gates. Although our approach is mainly oriented towards continuous-variable quantum computation, it may be used more generally whenever quantum states are to be transformed deterministically, e.g., in quantum control, discrete-variable quantum computation, or Hamiltonian simulation. We illustrate our scheme by presenting decompositions for various nonlinear Hamiltonians including quartic Kerr interactions. Finally, we conclude with two potential experiments utilizing offline-prepared optical cubic states and homodyne detections, in which quantum information is processed optically or in an atomic memory using quadratic light-atom interactions.

Demonstration of a controlled-phase gate for continuous-variable one-way quantum computation.

We experimentally demonstrate a controlled-phase gate for continuous variables using a cluster-state resource of four optical modes. The two independent input states of the gate are coupled with the cluster in a teleportation-based fashion. As a result, one of the entanglement links present in the initial cluster state appears in the two unmeasured output modes as the corresponding entangling gate acting on the input states. The genuine quantum character of this gate becomes manifest and is verified through the presence of entanglement at the output for a product two-mode coherent input state. By combining our gate with the recently reported module for single-mode Gaussian operations [R. Ukai et al., Phys. Rev. Lett. 106, 240504 (2011)], it is possible to implement any multimode Gaussian operation as a fully measurement-based one-way quantum computation.

Demonstration of unconditional one-way quantum computations for continuous variables.

One-way quantum computation is a very promising candidate to fulfill the capabilities of quantum information processing. Here we demonstrate an important set of unitary operations for continuous variables using a linear cluster state of four entangled optical modes. These operations are performed in a fully measurement-controlled and completely unconditional fashion. We implement three different levels of squeezing operations and a Fourier transformation, all of which are accessible by selecting the correct quadrature measurement angles of the homodyne detections. Though not sufficient, these linear transformations are necessary for universal quantum computation.

Synchronization of optical photons for quantum information processing.

A fundamental element of quantum information processing with photonic qubits is the nonclassical quantum interference between two photons when they bunch together via the Hong-Ou-Mandel (HOM) effect. Ultimately, many such photons must be processed in complex interferometric networks. For this purpose, it is essential to synchronize the arrival times of the flying photons and to keep their purities high. On the basis of the recent experimental success of single-photon storage with high purity, we demonstrate for the first time the HOM interference of two heralded, nearly pure optical photons synchronized through two independent quantum memories. Controlled storage times of up to 1.8 µs for about 90 events per second were achieved with purities that were sufficiently high for a negative Wigner function confirmed with homodyne measurements.

Demonstration of a quantum nondemolition sum gate.

The sum gate is the canonical two-mode gate for universal quantum computation based on continuous quantum variables. It represents the natural analogue to a qubit C-NOT gate. In addition, the continuous-variable gate describes a quantum nondemolition (QND) interaction between the quadrature components of two light modes. We experimentally demonstrate a QND sum gate, employing the scheme by R. Filip, P. Marek, and U. L. Andersen [Phys. Rev. A 71, 042308 (2005)10.1103/PhysRevA.71.042308], solely based on off-line squeezed states, homodyne measurements, and feedforward. The results are verified by simultaneously satisfying the criteria for QND measurements in both conjugate quadratures.

Universal quantum computation with continuous-variable cluster states.

We describe a generalization of the cluster-state model of quantum computation to continuous-variable systems, along with a proposal for an optical implementation using squeezed-light sources, linear optics, and homodyne detection. For universal quantum computation, a nonlinear element is required. This can be satisfied by adding to the toolbox any single-mode non-Gaussian measurement, while the initial cluster state itself remains Gaussian. Homodyne detection alone suffices to perform an arbitrary multimode Gaussian transformation via the cluster state. We also propose an experiment to demonstrate cluster-based error reduction when implementing Gaussian operations.

Experimental creation of a fully inseparable tripartite continuous-variable state.

A continuous-variable tripartite entangled state is experimentally generated by combining three independent squeezed vacuum states, and the variances of its relative positions and total momentum are measured. We show that the measured values violate the separability criteria based on the sum of these quantities and prove the full inseparability of the generated state.