Quantum Networks with Deterministic Spin-Photon Interfaces
QQuantum Networks with Deterministic Spin-Photon Interfaces
J. Borregaard, A. S. Sørensen, and P. Lodahl QMATH, Department of Mathematical Sciences,University of Copenhagen, 2100 Copenhagen Ø, Denmark Center for Hybrid Quantum Networks (Hy-Q), The Niels Bohr Institute,University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark (Dated: March 29, 2019)We consider how recent experimental progress on deterministic solid state spin-photon interfacesenable the construction of a number of key elements of quantum networks. After reviewing someof the recent experimental achievements, we discuss their integration into Bell state analyzers,quantum non-demolition detection, and photonic cluster state generation. Finally, we outline howthese elements can be used for long-distance entanglement generation and quantum key distributionin a quantum network.
I. INTRODUCTION
Over the past decades, the counterintuitive and mind-boggling features of quantum mechanics have movedfrom the stage of theoretical ”Gedanken Experiments”to being the effects underlying the development of awhole new range of quantum technologies. The promisesof ultra-sensitive metrology [1], powerful quantum com-puters [2], and new cryptographic primitives [3] havespurred substantial interest worldwide. A plethora ofexperimental platforms are currently being pursued aspossible hardware candidates each with their differentstrengths and weaknesses. The hardware of choicestrongly depends on the application in mind. Therehave been impressive developments of trapped ions [4],Rydberg atoms [5] and superconducting qubits [6] forquantum computation, while quantum communicationapplications such as quantum key distribution (QKD)require optical photons that could be generated bysingle atoms [7], solid-state defects [8], or quantumdots [9]. This has spurred substantial experimentalprogress towards deterministic solid-state spin-photon in-terfaces [10]. In this progress report, we discuss some ofthese experimental developments and consider how theymay enable the implementation of protocols for quantumkey distribution and quantum networks in general.
II. THE DETERMINISTIC SPIN-PHOTONINTERFACE
Optical photons are the carriers of choice for distribut-ing quantum information over long distances, since pho-tons can propagate with low-loss through optical fibersand encounter negligible thermal noise even at room tem-perature. On the other hand, for application in quan-tum information processing it is essential to be able tostore and process the information encoded in the pho-tons. As a consequence, an efficient interface betweenlight and matter is an essential building block. To thisend, the atomic-physics community has pioneered cav-ity quantum electrodynamics as an approach for inter- (a)(b)
FIG. 1. Scanning-electron microscope image of photonic crys-tal (PC) structures with single spins and photons sketched ontop. A PC waveguide is shown in (a), which can allow fornear-unity emission of single photons from a spin system intoa propagating guided mode. A PC cavity evanescently cou-pled to an optical fiber is shown in (b). The images are repro-duced (adapted) with permission (a): Ref. [9], 2015, AmericanPhysical Society and (b): Ref. [17], 2014, Springer Nature. facing single atoms and single photons by strongly en-hancing the electromagnetic field in a resonator [11–13].More recently, solid-state implementations have been de-veloped where single atoms are replaced by solid-statequantum emitters such as quantum dots [14, 15] or va-cancy centers in diamond [16]. Furthermore, the develop-ment of nanophotonics implies that advanced devices canbe fabricated where light-matter interaction is preciselytailored [9]. Consequently, it is today possible to engi-neer an almost deterministic interface between a singlephoton and a single quantum emitter, by creating con-ditions where the emitter is preferentially coupled to asingle mode of a cavity or a waveguide (see Fig. 1). If theground state of the quantum emitter consists of a coher-ent spin, the interface comprises a quantum memory en-abling advanced quantum functionalities. Various exper-imental implementations of spin-photon interfaces havebeen studied using, e.g., single trapped atoms in cavi- a r X i v : . [ qu a n t - ph ] M a r Notation Parameter Relevance γ rad Free-space radiative decay rate of emit-ter. - Loss of photons.- Efficiency of spin-photon interface. γ nonrad Non-radiative decay rate of emitter. - Efficiency of spin-photon interface. γ dp Pure dephasing rate. - Indistinguishability of photons.- Fidelity of protocols.Γ Total decay rate of emitter. - Bandwidth of spin-photon interface. T coh Spin coherence time. - Quantum memory time.∆ ω Inhomogeneous broadening. - Stability of optical transition.- Indistinguishability of photons.- Fidelity of protocols. β = Γ Γ Ratio between the optical decay rate intoa waveguide (Γ ) and the total decayrate of emitter. - Efficiency of spin-photon interface. β coh = Γ Γ+ γ dp Ratio between optical decay rate into awaveguide and the total decay rate in-cluding the pure dephasing rate of theemitter. - Efficiency of coherent photon generation.- Fidelity of protocols. C = | g | κ ( γ rad + γ nonrad ) Cooperativity of an emitter coupled to acavity, where g is the single photon Rabifrequency and κ is the total decay rateof the cavity field. - Ratio of decay into the cavity to unde-sired decay.- Efficiency of spin-photon interface. C coh = | g | κ ( γ rad + γ nonrad + γ dp ) Cooperativity of an emitter coupled to acavity including the pure dephasing rateof the emitter. - Ratio of coherent decay into the cavityto undesired decay.- Efficiency of coherent photon generation.- Fidelity of protocols. η in /η out Input/output coupling efficiency of light. - Efficiency of quantum operations.- Loss errors.TABLE I. Parameters characterizing a spin-photon interface and their relevance for the quantum-information protocols discussedin the main text. ties [17–20], Silicon (SiV) or Nitrogen (NV) vacancy cen-ters in diamond [21–23] or self-assembled quantum dotsin gallium arsenide [15, 24–28]. Here our main focus willbe on implementations where the photon-emitter cou-pling efficiency is near unity and highly coherent, whichis the limit where spin and photon become deterministi-cally coupled.Efficient spin-photon interfaces can be implemented ineither cavity or waveguide geometries, cf. Fig. 1, cor-responding to the case where the emitter is coupled toa localized or a travelling photon. Similar functionali-ties can in general be implemented on both platforms. In open waveguide geometries, however, the ability toengineer a chiral light-matter coupling can lead to newopportunities for spin-path photon entanglement, inte-grated quantum photonic circuits, and multi-emitter cou-pling [29–31]. The relevant figures-of-merit characteriz-ing the spin-photon interface are summarized in Table I.In essence, the radiative emitter decay time should beshort in order to rapidly generate photons, the spin co-herence time long to generate high-fidelity multi-photonentangled states, and any homogenous and inhomoge-neous broadening should be reduced in order to obtainindistinguishable photons. For waveguide implementa-tions, the β -factor is the essential parameter that charac-terizes the probability to generate a photon in the desiredmode. For instance, in QKD applications, the relevantfigure-of-merit is the probability to get a photon into anoptical fiber βη out , which is determined by the outcou-pling efficiency η out and the capture probability of thenanostructure β (see Tab. I).For applications requiring the interference of differentphotons or interaction of single photons with the quan-tum emitter, any decoherence processes become relevant.The pure dephasing rate γ dp signifies the broadening dueto fast decoherence processes and is important for char-acterizing an emitter. It is therefore often convenient tointroduce the coherent β coh factor, which includes thepure dephasing rate (see Tab. I). When considering pho-ton interference between different quantum emitters, theinhomogeneous broadening of the emitters ∆ ω needs tobe considered as well.For applications involving storage of photons, time re-versal arguments can be used to show [32, 33] that theprobability for an emitter to absorb an incoming photonis given by the same efficiency as the probability to emita photon into the desired mode. Hence the beta-factoralso plays an important role for storage.For several applications in quantum information pro-cessing it is also favorable to exploit the effective non-linear interaction between photons induced by the emit-ters. Ultimately such non-linearity arises from the factthat two photons cannot be absorbed by the same emittersimultaneously and is thus determined by the β factor.Note, however, that various decoherence processes mayleak qubit information to the environment and shouldtherefore also be carefully considered for such applica-tions.The above discussion has been phrased in the languageof wavequide interfaces where the β factor is the most im-portant quantity. For implementations based on opticalcavities, the figure of merit for the quality of the inter-faces is typically expressed in terms of the cooperativity C = 4 | g | / ( κ ( γ rad + γ nonrad )), where g is the single pho-ton Rabi frequency of the cavity coupled transition and κ is the total decay rate of the cavity field. The free spacedecay rate and non-radiative decay rate is denoted γ rad and γ nonrad , respectively. For broadband cavities, thecooperativity expresses the ratio of the cavity induceddecay rate to the decay rate in the absence of the cav-ity. Hence the equivalent of the β factor, the probabilityto decay through the cavity field, can be expressed as C/ (1 + C ). As with the β factor, it is also convenient tointroduce a coherent cooperativity C coh that includes thepure dephasing rate of the emitter (see Tab. I). With thisidentification, the functionalities of interfaces based onwaveguides and cavities becomes almost identical, withonly minor differences between them. The protocols de-scribed below are thus applicable for both implementa-tions, although there may be differences in the requiredlinear optical elements surrounding the interface. III. THEORETICAL BUILDING BLOCKS
The access to deterministic spin-photon interfacesopens up new routes to realize some basic elements ofa quantum network. In this section, we will discusshow optical Bell state analyzers, photonic quantum non-demolition detectors, and photonic cluster-state genera-tion may be realized with such hardware. In Sec. IV,we outline how these elements are crucial to a number ofproposals for entanglement and long-distance quantumkey distribution in quantum networks.
A. Optical Bell state analyzer
A Bell state analyzer is a device that allows to mea-sure two qubits in the Bell basis consisting of the fourBell states | φ ± (cid:105) = ( | (cid:105) ± | (cid:105) ) / √ | ψ ± (cid:105) = ( | (cid:105) ±| (cid:105) ) / √ | (cid:105) and | (cid:105) . Such a device canbe used for both heralded entanglement generation andentanglement swapping in quantum repeaters. In addi-tion, a Bell state analyzer can also be used for fusiongates in cluster state generation, as we discuss below andfor estimation of purity of a quantum state [34].A deterministic Bell analyzer is not possible with lin-ear optics elements [34] but probabilistic versions havebeen proposed and realized for photonic qubits [35–37].Without auxiliary photons, the maximum success prob-ability is 50% [36, 38] but allowing for auxiliary multi-photon states can enable near-deterministic Bell analyz-ers based on linear optics [39, 40]. While a success prob-ability of 75% is possible using only 4 auxiliary singlephotons [40], this approach, in general, requires the gen-eration of multi-photon entangled states. For instance,an entangled state of 30 photons is needed to reach a suc-cess probability of ∼
97% using the scheme of Ref. [39].An alternative strategy is to create strong optical non-linearities by coupling spin systems to optical resonatorsor waveguides. A controlled-phase gate (CZ-gate) canbe realized between two photons by sequential scatter-ing off a cavity or a waveguide coupled to a three-levelsystem [41] (see Fig. 2(a)). Together with single photonHadamard gates and detectors this enables a Bell stateanalyzer.The basic principle can be understood by consideringthe scattering of a single photon from a single sided cavitystrongly coupled to a three-level atomic system. Assumethat the atom is prepared in some arbitrary superposi-tion of two ground states | g (cid:105) and | s (cid:105) . The cavity fieldcouples state | s (cid:105) to an excited level | e (cid:105) with single pho-ton Rabi frequency g , while the other ground state | g (cid:105) is uncoupled. In the absence of intra-cavity losses, theannihilation operator describing the scattered light willbe [42] ˆ a out = − C ˆ N s C ˆ N s ˆ a in , (1) (a)(b) FIG. 2. (a) Schematic setup for the optical CZ-gate ofRef. [41]. The qubit information is encoded in the polarizationof the photons. A polarizing beam splitter (PBS) directs the h -polarized component to the spin system and the v -polarizedcomponent to a mirror M. The j ’th pulse needs to be incidenttwice to implement the gate. (b) Schematic setup of the ac-tive, error-proof optical Bell state analyzer of Ref. [43]. Thequbit information is assumed encoded in the path of the pho-tons (dual rail) and two spin systems are needed. The figuresare reproduced with permission (a): Ref. [41],2004 AmericanPhysical Society and (b): Ref. [43], 2012, IOP Publishing. assuming resonant (both with the cavity and the atomictransition) input light described by the annihilation op-erator ˆ a in . Here C = | g | / ( γκ ) is the cooperativity where κ is the intensity decay rate of the cavity field. Sponta-neous emission from the excited level is assumed to bedescribed by a Lindblad operator ˆ L = √ γ | s (cid:105)(cid:104) e | with γ being the spontaneous decay rate of the excited level.The quantity ˆ N s = | s (cid:105)(cid:104) s | is the projector onto state | s (cid:105) .Thus, ˆ N s = 1 if the atom is prepared in state | s (cid:105) suchthat ˆ a out ≈ ˆ a in for C (cid:29)
1. If the atom is prepared instate | g (cid:105) , we have that ˆ N = 0 and ˆ a out ≈ − ˆ a in . Conse-quently, the field experiences a π -phase shift dependingon the atomic state.If a qubit is encoded in the horizontal/vertical polariza-tion components of the photonic field and only horizontalpolarization couples to the atomic system, the scatteringwill amount to a CZ-gate between the photonic qubit andan atomic qubit encoded in the ground states: an arbi-trary state α | V (cid:105) | g (cid:105) + α | V (cid:105) | s (cid:105) + α | H (cid:105) | g (cid:105) + α | H (cid:105) | s (cid:105) is transformed to α | V (cid:105) | g (cid:105) + α | V (cid:105) | s (cid:105) + α | H (cid:105) | g (cid:105) − α | H (cid:105) | s (cid:105) up to a global phase in the limit C (cid:29)
1. Here | H (cid:105) ( | V (cid:105) ) dones a horizontal (vertical) polarized photon.A photon-photon CZ gate can then be obtained throughsequential scattering as described in Ref. [41].Comparing the single photon assisted linear opticalbell state analyzer to the non-linearity based approach, the latter seems most promising for integrated photon-ics. While efficient coupling of quantum dots and colordefects to nanophotonic resonators have already beendemonstrated in experiments [22, 24], a number of addi-tional requirements, however, have to be considered. Inthe original proposal [41], one of the photons has to scat-ter off the spin system twice requiring fast optical routingand delay lines. This can be circumvented by introducinga second spin system [43] (see Fig. 2(b)). Importantly,with a second spin system the Bell-state analyzer can alsobe made error-proof in the sense that limited couplingefficiency only reduces the success probability, but neverleads to the wrong outcome. Both proposals require con-trol pulses and/or measurements on the spin systems andare thus examples of active Bell state analyzers.A passive Bell state analyzer without the need for con-trol pulses was also proposed in Ref. [43]. Comparedto the active protocols, this protocol is based on photonsorting where the direct non-linearity associated with twophotons interacting with the same optical transition isused to distinguish between zero, one, or two photon in-puts (see Fig. 3(a)). It is, however, not possible to havea perfect and deterministic photon sorter with scatter-ing from a single two-level spin system [44]. As a result,the passive Bell state analyzer is inherently probabilisticalthough it can be made near-deterministic through con-catenated applications of it. In a similar manner, scatter-ing from multiple two-level systems can be used to boostthe fidelity of a passive and deterministic CZ gate [45]. Arecent proposal also shows that a deterministic Bell stateanalyzer can be realized when combining photon scatter-ing and active spectral-temporal mode selection [46] (seeFig. 3(b)). The main characteristics and requirementsof the different Bell state analyzers are summarized inTable II. B. Optical QND detection
Quantum Non-Demolition detection (QND detection)is another desirable primitive for quantum communica-tion. An optical QND detector makes it possible to de-termine the presence of a photon without destroying it.In a dual-rail encoding, where the logical states | (cid:105) and | (cid:105) correspond to a photon being in two different opticalmodes, this allows to repeatedly measure the qubit state,thereby increasing the measurement fidelity. A QND de-tection can also be used by a receiver to check whetherthe photon is present without disturbing the qubit infor-mation if another degree of freedom such as polarizationis used to encode the qubit. As a direct application,this can be used to perform device independent quantumkey distribution without the need for heralded entangle-ment [47].A QND detector can be realized using the same basicmechanisms underlying the Bell state analyzers: By scat-tering off a three-level spin system coupled to an opticalresonator or waveguide, the presence of a photon can be (a)(b) FIG. 3. (a) Passive optical Bell state analyzer of Ref. [43]based on photon sorters. The setup for the photon sorter (PS)is shown on top while its integration into a (probabilistic) Bellstate analyzer is shown in the bottom. BS and BS are lin-ear beam splitter arrays and the crossed squares are Faradaymirrors separating incoming and reflected modes. (b) OpticalBell state analyzer of Ref. [46]. A deterministic photon sorterwith active mode selection is shown in the top. TLS denotesa two-level emitter while SFG denotes sum frequency genera-tion, which converts the frequency of the single-photon com-ponent that is generated in an orthogonal spectral-temporalmode to the two-photon component after the scattering pro-cess. The dichroic beam splitter subsequently separates oneand two photon components. A deterministic Bell state an-alyzer can be constructed with four photon sorters, linearoptics, and photon counting (bottom). All operations in botha) and b) can be made error proof against finite coupling ef-ficiency, such that successful operation is heralded by clicksin the detectors and photon losses therefore only influencethe success probability, not the fidelity of the operation. Thefigures are reproduced with permission (a): Ref. [43], 2012,IOP Publishing and (b): Ref. [46], 2015, American PhysicalSociety. detected through detection of the spin levels. QND de-tection of optical photons has already been demonstratedexperimentally by scattering off an atom [48] or quan-tum dot [49] strongly coupled to an optical cavity. Inthe QND setup, the spin-system is initially prepared ina superposition ( | s (cid:105) + | g (cid:105) ) / √ | g (cid:105) , and a metastable state, | s (cid:105) . The ground state is as-sumed to be coupled through the waveguide or resonatormode to an excited state | e (cid:105) . The scattering of a photonon this transition will ideally result in a π phase shift on the | g (cid:105) state as described above. Consequently, the stateof the emitter will transform into ( | s (cid:105) − | g (cid:105) ) / √ | s (cid:105) + | g (cid:105) ) / √ ∼ β coh − ∼ C coh ). C. Photonic cluster generation
A key advantage of solid state emitters is that they canbe operated as very bright single photon sources. Thefast photon emission rates in the range of GHz makesit possible to emit many photons within the typical co-herence time of solid state spin states. This opens upthe possibility to create multi-photon entangled stateswith a single emitter [50] such as Greenberger-Horne-Zeilinger (GHZ) [51] and cluster states [52] Dependingon the specific state, different applications may be rel-evant. Photonic 1D cluster states and GHZ states mayserve as resources for quantum enhanced metrology [53],while 2D cluster states can serve as a resource for univer-sal measurement-based quantum computation [54]. Cer-tain loss-tolerant cluster states [55] can also be used forquantum repeater protocols as we will discuss below.It was shown in Ref. [52] that 1D cluster states canbe emitted from a single quantum emitter. The ba-sic mechanism behind this is the repeated excitationof the quantum emitter. Consider a quantum emitterwith two ground state levels | g (cid:105) , | s (cid:105) and two excitedlevels | e g (cid:105) , | e s (cid:105) . Assume the transitions | g (cid:105) ↔ | e g (cid:105) and | s (cid:105) ↔ | e s (cid:105) are both strongly coupled to a waveg-uide mode but to different polarizations (horizontal, | H (cid:105) and vertical, | V (cid:105) ). Initially, the emitter is prepared inthe state ( | g (cid:105) + | s (cid:105) ) / √
2. The protocol now excites theemitter with a laser pulse to make the transformation( | g (cid:105) + | s (cid:105) ) / √ → ( | e g (cid:105) + | e s (cid:105) ) / √
2. In the ideal limit,where emission is solely through the waveguide, the emit-ter coherently decay to the state1 √ | g (cid:105) | H (cid:105) + | s (cid:105) | V (cid:105) ) . (2)Repeating the procedure n times creates a GHZ state be-tween n photonic qubits [51] and the emitter. If a rota-tion of the emitter is performed in between emissions, theresulting state would be a ( n + 1)-qubit 1D cluster stateconsisting of n photonic qubits and the emitter [52]. Thisprotocol was recently realized in an experiment demon-strating a 1D cluster state with entanglement inferredtheoretically to last up to 5 photons [56].There have been a number of proposals for generat-ing 2D-cluster states based on a divide-and-conquer ap-proach where smaller (1D) states are fused together in Bell state analyser Characteristics RequirementsLinear optics [36, 39, 40] Success probability:- 50% - no auxiliary photons.- 75% - 4 single photon auxiliary states.- >
75% - multi-photon entangled auxiliarystates. - Beam splitters.- Single photon detectors.- Indistinguishable photons.Cavity CZ-gate [41] - Failure probability ∝ C .- Error from pulseshape distortion suppressedas ∼ σ ω κ .- Error from asymmetric spontaneous emis-sion loss ∝ C . - 3-level spin system with strong optical cou-pling.- Spin control and readout.- Single photon detection and Hadamardgates.- Optical routing and delay.Active Bell scheme [43] - Success probability ∼ (2 β − .- Error suppressed as ∼ (cid:16) σ ω Γ + γ (cid:17) . - Two 3-level spin system with strong opticalcoupling.- Spin control and readout.- Single photon detection and beam splitter.Passive Bell scheme [43] - Maximum success probability of ∼
75% fora single setup.- Success probability >
75% with many con-catenated setup.- Error from inhomogeneous coupling of emit-ters ∼ (cid:16) ∆Γ σ ω (cid:17) , where ∆Γ is the dif-ference in waveguide decay rate of the twospins. - Eight 2-level spin system with strong opticalcoupling.- Single photon detection and beam splitters.Passive CZ scheme [45] For perfect coherent scattering from N two-level systems:- Error ∼ . N − . .- Optimal width ∼ . N − . Γ. - Multiple coupled two-level spin systems.- Counter propagating wavepackets.- Chiral interations.Bell scheme with activeoptics [46] - Failure probability ∝ − ββ .- Errors from inhomogeneous coupling ofspins and non-perfect filtering. - Four 2-level spin systems with strong opticalcoupling.- Single photon detection and beam splitters.- Filtering through sum frequency generation.TABLE II. Main characteristics and requirements of the Bell state analyzers considered in Sec. III. In general, Gaussian pulseswith frequency width σ ω are assumed. The parameters used to characterize the performance of the schemes are defined inTab. I and perfect in/out coupling and photodetectors are assumed. parallel to make larger (2D) cluster states [57, 58]. Thefusion gates can be probabilistic, which makes these pro-posals suited for linear optics approaches. The divide-and-conquer approach enables an efficient (polynomial)scaling of resources (such as the number of single pho-ton sources and detectors) with the cluster size despitethe probabilistic operations. Nonetheless, the inherentprobabilistic nature of the fusion gates can still lead tosubstantial overhead.The access to non-linear quantum operations withquantum emitters opens up alternative routes to the gen-eration of 2D cluster states. Strings of 1D cluster statesbeing emitted from separate quantum emitters can bejoined by performing entangling gates between either the photons or the emitters. The former approach can be re-alized using optical Bell state analyzers based on opticalnon-linearities as described in Sec. III A. The latter ap-proach requires direct entangling gates between the emit-ters in between photon emissions [59] (see Fig. 4(A)) withthe number of quantum emitters scaling linearly with thesize of the 2D cluster state. The generation protocol mayhowever be optimized for other graph states than 2D clus-ters. One example is the loss-tolerant graph states con-sidered in the all-optical repeaters of Refs. [60, 61]. An ef-ficient scheme to generate such states has been proposedin Ref. [62] where the number of qubit spin systems scalelogarithmically with the size of the graph state. Refs. [59]and [62] both assume the availability of a deterministic, (A)(B) FIG. 4. (A) 2D cluster generation using multiple quan-tum emitters [59]. Entangling gates between two emitters(blue dots) in between photon emission allows to emit con-nected 1D cluster strings. Following two spin rotations (a)entanglement (solid line) is created between the two emitters(b) followed by the emission of two photons (white dots) (c).This procedure is then repeated leading to the states in (d)-(h). (B) Single-emitter proposal for generating 2D photonic-cluster states [63]. The introduced optical delay line allowsthe photons to interact with the emitter twice, which gener-ates the 2D cluster state. The figures are reproduced withpermission (A): Ref. [59], 2010, American Physical Societyand (B): Ref. [63], 2017, National Academy of Sciences. high-fidelity entangling gate between spin systems. Forsolid state systems this can be challenging to realize op-tically because of the inhomogeneity induced by the en-vironment. For diamond defects (NV and SiV) couplingthe electronic spin to a nuclear spin may be used to cir-cumvent this problem to some extent.Another approach was suggested in Ref. [63] where theidea is to route the photons emitted by a single emit-ter back to interact with the same emitter again suchthat they become entangled with photons emitted at latertimes (see Fig. 4(B)). This can be done by using a delayline for the emitted photons together with suitable exci-tation sequence of the emitter. In this way, a 2D clusterstate can be emitted in a sequential manner. Inhomo-geneity due to slow drifts of the optical lines is less ofa problem in this case since only a single emitter is ap-plied, which makes this proposal very promising for solidstate quantum emitters. Other cluster states, such asthe loss-tolerant tree-cluster state [55] can, in principle,be generated by performing single photon measurementstogether with feedforward on a 2D cluster state. Moreefficient generation schemes may, however, be envisioneddepending on the desired cluster state and the genera-tion scheme should, in general, be optimized based onthe desired target state.
IV. QUANTUM REPEATERS
The elements described in the previous section: Bellstate analyzers, QND detection, and cluster state gener-ation may be used to overcome a key challenge for theimplementation of long-distance quantum networks: pho-ton propagation loss, which limits the distance over whichquantum information can be distributed. In particular,the above mentioned resources may be used in differenttypes of quantum-repeater protocols to enable entangle-ment distribution or quantum key distribution over longdistances.The goal of a quantum repeater is to reliably trans-mit quantum information between two distant locationsin the presence of transmission loss and noise. Since thefirst idea of a quantum repeater was presented [64], nu-merous proposals for how to realize such devices havebeen formulated [60, 65–71]. The underlying structure ofa quantum repeater has also been subject to investiga-tion, resulting in proposals for repeater structures fun-damentally different from the original one. In general,two classes of repeaters have been considered: two-wayand one-way repeaters. The original repeater scheme [64]is a two-way implementation where information has tobe transmitted in both directions across the links. Incontrast to this, a one-way repeater only transmits in-formation in one direction and can therefore potentiallybe faster. While the two-way repeater relies on quantummemories and entanglement purification, the one-way re-peater uses quantum error-correction to battle transmis-sion loss and noise [68, 70, 72].The two forms of quantum repeaters may complementeach other in a quantum network depending on the ap-plication in mind. Two-way repeaters create entangle-ment between the stations establishing a quantum linkbetween them to be used for e.g. distributed quantumcomputing [73, 74] or metrology [75, 76]. To this end,they require the availability of long-term quantum mem-ories at the repeater stations. Other applications such asquantum key distribution (QKD) do not require entan-glement distribution but simply efficient transmission of aqubit from a sender to a receiver. In such cases, one-wayrepeaters can relax the memory requirement and boostthe rate. Alternatively, one-way quantum repeaters couldbe used for generating entanglement between end-nodescontaining large quantum memories, without the needfor memories at intermediate stations. The latter is rem-iniscent of current classical repeaters which provide highspeed connections between distant computers withouthaving large memory and processing power.We will consider how the elements described in Sec. IIImay be used in both architectures. In particular, Bellstate analyzers can be used both for entanglement swap-ping in two-way repeaters and for re-encoding informa-tion at the repeater stations for one-way repeaters. Pho-tonic cluster states can be used as photonic memories inall-optical repeaters and QND detection may be used fordevice-independent quantum key distribution [47] with
Bell meas. Error corr. (a) (b)
FIG. 5. (a) Basic elements of a two-way quantum repeater.First, entanglement is generated between two quantum mem-ories (indicated by blue boxes) over the elementary links ina heralded fashion. This requires the direct transmission ofa quantum signal (photon wave packet) together with classi-cal information (double arrows) signaling the success of theattempt. After neighboring links have succeeded, a Bell mea-surement swaps the entanglement to larger distances. Inter-mediate entanglement purification before the swap may benecessary and requires two-way classical communication. (b)The setup of a one-way quantum repeater. Quantum informa-tion is transmitted directly between the repeater stations inone direction. The qubit information is encoded in an error-correcting code such that transmission loss and noise can becorrected at the repeater stations. one-way repeaters.One technical aspect is that solid-state photonic sys-tems typically have the best optical properties at opti-cal wavelengths, which is shorter than the telecom C-band where low-loss optical fibers exist. Frequency con-version to the telecom band is therefore necessary forlong-distance quantum communication and quantum re-peaters. We will not go into any details about this here,but note that recent experiments have demonstrated effi-cient frequency conversion of single photons emitted fromquantum dots [77] and NV centers [78].
A. Two-way quantum repeaters
The general structure of a two-way quantum repeateris shown in Fig. 5(a). The total distance is divided into anumber of elementary links over which entanglement canbe created in a heralded fashion by direct transmissionof a quantum signal. There exist a number of proposalsfor entanglement generation schemes based on quantumemitters [79–82]. Two-way communication is necessaryregardless of whether the entanglement is generated usinga middle station [80–82], or by direct transmission overthe entire link [79]. In both cases, a quantum signal hasto be transmitted one way while classical informationabout the success of the transmission needs to be sentback to the sending station.Consider the scheme of Ref. [80]. In this scheme a quantum emitter strongly coupled to a cavity can coher-ently emit a horizontal (vertically) polarized cavity pho-ton | H (cid:105) ( | V (cid:105) ) through a decay to a ground state | (cid:105) ( | (cid:105) )from an exited state. The two transitions have equal cou-pling strengths and the emitter-cavity state after emis-sion will thus be ( | (cid:105) | H (cid:105) + | (cid:105) | V (cid:105) ) / √
2. The cavity pho-ton is now sent towards a middle station where it it isideally combined with a photon from a similar distantsystem. The (uncorrelated) quantum state of the twosystems is12 ( | (cid:105) | H (cid:105) + | (cid:105) | V (cid:105) ) ⊗ ( | (cid:105) | H (cid:105) + | (cid:105) | V (cid:105) ) (3)with subscript 1 (2) denoting system 1 (2). This statecan be re-written in the Bell state basis as12 (cid:0) (cid:12)(cid:12) φ + (cid:11) A (cid:12)(cid:12) φ + (cid:11) P + (cid:12)(cid:12) φ − (cid:11) A (cid:12)(cid:12) φ − (cid:11) P + (cid:12)(cid:12) ψ + (cid:11) A (cid:12)(cid:12) ψ + (cid:11) P + (cid:12)(cid:12) ψ − (cid:11) A (cid:12)(cid:12) ψ − (cid:11) P (cid:1) , (4)where subscript A ( P ) denotes a state of the two atomic(photonic) systems. It is seen from Eq. 4 that a Bell mea-surement of the photons will project the atomic systemsinto a Bell state and thus create entanglement betweenthe two distant atomic systems. In Ref. [80], the Bellmeasurement is performed with linear optics with a suc-cess probability of η /
2, where 0 ≤ η ≤ ∼ µ s) electronic spin states to thelong-lived ( ∼ ms-s) nuclear spin [23, 87].Realizing a long-term quantum memory with quantumdots is not straightforward since the typical spin coher-ence time is at best on the order of µ s [14]. To thisend, hybrid approaches have been proposed, e.g. involv-ing coupling photons from quantum dots to an atomicensemble [88]. Another approach is to generate loss-tolerant photonic cluster states for creating a photonicmemory for storing quantum information [62]. This hasbeen considered in proposals for all-optical quantum re-peaters where large loss-tolerant photonic cluster statesare generated at the repeater stations [60, 61]. The clus-ter states are connected to signal photons that are sent tothe middle stations to interfere with signal photons fromthe neighboring repeater station thereby entangling twoneighboring cluster states. The rate of such all photonicrepeaters can be boosted by also transmitting the clusterstates to the middle station [60] (see Fig. 6(b)).Optical Bell state analyzers may be employed in anumber of ways in two-way quantum repeaters. For NV-based repeaters, they can be employed to perform entan-glement swapping between two NV-memories by readingthem out and measuring the corresponding photonic sig-nals. For quantum repeaters based on photonic memo-ries, they can be used in the generation of the clusterstates as outlined in Sec. III C. For both repeater types,the Bell state analyzers can also be used for the her- m = 3 BC C C I JHG encoded C A C G c3 Bell M (a)(b)
FIG. 6. (a) Two-way quantum repeater with minimal re-sources based on NV centers in diamond with both nuclear(upper black circle) and electronic (lower red circle) spin sys-tems. Entanglement between nodes is represented by bothdashed and solid lines. Entanglement purification and swap-ping is represented by rectangles and ovals, respectively. (b)Sketch of an all-photonic quantum repeater. Alice (Bob)sends one half of m entangled pairs to receiver nodes C r ( C rn +1 ). At the same time all source nodes C si creates en-coded cluster states and transmits one half of the cluster to C ri and the other half to C ri +1 . The receiver nodes attemptBell measurements on the incoming photons. If at least asingle Bell measurement is successful, they measure out re-dundant photons in the encoding to establish entanglementbetween adjacent receiver nodes. If no Bell measurementsare successful they report a failure. At the end, the repeaternodes all announce their measurement results and entangle-ment between Alice and Bob is established if no node reporteda failure. The figures are reproduced with permission (a):Ref. [86], 2005, American Physical Society and (b): Ref. [60],2015, Nature Publishing Group. alded entanglement generation in the elementary links.In entanglement generation schemes with a station in themiddle, a full Bell state measurement of the transmit-ted photons deterministically projects the correspondingmemories into an entangled state. As noted above, thisresults in an increase of the entanglement generation rateby a factor of two compared to Bell measurements basedon linear optics.While large photonic cluster states generated by quan-tum dots may function as quantum memories for two-wayrepeaters, the NV systems arguably seem more suited fortwo-way repeaters due to the availability of nuclear spinmemories. Notably, this also opens up the possibilityof performing both entanglement purification [23] andentanglement swapping within the same diamond in aminimum resource setup [66, 86] (see Fig. 6(a)). Havingaccess to more than a single NV system will, however,allow the repeater to boost its rate through parallel en-tanglement generation attempts [71]. The possibility toperform Bell measurements on different NV systems willalso allow for multiplexed schemes, which can lower thememory time requirements [89].0 B. One-way quantum repeaters
Quantum dots have limited memory time but do emitphotons very rapidly. They may therefore be well suitedfor the construction of one-way quantum repeaters. Ina one-way repeater, the quantum information is encodedinto an error-correcting code and transmitted from onerepeater station to the next [90] (see Fig. 5(b)). At eachrepeater station, the errors are corrected and the quan-tum information is re-encoded. This circumvents theneed for long-term quantum memories since there is nowaiting for a heralding signal. Consequently, the repeti-tion rate of a one-way repeater is solely determined bythe local repetition rate, i.e. how fast the errors can becorrected at the repeater stations, instead of the signal-ing time between repeater stations. This is, however, nottrue if the task of the repeater is to generate an entan-gled link between two remote parties. In that case, thefirst party has to store one part of the entangled pairwhile waiting for the second party to communicate thatthe other part was received. Quantum memories willthus still be required at the end-nodes, but the one-wayrepeater alleviates the requirements for quantum memo-ries at the intermediate repeater stations. On the otherhand one-way quantum repeaters are highly suited fortasks such as quantum key distribution (QKD) wheresecret bits of quantum information are transmitted. Inthis case, only classical information has to be stored atthe two locations and no long-time quantum memory isrequired.In combination with photonic QND detection, device-independent QKD (DI-QKD) could also be achieved insuch a memory-less setting. The underlying assumptionof DI-QKD is that the two parties do not trust their ownmeasurement devices. Nonetheless, they can still obtaina secret key if they can verify that their shared correla-tions are strong enough to violate a Bell-inequality [47].The two parties have to perform loop-hole free Bell testsin order to assure that they can share a secret key. TheQND detection allows the receiving party to determinewhether the transmitted qubit was lost or not before per-forming a measurement and can thus be used to closethe detection efficiency loophole [91, 92] in a Bell testscenario despite transmission loss.A number of one-way repeater schemes have been pro-posed based on the quantum parity code [68, 90, 93–95].While this code is able to correct for up to 50% loss,the optimal spacing of repeater stations is often foundto be 1-2 km corresponding to around 10% transmissionloss at telecom wavelengths [90, 93, 95]. The parity codeinvolves encoding a single qubit into a multi-photon en-tangled state and performing teleportation-based errorcorrection at the repeater nodes (see Fig. 7). The codeoperates with the following logical states | (cid:105) L = 1 √ | + (cid:105) L + |−(cid:105) L ) , | (cid:105) = 1 √ | + (cid:105) L − |−(cid:105) L ) , (5) with |±(cid:105) L = √ n (cid:16) | (cid:105) ⊗ m ± | (cid:105) ⊗ m (cid:17) ⊗ n . Such a state canbe generated by fusing smaller entangled states togetherusing optical Bell analyzers. The fundamental buildingblock of this is photonic GHZ states [61]. Spin-photoninterfaces may be used both as Bell state analyzers andfor the generation of photonic GHZ states as outlined inSec. III.The teleportation-based error correction also requiresBell measurements to re-encode the quantum informa-tion at the repeater stations. At the repeater stations, alogical Bell state of the form ( | (cid:105) L | (cid:105) L + | (cid:105) L | (cid:105) L ) / √ α | (cid:105) L + β | (cid:105) L )and one part of the Bell pair is performed. This can bedone using spin-photon gates as considered in Ref. [90].The structure of the parity code, however, also allowsthese Bell measurements to be done efficiently with lin-ear optics. This was shown in Refs. [93, 94] and usedto construct a linear-optics, one-way quantum repeaterwithout feedforward. Employing feedforward allows toreach the fundamental efficiency limit of the logical Bellmeasurement set by linear optics and the no-cloning the-orem [95].One key challenge will be to ensure that photons emit-ted from different emitters are indistinguishable such thathigh-quality Bell measurements can be performed. Inparticular, the parity code requires a number of Bell mea-surements that increases linearly with the size ( nm ) ofthe code in order to re-encode the information at therepeater stations. This means that for spin-based imple-mentations hundreds of matter qubits per repeater sta-tion will be required [90].To decrease the complexity of the quantum repeater,alternative loss-tolerant codes may be considered. Inparticular, matter based qudits have been considered todecrease the number of matter based quantum systemsat the repeater stations [96, 97]. These works considergeneral Calderbank-Shor-Steane (CSS) codes in order tominimize the number of required matter based quditsat the repeater stations to on the order of ∼
10 [96].As with parity code quantum repeaters, spin-photon CZgates and QND detection are also highly desirable oper-ations for these repeater schemes.A significant challenge of the one-way quantum re-peaters is the necessary level of noise suppression. Therelevant error-correcting codes have high tolerance forloss since this is an easily detectable error. Other errorssuch as dephasing and depolarizing noise are harder tocorrect and one-way repeaters often require these to beat the 0 .
1% level [70, 96, 98].
V. CONCLUSION AND DISCUSSION
In conclusion, we have considered how the recentexperimental progress towards deterministic solid statespin-photon interfaces enables implementing a number1
FIG. 7. (a) One-way repeater of Ref. [90] where the encodedphotonic state is transferred to matter qubits at the repeaterstations for teleportation based error correction (TEC). (b)The TEC procedure involves the generation of an encodedBell state, which is used for a teleportation at the logical level.The teleportation operation results in an error-corrected tele-ported logical state. The figure is reproduced with permissionfrom Ref. [90], 2015, American Physical Society. of proposals relevant for the construction of quantumnetworks. Specifically, we have discussed how opticalBell state analyzers, QND detectors, and photonic clusterstate generation can be realized based on such hardware.We have discussed the integration of these elements intoboth two-way and one-way quantum repeater architec-tures, which are necessary to battle transmission loss andnoise in future quantum networks.As outlined in the article, the performance of thedevices depends on key parameters such as couplingstrength between the spin system and the optical cav-ity/waveguide, collection efficiency of emitted photons,spin coherence and inhomogeneous broadening. Theseparameters vary substantially for the range of systemscurrently being developed for spin-photon interfaces.Systems such as neutral atoms and diamond vacancies with nuclear spin coupling have long coherence times andthus seem very suited for applications such as two-wayrepeaters. In such cases, the limited collection efficiencieswill limit the rate of entanglement distribution but notnecessarily the fidelity in heralded entanglement gener-ation schemes. The limited coherence time of quantumdots makes them not well suited for this purpose buttheir very fast photon emission rates compared to thespin coherence rates makes them very strong candidatesfor multi-photon entanglement sources that could be usedin one-way repeaters or as elements in optical Bell stateanalyzers.A number of daunting challenges still remain in or-der to realize full scale quantum repeaters based on theconsidered systems. These include fast optical routing,efficient single-photon detectors, and the suppression ofthe general noise level to the 0 .
1% level for one-way re-peaters [70, 96, 98] and to the 1% level for two-way re-peaters [64, 66, 71]. Furthermore, high-performance en-tangling gate operations between emitters and conversionof optical signals to telecom wavelengths have to be de-veloped. Nonetheless, it seems that solid state emittersare already at a level of maturity where proof-of-principleexperiments of the elementary building blocks discussedin this work may be realized in the near future. Besidesdemonstrating interesting quantum mechanical phenom-ena, such experiments could also help further establishthe route towards the realization of full scale quantumrepeaters for future quantum networks.
ACKNOWLEDGMENTS
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