Quantum tele-amplification with a continuous variable superposition state
Jonas S. Neergaard-Nielsen, Yujiro Eto, Chang-Woo Lee, Hyunseok Jeong, Masahide Sasaki
QQuantum tele-amplification with a continuous variable superposition state
Jonas S. Neergaard-Nielsen,
1, 2
Yujiro Eto,
1, 3
Chang-Woo Lee,
4, 5
Hyunseok Jeong,
4, 6 and Masahide Sasaki National Institute of Information and Communications Technology (NICT),4-2-1 Nukui-kitamachi, Koganei, Tokyo 184-8795, Japan Department of Physics, Technical University of Denmark, Fysikvej, 2800 Kgs. Lyngby, Denmark Department of Physics, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan Center for Macroscopic Quantum Control, Department of Physics and Astronomy,Seoul National University, Seoul, 151-747, Korea Department of Physics, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar Centre for Quantum Computation and Communication Technology, School ofMathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia (Dated: November 1, 2018)Optical coherent states are classical light fields with high purity, and are essential carriers ofinformation in optical networks. If these states could be controlled in the quantum regime, allowingfor their quantum superposition (referred to as a Schr¨odinger cat state), then novel quantum-enhanced functions such as coherent-state quantum computing (CSQC) , quantum metrology ,and a quantum repeater , could be realized in the networks. Optical cat states are now routinelygenerated in the laboratories. An important next challenge is to use them for implementing theaforementioned functions. Here we demonstrate a basic CSQC protocol, where a cat state is usedas an entanglement resource for teleporting a coherent state with an amplitude gain. We also showhow this can be extended to a loss-tolerant quantum relay of multi-ary phase-shift keyed coherentstates. These protocols could be useful both in optical and quantum communications.
Among various optical implementations of quantuminformation processing (QIP), coherent-state quantumcomputing (CSQC) is of special interest for enhancingthe performance of optical communications, where in-formation is encoded into coherent states. These arethe only states that can be transmitted preserving thestate purity even through a lossy channel since they areeigenstates of the annihilation operator, ˆ a | α (cid:105) = α | α (cid:105) .Hence, a simple classical encoding with coherent statescan be the optimal strategy of the transmitter to achievethe ultimate capacity of a lossy optical channel . Onthe receiver side, the sequence of coherent-state pulsesshould be decoded fully quantum mechanically by em-ploying a collective measurement with CSQC . Thisscheme can realize communication with larger capac-ity, beating the conventional homodyne limit of opti-cal communications . Although practical implementa-tion of CSQC remains a big challenge, recent progress ingenerating and manipulating optical cat statesmakes it realistic to implement its basic building blocks.In this paper, we propose and demonstrate the first op-erational application of cat states for QIP, where a catstate is used as the entanglement resource for teleportinga coherent state with an amplitude gain. We also proposeits new application to quantum key distribution (QKD),namely a loss-tolerant quantum relay of multi-ary phase-shift keyed (M-PSK) coherent states that does not as-sume a trusted node. We present its proof-of-principledemonstration with binary PSK states.The basic scheme of teleportation from Alice to Bobof a cat state qubit | ψ (cid:105) A = c + | α (cid:105) A + c − |− α (cid:105) A , which isa variation of the schemes in refs. 20 and 21, is depictedin Fig. 1 a . Bob prepares an odd cat state | Φ − ( β ) (cid:105) B = N − (cid:0) | β (cid:105) B − |− β (cid:105) B (cid:1) with normalization N − and splits itinto an entangled cat state over paths B and C via abeam-splitter (BS) ˆ V BC with reflectivity R B . He sendsone part of it to Alice at port C. She then combines it onan R A -reflectivity BS with her input | ψ (cid:105) A at port A as | Ψ (cid:105) ABC = ˆ V AC | ψ (cid:105) A ˆ V BC | Φ (cid:105) B | (cid:105) C . (1)She finally measures modes A and C by single-photondetectors. By conditioning port B on her measurementresult, Bob can restore Alice’s input.The amplitude of the resource cat state is set as β = α (cid:112) (1 − R A ) /R A R B , such that the components at portA turn into either the vacuum or a non-vacuum state.Then, when Alice’s detectors register a single photon atport A and nothing at port C – denoted (1,0) – Bobunambiguously obtains the state (see Appendix A) AC (cid:104) , | Ψ (cid:105) ABC ∝ c + |− gα (cid:105) B + c − | gα (cid:105) B , (2)where g = (cid:112) (1 − R A )(1 − R B ) /R A R B is the gain pa-rameter. By a simple π -phase shift, it can be trans-formed to Alice’s input | ψ (cid:105) A , but with a modified am-plitude α (cid:48) = gα . This process, previously suggested inref. 3, we will call tele-amplification .Unfortunately this tele-amplification is vulnerable tolosses. Suppose the channel between Alice and Bob issubject to a linear loss R E . The amplitude of the resourcecat state should then be chosen as β = (cid:115) − R A R A R B (1 − R E ) α. (3)After conditioning on Alice’s detection event (1,0), Bob’sstate gets entangled with an external mode E as | ψ (cid:105) A | (cid:105) E (cid:55)→ c + |− gα (cid:105) B | ε (cid:105) E + c − | gα (cid:105) B |− ε (cid:105) E (4) a r X i v : . [ qu a n t - ph ] N ov BACBACE AC C'A' c C ba Alice's measurementin the 4PSK case C Single photon detectionLossy channel R E R A R B R A R B R A FIG. 1: Scheme of quantum tele-amplification and quantumrelay. a Tele-amplification of binary cat-state in an ideal loss-less channel. R A and R B are the reflectivities of the BSs. b Loss tolerant quantum relay. R E is the reflectivity of the BSwhich models the lossy channel. c Alice’s four-port measure-ment for the case of 4PSK states. with the modified gain including the loss rate R E g = (cid:115) (1 − R A )(1 − R B ) R A R B (1 − R E ) . (5)Here ε = (cid:112) (1 − R A ) R E /R A (1 − R E ) α . Thus, the out-put at Bob is generally in a decohered state.One can, however, see that if Alice’s inputs are re-stricted to classical components, | α (cid:105) or |− α (cid:105) , as in Fig.1 b , the output state can be completely disentangled fromthe external mode. This means that the coherent statescan be tele-amplified faithfully to the target states eventhrough the lossy channel as |± α (cid:105) A (cid:55)→ |± gα (cid:105) B . (6) This is referred to as loss-tolerant quantum relay . In thiscontext, Bob plays the role of an intermediate node, re-stores the target states |± gα (cid:105) B , and sends them to theterminal node, Charlie.This simplest binary case can be extended into M-PSK coherent states. Let us show it for the 4-PSK case, | α m (cid:105) , ( α m = i m α, m = 0 , , , | Φ (cid:105) B = N (cid:88) k =0 i k (cid:12)(cid:12) i k β (cid:11) B (7)as a resource. This state is beam-split, and is shared withAlice. We set R A = 0 .
5. As in Fig. 1 c , Alice performs afour-port single-photon detection at paths A, A’, C, andC’ on this state. Depending on the set of results at thefour ports, (A, A’, C, C’), the inputs are tele-amplifiedas | α m (cid:105) (cid:55)→ | gα m (cid:105) , for (0 , , , , | α m (cid:105) (cid:55)→ | igα m (cid:105) , for (1 , , , , | α m (cid:105) (cid:55)→ |− gα m (cid:105) , for (1 , , , , | α m (cid:105) (cid:55)→ |− igα m (cid:105) , for (1 , , , . (8)Thus, the simple tele-amplification is performed for theresult (0,1,1,1). Moreover the output state can beswitched to another element by choosing an appropriateclick pattern at Alice (see Appendix B).The faithful relay itself can also be realized in a clas-sical way, where Bob at the intermediate node performsan unambiguous state discrimination on the signal state,reproduces an amplified state for his confident result, andfinally resends it to Charlie. The success probabilities ofthe two methods are compared in Appendix C. This clas-sical relay cannot, however, be applied to a QKD relaynode without the trusted node assumption. In contrast,a quantum relay can be carried out in the fully quantumdomain, without Bob’s knowing the signal state itself,though at the expense of preparing the entangled catstate, and an appropriate entanglement verification ses-sion. Similar ideas for single-photon QKD were presentedin refs. 22 and 23.Our loss-tolerant quantum relay is particularly usefulfor extending the distance of QKD which uses PSK co-herent states, such as B92 and BB84 . Althoughthe secure key generation probabilities at short distancesslightly degrades from the original PSK-BB84, they canremain at reasonable levels up to much longer distancesby the loss-tolerant quantum relay (see Appendix G).We carried out an experimental demonstration of thetele-amplification in the simplest case of binary PSK asin Fig. 1 b to realize Eq. (6). The resource cat state | Φ − (cid:105) was generated by photon-subtraction from a squeezedvacuum with anti-squeezing along the real axis in phasespace (Methods). Bob’s BS was set to R B = 0 .
1. Fora given desired gain g , we varied R A according to Eq.(5). The resource cat-state amplitude β , experimentallytuned by the squeezing level, was then set by Eq. (3). g = 3.0 g = 2.2 g = 1.5 g = 1.0g = 0.76g = 0.5 xp 1 2010.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.60.81.01.2 FIG. 2: Measured results for the twelve cases. The blue straight lines represent α (cid:48) = gα . The open circles represent thesets of α and g tg in Table I. The fidelities between the measured states and the targeted | g tg α (cid:105) are indicated by the numbersin brackets. Filled circles represent the amplitudes α ( α (cid:48) ) of the coherent states that have the maximum fidelity with themeasured input (output) states. Insets:
Output state Wigner functions are shown in blue contour plots. The red solid anddashed circles are for the input and ideal targeted states, respectively.
The detector at port C was omitted with negligible effecton the outcomes since the events (A,C)=(1,1) would berare. Bob’s output state was characterized by homodynetomography.We tested twelve settings as summarized in TableI. The five different input amplitudes α were real andranged between 0.35 and 1.4. The protocol was carriedout only for | α (cid:105) because the outcome for |− α (cid:105) would betrivially identical. The measured results are shown inFig. 2. The blue straight lines are gain curves α (cid:48) = gα in the ( α, α (cid:48) ) diagram. The open circles plotted alongthese lines represent sets of α and g tg in Table I. TheWigner functions of the tomographically reconstructedtele-amplified output states ˆ ρ out are shown as blue con-tour plots in the insets. One contour level is high-lighted for comparison with the targeted states | g tg α (cid:105) (red dashed) and the actual input states ˆ ρ in ≈ | α (cid:105)(cid:104) α | (red solid, also characterized by homodyne tomography).The discrepancies between ˆ ρ out and | g tg α (cid:105) are due toimperfections, including the deviation of the photon-subtracted state from the ideal resource cat, losses, im-purity, and Alice’s use of an on/off detector instead oftwo single-photon detectors (see Appendix C). For eachsetting, we calculate which coherent states | α (cid:105) , | α (cid:48) (cid:105) havethe highest fidelity with the measured input and out-put states, respectively, that is, α = argmax γ (cid:104) γ | ˆ ρ in | γ (cid:105) and α (cid:48) = argmax γ (cid:48) (cid:104) γ (cid:48) | ˆ ρ out | γ (cid:48) (cid:105) . These ( α, α (cid:48) ) pairs aremarked as filled circles.Despite the imperfections, the tele-amplification suc-ceeded with high fidelities F = (cid:104) g tg α | ˆ ρ out | g tg α (cid:105) between0.89 and 0.95 as shown next to each inset. The obtainedamplitudes (filled circles) are close to the targeted ones(open circles) in almost all cases. The success probabil-ities were in the range 0.3% to 0.65% (Methods). Forlarger α (cid:48) , the Wigner function shapes are slightly elon-gated due to the larger squeezing needed to produce thosestates. We note that our experimental settings were notfully optimized by taking into account spectral mode mis-match between the resource cat and input coherent statesas well as between the APD and homodyne detectors.Had we done it, we estimate the achieved fidelities tohave been 0.94–0.99.The settings R E = 0 .
8) in the channel from Bob to Alice. In R A = 0 . R A = 0 .
83 as optimized according to Eq. (5), re-sulting in success probabilities of 0.17% and 0.11%. Thefidelities with the target state are as high as 0.839 and0.872, respectively, as compared with 0.901 in the loss-less case. This demonstrates the loss tolerance of theprotocol. α g tg β R A R E F g tg is the targeted gain. Thelast column shows the obtained teleportation fidelities. FIG. 3: Simulated average qubit teleportation fidelity as afunction of input ( α ) and output state ( α (cid:48) ) amplitudes. Allrelevant practical imperfections in our experimental setup, asdescribed in Appendix F, are taken into account. The redcurve labelled “F = 2/3” indicates the classical teleportationbound. Teleportation of a cat-state qubit as in Eqs. (1-A5) is a prerequisite for CSQC. Interestingly, the tele-amplification allows to convert between different ampli-tude qubit bases. Although we previously generatedsuch arbitrary cat qubits , it was not feasible to tele-amplify them with the current setup since three simulta-neous APD clicks would be needed. Instead we simulatedthis protocol by accurately modelling the current exper-iment including all relevant practical imperfections (seeAppendix F). Figure 3 shows the average fidelities be-tween the teleported state for an input cat-state qubitand an output state from the model (Methods). For awide range of input amplitudes α and output amplitudes α (cid:48) , it is possible to surpass the classical limit of 2/3. Finally we make a brief comparison between ourscheme and the quantum noiseless amplifier with single-photon ancilla . The latter is intended to noiselesslyamplify a coherent state with an unknown amplitudeat the cost of the success probability. In contrast, ourscheme assumes the known amplitude α but instead en-ables one to tele-amplify PSK coherent states over a lossychannel with perfect fidelity and high success probability.It can also implement, in principle, the teleportation oftheir arbitrary superpositions.In summary, we presented tele-amplification and loss-tolerant quantum relay of coherent states as the first op-erational application of optical cat states. The scheme isan essential building block for CSQC as well as quantumcommunications. Methods
Experiment
We generated the squeezed vacua at 860nm wavelength from an OPO (optical parametric oscil-lator) continuously pumped with pump parameters be-tween 0.15 and 0.31, corresponding to β values of 0.78 to1.15. We tapped off 5% of the squeezed beam on a BSand guided it to an APD. A click of the APD heraldedthe subtraction of a photon from the main beam .The state thus generated is a close approximation to theodd cat state | Φ − (cid:105) , and has been shown to provide near-perfect teleportation performance .Whenever Alice’s APD clicked simultaneously with theheralding signal of the single-photon subtraction for theresource cat-state generation, the tele-amplification wassuccessful, and we recorded a trace of the homodyne sig-nal of Bob’s output state. The success probability isgiven by the ratio of the simultaneous click rate ( ∼ s − ) to the photon subtraction click rate ( ∼ s − ). It is mainly limited by detector and spectral fil-tering efficiency. To build the homodyne tomogram, werepeated this procedure 6000–24000 times for each fixedinput state, with the local oscillator of the homodyne de-tector locked at phases − ◦ , − ◦ , . . . , ◦ with re-spect to the input state. Note that the protocol succeedsas a single shot for an unknown input state – the repeatedmeasurements with identical inputs are only needed forcharacterizing the process by homodyne tomography.Alice’s input states were independently characterizedby homodyne tomography at port C by setting R A = 1.To determine the input states accurately just at Alice’sBS, we correct their reconstruction for the detection ef-ficiency and the propagation losses from that point tothe homodyne detector. This total efficiency amountsto 88%. Likewise, in the reconstruction of Bob’s outputstates we correct for the overall detection efficiency of94% but not for any propagation losses.A more detailed description of the experimental setupand the state characterization can be found in AppendixE. Simulation of cat-state qubit teleportation
A cat-state qubit can be represented on a Bloch sphere as | ψ ( α, θ, φ ) (cid:105) = c + | α (cid:105) + c − |− α (cid:105) = cos θ | Φ + ( α ) (cid:105) + e iφ sin θ | Φ − ( α ) (cid:105) , where | Φ ± ( α ) (cid:105) = N ± ( | α (cid:105) ± |− α (cid:105) ) are the even/odd catstates with N ± = 1 / (cid:112) ± e − α ) and c ± = N + cos θ ±N − e iφ sin θ . Given an input state | ψ ( α, θ, φ ) (cid:105) , our model of theexperiment, described in Appendix F, returns a tele- ported output state ˆ ρ α,θ,φ . To quantify the performanceof the qubit teleportation for specific settings of α and α (cid:48) = g tg α , we calculate the average fidelity of the tele-ported state with the target state by integrating over theBloch sphere: F avg α → α (cid:48) = (cid:90) dφdθ sin θ π (cid:104) ψ ( α (cid:48) , θ, φ ) | ˆ ρ α,θ,φ | ψ ( α (cid:48) , θ, φ ) (cid:105) . The results for a range of amplitude settings are plottedin Fig. 3.
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We acknowledge helpful discus-sions with K. Wakui, M. Takeoka, K. Hayasaka, M. Fu-jiwara, T. C. Ralph, A. P. Lund, K. Tamaki, and M.Koashi. This work was partly supported by the Quan-tum Information Processing Project in the Program forWorld-Leading Innovation Research and Development onScience and Technology (FIRST) and by a National Re- search Foundation of Korea (NRF) grant funded by theKorean Government (Ministry of Education, Science,and Technology) (No. 2010-0018295).
Author contributions
M.S. and J.S.N-N. formulatedthe basic protocol of tele-amplification and loss-tolerantquantum relay, inspired by a teleportation scheme by C.-W.L. and H.J.. J.S.N-N. and Y.E. carried out the ex-periment. J.S.N-N., C.-W.L., M.S. and H.J. performedthe theoretical calculations. J.S.N-N. and M.S. wrote thepaper with discussions and input from all the authors.
Appendix A: Tele-amplification of a binarycomponent cat-state
We first describe the tele-amplification of a binarycomponent cat-state | ψ ( α ) (cid:105) A = c | α (cid:105) A + c |− α (cid:105) A through a lossless channel. The whole three-mode stateafter the beam-splitting operation in Fig. 1 a is given by | Ψ (cid:105) ABC = ˆ V AC | ψ ( α ) (cid:105) A ˆ V BC | Φ (cid:105) B | (cid:105) C = N c (cid:12)(cid:12)(cid:12)(cid:112) − R A α − (cid:112) R A R B β (cid:69) A (cid:12)(cid:12)(cid:12)(cid:112) − R B β (cid:69) B (cid:12)(cid:12)(cid:12) − (cid:112) R A α − (cid:112) (1 − R A ) R B β (cid:69) C − N c (cid:12)(cid:12)(cid:12)(cid:112) − R A α + (cid:112) R A R B β (cid:69) A (cid:12)(cid:12)(cid:12) − (cid:112) − R B β (cid:69) B (cid:12)(cid:12)(cid:12) − (cid:112) R A α + (cid:112) (1 − R A ) R B β (cid:69) C + N c (cid:12)(cid:12)(cid:12) − (cid:112) − R A α − (cid:112) R A R B β (cid:69) A (cid:12)(cid:12)(cid:12)(cid:112) − R B β (cid:69) B (cid:12)(cid:12)(cid:12)(cid:112) R A α − (cid:112) (1 − R A ) R B β (cid:69) C − N c (cid:12)(cid:12)(cid:12) − (cid:112) − R A α + (cid:112) R A R B β (cid:69) A (cid:12)(cid:12)(cid:12) − (cid:112) − R B β (cid:69) B (cid:12)(cid:12)(cid:12)(cid:112) R A α + (cid:112) (1 − R A ) R B β (cid:69) C (A1)where N = 1 / (cid:112) − exp( − | β | )] is the normalizationof the resource cat state | Φ (cid:105) B . We now impose a condi-tion on the amplitude of the resource cat state β = (cid:114) − R A R A R B α (A2)such that the components at port A turn into either ofthe vacuum or non-vacuum states as | Ψ (cid:105) ABC = N c | (cid:105) A | gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) − √ R A α (cid:29) C − N c (cid:12)(cid:12)(cid:12) (cid:112) − R A α (cid:69) A |− gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) − R A √ R A α (cid:29) C + N c (cid:12)(cid:12)(cid:12) − (cid:112) − R A α (cid:69) A | gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) − − R A √ R A α (cid:29) C − N c | (cid:105) A |− gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) √ R A α (cid:29) C (A3)with the gain g = (cid:112) (1 − R A )(1 − R B ) /R A R B . (A4)Alice then performs single photon detection on pathsA and C as shown in Fig. 1 a , and selects single photon atport A and nothing at port C – denoted (1,0). Then Bobcan unambiguously exclude the first and fourth terms inEq. (A3), and has the state AC (cid:104) , | Ψ (cid:105) ABC ∝ | ψ ( − gα ) (cid:105) B . (A5)In the case where the channel between Alice and Bob issubject to a linear loss with the rate R E , one can consideran external mode E. Bob chooses the cat-state amplitudeas β = (cid:115) − R A R A R B (1 − R E ) α. (A6) The whole state before Alice’s measurement is | Ψ (cid:105) ABCE = ˆ V AC | ψ (cid:105) A ˆ V EC ˆ V BC | Φ (cid:105) B | (cid:105) C | (cid:105) E = N c | (cid:105) A | gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) − √ R A α (cid:29) C |− ε (cid:105) E −N c (cid:12)(cid:12)(cid:12) (cid:112) − R A α (cid:69) A |− gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) − R A √ R A α (cid:29) C | ε (cid:105) E + N c (cid:12)(cid:12)(cid:12) − (cid:112) − R A α (cid:69) A | gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) − − R A √ R A α (cid:29) C |− ε (cid:105) E −N c | (cid:105) A |− gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) √ R A α (cid:29) C | ε (cid:105) E (A7)with the gain g = (cid:115) (1 − R A )(1 − R B ) R A R B (1 − R E ) (A8)and ε = (cid:112) (1 − R A ) R E /R A (1 − R E ) α . Appendix B: Extension to multi-ary coherent states
The binary case can be extended to M -ary phase-shift-keyed coherent states | α m (cid:105) , where α m = αu m , u = e πi/M . (B1)Here α = α is taken to be real. The states are generatedas | α m (cid:105) = ˆ V m | α (cid:105) (B2)by modulating the phase of the coherent state | α (cid:105) withˆ V = exp (cid:18) πiM ˆ n (cid:19) . (B3)An input state at Alice is generally a superposition state | ψ (cid:105) A = M − (cid:88) m =0 c m | α m (cid:105) A . (B4)Bob prepares a cat state for the entanglement resource, | Φ (cid:105) B = M − (cid:88) k =0 b m | β m (cid:105) B (B5)where β m = βu m with β real. In order to analyzethe scheme and its performmance, we introduce the or-thonormal basis {| ω m (cid:105)} in the space spanned by {| β m (cid:105)} as follows | ω m (cid:105) = 1 √ M λ m M − (cid:88) k =0 u − mk | β k (cid:105) (B6)where λ m = M − (cid:88) k =0 u − km (cid:104) β | β k (cid:105) . (B7)The orthonormality (cid:104) ω m (cid:48) | ω m (cid:105) = δ m (cid:48) ,m can be verified bya relation M − (cid:88) k =0 u ( m − n ) k = M δ m,n + lM ( ∀ integer l ) (B8)and that the Gram matrix [ (cid:104) β k (cid:48) | β k (cid:105) ] is cyclic. The co-herent states can be expanded as | β m (cid:105) = 1 √ M M − (cid:88) k =0 (cid:112) λ k u mk | ω k (cid:105) . (B9)Then one can see thatˆ ρ = M − (cid:88) m =0 | β m (cid:105) (cid:104) β m | = M − (cid:88) m =0 λ m | ω m (cid:105) (cid:104) ω m | . (B10)Thus λ m are the eigenvalues of the density operator ofthe ensemble {| β m (cid:105)} . The mean photon number of thebasis states is (cid:104) ω m | ˆ n | ω m (cid:105) = λ m − λ m | β | . (B11)To maximize the success probability of Alice’s measure-ment, one should use the | ω m (cid:105) which has the maximumphoton number for the entanglement resource. For rel-atively smaller | β | , it is the | ω M − (cid:105) . In fact, the basisstates can be represented by the number states as | ω m (cid:105) = (cid:114) Mλ m e − β / ∞ (cid:88) l =0 β m + lM (cid:112) ( m + lM )! | m + lM (cid:105) . (B12) Thus | ω m (cid:105) consists of a set of the photon number states {| m + lM (cid:105) ; l = 0 , , ... } .Now let us see the case of M = 4 ( u = i ). The basisstates are explicitly given by | ω (cid:105) = 2 e − β / √ λ (cid:16) | (cid:105) + β √ | (cid:105) + · · · (cid:17) | ω (cid:105) = 2 e − β / √ λ (cid:16) β | (cid:105) + β √ | (cid:105) + · · · (cid:17) | ω (cid:105) = 2 e − β / √ λ (cid:16) β √ | (cid:105) + β √ | (cid:105) + · · · (cid:17) | ω (cid:105) = 2 e − β / √ λ (cid:16) β √ | (cid:105) + β √ | (cid:105) + · · · (cid:17) (B13)with the eigenvalues λ = 2 e − β (cosh β + cos β ) ,λ = 2 e − β (sinh β + sin β ) ,λ = 2 e − β (cosh β − cos β ) ,λ = 2 e − β (sinh β − sin β ) . (B14)The cat state for the entanglement resource is chosen as | Φ (cid:105) B = | ω (cid:105) B = 1 √ λ M − (cid:88) k =0 u k | β k (cid:105) B . (B15)The above state is beam-split into paths B and C, thecomponent of mode C is sent to Alice through a lossychannel, and then combined with the input at path A.Bob chooses the cat-state amplitude as Eq. (A6). Thewhole state before the measurement is given by | Ψ (cid:105) BACE = (cid:88) m =0 c m (cid:88) k =0 u k √ λ (cid:12)(cid:12) gαu k (cid:11) B ⊗ (cid:12)(cid:12)(cid:12)(cid:112) − R A α ( u m − u k ) (cid:69) A ⊗ (cid:12)(cid:12)(cid:12)(cid:12) − α √ R A (cid:104) R A u m + (1 − R A ) u k (cid:105)(cid:29) C ⊗ (cid:12)(cid:12) εu k (cid:11) E (B16)with the gain given by Eq. (A8).We set R A = 0 .
5. Alice further introduces additionalmodes A’ and C’ to implement the four-port single pho-ton detection as shown in Fig. 1 b . The state before thedetection is given by | Ψ (cid:105) BAA (cid:48) CC (cid:48) E = (cid:88) m =0 c m (cid:88) k =0 u k (cid:12)(cid:12) gαu k (cid:11) B ⊗ (cid:12)(cid:12)(cid:12)(cid:12) α u m − u k (cid:29) A ⊗ (cid:12)(cid:12)(cid:12)(cid:12) − α u m + u k (cid:29) C ⊗ (cid:12)(cid:12)(cid:12)(cid:12) α u m − u m +1 + u k + u k +1 √ (cid:29) A (cid:48) ⊗ (cid:12)(cid:12)(cid:12)(cid:12) α u m + u m +1 + u k − u k +1 √ (cid:29) C (cid:48) ⊗ (cid:12)(cid:12)(cid:12)(cid:12) R E √ − R E αu k (cid:29) E (B17)When the loss can be neglected ( R E = 0), the input | ψ (cid:105) of Eq. (B4) can be faithfully tele-amplified to thetarget state | gψ (cid:105) B = M − (cid:88) m =0 c m | gα m (cid:105) B , (B18)by selecting a set of Alice’s measurement result as (A,A’, C, C’)=(0,1,1,1), namely no count at port A whilesingle-photon counts at port A’, C, and C’.In the lossy case, it is impossible to teleport a super-position state faithfully. However, when an input is re-stricted to a classical state drawn from the set | α m (cid:105) , thenthe tele-amplification to the target pure state is possible.Actually depending on a set of the results at the fourports, (A, A’, C, C’), the inputs are tele-amplified as | α m (cid:105) (cid:55)→ | gα m (cid:105) , for (0 , , , , | α m (cid:105) (cid:55)→ | igα m (cid:105) , for (1 , , , , | α m (cid:105) (cid:55)→ |− gα m (cid:105) , for (1 , , , , | α m (cid:105) (cid:55)→ |− igα m (cid:105) , for (1 , , , . (B19) Appendix C: On/off detection at Alice
In our experiment, Alice’s measurement is imple-mented by avalanche photodiodes (APDs) instead ofideal “single-photon detectors” that discriminate be-tween “0”, “1” and “2 or more” photons. APDs can-not, however, discriminate photon numbers, but dis-tinguish merely the vacuum or non-vacuum state, i.e.“off” or “on”. They are represented by the operatorsˆΠ off = | (cid:105) (cid:104) | and ˆΠ on = ˆ I − | (cid:105) (cid:104) | . Then the tele-amplification described in the previous section should becorrected slightly. For example, Eq. (A5) for the binarycase becomes ABC (cid:104) Ψ | ˆΠ A on ˆΠ C off | Ψ (cid:105) ABC = | ψ ( − gα ) (cid:105) B (cid:104) ψ ( − gα ) | + tanh (cid:16) ˜ α (cid:17) | ˜ ψ ( − gα ) (cid:105) B (cid:104) ˜ ψ ( − gα ) | (C1) where | ˜ ψ ( − gα ) (cid:105) = c | α (cid:105) − c |− α (cid:105) and ˜ α = 2 √ − R A α .The second term is the correction. When α is small, thecoefficient of the second term is small as tanh(˜ α/ ∼ ˜ α/
2. In this regime, the tele-amplification would approx-imately work with on/off detection. However, in general,the second term cannot be ignored.If the input state was a coherent state, i.e., c = 0or c = 0, the above state would become a pure coher-ent state with the gain g , and on/off detection would besufficient. Appendix D: Success probability1. Loss tolerant quantum relay
In the binary case, the input is either of | α (cid:105) and |− α (cid:105) .The whole state before Alice’s measurement is | Ψ ± (cid:105) = ˆ V AC |± α (cid:105) A ˆ V EC ˆ V BC | Φ (cid:105) B | (cid:105) C | (cid:105) E = ±N | (cid:105) A |± gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) ∓ √ R A α (cid:29) C |∓ ε (cid:105) E ∓N (cid:12)(cid:12)(cid:12) ± (cid:112) − R A α (cid:69) A |∓ gα (cid:105) B (cid:12)(cid:12)(cid:12)(cid:12) ± − R A √ R A α (cid:29) C |± ε (cid:105) E (D1)The success probability of the tele-amplification |± α (cid:105) A (cid:55)→ |± gα (cid:105) B is given by the expectation value ofˆΠ ≡ ˆΠ A on ⊗ ˆΠ C off as P (2)Tele − amp = 12 (cid:104) Ψ + | ˆΠ | Ψ + (cid:105) + 12 (cid:104) Ψ − | ˆΠ | Ψ − (cid:105) = exp (cid:104) − (1 − R A ) R A α (cid:105) − exp (cid:104) − α R A (cid:105) (cid:16) − exp (cid:104) − − R A ) R A R B (1 − R E ) α (cid:105)(cid:17) . (D2)In the case of 4-PSK states, the state before Alice’smeasurement is given by | Ψ m (cid:105) = (cid:88) k =0 u k (cid:12)(cid:12) gαu k (cid:11) B ⊗ (cid:12)(cid:12)(cid:12)(cid:12) α u m − u k (cid:29) A (cid:12)(cid:12)(cid:12)(cid:12) − α u m + u k (cid:29) C ⊗ (cid:12)(cid:12)(cid:12)(cid:12) α u m − u m +1 + u k + u k +1 √ (cid:29) A (cid:48) ⊗ (cid:12)(cid:12)(cid:12)(cid:12) α u m + u m +1 + u k − u k +1 √ (cid:29) C (cid:48) ⊗ (cid:12)(cid:12)(cid:12)(cid:12) R E √ − R E αu k (cid:29) E (D3)The success probability of | α m (cid:105) A (cid:55)→ | gα m (cid:105) B is given bythe expectation value ofˆΠ ≡ ˆΠ A off ⊗ ˆΠ A (cid:48) on ⊗ ˆΠ C on ⊗ ˆΠ C (cid:48) on (D4)0 FIG. 4: Success probabilities for the case of the BPSK coher-ent states. The channel loss is assumed be 80% ( R E = 0 . R E = 0 . as P (4)Tele − amp = 14 (cid:88) k =0 (cid:104) Ψ m | ˆΠ | Ψ m (cid:105) = (cid:104) Ψ | ˆΠ | Ψ (cid:105) = (1 − e − α / ) (1 − e − α )4 λ ( α R B (1 − R E ) ) (D5)where λ ( x ) = 2 e − x (sinh x − sin x ) . (D6)
2. Measure-resend strategy
The task to relay attenuated coherent states to the re-ceiver, converting them faithfully to the target amplifiedstates, can also be realized by a classical strategy. Atypical one is a measure-resend strategy. In the interme-diate node, Bob has attenuated states { (cid:12)(cid:12) √ − R E α m (cid:11) } . He tries to discriminate them unambiguosly without er-rors, but at a finite success rate, referred to as unam-biguous state discrimination (USD), and then prepare atarget amplified state (cid:12)(cid:12) g √ − R E α m (cid:11) for the measure-ment result m . The success rate is well known for thiskind of equally probable symmetric states . Denoting | γ m (cid:105) = (cid:12)(cid:12) √ − R E α m (cid:11) and using the eigenvalues and thediagonalizing vectors of the density operatorˆ ρ = M − (cid:88) m =0 | γ m (cid:105) (cid:104) γ m | = M − (cid:88) m =0 λ m | ω m (cid:105) (cid:104) ω m | , (D7)the success rate is given by P USD = min k λ k . (D8)The signal states are represented as | γ m (cid:105) = (cid:12)(cid:12)(cid:12)(cid:112) − R E α m (cid:69) = 1 √ M M − (cid:88) k =0 (cid:112) λ k u mk | ω k (cid:105) . (D9)The detection operators are given byˆΠ m = Λ M P
USD (cid:12)(cid:12) γ ⊥ m (cid:11) (cid:10) γ ⊥ m (cid:12)(cid:12) (D10)for the signal state | γ m (cid:105) , using the reciprocal states (cid:12)(cid:12) γ ⊥ m (cid:11) = 1 √ Λ M − (cid:88) k =0 u mk √ λ k | ω k (cid:105) (D11)where Λ = (cid:80) k λ − k . They satisfy the orthogonality rela-tion (cid:10) γ ⊥ m | γ m (cid:48) (cid:11) = (cid:114) M Λ δ m,m (cid:48) . (D12)The operator for the inconclusive result is given byˆΠ F = ˆ I − M − (cid:88) m =0 ˆΠ m . (D13)
3. Numerical results
Numerical results of the success probabilities in thecase of R E = 0 . FIG. 6: Sketch of the experimental setup for tele-amplification of a coherent state from Alice to Bob using shared entanglementin the form of a photon-subtracted squeezed vacuum state.
Appendix E: Experimental details1. Setup
The most relevant elements of the experimental setupare sketched in Fig. 6 and described in the following.
Resource state generation
The output of an 860 nm continuous-wave Ti:Sapphlaser (a) is phase-modulated by an EOM (b) for Pound-Drever locking of the SHG and OPO cavities. Parts ofthe beam are tapped off for use as local oscillators, coher-ent input beam, probe beam and cavity locking beams,but the main part is frequency-doubled in the secondharmonic generator (c). The 430 nm output of this SHGpumps a bow-tie configuration optical parametric oscil-lator (OPO, d) with a PPKTP crystal and a HWHMbandwidth of γ/ π = 4 . <
1% tap-off of theOPO output (h). To obtain an error-signal, the probe’sphase is dithered on a piezo-mounted mirror by a micro-controller unit (Arduino) which also processes the de-tected signal and provides feedback to lock the phase (i).A photon is subtracted from the squeezed vacuum atrandom times by the detection on an avalanche photo-diode (APD) detector, placed after a 5% tapping beam-splitter and two frequency-filtering cavities (j). The APDis protected from the strong probe beam by an AOM thatdirects the OPO output to the APD only during the in-tervals when the probe beam is switched off. The result-ing photon-subtracted squeezed vacuum (PSSV) state (k)is the resource of entanglement in our protocol, after itis split into two modes propagating towards the Aliceand Bob sections of the setup. In the description of theprotocol in the main section, a fraction R B = 0 . R B is actually takenas the transmitted part of a variable beam-splitter fixedat 90% reflection (l). Bob’s share of the entangled stateis directed towards a homodyne detector for output stateanalysis.2 FIG. 7: Wigner function for one of the PSSV states usedas approximations for the odd cat resource state, in this casewith (cid:15) = 0 .
20. The upper is experimentally generated and to-mographically reconstructed, while the lower one is obtainedfrom our model. The fidelity between them is above 98%,showing the validity of the model.FIG. 8: Relation between OPO pump parameter (cid:15) for theproduction of realistic PSSV states and the amplitude of thecat state | Φ − ( β ) (cid:105) that maximizes the mutual fidelity. Theblue curve indicates these optimal β amplitudes, while thedashed red curve shows the corresponding fidelities. In theexperiment, we used (cid:15) in the range 0.15–0.31. The amount of squeezing produced by the OPO deter-mines the amplitude of the cat-like PSSV state. It is reg-ulated by the pump parameter (cid:15) = (cid:112) P pump /P threshold .To find what cat amplitude β a given pump parame-ter corresponds to, we model the PSSV as in Ref. [23]of the main text. Here, we include the OPO temporalcorrelations, the temporal modes of the APD and homo-dyne detection (described in the following section), the96% escape efficiency of the OPO, the 95% propagationefficiency towards the beam-splitter (l), the 5% tappingratio, the ∼
10% overall APD detection efficiency, and thefiltering bandwidth to find a ˆ ρ PSSV that closely emulatesthe actually produced states. Fig. 7 shows an exampleof an experimentally generated PSSV state and, for com-parison, the modelled state with equivalent parameters.We then maximize for β the state’s fidelity with a truecat state, (cid:104) Φ − ( β ) | ˆ ρ PSSV | Φ − ( β ) (cid:105) , and get the (cid:15) → β cor-respondence plotted in Fig. 8 and thereby the β valuesof Table 1 in the main text. Input state and teleportation
Alice’s input coherent state (m) is prepared in a con-figuration of two double-pass AOMs similar to that usedfor the OPO probe beam, with a strong phase lockingbeam switched on during the 20% locking part of the 10kHz cycle. However, instead of switching the light com-pletely off during the remaining 80% of the cycle, a weakamplitude beam is generated instead. This is done byswitching to a lower voltage RF driving signal for theAOM.The share of the entangled PSSV state propagatingfrom Bob to Alice in mode C is optionally subjected to aloss at a variable beam-splitter (n) before it is overlappedwith her input state on a polarizing beam splitter (PBS)in orthogonal polarizations (o). A half-waveplate fol-lowed by another PBS (p) then interferes the two modesas the protocol’s R A reflectivity beam-splitter. Whencharacterizing the input state, R A is set to 1, whichmeans that all of the input state is sent towards the ho-modyne detector (q). Otherwise, when running the tele-amplification protocol, R A is set to its appropriate value,and the output of the beamsplitter in mode A is sent to-wards an APD (r) with the same frequency filtering andchopping configuration as that in (j).The detection events from the two APDs are correlatedwith digital timing electronics that pick out simultane-ous events and triggers the acquisition of Bob’s homo-dyne signal at a fast digital oscilloscope (s). The detectedphoto-currents are subsequently temporally filtered on aPC to extract the measured quadrature values, as de-scribed in the following section.3 Phase locking
For the tele-amplification protocol to work, the inputstates |± α (cid:105) should be interfered in-phase with the anti-squeezed quadrature of the entangled PSSV state. Wedo this by putting a normal intensity detector (t) at themode C output port of the beamsplitter instead of Alice’shomodyne detector (q). The detected interference signalbetween the probe portions of the input coherent statebeam and the PSSV state beam is used as the input toan FPGA-based lock unit, which provides feedback to thephase of the input beam (u). When the two beams areinterfered at 90 ◦ , we get the desired phase relation, sincethe PSSV probe was locked to the squeezed quadrature.The phase of the local oscillators (LO) in the two ho-modyne detectors can be locked to arbitrary phases rel-ative to the PSSV probe by using a combination of DCand side-band detection of the interference between theLO and the probe beam. An 8 MHz phase modulationis applied to the LO (v), and the interference signals ob-served by the fast homodyne detectors (from a separatelow-gain amplification output) are demodulated at thatsame frequency. This provides an interference signal thatis 90 ◦ out of phase with the DC signal. In the FPGA lockunits (u,w), the two signals are added with weightingfactors corresponding to the desired LO phase in phasespace, resulting in an error signal for the feedback topiezo-mounted mirrors in the LO beam path (in the caseof Bob’s output homodyner) or in the input beam path(for Alice).All phase locks are engaged only during the intervalsof the 10 kHz experiment cycle in which the probe beamsare turned on. For the remaining time, the feedbacksignals are just held at their last actively set value.
2. Quantum state tomography
Temporal modes
The squeezed vacuum has a bandwidth given by theOPO’s HWHM of γ/ π = 4 . t , the continuous-wave squeezedvacuum is converted into a temporally localized PSSVstate in a temporal mode around t which, in the low-squeezing limit, has the form exp( − γ | t − t | ). The filtercavities in front of the APD, needed to remove the pho-tons down-converted into the many non-degenerate OPOresonances, modify the temporal mode to be f ( t ) ∝ γ − e − γ | t − t | − κ − e − κ | t − t | , (E1)where κ/ π ≈
25 MHz is the combined bandwidth of thetwo filters, approximated by a single Lorentzian spectralprofile. This will also be the temporal mode of the tele-ported output state in the low-squeezing limit. For thestate tomography, we therefore extract a single quadra-ture value from the continuous photo current signal of the homodyne detector by integrating it over a mode f HD , out ( t ) equal to the one in Eq. (E1). At higher squeez-ing levels, the optimal mode function is not that simple ,but in this work we stick to the simple expression for allsqueezing levels.An interesting, but also complicating aspect of our cur-rent implementation of the tele-amplification protocol isthat the input and output states are in rather differentspectral modes: The input coherent state is derived di-rectly from the narrow-band laser, whereas the entangledPSSV state is in the broadband mode described above.At a first glance it would appear like the two modes willnot interfere well and the teleportation will fail. How-ever, the spectral response of Alice’s APD is very broad,so it is unable to distinguish the modes. Thereby it canbe said that the detection itself induces the interferencebetween the input and the entangled state. Another wayto see it is in the time domain: compared to the cw in-put beam and the ∼ /γ extent of the PSSV state, thetemporal response of the APD ( ∼
350 ps jitter) is essen-tially delta function-like. Within this short time window,almost no phase shift will occur between the different fre-quency components, so interference will not be destroyed.One problem we do get from this spectral mismatch,however, is the issue of which temporal mode, f HD , in ( t )to use for the definition of the coherent input state. Asthe beam is continuous, the choice of temporal mode canbe done arbitrarily. The photon number n in within thechosen mode will be proportional to the width of themode function, so to obtain a desired coherent state theintensity of the beam should be adjusted inversely pro-portional to that width. In our experiment we make therather natural choice to use the f HD , out ( t ) mode, suchthat we observe the same temporal mode in both theinput and the output homodyne tomography. The mea-sured α and α (cid:48) values in Fig. 2 of the main paper aretherefore directly comparable.However, this mode is not the one detected by theAPD. Since the APD with its delta function-like responseis preceded by filtering cavities which act as delays forincoming fields, its temporal mode can be approximatelydescribed as a single-sided exponential decay, with timeconstant given by the filter bandwidth, f APD ( t ) ∝ e − κ | t − t | H ( t − t ) , (E2)with H ( t ) being the Heaviside step function. Because thePSSV entangled state is temporally localized, as opposedto the input state, the ratio of the photon numbers of thetwo states, n in /n PSSV will be different within the differ-ent modes f HD , in ( t ) and f APD ( t ). There is therefore amismatch between the input state amplitude, α , that weexpect to have and the amplitude actually seen by Alice’sAPD, which is the one to induce the teleportation. Thus,the β and R A values that we experimentally adjusted tomatch a given α were actually not optimal, and this re-sulted in output-to-target fidelities that were lower thanwe could have otherwise obtained. In a possible follow-upexperiment, it would be advisable to consider this issue of4 FIG. 9: Example tomogram, showing the 12 × the input state amplitude in more detail. Simulations in-dicate that with optimized settings, fidelities could havereached 0.94–0.99. State reconstruction
For a given realization of the tele-amplification, we con-struct a homodyne tomogram of the output (and input)state by repeating the state preparation, on/off detec-tion and conditional homodyne detection multiple times,with the LO phase of the homodyne detector fixed atvarious angles. After filtering the oscilloscope traceswith the chosen temporal mode, as described above, theobtained quadrature values are normalized by vacuumtraces recorded under the same experimental conditionswhile we also pay attention to proper offset correctionof the traces, which can be particularly tricky for themeasurement of the input coherent state. That gives usa homodyne tomogram like the one shown in Fig. 9.From this we reconstruct an estimate of the underlyingquantum state using the maximum likelihood method .As mentioned in the Methods section, we correct for thenon-perfect detector efficiencies in order to get the mostaccurate characterization of the protocol.The phase values in the figure indicate the relativephase between the local oscillator and the OPO-injectedprobe beam. The probe beam is locked to the squeezedquadrature of the PSSV state, and Alice’s input coherentbeam is locked at 90 ◦ to the probe beam. Since we defineour phase space in such a way that the anti-squeezing isaligned along the x -axis and the input states have realamplitudes (i.e. also along the x -axis), the 90 ◦ phase ofthe LO should correspond to the x -quadrature. We there-fore rotate the reconstructed quantum state by − ◦ inphase space - the free choice of global phase. FIG. 10: Bloch sphere map of the fidelities between modelledand targeted outcomes of the | ψ ( α, θ, φ ) (cid:105) → | ψ ( α (cid:48) , θ, φ ) (cid:105) tele-amplification, in this case for α = 0 . α (cid:48) = 0 .
6. Theaveraged fidelity here is 77%.
Appendix F: Modelling of qubit teleportation
To simulate the performance of our tele-amplifier setupin the case where the input is an arbitrary coherent statequbit | ψ ( α, θ, φ ) (cid:105) = cos θ | Φ + ( α ) (cid:105) + e iφ sin θ | Φ − ( α ) (cid:105) , (F1)we set up a model for the protocol, using Wigner functionformalism.As the input state to be teleported, we took a purequbit state of the above form. The initial resource statewas a squeezed vacuum state with appropriate squeezinglevels. In the experiment, the squeezed vacuum statewithin the homodyne-observed mode f HD , out ( t ) is notpure. The impurity due to this mode selection can bemodelled quite well by propagating the initially puresqueezed vacuum through a 92% transmission beam-splitter. The losses suffered by the photon-subtractedsqueezed vacuum were similarly modelled by virtualbeam-splitters, taking account of the 96% escape effi-ciency of the OPO, the 5% tapping ratio for the photonsubtraction, and the 95% propagation efficiency towardsthe separating beam-splitter. Alice’s detector was mod-elled as an on/off detector with 10% efficiency, roughlycorresponding to our APD’s detection efficiency and thetransmission of the spectral filters.For a given input amplitude α and desired output am-plitude α (cid:48) , we simulate the tele-amplification process for168 evenly distributed qubit states on the ( θ, φ ) Blochsphere and calculate the fidelity between the outputstates and the targeted states | ψ ( α (cid:48) , θ, φ ) (cid:105) . This resultsin a “fidelity map” like the one in Fig. 10 for every ( α, α (cid:48) )setting. It is clear that the teleportation works best forcoherent state inputs (near 100% fidelity) and for statesnear the North Pole (which is the even cat) and not verywell for states near the South Pole (odd cat). By averag-ing over the Bloch sphere, we obtain an average fidelityfor the outcomes of the protocol for the given ( α, α (cid:48) ) pair,giving one value for the average fidelity plot in Fig. 3 ofthe main text.5 Appendix G: Application to quantum keydistribution
The tele-amplification scheme would be useful to im-prove the performance of quantum key distribution(QKD) schemes which use phase-shifted coherent-statesignals, such as B92 protocol with BPSK states and BB84 protocol with 4PSK states . The tele-amplifications of BPSK and 4PSK states could be appliedto B92 and BB84, respectively. The B92 with BPSKstates would be more interesting from the viewpoint ofpractical implementation because the necessary cat-stateresources are readily available in laboratories. Unfortu-nately, however, its security proof and performace eval-uation when the tele-amplification is included are moreinvolved. In contrast, BB84 protocol with 4PSK statescan be analyzed more clearly with the tele-amplification.Therefore we consider the BB84 scheme where the keyinformation is encoded in the relative phase of a coherent-state reference pulse and a coherent-state signal pulse as (cid:12)(cid:12) ˜0 X (cid:11) = | α (cid:105) R ⊗ | α (cid:105) A (G1a) (cid:12)(cid:12) ˜1 X (cid:11) = | α (cid:105) R ⊗ |− α (cid:105) A (G1b) (cid:12)(cid:12) ˜0 Y (cid:11) = | α (cid:105) R ⊗ | iα (cid:105) A (G1c) (cid:12)(cid:12) ˜1 Y (cid:11) = | α (cid:105) R ⊗ |− iα (cid:105) A (G1d)where mode A is for the signal pulse while mode R forthe reference pulse . This scheme is referred to as the4PSK-BB84 with reference pulse. The amplitude α isunderstood as the one at the receiver Bob. It reducesfrom α in at Alice by the channel loss as α = η ( L ) α in (G2)where η ( L ) = 10 − ξL/ (G3)with the distance L and the channel loss rate ξ . Thephase of α is defined relative to a fixed classical phasereference frame that Eve can access. Alice emits one ofthe four states. Bob randomly chooses one of two mea-surement apparatuses, the X-basis or the Y-basis mea-surement, and measures the signal. In the X-basis mea-surement, the two modes are first combined on a balancedbeam splitter asˆ V | α (cid:105) R | α (cid:105) A = (cid:12)(cid:12)(cid:12) √ α (cid:69) R | (cid:105) A (G4a)ˆ V | α (cid:105) R |− α (cid:105) A = | (cid:105) R (cid:12)(cid:12)(cid:12) −√ α (cid:69) A (G4b)then directed to two on/off detectors described by oper-ators ˆΠ off = e − ν ∞ (cid:88) m =0 (1 − η B ) m | m (cid:105) (cid:104) m | (G5a)ˆΠ on = ˆ I − ˆΠ off (G5b) where ν is the dark count probability and η B is the de-tection efficiency. We define a POVM for making rawkeys “0” and “1”, and an inconclusive outcome “2” byˆΠ X = ˆ V † (cid:18) ˆΠ R on ⊗ ˆΠ A off + 12 ˆΠ R on ⊗ ˆΠ A on (cid:19) ˆ V (G6a)ˆΠ X = ˆ V † (cid:18) ˆΠ R off ⊗ ˆΠ A on + 12 ˆΠ R on ⊗ ˆΠ A on (cid:19) ˆ V (G6b)ˆΠ X = ˆ V † ˆΠ R off ⊗ ˆΠ A off ˆ V . (G6c)We have three kinds of probabilities, P c for correctly out-putting the bit 0 (1) given the signal ˜0 X (˜1 X ), P e forincorrectly outputting the bit 0 (1) given the signal ˜1 X (˜0 X ), and P i for inconclusive outcome P c = (cid:10) ˜0 X (cid:12)(cid:12) ˆΠ X (cid:12)(cid:12) ˜0 X (cid:11) = (cid:10) ˜1 X (cid:12)(cid:12) ˆΠ X (cid:12)(cid:12) ˜1 X (cid:11) = 12 (cid:16) − e − ν − η B | α | (cid:17) (1 + e − ν ) (G7) P e = (cid:10) ˜0 X (cid:12)(cid:12) ˆΠ X (cid:12)(cid:12) ˜0 X (cid:11) = (cid:10) ˜1 X (cid:12)(cid:12) ˆΠ X (cid:12)(cid:12) ˜1 X (cid:11) = 12 (cid:16) e − ν − η B | α | (cid:17) (1 − e − ν ) (G8) P i = (cid:10) ˜0 X (cid:12)(cid:12) ˆΠ X (cid:12)(cid:12) ˜0 X (cid:11) = (cid:10) ˜1 X (cid:12)(cid:12) ˆΠ X (cid:12)(cid:12) ˜1 X (cid:11) = e − ν − η B | α | . (G9)Similarly, the Y-basis measurement is described by aPOVM ˆΠ Y = e i π ˆ n A ˆ V † (cid:16) ˆΠ R on ⊗ ˆΠ A off + 12 ˆΠ R on ⊗ ˆΠ A on (cid:17) ˆ V e − i π ˆ n A (G10a)ˆΠ Y = e i π ˆ n A ˆ V † (cid:16) ˆΠ R off ⊗ ˆΠ A on + 12 ˆΠ R on ⊗ ˆΠ A on (cid:17) ˆ V e − i π ˆ n A (G10b)ˆΠ Y = e i π ˆ n A ˆ V † ˆΠ R off ⊗ ˆΠ S off ˆ V e − i π ˆ n A . (G10c)where the factor e i π ˆ n A is for shifting the phase. Elimi-nating the inconclusive outcomes, the filtered fraction forsifted keys is defined by Q = 1 − P i (G11)and the bit error rate (BER) is defined by δ = P e Q (G12)Then the upper bound for the phase error rate (PER) isgiven by δ ph = δ + 4∆ (cid:48) (1 − ∆ (cid:48) )(1 − δ )+ 4(1 − (cid:48) ) (cid:112) ∆ (cid:48) (1 − ∆ (cid:48) ) δ (1 − δ ) (G13)where ∆ (cid:48) = ∆ Q (G14a)∆ = 12 (cid:104) − e − α (cid:0) cos α + sin α (cid:1)(cid:105) . (G14b)6The quantity ∆ specifies the imbalance of the “coin” forthe choice of X and Y bases depending on the statesoverlap among {| α in (cid:105) , |− α in (cid:105) , | iα in (cid:105) , |− iα in (cid:105)} . The se-cure key generation probability is given by G = 12 Q [1 − H ( δ ) − H ( δ ph )] (G15)where H ( x ) is the binary Shannon entropy H ( x ) = − x log x − (1 − x ) log (1 − x ) . (G16)Now let us consider an extention of this BB84 imple-mentation to a tele-amplification assisted scheme. Thesignals sent by Alice are α in m = α in u m where u = i and m =0, 1, 2 and 3. The signals first arrive at the re-lay node, referred to as Amy, and are then relayed tothe receiver at the terminal node, referred to as Bob.The total distance between Alice and Bob is L . Therelay node Amy is located at the distance xL from Al-ice, where 0 < x <
1. The input coherent-state ampli-tude to the tele-amplifier at Amy is α m = (cid:112) η ( xL ) α in u m .Bob prepares the resource cat state, beam-splits it, andsends Amy one half of the split cat-state over the distance(1 − x ) L . At the relay node, Amy combines it with thesignal state | α m (cid:105) A on the beam splitter, and measuresthem by the four-port interferometric receiver. For sim-plicity, we assume that the single photons can be detectedwith perfect efficiency at the four ports. Three-photoncoincidence counts at these ports herald the successfultele-amplification events. The dark count effect can benegligible by this multi-photon coincidence filtering. Bobfinally receives the tele-apmplified signal | gα m (cid:105) B , whichis subject to the X or Y basis measurement. The gain isnow a function of x and Lg ( x, L ) = (cid:115) − R B R B η [(1 − x ) L ] . (G17)Then the security proof in ref. 7 can be appliedto the tele-amplification assisted BB84 provided thatthe reference pulse arrives at Bob so as to be in (cid:12)(cid:12)(cid:12) g ( x, L ) (cid:112) η ( xL ) α in (cid:69) R ⊗ (cid:12)(cid:12)(cid:12) g ( x, L ) (cid:112) η ( xL ) α in u m (cid:69) B . Thefiltering fraction Q , the BER δ , the PER δ ph and ∆ (cid:48) aregiven by replacing α in Eqs. (G11), (G12), (G13), and(G14) with the new one g ( x, L ) (cid:112) η ( xL ) α in . Here notethat the signal attenuation occurs only for the channelinterval between Alice and Amy over a distance xL . Inthe remaining channel with a distance (1 − x ) L , the signalattenuation is compensated by the cat-assisted amplifi-cation with the gain g ( x, L ).The secure key generation probability is finally givenby multiplying the expression in Eq. (G15) by the successprobability of the tele-amplification as G = 12 P Suc Q [1 − H ( δ ) − H ( δ ph )] . (G18)Some numerical results are shown in Fig. 11 as a func-tion of the transmission distance. In the following, the dark count probability is ν = 10 − , and the detectionefficiency of the receiver Bob is η B = 0 .
2. The dashedand dotted lines correspond to the 4PSK-BB84 with ref-erence pulse. In this PSK coherent-state scheme, aneavesdropper Eve can effectively perform the photon-number splitting attack with phase information. So theinput coherent-state amplitude should be set small. InFig. 11, the dashed and dotted lines correspond to | α in | = 0 .
008 and 0.001, respectively. The key genera-tion probabilities decrease more rapidly than that of thephase-randomized decoyed scheme. The solid lines rep-resent the performances of the tele-amplification assistedBB84 with R B = 0 .
2. It can be seen that the securekey generation probabilities are smaller than those with-out tele-amplification at short distances, however, theycan remain at reasonable levels up to longer distances.The input coherent-state amplitude α in is allowed to belarger in the tele-amplification assisted BB84. The redand brown lines are the cases where the relay node Amyis located closer to Bob, namely x = 0 . x = 0 .
6, re-spectively. The blue and black lines are the cases whereAmy is located at 0 . L from Alice, with | α in | = 0 . x = 0 . | α in | = 0 . FIG. 11: The secure key generation probabilities as a func-tion of distance. The dashed and dotted lines correspond toBB84 without tele-amplification. The solid lines correspondto tele-amplification assisted BB84 where R B = 0 .
2. For allthe cases, the dark count probability is ν = 10 − , and thedetection efficiency of the receiver Bob is η B = 0 . The performance of the green line is remarkable, how-ever, one should prepare the resource cat-state with amuch larger amplitude. The mean photon number of theresource cat state is given by β ( x, L ) = η ( xL ) α η [(1 − x ) L ] R B . (G19)It is shown in Fig. 12. To extend the distance beyond200 km, the resource cat-state should include more thana hundred photons.7 FIG. 12: The mean photon numbers of the required catstates as a function of distance.FIG. 13: The channel transmittance as a function of dis-tance. The dashed line corresponds to the original channelwithout the tele-amplificcation.
The essential effect brought by the tele-amplificationis simply the improvement of the channel transmittance η ( L ) → g ( x, L ) η ( xL ) (G20)but on the other hand also the reduction of the detectionrate due to the additional filtering at the relay node. Weplot the channel transmittance in Fig. 13. For x < . x > .
5, on the other hand, the BER andPER increase with the distance. In particular, the PERis the dominant error.The success rate of the tele-amplification decreases with the distance as shown in Fig. 16. This directlyleads to the decrease of the key generation probability.
FIG. 14: The BER as a function of distance.FIG. 15: The PER as a function of distance.FIG. 16: The success rate of the tele-amplification as a func-tion of distance.
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