# Quantum teleportation of propagating quantum microwaves

- R Di Candia
^{1}Email author, - KG Fedorov
^{2, 3}, - L Zhong
^{2, 3, 4}, - S Felicetti
^{1}, - EP Menzel
^{2, 3}, - M Sanz
^{1}, - F Deppe
^{2, 3, 4}, - A Marx
^{2}, - R Gross
^{2, 3, 4}and - E Solano
^{1, 5}

**2**:25

**DOI: **10.1140/epjqt/s40507-015-0038-9

© Di Candia et al. 2015

**Received: **8 July 2015

**Accepted: **29 November 2015

**Published: **15 December 2015

## Abstract

Propagating quantum microwaves have been proposed and successfully implemented to generate entanglement, thereby establishing a promising platform for the realisation of a quantum communication channel. However, the implementation of quantum teleportation with photons in the microwave regime is still absent. At the same time, recent developments in the field show that this key protocol could be feasible with current technology, which would pave the way to boost the field of microwave quantum communication. Here, we discuss the feasibility of a possible implementation of microwave quantum teleportation in a realistic scenario with losses. Furthermore, we propose how to implement quantum repeaters in the microwave regime without using photodetection, a key prerequisite to achieve long distance entanglement distribution.

## 1 Introduction

In 1993, CH Bennett et al. [1] proposed a protocol to disassemble a quantum state at one location (Alice) and to reconstruct it in a spatially separated location (Bob). They proved that, if Alice and Bob share quantum correlations of EPR type [2], then Bob can reconstruct the state of Alice by using classical channels and local operations. This phenomenon is called ‘quantum teleportation’, and it has important applications in quantum communication [3]. The result inspired discussions among physicists, in particular, on the experimental feasibility of the protocol. Despite some controversies in technical issues, the first experimental realisation of quantum teleportation was simultaneously performed in 1997 in two groups, one led by A Zeilinger in Innsbruck [4], and the other by F De Martini in Rome [5]. In both experiments, the polarisation degrees of freedoms of the photons were teleported. It was shown that, even within the unavoidable experimental errors, the overlap between the input state and the teleported one exceeded the classical threshold achievable when quantum correlations are not present. After the success of the first experiments, alternatives for a variety of systems and degrees of freedom emerged. Of particular interest is the continuous-variable scheme studied by L Vaidman [6] and SL Braunstein et al. [7], whose experimental implementation was realised by A Furusawa et al. [8] in the optical regime. This experiment consisted in teleporting the information embedded in the continuous values of the conjugated variables of a propagating electromagnetic signal in the optical regime. Optical frequencies were preferred because of their higher detection efficiency, essential to achieve a high fidelity performance [7, 8], and because propagation losses are almost negligible. During the last years, an impressive progress in teleporting quantum optical states to larger distances, first in fibers [9, 10], and afterwards in free-space [11–13], was made. This rapid progress may even allow us to realise quantum communication via satellites in near future with corresponding distances of about 150 km. In optical systems, the long-distance teleportation is, to some extent, straightforward, because of the high transmissivity of optical photons in the atmosphere. Nevertheless, unavoidable losses are setting an upper limit for the teleportation distance. However, there were fundamental theoretical studies on how to allow for a long-distance entanglement distribution. The underlying concepts are based on quantum repeaters [14, 15], whose implementation on specific platforms needs an individual study. So far, the entanglement sharing and quantum teleportation was reported for cold atoms [16–18], and even for macroscopic systems [19].

In this article, we discuss the possibility of implementing the quantum teleportation protocol of propagating electromagnetic quantum signals in the microwave regime. This line of research is justified by the recent achievements of circuit quantum electrodynamics (cQED) [20, 21]. In cQED, a quantum bit (qubit) is implemented using the quantum degrees of freedom of a macroscopic superconducting circuit operated at low temperatures, i.e. \(\hbox{$<$}\kern-0.7em\raise0.82 ex\hbox{$\sim$}\)10-100 mK, in order to suppress thermal fluctuations. Superconducting Josephson junctions are used to introduce non-linearities in these circuits, which are essential in both quantum computation and the engineering of qubits. Typical qubits are built to have a transition frequency in the range 5-15 GHz (microwave regime), and they are coupled to an electromagnetic field with the same frequency. This choice is determined by readily available microwave devices and techniques for this frequency band, such as low noise cryogenic amplifiers, down converters, network analysers, among others. We note that apart from its relevance in quantum communication, quantum teleportation is also crucial to perform quantum computation, e.g. it can be used to build a deterministic CNOT gate [22].

Recently, path-entanglement between propagating quantum microwaves has been investigated in Refs. [23–25]. Following what was previously done in the optical regime [26], a two-mode squeezed state, in which the modes were spatially separated from each other, was generated. The two entangled beams could be used to perform with microwaves a protocol equivalent to the one used in optical quantum teleportation [7, 8]. These articles represent the most recent of a large amount of results presented during the last years [24, 27–42], which are the building blocks of a quantum microwave communication theory. Inspired by the last theoretical and experimental results, we want to discuss the feasibility of a quantum teleportation realisation for propagating quantum microwaves. The article is organised in the following way: In Section 2, we introduce the continuous-variable quantum teleportation protocol and its figures of merit. In Section 3, we describe the preparation of a propagating quantum microwave EPR state. In Section 4, we show how to implement a microwave equivalent of an optical homodyne detection, by using only linear devices. The Section 5 is focused on the analysis of losses. In particular, we consider an asymmetric case in which the losses in Alice’s and Bob’s paths are different. In Section 6, we discuss the feedforward part of the protocol in both a digital and an analog fashion. Finally, as the entanglement distribution step is affected by losses, we present in Section 7 how to implement a quantum repeater based on weak measurements in a cQED setup, in order to allow the entanglement sharing at larger distances.

## 2 The protocol

*A*, wants to send a quantum state \(|\phi\rangle_{T}\), whose corresponding system is labelled by

*T*, to Bob, denoted by

*B*. Additionally, let them share an ancillary entangled state \(|\psi\rangle_{AB}\) given by

*x̂*and

*p̂*are quantum conjugate observables obeying the standard commutation rule \([\hat{x},\hat{p}]=i\). After Alice performs a Bell-type measurement on the system

*T*-

*A*,

*a*and

*b*are the outcomes of the measurement. The resulting values of Bob’s quadrature would be

*a*and \(p_{B}\) by

*b*, we finally have \(\hat{x}_{B}|\phi\rangle_{B}=\hat{x}_{T}| \phi\rangle_{T}\) and \(\hat{p}_{B} |\phi\rangle_{B}=\hat{p}_{T}| \phi\rangle _{T}\), where \(|\phi\rangle_{B}\) is the final state of Bob. Therefore, the final state of Bob is the state of the system

*T*. Note that Bob needs to perform local operations conditioned to Alice’s measurement outcomes. As the outcomes are two numbers, we may allow Alice and Bob to communicate throughout a classical channel, see Figure 1. Bennett et al. [1] called this protocol a

*quantum teleportation*[45].

*r*is a squeezing parameter [46] and, also, we have considered an asymptotic behaviour for large

*r*. For finite

*r*, the state of the system

*A*-

*B*fulfils

*r*, \(P_{c}\) approaches to the delta function, and then \(W_{B}=W_{T}\). In the following, we will refer only to the variance of the quadratures, regardless of whether they are noisy or not. Therefore, our treatment is general, and it includes also the lossy case, in which we do not have a perfect two-mode squeezed state as a resource. In order to evaluate the performance of the protocol, entanglement fidelity [47] can be used. If

*T*is in a pure state, the entanglement fidelity is given by

## 3 Generation of EPR state

*ĉ*with an environment, as depicted in Figure 2:

*â*. One can write down these equations in the Heisenberg picture, and look at input-output relations of the fields:

## 4 Quadrature measurement

*A*and the state

*T*to a hybrid ring, obtaining

*x*-quadrature of the mode 1 and the

*p*-quadrature of the mode 2. If we amplify this signal with a HEMT and then measure it afterwards, the state of Bob after the local displacement is

*s*are small. By defining the JPA quadrature noise \(A_{J}\equiv\frac{s}{g_{J}} \Delta\hat{x}_{h_{J}}^{2}\), and the HEMT quadrature noise \(A_{H}\equiv\frac {g_{H}-1}{g_{H}}\Delta\hat{x}_{h_{H}}^{2}\), we have to analyse for which experimental values the total noise

## 5 Protocol with losses

So far, we have not taken into account possible losses in the protocol. Typically, losses in the microwave domain are much larger than in the optical domain, and therefore can significantly affect the quality of the teleportation protocol. In the following, we analyse the protocol with all possible loss mechanisms, see Figure 1. Note that losses after the HEMT amplification are negligible, and therefore omitted.

*α*is the transmission coefficient from the output of the hybrid ring to the JPA, taking into account the hybrid ring losses. Moreover,

*β*is the transmission coefficient from the JPA to HEMT amplifier. Hence, the total noise is

**Tables with the maximum value of**
\(\pmb{A_{J}^{\mathrm{max}}}\)
**allowed in order for the quantum teleportation protocol to work**

| \(\boldsymbol{\Delta\xi^{\prime2}}\) | \(\boldsymbol{A_{J}^{\mathrm{max}}}\) |
---|---|---|

| ||

≪1 | 0.482 | 0.164 |

1 | 0.500 | 0.155 |

10 | 0.734 | 0.046 |

20 | 0.867 | unf. |

| ||

≪1 | 0.476 | 0.166 |

1 | 0.486 | 0.162 |

10 | 0.623 | 0.098 |

20 | 0.737 | 0.045 |

| ||

≪1 | 0.789 | 0.020 |

1 | 0.802 | 0.014 |

10 | 0.895 | unf. |

| ||

≪1 | 0.666 | 0.078 |

1 | 0.677 | 0.073 |

10 | 0.767 | 0.031 |

## 6 Analog vs. digital feedforward

*classical*channel. Then, Bob uses this information to apply a displacement in his system. This process is called a feedforward, and is considered tough to implement, independently of the considered system. In particular, in the microwave case, the measurement process may be slow, resulting in an ultimate loss of fidelity. In realistic experiments, a quantum microwave signal has to be amplified before detection. If the amplification is large, the signal becomes insensitive to losses at room temperature. Therefore, an idea is to use the output signal of Alice to perform classical communication without digitally measuring it. This analog feedforward is depicted in Figure 4, and it works in the following way. Let us assume the lossless case, and send the two amplified signals of Alice to a hybrid ring. One of the two outputs of the latter provides us with

*F*’ stands for the feedforward. Indeed, Bob may use this signal to perform the displacement.

*α*is without a hat because it represents a coherent state. If we choose \(\tau\sim1\) and \(|\alpha|\gg1\) such that \(\sqrt{1-\tau} \alpha=z\), we obtain

*B*and

*F*as inputs to a directional coupler with transmissivity \(\tau\simeq1-\frac{4}{g_{J}g_{H}}\), the corresponding output is

*τ*would result in a large error in the displacement operator. This problem can be overcome by attenuating at low temperatures the signal

*F*before the directional coupler, in order to neglect the attenuator noise. In this case, setting \(\tau=1-\frac{4}{\eta _{\mathrm{att}}g_{J}g_{H}}\), the transmitted signal is the same as in (32)-(33). For instance, if we choose \(\eta _{\mathrm{att}}\sim10^{-3}\), we derive a reasonable value for the reflectivity: \(1-\tau\sim10^{-3}\).

The described analog method allows us to perform the feedforward without an actual knowledge of the result of Alice’s measurement. Indeed, the JPA and HEMT amplifiers work as measurement devices. On the one hand, the advantage is that we save the time required to digitalised the signal. On the other hand, the disadvantage is that all the noise sources in Alice are mixed, resulting in a doubling of the noise *A*, as we see in Eqs. (32)-(33) (the same claim holds for the lossy case). Therefore, one should carefully evaluate whether the digital feedback is convenient against the analog one, by comparing *A*, which quantify the loss of fidelity in the analog feedforward case, with the noise added due to the delay line added in Bob in the digital feedforward case. This can be done by estimating the digitisation time and the corresponding losses in the Bob delay line, which strongly depends on the available technology. Indeed, currently available IQ mixers and FPGA technology requires \(t_{p}\sim200\mbox{-}400\) ns for measuring and processing the information. During this time, the signal needs to be delayed in Bob’s channel. If we consider a delay line where the group velocity of the electromagnetic field is \(v\simeq2 \times10^{8} \) m/s, \(t_{p}\) corresponds to a delay line in Bob of 40-80 m. Comparing the values of \(\Delta\xi^{\prime2}\) for the zero measurement time and the realistic 200 ns measurement time, we see a change in \(\Delta\xi^{\prime2}\) of ∼0.30 in the case of 1 m distance (assuming 0.1 dB per meter of power cable losses), which is considerably lower than the current values achievable for *A*. Notice that this discrepancy decreases with the distance between Alice and Bob. This means that the digital feedforward is currently preferable to the analog one, but the analog feedforward can become a useful technological tool when the JPA technology will reach a reasonable noise level.

## 7 Quantum repeaters

As we have discussed in the previous sections, the entanglement distribution between the two parties, Alice and Bob, is particularly challenging due to the large losses involved. Moreover, while in the optical case the noise added by a room temperature environment corresponds to the vacuum, in the microwave regime, this noise would correspond to a thermal state containing ∼10^{3} photons. Even in the most favourable situation in which we build a cryogenic setup to share the entanglement, we would have a collapse of the correlations after ∼10 m due to the detection inefficiency and losses. The implementation of quantum repeaters in the microwave regime could potentially solve this issue. A quantum repeater is able to distillate entanglement and to share it at larger distance, at the expense of efficiency. A protocol for distributing entanglement at large distance in the microwave regime has been recently proposed in [57], but it relies on the implementation of an optical-to-microwave quantum interface [58], which has not yet been realised experimentally.

*r*. If we are able to implement the operator \(g^{\hat{n}}\), with \(g>1\) on one mode, say Bob, we have

The operator \(g^{\hat{n}}\) corresponds to a noiseless phase-insensitive linear amplifier, and it cannot be implemented deterministically. However, there exist probabilistic methods to realise it approximately. A probabilistic noiseless linear amplification scheme has already been demonstrated in the optical regime [60, 61], but it relies on the possibility of counting photons. In contrast, the weak measurement scheme [59] requires quadrature measurements that can be applied in the microwave regime.

*k*is a coupling constant. Let us further consider low-time interaction, i.e. \(k\Delta t\ll1\). If we postselect the ancilla in the state \(|p\rangle\), i.e. the eigenstate of the

*p̂*quadrature corresponding to the eigenvalue

*p*, the whole final state is

*α*and

*p*, we can induce a value of \(A_{w}\), whose imaginary part is positive. If we set \(g\equiv e^{k\Delta t \operatorname {Im}(A_{w})}\), we have a scheme to implement \(g^{\hat{n}_{B}}\) up to a known phase-shift \(e^{-ik\Delta t \operatorname {Re}(A_{w})\hat{n}_{B} }\), with success probability density \(|\langle p|\alpha \rangle|^{2}=\frac{1}{\sqrt{\pi}}e^{- (p-\operatorname {Im}(\alpha) )^{2}}\). For instance, by choosing \(\operatorname {Im}(\alpha)=0\) and \(\operatorname {Re}(\alpha) <0\), we have a gain for any \(p>0\), which happens with a 50% probability. In this case, an imperfect quadrature measurement can be corrected by just shifting the allowed results of the ancilla measurement, with a consequent lost of efficiency. Note that, due to the probabilistic nature of the scheme, Alice and Bob need to communicate classically in order to distillate the entanglement, see Figure 5. However, this classical communication can be performed at the end, in a post-selection fashion, as Alice does not need to perform any operation on her system.

*a*represents the ancillary mode and

*b*Bob’s mode. In the interaction picture with respect \(H_{0}=\hbar\omega_{a} a^{\dagger}a+\hbar\omega_{b} b^{\dagger}b+\hbar\omega_{a} |2\rangle\langle2|+\hbar \omega_{b}|1\rangle\langle1| \), the new Hamiltonian is

## 8 Conclusions

We have considered a quantum teleportation protocol of propagating quantum microwaves. We have analysed its realisation by introducing figures of merit (i.e. Ξ and *A*) that takes into account losses and detector efficiency. In particular, we have underlined the difference between the optical case (where photodetectors are available, and losses are negligible) and the microwave regime. Indeed, we have considered JPAs in order to perform single-shot quadrature measurements, and we have proposed an analog feedforward scheme, which does not rely on digitisation of signals. Moreover, we have discussed the losses mechanisms, highlighting in which measure they limit the realisation of the protocol. We have used typical parameters of present state-of-art experimental setups in order to identify the required improvements of these setups to allow for a first proof-of-principle experiment. Finally, we have introduced a quantum repeater scheme based on weak measurements and postselection.

## Declarations

### Acknowledgements

This work is supported by the German Research Foundation through SFB 631, and the grant FE 1564/1-1; Spanish MINECO FIS2012-36673-C03-02; UPV/EHU UFI 11/55; Basque Government IT472-10; CCQED, PROMISCE, and SCALEQIT EU projects.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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