Secure quantum remote state preparation of squeezed microwave states

TL;DR

This paper demonstrates an experimental method for secure, deterministic remote state preparation of squeezed microwave states over 35 cm, advancing microwave quantum networks with potential applications in quantum communication.

Contribution

It presents the first experimental realization of continuous-variable remote state preparation in the microwave regime using propagating two-mode squeezed states.

Findings

Achieved remote preparation of squeezed states with up to 1.6 dB squeezing.

Quantified security using the one-time pad concept.

Demonstrated feasibility over a 35 cm distance in microwave quantum communication.

Abstract

Quantum communication protocols based on nonclassical correlations can be more efficient than known classical methods and offer intrinsic security over direct state transfer. In particular, remote state preparation aims at the creation of a desired and known quantum state at a remote location using classical communication and quantum entanglement. We present an experimental realization of deterministic continuous-variable remote state preparation in the microwave regime over a distance of 35 cm. By employing propagating two-mode squeezed microwave states and feedforward, we achieve the remote preparation of squeezed states with up to 1.6 dB of squeezing below the vacuum level. We quantify security in our implementation using the concept of the one-time pad. Our results represent a significant step towards microwave quantum networks between superconducting circuits.

Figures

Equations

$${\sigma }_{\mathrm{s}}^2=\frac{1}{4}\left[2(1+2n)e^{-2r}(1-\eta )\tau +2(\eta +n_{\mathrm{f}})\tau \right],$$

$$H(M)-H(M|C)=0,$$

Full text

In quantum technology, an efficient and secure exchange of quantum information between quantum nodes plays a crucial rolecirac1997 . One of the first protocols realizing such a task was quantum teleportation, where an unknown quantum state is safely transferred from one party to another by using a shared entangled resource and classical feedforwardBennett1993 ; bouwmeester1997 . In a different scenario, where one party has full classical knowledge about a to-be-communicated quantum state, remote state preparation (RSP) can be used to remotely create this quantum state by employing similar tools as in quantum teleportationBennett2001 . Compared to known classical methods, both protocols provide a quantum advantage as they require a smaller amount of classical information in the feedforward signal in order to communicate a desired quantum statekurucz2005 . However, in contrast to quantum teleportation, RSP allows for a nontrivial trade-off between the amount of classical communication and entanglement necessary for a successful protocolBennett2001 . Furthermore, the use of an entangled resource allows RSP to operate perfectly securekurucz2005 . Even though RSP is extensively investigated both theoretically and experimentally for discrete-variable systemsDakic2012 ; Peters2005 ; Ye2004 , deterministic implementations with continuous-variable systems are still lackingLeJeannic2018 ; Laurat2003 . At the same time, quantum communication based on continuous-variables is a field of intense researchWeedbrook2012 ; Braunstein2005 investigating, e.g., quantum key distributionJouguet2013 , quantum teleportationCVTele1998 ; Braunstein1998 , dense codingMattle1996 , and free-space quantum communicationHeim2014 .

Quantum communication in the microwave domain is motivated by the tremendous progress in the area of quantum information processing with superconducting circuits. In particular, the development of superconducting multi-qubit processorsQSuprem2018 ; QED2016 , operated at gigahertz frequencies has been highly successful. We promote an approach of quantum communication directly in the microwave regime based on propagating squeezed states. Since these states have the same frequency and are generated by technology platforms already used for superconducting quantum computers, there is no mismatch between communication and data processing units. This approach is expected to be useful for short and medium distances, where superconducting waveguides can be used.

In this work, we realize deterministic continuous-variable RSP by creating Gaussian squeezed states over a distance of $35\text{\,}\mathrm{c}\mathrm{m}$. We investigate the phase space of remotely preparable squeezed states and obtain good agreement with our model calculations based on the input-output formalism. Additionally, we find that our scheme corresponds to an extension of the one-time pad cryptographic protocolVernam1926 into the quantum regime which allows for information-theoretic security. In contrast to already demonstrated quantum state transfer protocols between superconducting circuitsKurpiers2018 ; Axline2018 , our protocol does not directly transmit the target states to the receiving party. Moreover, it can be operated in the continuous regime and utilizes preshared entanglement to enable secure communication between parties. Since the generation and manipulation of Gaussian states is well understoodWeedbrook2012 , they offer a viable option for building future intracity low-temperature quantum networks Xiang2016 .

The general idea behind the RSP protocol and our experimental implementation using continuous-variable microwave states are described in Fig. 1. We use flux-driven Josephson parametric amplifiers (JPAs) as the key elements for the generation and manipulation of squeezed microwave statesYamamoto2008 ; Pogorzalek2017 ; Goetz2016 . We operate all JPAs in the degenerate regime at the frequency $f_{0}\,{=}\,$5.435\text{,}\mathrm{G}\mathrm{H}\mathrm{z} with a pump frequency $f_{\mathrm{p}}\,{=}\,2f_{0}$. The task of JPA 1 and JPA 2 is to generate propagating squeezed states which are incident to an entangling beam splitter. The resulting symmetric two-mode squeezed (TMS)[Fedorov2018 ](#bib.bib27); [Flurin2012 ](#bib.bib28) states have a two-mode squeezing level[Eichler2011b ](#bib.bib29) of $S_{\mathrm{TMS}}\,{=}\,$7.1\text{\,}\mathrm{d}\mathrm{B} and an entanglement strength characterized by the negativity criterionFedorov2018 ; Menzel2012 of $N\,{=}\,2.2$. This number quantifies the strength of nonlocal correlations present between field quadratures of signals propagating along different beam splitter output paths. Additionally, the symmetric TMS states have negligible local squeezing within each path. In other words, the microwave signals propagating on the two paths locally look like thermal states with, nevertheless, strong entanglement between them.

In the next step, we employ the symmetric propagating TMS states as a resource for remotely preparing the target squeezed microwave states. For this purpose, the TMS states are continuously distributed between two parties, Alice and Bob, who are separated by $35\text{\,}\mathrm{cm}$ of superconducting cable. Alice generates a feedforward signal carrying the classical information about her choice on what quantum state is to be remotely prepared at Bob’s side. Finally, Bob displaces his part of the resource state proportionally to the communicated signal by using a directional coupler with a fixed coupling of $\beta\,{\simeq}\,$-15\text{,}\mathrm{d}\mathrm{B}$$Fedorov2016 ; DiCandia2015 . We experimentally implement the feedforward by operating JPA 3 as a phase-sensitive amplifier. Alice uses it to choose and strongly amplify a certain quadrature of the incoming TMS states. Note that, in contrast to the other JPAs, it does not matter whether the outgoing feedforward signal from JPA 3 is squeezed or not (Supplementary Note 5). The essential classical information, as required for a successful RSP, is encoded in the large instantaneous amplitude of the phase-sensitively amplified field quadrature.

Figure 2a shows the experimental performance of the RSP scheme as a function of the JPA 3 degenerate gain $G_{\mathrm{f}}$ for a fixed JPA 3 amplification angle $\gamma_{\mathrm{f}}\,{=}\,$0\text{,}\mathrm{\SIUnitSymbolDegree}. The latter is defined as the deviation from the angle of the optimal working point at which we achieve the highest purity in the remotely prepared states. We fully characterize these states in terms of their squeezing level $S_{\mathrm{rp}}$, antisqueezing level $A_{\mathrm{rp}}$, and squeezing angle $\gamma_{\mathrm{rp}}$ (see Methods). We clearly observe squeezing up to $S_{\mathrm{rp}}\,{=}\,$1.6\pm 0.1\text{\,}\mathrm{d}\mathrm{B} in the final states at the output of the displacer near the optimal JPA 3 gain $G_{\mathrm{f}}\,{\simeq}\,$13\text{,}\mathrm{d}\mathrm{B}$$ (see Fig. 2b). $S_{\mathrm{rp}}$ decreases and the states even become non-squeezed upon deviation from the optimal JPA 3 gain as shown in Fig. 2c. The remotely prepared states can be encoded not only by varying $G_{\mathrm{f}}$ but also by changing $\gamma_{\mathrm{f}}$. The latter leads to a different quadrature in the resource TMS states being projected, and accordingly, to a different state being remotely prepared. The squeezing level and squeezing angle of the remotely prepared states obtained by sweeping both $G_{\mathrm{f}}$ and $\gamma_{\mathrm{f}}$ are shown in Fig. 2d,e. The results for the antisqueezing level $A_{\mathrm{rp}}$ can be found in Supplementary Fig. 3.

Our experiment can be theoretically described by a model based on the input-output transformations for every component in the setup including transmission losses. Additionally, we assume imperfect JPAs adding a certain amount of noise with mean thermal photon numbers $n_{i}$ ($i\,{\in}\,\{1,2,\mathrm{f}\}$) to the JPA input signal. The RSP protocol is expected to work optimally for $G_{\mathrm{f}}\,{=}\,\tau/(1-\tau)$ and $\gamma_{\mathrm{f}}\,{=}\,$0\text{,}\mathrm{\SIUnitSymbolDegree}, where $\tau\,{=}\,1-10^{\beta/10}$ is the transmissivity of the directional coupler. At this optimal point and under the condition $S_{\mathrm{TMS}}\,{\geq}$3\text{\,}\mathrm{d}\mathrm{B}, we obtain for the squeezed variance of the remotely prepared state

[TABLE]

where we assume equal noise photon numbers $n_{1}\,{=}\,n_{2}\,{=}\,n$ and squeezing factors $r_{1}\,{=}\,r_{2}\,{=}\,r$ of JPA 1 and JPA 2 as well as equal losses $\eta$ after the beam splitter on both Alice’s and Bob’s side. Furthermore, $n_{\mathrm{f}}$ is the noise photon number of JPA 3 (Supplementary Note 3). Equation (1) indicates that the prepared squeezing level $S_{\mathrm{rp}}$ at the optimal point is at least $3\text{\,}\mathrm{d}\mathrm{B}$ below the squeezing level of the used resource. In order to correctly model the experiment, we additionally include a finite crosstalk between JPA 3 and the JPAs creating the TMS states as well as losses before the beam splitter. Figure 2f,g depicts a joint fit to the corresponding data. We observe a very good coincidence between the experimental results and our model for the following parameters: JPA 1,2 squeezing levels $S\,{=}\,$10.1\text{,}\mathrm{d}\mathrm{B}, $n\,{=}\,0.04$, a total loss between JPA 1,2 and the directional coupler input $\chi_{1}\,{=}\,$1.0\text{\,}\mathrm{d}\mathrm{B} and a total loss between JPA 1,2 and JPA 3 $\chi_{2}\,{=}\,$1.1\text{,}\mathrm{d}\mathrm{B}$$. All values nicely agree with those obtained from independent JPA characterization measurements and loss estimations.

The quantum advantage of the RSP protocol consists in a smaller amount of classical information sent through the feedforward channel in order to prepare a desired state as compared to a purely classical protocolkurucz2005 ; Killoran2010 . In the current experiment, this becomes evident by considering that only the amplified quadrature of the feedforward signal will affect the signal at Bob’s side due to the low coupling $\beta\,{\ll}\,$0\text{,}\mathrm{d}\mathrm{B}$$ of the displacer. Consequently, we only send two real numbers while we are able to prepare different undisplaced mixed squeezed states which are fully described by three real numbers ($S_{\mathrm{rp}}$, $A_{\mathrm{rp}}$, $\gamma_{\mathrm{rp}}$).

The manifold of undisplaced Gaussian states we can prepare is intuitively understood by plotting the results from Fig. 2d,e in the phase space of the prepared squeezing level and angle, as it is shown in Fig. 3. The purity $\mu\,{=}\,1/(4\sqrt{\mathrm{det}\sigma})$, where $\sigma$ is the covariance matrix of the remotely prepared state, incorporates the information about the antisqueezed quadrature and is a measure of how close the state is to a pure state. Compared to a perfectly prepared state with unit purity, we achieve the highest uncorrected purity $\mu\,{=}\,0.54\,{\pm}\,0.01$ at the optimal point which is sufficient for many applications of squeezed states such as entanglement generationMenzel2012 , sideband cooling of optomechanical systemsClark2017 , and quantum illuminationLasHeras2017 . Using the model of the experiment, we identify that the observed purity is limited by the added noise of the JPAs and the losses in the setup. Upon reducing the JPA noise photon number by one order of magnitude as well as the total losses to $\chi_{1}\,{=}\,\chi_{2}\,{=}\,$0.2\text{,}\mathrm{d}\mathrm{B}$$, we expect an optimized purity $\mu_{\mathrm{opt}}\,{=}\,0.80$ for the prepared state at the optimal point. The reduction of losses can be achieved by using a superconducting hybrid ring, optimized cable connectors, and improved circulators. In this context, one should remember that our protocol allows for the preparation of continuous-variable squeezed states with a degree of squeezing $S_{\mathrm{rp}}$ that is fundamentally related to the initial two-mode squeezing of the resource state. In the current implementation, even for a fixed resource TMS state, $S_{\mathrm{rp}}$ and $\gamma_{\mathrm{rp}}$ can be changed at the expense of a reduced purity $\mu$. By adding a phase shifterKokkoniemi2017 on her side, Alice could prepare squeezed states with arbitrary $\gamma_{\mathrm{rp}}$ while keeping $S_{\mathrm{rp}}$ and $\mu$ constant.

Finally, we relate our experimental RSP scheme to the cryptographic protocol known as the one-time pad by extending the latter to the quantum regime kurucz2005 ; Leung2002 . Here, Alice securely sends a quantum state $M$ to Bob over an insecure channel. We identify the transmitted message $M$ as the remotely prepared state on Bob’s side and the openly communicated cipher $C$ as the feedforward signal (see Fig. 4a). The entangled TMS states provide the random key $K$ in the form of quantum fluctuations to both parties. Note that $K$ is essential for the one-time pad since it is used by Alice and Bob to encode and decode $M$. For secure communication, $K$ needs to be a uniform random variable, such that an eavesdropper with knowledge about $C$ does not gain any information about $M$Shannon1949 . Formally, we can write

[TABLE]

where $H(M)$ is the von Neumann entropy of the remotely prepared state and $H(M|C)\,{=}\,H(M,C)\,{-}\,H(C)$ is the conditional entropy of $M$ given the feedforward signal (Supplementary Note 6). We experimentally investigate the quantum one-time pad by measuring the prepared states as a function of the JPA 3 parameters while additionally detecting the signal $C^{\prime}$ from the second directional coupler output. We compute $\delta\,{=}\,H(M)-H(M|C^{\prime})$ to verify equation (2) under the reasonable approximation $C^{\prime}\,{\approx}\,C$ (due to $\tau\,{\simeq}\,1$) using state tomography. In Fig. 4b, we observe a decrease in $\delta$ when moving towards the optimal point where the smallest value $\delta\,{=}\,0.06\,{\pm}\,0.04$ is reached and the entropy of the prepared state is $H(M)\,{=}\,0.80\,{\pm}\,0.02$. The observation $\delta\ll H(M)$ indicates that the transfer of a quantum state from Alice to Bob is close to perfect security when approaching the optimal point.

To conclude, we have successfully implemented a deterministic RSP protocol over a distance of $35\text{\,}\mathrm{c}\mathrm{m}$ in the microwave regime with continuous variables and explored the influence of different parameters on the remotely prepared states. We have remotely prepared squeezed states with a squeezing level of up to $1.6\text{\,}\mathrm{d}\mathrm{B}$ below the vacuum limit. In our specific RSP implementation, Alice can control the squeezing level and, to some extent, the squeezing angle of the remotely prepared state at the expense of a reduced purity. Additionally, we demonstrate that the protocol can be interpreted as a secure one-time pad near the optimal point. The operational range of both the RSP and quantum one-time pad protocols can be extended to any angle $\gamma_{\mathrm{rp}}$ by an additional phase shifter on Alice’s side. The demonstrated protocol opens a way to a multitude of intriguing experiments with quantum microwaves such as probing the Holevo bound limitsZwolak2013 , studying the role of quantum discord in quantum communication protocolsGu2012 , exploring hybrid continuous-discrete schemes of quantum information processingAndersen2015 , and implementing quantum illumination protocolsLasHeras2017 . Our experiment proves that prototypical local quantum networks using continuous-variable quantum microwaves are within experimental reach.

Methods

JPA 1 and JPA 2 perform a squeezing operation $\hat{S}(\xi)|0\rangle$, where $\hat{S}(\xi)\,{=}\,\exp(\frac{1}{2}\xi^{*}\hat{a}^{2}\,{-}\,\frac{1}{2}\xi(\hat{a}^{\dagger})^{2})$ is the squeezing operator, $\hat{a}^{\dagger}\,{=}\,\hat{q}-i\hat{p}$ and $\hat{a}\,{=}\,\hat{q}+i\hat{p}$ are the creation and annihilation operators with $\left[\hat{a},\hat{a}^{\dagger}\right]\,{=}\,1$ of the $f_{0}$ mode with quadratures $\hat{q}$ and $\hat{p}$, and $\xi\,{=}\,re^{i\phi}$ is the complex squeezing amplitude. Here, the phase $\phi\,{=-}2\gamma$ determines the squeezing angle $\gamma$ between the antisqueezed quadrature and the $p$-axis in the phase space, while the squeezing factor $r$ parameterizes the amount of squeezing. We define the degree of squeezing in decibels as $S\,{=}\,{-}10\,\log_{10}[\sigma_{\mathrm{s}}^{2}/0.25]$, where $\sigma_{\mathrm{s}}^{2}$ is the variance of the squeezed quadrature and the vacuum variance is $0.25$. Positive values of $S$ indicate squeezing below the vacuum level. The antisqueezing level is defined as $A\,{=}\,10\,\log_{10}[\sigma_{\mathrm{a}}^{2}/0.25]$, where $\sigma_{\mathrm{a}}^{2}$ is the variance of the antisqueezed quadrature. We generate symmetric two-mode squeezed states at the output of the hybrid ring by pumping JPA 1 and JPA 2 with strong quasi-continuous microwave drives so that they produce squeezed states with the same squeezing level and orthogonal squeezing angles $\gamma_{2}\,{=}\,\gamma_{1}\,{+}\,\pi/2$. These angles are stabilized by controlling the respective pump phases employing a phase-locked loopFedorov2016 ; Fedorov2018 . The stability of these TMS states in terms of two-mode squeezing and symmetry is of paramount importance in our experiments. Only by utilizing the nonclassical correlations between Alice and Bob, it is possible to demonstrate the successful RSP protocol. In order to reconstruct the quantum states in the experiment, we employ a well-tested reference state tomography based on statistical moments of the detected field quadraturesMenzel2012 ; Eichler2011b .

Acknowledgments

We acknowledge support by the German Research Foundation through FE 1564/1-1 and the Munich Center for Quantum Science and Technology (MCQST), Elite Network of Bavaria through the program ExQM, EU Flagship project QMiCS and OpenSuperQ, Spanish MINECO/FEDER FIS2015-69983-P, Basque Government Grant No. IT986-16 and PhD Grant No. PRE-2016-1-0284, projects JST ERATO (Grant No. JPMJER1601) and JSPS KAKENHI (Grant No. 26220601 and Grant No. 15K17731). This material is also based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advance Scientific Computing Research (ASCR), under field work proposal number ERKJ335. We would like to thank K. Kusuyama for assistance with part of the JPA fabrication and S. B. Ghaffari for help in early stages of the experiment.