Super Sensitivity and Super Resolution with Quantum Teleportation

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ARTICLE OPEN Super sensitivity and super resolution with quantum teleportation

J. Borregaard 1, T. Gehring2, J. S. Neergaard-Nielsen2 and U. L. Andersen2

We propose a method for quantum enhanced phase estimation based on continuous variable (CV) quantum teleportation. The phase shift probed by a coherent state can be enhanced by repeatedly teleporting the state back to interact with the phase shift again using a supply of two-mode squeezed vacuum states. In this way a sequential protocol exhibiting both super-resolution and super-sensitivity can be obtained due to the coherent addition of the phase shift. The protocol enables Heisenberg-limited sensitivity and super-resolution given sufﬁciently strong squeezing. The proposed method could be implemented with current or near-term technology of CV teleportation. npj Quantum Information (2019) 5:16 ; https://doi.org/10.1038/s41534-019-0132-4

INTRODUCTION multiple times (see Fig. 1), thus effectively realizing a multi-pass Quantum correlations can be used in a number of ways to protocol. This circumvents the need for physically redirecting a enhance metrological performance.1–4 Highly entangled states probe state to the same phase shift multiple times and allows to such as NOON and GHZ states can enable Heisenberg-limited keep the entangled resources separate from the potentially lossy sensitivity yielding a square root improvement with the number of phase shifting system. Compared to the mirror-based approach, probes over the standard quantum limit (SQL).5–7 This kind of no fast optical routing is required and the setup is more flexible in improvement is particularly useful when probing fragile systems not having to be confined in between mirrors. We describe how where photon damage limits the allowed number of probe this protocol can be implemented with current technology of photons. This can be the case in, e.g., measuring of biological continuous variable teleportation using two-mode squeezed systems8–10 or cold trapped atoms.11 While this effect has been vacuum states and an initial coherent state as a probe. demonstrated in experiments for small probe sizes,11–14 scaling up the size of the entangled states remains a technological barrier due to their fragility to loss and noise. Other strategies based on RESULTS the more experimentally accessible squeezed vacuum states have In the general setup, we consider some initial probe in a state |ψ 〉 – 0 also shown to beat the SQL in various settings.15 20 An alternative which is subject to an unknown phase shift described by a unitary strategy is to perform multi-pass protocols with a single probe. UðϕÞ¼eiϕn^, where n^ is the photon number operator. The goal is This enables both Heisenberg-limited sensitivity and super- to estimate the phase ϕ. After the interaction, an entangled state resolution21 for phase estimation without entangled resources is used to teleport the output state U(ϕ)|ψ 〉 back to interact with – 0 by applying the phase shift to the same probe multiple times.22 24 the phase shift again. This process is then iterated m times. If the Its experimental demonstration was realized by surrounding the teleportation is perfect, this would correspond to the transforma- (m+1) phase shift system with mirrors to measure a transversally tion |ψ0〉 → (U(ϕ)) |ψ0〉 = U((m + 1)ϕ)|ψ0〉 of the input state distributed phase shift25 or an image26 with Heisenberg-limited where m is the number of teleportations. By coherently applying sensitivity. While these approaches have demonstrated the effect the phase (m + 1) times, the signal can have both super-resolution of sub-shot noise scaling without entanglement, both demonstra- and super-sensitivity since it will now depend on (m + 1)ϕ instead tions were based on post-selection, rendering the efficiency very of just ϕ.22,23 low. In addition, the former demonstration could only measure a As a physical realization of this protocol, we consider the setup transversally distributed phase shift requiring a sample size much illustrated in Fig. 1 where consecutive two-mode squeezed larger than the beam size. The key problem is that the number of vacuum states are supplied by interferring the output of two passes through the sample needs to be carefully controlled. For single-mode squeezed vacuum sources on a balanced beam mirror-based approaches, this may be obtained with fast splitter.27 One mode is delayed in a fiber and will subsequently be integrated optical routing on the timescale of the roundtrip time subject to an unknown phase shift described by the unitary U(ϕ). between mirrors, which can be very challenging experimentally. Feedback based on previous measurements is applied before the Here we propose a fundamentally different method based on phase shift. This feedback can be implemented by mixing in an quantum teleportation for realizing quantum enhanced phase auxiliary laser field using a high transmission beam splitter.27 The measurements. The essence of our proposal is to repeatedly delay (T) is chosen such that the phase shifted mode is interfered teleport back the probe to coherently apply the phase shift with the first mode of the subsequent two-mode squeezed

1QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark and 2Center for Macroscopic Quantum States (bigQ), Department of Physics, Technical University of Denmark, Fysikvej, 2800 Kgs. Lyngby, Denmark Correspondence: J. Borregaard ([email protected]) Received: 27 July 2018 Accepted: 21 January 2019

Published in partnership with The University of New South Wales J. Borregaard et al. 2 DE ^ ^ 2 À2r= ^ (a) (b) (c) (d) (e) (f) (g) squeezing parameter r such that ðÞx2 À x3 ¼ e 2, where x2, ^x3 are the position quadratures for the two modes. The first mode of the two-mode squeezed vacuum state is mixed with the probe state on the balanced beam splitter before measurement. The 50:50 ^0 50:50 output modespffiffiffi of the beam splitterp haveffiffiffi position quadratures x1 ¼ Fibre delay ^ ^ = ^0 ^ ^ = ðx1 þ x2Þ 2 and x2 ¼ðx1 À x2Þ 2 with similar expressions for the momentum quadratures. Here ^x1 is the position quadrature of ^0 ^0 the probe state. The quadratures p1 and x2 are measured giving 0 ; 0 measurement outcomes fp1 x2g. The feedback thenpffiffiffi implements ^ ^0 ^ 0 the displacementspffiffiffi x3 ! x3 ¼ x3 þ gx 2x2 and 99:1 LO ^ ^0 ^ 0 OPO p3 ! p3 ¼ p3 þ gp 2p1, which concludes the teleportation protocol of ref. 29 50:50 The feedback displaces the quadratures such that the AM - iϕ teleported state, ψ1 will be close to |−iαe 〉. The quality of the

PM teleportation will depend on the amount of squeezing contained Feedback in the two-mode squeezed vacuum state and the feedback strength quantified by the gains gx and gp. In the limit of high = = Fig. 1 Sketch of the setup. Two optical parametric oscillators (OPO) squeezing, perfect teleportation is obtained for gx gp 1. The output single-mode squeezed vacuum states, which can be viewed protocol now repeats itself m times corresponding to m as a consecutive train of independent squeezed temporal modes teleportations being performed. At the end of the protocol, the illustrated by black dots (a). Interfering the modes on a 50:50 beam squeezed light sources should be switched off such that the final splitter results in two-mode squeezed vacuum states shown as teleported state |ψm〉 is not mixed with any two-mode squeezed connected black dots (b). The bottom modes are input to a fiber states before measurement. The teleported state |ψm〉 is subject to delay so they coincide in time with the top mode of the subsequent the phase shift resulting in state ψ0 U ϕ ψ . This state is pair (c). The bottom mode is subject to feedback based on previous m ¼ ð Þjim measurements (d). This feedback can be implemented by mixing in split by the 50:50 beams splitter (with vacuum in the other input an auxiliary laser subject to amplitude (AM) and phase (PM) port) and the position quadratures of the output modespffiffiffi are modulation using a high transmission beam splitter (here 99:1). measured and classically added with equal weight ofÀÁ 1= 2. This is

1234567890():,; ψ 〉 ^0 ψ0 The feedback is part of teleporting back the initial probe state | 0 . equivalent to measuring the position quadrature xm of m The teleported state undergoes a phase shift U(ϕ)(e) before being before the beam splitter and we can therefore simply consider this mixed with the top mode of a two-mode squeezed vacuum pair on situation. Assuming gains of gx = gp = 1, the mean and variance of a 50:50 beam splitter (f). After the beam splitter, the quadratures of ^x0 is the output states are measured with homodyne detection as part of m ^0 ψ0 ^0 ψ0 α ϕ the repeated teleportation protocol giving the classical feedback xm ¼ m xm m ¼ sinððm þ 1Þ Þ (1) shown with double lines (g) ÀÁ 1 2meÀ2r ^0 ψ0 ^0 2 ^0 2 ψ0 þ : (2) vacuum state on a balanced beam splitter before measurement. Var xm ¼ m ðxmÞ À xm m ¼ The setup, which is similar to a Mach–Zehnder interferometer with 4 feedback, is inspired by ref. 28 where the generation of continuous It is clear from Eq. (1) that the signal exhibits super-resolution in variable (CV) cluster states is demonstrated. As demonstrated in ϕ by a factor of (m + 1). The sensitivity of the measurement can be ref. 28, the squeezed light can be produced using optical quantified as6 parametric oscillation (OPO). Binning the temporal modes such pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varð^x0 Þ 1 þ 2meÀ2r that T is larger than the inverse bandwidth of the OPO results in σ ¼ m ¼ : (3) m δ ^0 =δϕ α ϕ independent squeezed temporal modes. We choose the measure- j hxmi j 2ðm þ 1Þ jcosððm þ 1Þ Þj ments and the feedback such that the CV teleportation protocol of 29 Note that the sensitivity exhibits a linear decrease in the ref. is realized. In this teleportation protocol, the momentum number of teleportations m as long as |cos((m + 1)ϕ)| ≈ 1 and the quadrature of one of the output modes and the position squeezing is sufficiently strong such that 2meÀ2r 1. For a quadrature of the other is measured, which can be achieved with − α〉 classical (SQL) strategy with m independent coherentpffiffiffiffi states | i , homodyne detection in a single-shot measurement. The feedback the sensitivity would have a scaling of / 1=ð mαÞ. The average consists of quadrature displacement based on the measurement number of probe photons, n contained in the state |ψ 〉 is outcomes. For perfect teleportation, infinitely many photons are, j j 2 À2r in principle, needed in the two-mode squeezed vacuum states. nj ¼ α þ je ; (4) The number of photons actually obtaining the phase shift will nonetheless only depend on the initial input state. For situations thus the total average number of probe photons that have where the phase shift is obtained by interaction with a interacted with the phase shift system will be – photosensitive system,8 11 the effective number of probe photons Xm 2 1 À2r interacting with the system is the limited resource. This number ntotal ¼ nj ¼ðm þ 1Þα þ mðm þ 1Þe : (5) j 0 2 will be ðm þ 1Þn0 where n0 is the number of photons in the ¼ initial state. We will show that Heisenberg-limited sensitivity in If the coherent state containsÀÁ one photon (α = 1) on average, terms of probe photons can be reached with a simple coherent we have that n ¼ðm þ 1Þ 1 þ 1 meÀ2r and the sensitivity is state as input state. Furthermore, the phase resolution can be total 2 qÀÁffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + 2 enhanced by a factor of m 1. 1 þ 1 meÀ2r ðÞ1 þ 2meÀ2r We consider a coherent state |−iα〉 as the initial probe state |ψ 〉 σ 2 : (6) 0 m ¼ ϕ ðÞα 2 R . In the setup of Fig. 1, we can input the initial state by 2ntotaljcosððm þ 1Þ Þj displacing the initial vacuum mode of the lower arm using the Thus, if meÀ2r 1, the sensitivity exhibits Heisenberg scaling in feedback laser. After the interaction of U(ϕ), the state will be | the number of probe photons for |cos((m + 1)ϕ)| ≈ 1. This ϕ −iαei 〉. This state is now teleported back to the second mode of sensitivity is similar to what could be obtained using NOON states the first two-mode squeezed vacuum state following the CV of (m + 1) photons and expresses the ultimate scaling allowed by protocol of ref. 29 The two-mode squeezed vacuum state has quantum mechanics.1

npj Quantum Information (2019) 16 Published in partnership with The University of New South Wales J. Borregaard et al. 3 (a) performance relative to a standard coherent state protocol with matched average photon number.ÀÁpffiffiffiffiffiffiffiffiffiffi For such an approach, the sensitivity is simply σcoh ¼ 1= 2 ntotaljcosðϕÞj where ntotal is the n =10 total average number of probe photons. For |cos(ϕ)| ≈ 1, the coherent n =102 state approach exhibits sensitivity at the SQL. Figure 2 shows the 101 total two effects of the imperfect teleportation; noise is added in the n =103 ^ total x-quadrature (see Eq. (2)) and more photons are added to the m 4 probe state (see Eq. (5)). In the minimization, the error from the

/ n =10 total extra photons added by an imperfect teleportation has smaller

coh 2r weight for higher ntotalp.ffiffiffi In the limit where ntotal e , the r enhancement is e = 2 and equal gains of gx = gp = 1 are optimal. This is the limit where the extra photons added to the probe state do not have any significant effect on the optimum performance. We note that a similar enhancement in sensitivity could be obtained by using a squeezed coherent state as 100 probe.15,30 For such protocols, the squeezed photons, however, 0 0.5 1 1.5 2 2.5 3 3.5 interact with the phase shift system, which is not the case here. r 2r Consequently, our protocol also works1 in the limit ntotal e 2r 4 where an enhancement of ðÞntotale =2 can be obtained for gx = = (b) gp 1. Note that our numerical optimization shows that larger enhancement can also be obtained for optimized gains in this n =10 limit (see Fig. 2b). Finally, we note that no enhancement is total possible when rt0:35 for gx = gp = 1 since the extra amplitude n =102 1 total noise added by the teleportation cancels the gain in the signal 10 3 (see Eq. (3)). For optimized gains, however, there can be a small n =10 total

m enhancement. n =104 / total Photon loss coh One of the technological challenges of using highly entangled quantum states for enhanced phase measurements is that they are very fragile to losses. Multi-pass protocols share this fragility since losses grow exponentially with the number of passes through the sample.24 This means that if the losses are too high, 100 the sensitivity enhancement of the multi-pass protocol proposed 0 0.5 1 1.5 2 2.5 3 3.5 here will vanish. Note however that while approaches based on r NOON states rely on single photon detection, this protocol is based on homodyne detection, which in practice is much more Fig. 2 Performance for finite squeezing. Maximum gain in sensitivity efficient. Since imperfect photon detection will add to the overall by using the teleportation scheme compared to a classical coherent fi loss, this means that the effective loss may be substantially state protocol for limited amount of squeezing (r)and xed average reduced with this protocol. number of total photons n . We have assumed that |cos((m + 1) total We investigate the performance of the proposed protocol in ϕ)| ≈ 1. The performance is better for high ntotal becauseherethe photons added to the probe states by imperfect teleportation have the presence of both loss acting on the probe state smaller weight compared to the extra noise added to the quadrature. corresponding to a lossy phase shift system and loss acting We have assumed gains of gx = gp = 1in(a) while we have numerically on the two-mode squeezed vacuum states. Any (symmetric) 2r optimized the gains in (b). It is seen that in the limit ntotalte the detection loss would add directly to both of these losses. We = = optimal gains are different from gx gp 1. The optimal number of model the losses with fictitious beam splitters where the teleportations m found in the optimizations are indicated with circles, unused output port is traced out. To model the lossy phase shift squares and diamonds. These indicate the transitions to m ≥ 10, 100, system, a fictitious beam splitter of transmission η1 is inserted and 1000, respectively, on the curves after the phase shift U(ϕ)(seeFig.1). For the loss in the two- mode squeezed vacuum state, fictitious beam splitters both with transmission η2 are inserted for each of the modes. For simplicity, we have assumed equal losses for both modes. Limited squeezing Assuming equal gains of gx = gp = 1, the signal and sensitivity η η One of the dominant experimental limitations of the proposed after m teleportations for { 1, 2}<1is mþ1 protocol will arguably be the amount of squeezing in the two- h^x0 i¼αη 2 sinððm þ 1ÞϕÞ (7) mode squeezed vacuum states. This will limit how many m 1 teleportations can be performed before the extra noise from the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1Àηm imperfect teleportations will dominate the signal. We therefore η 1 η À2r η 1 þ 2 1 η ðÞ2e þ 1 À 2 σ 1À 1 : (8) consider what the optimum strategy is given a constraint on m ¼ mþ1 αη 2 ϕ the amount of squeezing. In addition, we also limit the total 2ðm þ 1Þ 1 jcosððm þ 1Þ Þj average number of photons that can interact with the phase shift As expected, the loss on the probe state (η1) enters in the system. We then optimize over the number of teleportations m expression for the sensitivity exponentially in m, while loss on the and the size of the coherent probe state α to find the strategy that two-mode squeezed vacuum states (η2) only has a linear effect in provides the maximum sensitivity for these limitations. Further- m. The effect of η2 < 1 on the sensitivity is equivalent to having a 1 2r more, we also allow for arbitrary gains gx and gp. The result of the η À η limited squeezing of rlim ¼À2 lnðÞ2e þ 1 À 2 . This also holds optimization is shown in Fig. 2 where we illustrate the when considering the average number of total probe photons

Published in partnership with The University of New South Wales npj Quantum Information (2019) 16 J. Borregaard et al. 4 (a) found by optimizing the gains, near optimal performance is reached for gx = gp = 1. The error from losses in the two-mode 7 squeezed vacuum state limits the gain in the same way as finite 6 r=1.0 squeezing does for the lossless case. Consequently, whenpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi these r=1.5 = η 5 losses dominate the error, the enhancement is 1 2ð1 À 2Þ r=2.5 and no enhancement is possible for η ~1/2. When losses in the 4 r=3.5 2 probe state limit the enhancement, the optimumpffiffiffiffiffiffiffiffiffiffiffiffiffi performance is m effectively found as a tradeoff between the m þ 1 enhancement / 3 due to the teleportation and the exponential reduction due to the fi coh loss.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi As a result, we nd that the enhancement is 2= 3 1 η and no enhancement is possible for η ~ 1/3. 2 ð ð À 1ÞÞ 1 We note that while losses quickly reduce the enhancement, the scheme still exhibits enhanced sensitivity compared to the standard coherent state probe even for substantial losses.

1 DISCUSSION 10-2 10-1 100 In conclusion, we have shown how both super-sensitivity and super- 1- resolution can be obtained for an optical phase measurement using 1 continuous variable quantum teleportation based on two-mode squeezed vacuum states. For negligible losses, the protocol (b) can exhibit Heisenberg-limited sensitivity (~1/N) for squeezing 5 2NeÀ2r 1 and increase the resolution by a factor of N,where N is the number of probes. While this is equivalent to the 4 r=1.0 enhancement possible with N-photon NOON states and single r=1.5 photon detection,6 the protocol proposed here relies on homodyne ﬁ 3 r=2.5 detection, which generally is more ef cient than single photon r=3.5 detection. As a consequence of the super-resolution, the phase to m be estimated should, in principle, be localized within a window of / 1/N to reach the Heisenberg limit as for a NOON or GHZ state 1,5

coh 2 approach. However, methods developed to estimate arbitrary phases,19,31 in particular, for NOON32 and GHZ states,33 might also be employed in a straightforward way to this scheme. The latter method uses GHZ states of different sizes in order to estimate the digits of the phase allowing for arbitrary phase estimation.33 The same technique could be employed here by operating with different 1 number of teleportations before readout, which effectively corre- 10-2 10-1 100 sponds to sending entangled states of varying sizes. We have also studied the effect of photon loss on the scheme both for loss in the 1- two-mode squeezed vacuum states used for teleportation (limits the 2 effective squeezing) and for loss on the probes corresponding to a Fig. 3 Performance with loss. Maximum gain in sensitivity by using lossy phase shift system. While loss quickly reduces the perfor- the teleportation scheme compared to a classical coherent state mance, the protocol may still provide super-sensitivity for loss on protocol for limited amount of squeezing (r) and fixed average the order of several percent. For an effective squeezing of 13 dB = σ2 number of total photons ntotal = 100 in the presence of a loss on the (r 1.5), a 6 dB enhancement of the sensitivity ( )maybeobtained probe state (η1 <1, η2 = 1) and b loss on the two-mode squeezed even with 10% loss in the phase shift system for ntotal = 100 using vacuum state (η1 = 1, η2 < 1). The optimal number of teleportations m = 12 teleportations. For a more modest squeezing of 8 dB m decreases as the loss increases. The circles (squares) indicate the (r = 0.92), a 3.5 dB enhancement may be obtained under the same transition where m ≥ 10 (m ≥ 100). We have assumed that |cos((m + conditions with m = 6 teleportations. ϕ ≈ 1) )| 1 While the specific protocol studied here employed CV teleportation of a coherent state with two-mode squeezed incident on the phase shift system. For m teleportations and gains vacuum states, the generic setup of teleporting back a probe = = of gx gp 1, we have that state to interact with the phase shift system multiple times may be extended to other scenarios. In particular, our method can be 1 À ηmþ1 mð1 À η ÞÀη ð1 À ηmÞ ÀÁ n ¼ 1 α2 þ 1 1 1 η eÀ2r þ 1 À η : easily extended to a multi-mode scheme to demonstrate total η η 2 2 2 1 À 1 ð1 À 1Þ Heisenberg-limited imaging. This can be realized by replacing (9) the single-mode teleportation scheme with a multi-mode scheme in which multiple higher-order spatial modes are simultaneously η → η = – 34 Note that by taking the limit 1 1 for 2 1, Eqs. (7) (9) teleported. Using such a multi-mode approach, sub-shot noise reduce to Eqs. (1), (3) and (5). If excess noise on the squeezed and eventually Heisenberg-limited microscopy can be realized. states is included by mixing in thermal states of average photon Non-Gaussian states such as photon subtracted two-mode 35 number n instead of vacuum in the fictitious beam splitters (η2), squeezed states may also be considered for enhanced 1 η À2r η one would have that rlim ¼À2 lnðÞ2e þð1 þ 2nÞð1 À 2Þ .We teleportation performance or different probe states providing will assume that n 1 such that excess noise can be neglected. better single-shot estimation.7 A discrete variable variant of the To see the effect of finite losses, we again compare the protocol to protocol could also be envisioned using one-dimensional cluster σ 36 theÀÁffiffiffiffiffiffiffiffiffiffiffiffiffiffi simple coherent strategy for which coh ¼ states emitted by single quantum emitters. Here every second = pη ϕ 1 2 1ntotaljcosð Þj in the presence of loss. The result of the qubit could probe the phase shift while the remaining qubits are optimization is shown in Fig. 3. While a small improvement was used to teleport the phase information.

npj Quantum Information (2019) 16 Published in partnership with The University of New South Wales J. Borregaard et al. 5 METHODS Numerical optimization

We evaluate the performance of the teleportation protocol using the For arbitrary gains gx and gp, it becomes infeasible to find a closed Wigner functions37 of the photonic modes. The two-mode squeezed analytical expression for the sensitivity for an arbitrary number of vacuum state has Wigner function teleportations m. For Figs. 2 and 3, we have therefore numerically minimized the sensitivity in both the number of teleportations (m), the 2 2 À2r 2 2 2r 2 2 fi α Àe ðÞðx2 þx3 Þ þðp2Àp3 Þ Àe ðÞðp2 þp3Þ þðx2 Àx3 Þ gains (gx, gp) (assumed xed for each iteration), and for given values of Wsqðx2; p2; x3; p3Þ¼ e e ; π the squeezing parameter (r) and photon loss (η1, η2). In the optimization, (10) we use the analytical expressions for the teleported state derived as described above. By inserting the numerical values for the constants (c, αx, with (x2, p2) and (x3, p3) describing the position and momentum αp, β, λx, λp, γ), we can repeatedly iterate the protocol and get the values of quadratures of the two modes, respectively. Photon loss and excess noise fi out ; the constants for the nal Wigner function Wp;mðx1 p1Þ after m is included by mixing both modes with thermal fields of average photon teleportations. From this, the sensitivity and the mean number of probe fi η number n on ctitious beam splitters of transmission 2 (assumed equal for photons can be calculated. The optimal m, gx,gp, and α are then identified both modes). The input modes before the beam splitters are described by as the ones that minimize the sensitivity under the constraint of a fixed the Wigner function number of total probe photons.

Wnoise;in ¼ Wsqðx2; p2; x3; p3ÞWthðx2t; p2tÞWthðx3t; p3tÞ; (11) where DATA AVAILABILITY The datasets generated during and/or analyzed during the current study are available 1 À 1 ðÞx2 þp2 W ðx; pÞ¼ e nþ1=2 : (12) from the corresponding author on reasonable request. th πðÞn þ 1=2 The action of the beam splitters corresponds to making the transformation ACKNOWLEDGEMENTS ffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi We would like to thank Matthias Christandl for valuable feedback on the manuscript pη p η x2 ! 2x2 þ 1 À 2x2t and helpful discussions. We also acknowledge funding from Center for Macroscopic pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi p2 ! η p2 þ 1 À η p2t Quantum States (bigQ DNRF142) and Qubiz - Quantum Innovation Center. J.B. ffiffiffiffiffi2 ffiffiffiffiffiffiffiffiffiffiffiffiffi2 (13) pη p η acknowledges financial support from the European Research Council (ERC Grant x2t ! 2x2t À 1 À 2x2 ffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi Agreements no 337603), the Danish Council for Independent Research (Sapere Aude), pη p η ; p2t ! 2p2t À 1 À 2p2 VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). in Eq. (11) together with a similar transformation for x3, p3, x3t, and p3t. The noise modes (x2t, p2t) and (x3t, p3t) are then traced out by integration. The resulting Wigner function describes the noisy two-mode squeezed input AUTHOR CONTRIBUTIONS in ; ; ; states for the teleportation protocol and is denoted Wsqðx2 p2 x3 p3Þ. The project was conceived by J.B., T.G., J.S.N.-N. and U.L.A. The theory was developed The probe state is in general described by a Gaussian Wigner function of by J.B. with details on experimental noise and considerations supplied by T.G., J.S.N.- the form N., and U.L.A. All authors contributed in the writing of the manuscript.

α 2 α 2 β λ λ γ in ; À x x1 À pp1 þ x1 p1 þ x x1þ pp1 þ : Wp;iðx1 p1Þ¼ce (14) ADDITIONAL INFORMATION where c, αx, αp, β, λx, λp, γ are constants. For the initial coherent state probe − α〉 in Competing interests: The authors declare no competing interests. (| i with Wigner function Wp;0), we have that 2 2 c ¼ π ; αx ¼ αp ¼ 2; β ¼ 0; λx ¼ 0; λp ¼À4α; γ ¼À2α . The phase shift of ’ the probe state makes the (unknown) transformation x1 → cos(ϕ)x1 + sin Publisher s note: Springer Nature remains neutral with regard to jurisdictional claims fi (ϕ)p1, p1 → cos(ϕ)p1 − sin(ϕ)x1 in Eq. (14). Loss is modeled by subsequently in published maps and institutional af liations. mixing with a vacuum mode (Wth with n ¼ 0) on a fictitious beam splitter of transmission η1 and tracing out the output noise mode as with the two- mode squeezed vacuum states. The resulting state is described by a REFERENCES out ; Wigner function Wp;i ðx1 p1Þ. 1. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: in ; ; ; The noisy two-mode squeezed input states, Wsqðx2 p2 x3 p3Þ are now beating the standard quantum limit. Science 306, 1330–1336 (2004). out ; used to teleport the probe state Wp;i ðx1 p1Þ back to interact with the 2. Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nat. phase shift again. The mode (x3, p3) is assumed to be the delayed mode Photonics 5, 222–229 (2011). and the goal is thus to teleport the mode (x1, p1) back to this mode. This is 3. Escher, B. M., de Matos Filho, R. L. & Davidovich, L. General framework for esti- done by mixing the modes (x2, p2) and (x1, p1) on a 50:50 beam splitter, mating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. measuring the output quadratures and providing feedback on the modes Phys. 7, 406–411 (2011). (x3, p3). The input state before the 50:50 beam splitter has the Wigner 4. Degen, C. L., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, function 035002 (2017). in ; ; ; out ; : 5. Pan, J.-W. et al. Multiphoton entanglement and interferometry. Rev. Mod. Phys. Wtel;in ¼ Wsqðx2 p2 x3 p3ÞWp;i ðx1 p1Þ (15) 84, 777–838 (2012). ń ł ń The beam splitter makes the transformation 6. Demkowicz-Dobrza ski, R., Jarzyna, M. & Ko ody ski, J. Chapter four - quantum – pffiffiffi pffiffiffi limits in optical interferometry. Progress. Opt. 60, 345 435 (2015). = ; = 7. Bouchard, F. et al. Quantum metrology at the limit with extremal majorana x1 !ðx1 þ x2Þ ffiffiffi2 p1 !ðp1 þ p2Þ ffiffiffi2 p p (16) constellations. Optica 4, 1429–1432 (2017). x !ðx À x Þ= 2; p !ðp À p Þ= 2 2 1 2 2 1 2 8. Frigault, M. M., Lacoste, J., Swift, J. L. & Brown, C. M. Live-cell microscopy – tips – in Eq. (15). The measurements of the quadratures p1 and x2 and and tools. J. Cell Sci. 122, 753 767 (2009). fi 9. Crespi, A. et al. Measuring protein concentration with entangled photons. Appl. corresponding feedback on mode (x3,pp3)ffiffiffi are then describedp byffiffiffi rst making the transformation x3 ! x3 À gx 2x2, and p3 ! p3 À px 2p2 in Phys. Lett. 100, 233704 (2012). the Wigner function and then integrating out modes (x1, p1) and (x2, p2). 10. Piliarik, M. & Sandoghdar, V. Direct optical sensing of single unlabelled proteins The corresponding teleported state corresponds to probe state and super-resolution imaging of their binding sites. Nat. Commun. 5, 4495 (2014). in ; 11. Wolfgramm, F., Vitelli, C., Beduini, F. A., Godbout, N. & Mitchell, M. W. Wp;iþ1ðx1 p1Þ. The process can then be repeated to get the next iteration. After a fixed number of teleportations m, the phase shift is estimated from Entanglement-enhanced probing of a delicate material system. Nat. Photonics 7, – measurement of 〈x1〉 for the light described by Wigner function 28 32 (2012). out ; 12. Mitchell, M. W., Lundeen, J. S. & Steinberg, A. M. Super-resolving phase mea- Wp;mðx1 p1Þ. All moments of x1 and p1 can be extracted from out ; surements with a multiphoton entangled state. Nature 429, 161–164 (2004). Wp;mðx1 p1Þ, which makes the calculation of the sensitivity straightforward. Note that the mean number of photons contained in the state is 13. Afek, I., Ambar, O. & Silberberg, Y. High-noon states by mixing quantum and 2 2 = classical light. Science 328, 879–881 (2010). nm ¼hx1 iþhp1iÀ1 2.

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