Quantum Nondemolition Gate Operations and Measurements in Real Time on Fluctuating Signals
Yu Shiozawa, Jun-ichi Yoshikawa, Shota Yokoyama, Toshiyuki Kaji, Kenzo Makino, Takahiro Serikawa, Ryosuke Nakamura, Shigenari Suzuki, Shota Yamazaki, Warit Asavanant, Shuntaro Takeda, Peter van Loock, Akira Furusawa
QQuantum Nondemolition Gate Operations and Measurementsin Real Time on Fluctuating Signals
Yu Shiozawa, Jun-ichi Yoshikawa,
1, 2
Shota Yokoyama,
1, 3
Toshiyuki Kaji, KenzoMakino, Takahiro Serikawa, Ryosuke Nakamura, Shigenari Suzuki, Shota Yamazaki, Warit Asavanant, Shuntaro Takeda,
1, 4
Peter van Loock, and Akira Furusawa ∗ Department of Applied Physics, School of Engineering,The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Quantum-Phase Electronics Center, School of Enginerring,The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Centre for Quantum Computation and Communication Technology,School of Engineering and Information Technology,University of New South Wales Canberra, ACT 2600, Australia JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan Institute of Physics, Johannes Gutenberg-Universit¨at Mainz, Staudingerweg 7, 55099 Mainz, Germany
We demonstrate an optical quantum nondemolition (QND) interaction gate with a bandwidth ofabout 100 MHz. Employing this gate, we are able to perform QND measurements in real time onrandomly fluctuating signals. Our QND gate relies upon linear optics and offline-prepared squeezedstates. In contrast to previous demonstrations on narrow sideband modes, our gate is compatiblewith non-Gaussian quantum states temporally localized in a wave-packet mode, and thus opens theway for universal gate operations and realization of quantum error correction.
Introduction .—A quantum nondemolition (QND) in-teraction enables indirect and nondestructive measure-ments of quantum systems [1, 2]. It couples two quan-tum systems so that a signal observable of one systemis measured without disturbance by measuring a probeobservable of the other system. This is also the essenceof error syndrome identification in general quantum er-ror correction, including the special case of continuous-variable (CV) error correction schemes [3–5]. In the con-text of CV quantum information processing, a QND in-teraction is also referred to as a sum gate, transformingthe position quadratures of two modes as ˆ x → ˆ x andˆ x → ˆ x + ˆ x in the Heisenberg picture [6, 7]. This isa direct analogue of a controlled-NOT gate for qubitsand considered as an elementary two-mode entanglinggate for universal quantum computation over continu-ous variables [8, 9]. The QND gate is an integral partof many important CV quantum information protocols,such as generation of cluster states for one-way quantumcomputing [10, 11], realization of non-Gaussian gates viagate teleportation [5, 12], as well as CV coherent com-munication [13].Thus far, QND interactions and measurements havebeen demonstrated in several optical experiments. Onescheme to implement the QND gate is to directly coupletwo input optical fields and pump fields via paramet-ric amplification using a nonlinear optical media [14].However, this direct scheme typically induces couplinglosses on fragile quantum states, degrading the gate fi-delity. Another scheme which does not require direct cou-pling and instead uses linear optics and offline-preparedsqueezed states was proposed [15] and demonstrated with ∗ [email protected] high precision [16]. This offline scheme is, in principle,applicable to arbitrary optical quantum states. However,previous QND gates [16–18] only work on quantum statesin narrow sideband modes in the frequency domain.That is, they are not applicable to general non-Gaussianquantum states generated in wave-packet modes, suchas single-photon states [19, 20] and Schr¨odinger’s catstates [21, 22], although such states are included in manyCV protocols and are also required for universal quantumcomputing [5, 8, 9].In this Letter, we demonstrate QND gate operationsand measurements in real time on continuously fluctuat-ing signals with a bandwidth of about 100 MHz. Un-like previous experiments [16–18], the input signal israndomly fluctuating with a short autocorrelation time,and thus the success of QND interactions on this sig-nal is a proof that our gate correctly operates instantsignals without memory-like effects. The time-domaintraces of quadrature values are obtained in real time byjust applying electric filters [23], and thus can be inter-preted as results of real-time QND measurements withrespects to time-shifted wave-packet modes determinedby the electric filters. Since our QND gate works on anywave-packet modes for up to about 100 MHz, our gateis compatible with non-Gaussian quantum states local-ized in a wave-packet mode. Note that, for CV single-mode squeezing and teleportation gates, the bandwidthhas been widened to about 10 MHz and operations onnon-Gaussian quantum states have already been demon-strated [24–29]. Here we demonstrate for the first time abroadband interaction gate, and furthermore the band-width is widened to about 100 MHz. Our gate is a cru-cial component for future realizations of non-Gaussiangates [5, 12], time-domain multiplexed cluster states [11],error syndrome measurements of a qubit encoded in an a r X i v : . [ qu a n t - ph ] D ec oscillator [5], and CV gate sequences in a loop-based ar-chitecture [30]. Theory .—Let us define quadratures of a quantum op-tical field mode k as ˆ x k and ˆ p k with ˆ x k ≡ (ˆ a † k + ˆ a k ) andˆ p k ≡ i (ˆ a † k − ˆ a k ), where ˆ a k and ˆ a † k are annihilation and cre-ation operators, respectively ( (cid:126) = 2, [ˆ x k , ˆ p k (cid:48) ] = 2 iδ kk (cid:48) ).The QND interaction is a two-mode unitary operationˆ U QND = exp (cid:0) − i G ˆ x ˆ p (cid:1) , where G is a QND gain, i.e.the strength of the interaction of two optical modes. Thisinteraction transforms the quadrature operators as (cid:20) ˆ x out1 ˆ x out2 (cid:21) = (cid:20) G (cid:21) (cid:20) ˆ x in1 ˆ x in2 (cid:21) , (cid:20) ˆ p out1 ˆ p out2 (cid:21) = (cid:20) − G (cid:21) (cid:20) ˆ p in1 ˆ p in2 (cid:21) . (1)Since the QND interaction belongs to the class of Gaus-sian operations, it is decomposable into beam-splitterinteractions and single-mode squeezing operations [SeeFig. 1(a)] [15, 31]. Furthermore, squeezing is also a Gaus-sian operation, and is realized by an offline scheme witha beam splitter and ancillary squeezed light [15], wherethe squeezing degree is tunable via the reflectivity ofthe beam splitter R . The QND gate is implemented bychoosing the beam-splitter reflectivities before and afterthe squeezing gates as 1 / (1 + R ) and R/ (1 + R ), respec-tively [See Fig. 1(a)]. We obtainˆ x out1 = ˆ x in1 − (cid:114) − R R ˆ x (0)A e − r A , (2a)ˆ x out2 = 1 − R √ R ˆ x in1 + ˆ x in2 + (cid:114) R − R R ˆ x (0)A e − r A , (2b)ˆ p out1 = ˆ p in1 − − R √ R ˆ p in2 + (cid:114) R − R R ˆ p (0)B e − r B , (2c)ˆ p out2 = ˆ p in2 + (cid:114) − R R ˆ p (0)B e − r B , (2d)where ˆ x (0)A e − r A and ˆ p (0)B e − r B are quadratures of ancillarysqueezed vacua of squeezing gates A and B with finitesqueezing parameters r A and r B . In the ideal limit of r A , r B → ∞ , both ˆ x (0)A e − r A and ˆ p (0)B e − r B terms vanish,and Eq. (2) becomes equivalent to Eq. (1), where theQND gain is G = (1 − R ) / √ R . In the experiment, wechoose the QND gain G = 1. In this case, R = (3 −√ / ≈ .
38, 1 / (1 + R ) ≈ .
72 and R/ (1 + R ) ≈ . x k or ˆ p k ( k = 1 ,
2) by homodyne detection using alocal oscillator (LO). Generally, in the case that the LOis a continuous coherent light, the detected homodynesignal is also continuous. The quadrature of a quan-tum state in a wave-packet mode g mode ( t ) is obtainedfrom the original homodyne signal ˆ X k ( t ) by an integra-tion ˆ x k = (cid:82) g mode ( τ ) ˆ X k ( τ ) dτ . On the other hand, whena continuous signal ˆ X k ( t ) passes through a filter with aresponse function g filter ( t ), the resulting continuous sig-nal becomes ˆ x k ( t ) = (cid:82) g filter ( t − τ ) ˆ X k ( τ ) dτ . Thereforewe obtain quadrature values in real time just by in-serting an electric filter, where the mode function that (a)(c) Squeezinggate-AQND gate1/(1+R) R/(1+R)Input-1Input-2 Output-1Output-2Squeezinggate-B (b)
QND gate
HD-1HD-2Output-1Output-2 HPF LPFHPF LPFHPF LPF O sc ill o sc ope FluctuatingsignalsInputpreparation 99%
Input-1Input-2 Output-1Output-2
Squeezinggate-ASqueezinggate-B1/(1+R) RR for LOs etc.CW 860 nmPump BeamCW 430 nm
HD-AHD-B
OPOOPO
HWP-1PBS-1HWP-2PBS-2 O p t i c a l D e l a y L i ne EOM-AEOM-B HWP-3PBS-3
Ti:SaLaserSHG (i)(ii)(iii)
FIG. 1. Experimental setup. (a) Decomposition of a QNDgate. (b) Peripheral systems for the input signal preparationand the output measurements. Input optical signal is sent toeither Input-1 or Input-2. (c) Experimental setup of the QNDgate. Squeezing gate-A and B share an optical delay line inorthogonal polarizations. SHG, second harmonic generator;HD, homodyne detector. corresponds to the quadrature value ˆ x k ( t ) obtained attime t is g mode ,t ( t ) = g filter ( t − t ) [23]. Note thatreal-time measurements are necessary for nonlinear feed-forward operations in measurement-based quantum com-putation [23]. We choose a low-pass filter (LPF) whichhas a flat passband and a steep edge with a cutoff fre-quency of 100 MHz in order to treat the bandwidth of100 MHz equally. However, the QND gate itself can workon arbitrary wave-packet modes for up to the bandwidthof 100 MHz, enabling operations on non-Gaussian states.As already noted, in order to show memoryless fea-tures of our gate, we use random white signals as inputs.From the signal-to-noise ratio of this random signal, wecan evaluate the conventional QND quantities T S and T P [32]. However, unlike previous experiments [2, 14, 16–18], it may not be appropriate to evaluate T S and T P justby transfer of signal powers. If the signal is modified un-expectedly by irregular gate responses, a part of the inputsignal is considered to be converted to noise at the out-put, by which the effective T S and T P degrades. In orderto exclude such a possibility, we check the cancellation ofthe output signals by using the input signal. The setup isshown in Fig. 1(b). The random signal is split into two;one is utilized for generating the input optical signal, andthe other is stored for reference. Here we set the targetof the QND measurement to the quadrature amplitudeproduced by the random signal in the wave-packet modedefined by the electric filters. This input amplitude is di-rectly stored by applying the same electric filters to therandom signal before storage. Therefore, we can cancelthe produced output signals [(ii) and (iii) in Fig. 1(b)]by using the stored signal [(i) in Fig. 1(b)] with an ap-propriate shift of the time origin. This is also a newachievement of this research. Experimental setup .—We use a continuous-wave (CW)Ti:Sa laser at a wavelength of 860 nm. Input states ofthe QND gate are vacuum states and coherent states.We generate a random optical signal using a waveguideelectro-optics modulator (EOM) and an amplified John-son electric noise, which is applied to each of the inputquadratures ( x in1 , x in2 , p in1 , p in2 ). For the frequency charac-teristic of the random signal and the scheme of gener-ating the coherent state, see the Supplemental Material(SM) [33]. The other three input quadratures are at vac-uum levels. This is sufficient to characterize the gate-response matrix on the assumption of the linearity of thegate.The QND gate consists of a Mach-Zehnder interfer-ometer containing two squeezing gates in it as shownin Fig. 1(a). The squeezing gate has an optical delayline to compensate the delay of electronic circuits forfeed-forward operations. In order to match the delays oftwo squeezing gates, we implement a common delay line(about 3 m) by utilizing the optical polarization degreesof freedom as shown in Fig. 1(c). We insert a half-waveplate (HWP) before a polarizing beam splitter (PBS) toseparate the two outputs, by which the latter beam split-ter R/ (1 + R ) is implemented. The ancillary squeezedvacua are generated from triangle-shaped optical para-metric oscillators (OPOs) [36]. For the broadband spec-tra of ancillary squeezed vacua and homodyne detectors,see the SM [33].We apply, in addition to the 100-MHz LPF mentionedabove, a high-pass filter (HPF) with a cutoff frequencyof 1 MHz to the output homodyne signals for rejectionof low-frequency noises. The mode function is mainlydetermined by the LPF, and the deformation of it by theHPF is negligible. The frequency characteristic of thesefilters are shown in the SM [33]. We acquire the filteredhomodyne signals, together with the filtered input signal,by an oscilloscope at the sampling rate of 1 GHz. For theQND quantities T S , T P and V S | P [32], we use 1,000 setsof sequential 10,000 data points. For the power spectra,we use 9,000 sets of sequential 1,024 data points. Experimental results .—First, as an example, we showthe time-domain traces for the case where the white sig-nal is applied to ˆ x in1 . The other three cases are shown −100−50 −9−6−3 −9−6−3 −9−6−3 −9−6−3 Time [ns] V o l t age [ m V ] Q uad r a t u r e Q uad r a t u r e Q uad r a t u r e Q uad r a t u r e (a)(b)(c)(d)(e) FIG. 2. Time-domain traces for 300 ns. (a) The filtered inputwhite signals. (b) The filtered output homodyne signals ofˆ x out1 (red) and ˆ x out2 (blue). (c) Results of the cancellation. (d)The filtered output homodyne signals of ˆ x out1 (red) and ˆ x out2 (blue) for vacuum inputs. (e) The filtered output homodynesignals of ˆ p out1 (red) and ˆ p out2 (blue) for vacuum inputs. in the SM [33]. In Fig. 2, we show typical time-domaintraces of the filtered white signals and the filtered homo-dyne signals for 300 ns. Figures 2(a) and (b) show thetraces of the input white signal and the output quadra-tures ˆ x out1 and ˆ x out2 [(i), (ii), and (iii) in Fig. 1(b)], re-spectively. We can see that the output quadratures ˆ x out1 and ˆ x out2 follow the input white signal with a time de-lay of 36 ns, which is shown by gray backgrounds anddotted lines in Figs. 2(a) and (b). This means that thesignal input ˆ x in1 is transmitted non-destructively to thesignal output ˆ x out1 , and simultaneously the signal infor-mation is copied to the probe output ˆ x out2 . Then wesubtract the input white signal from the output respec-tive quadratures ˆ x out1 and ˆ x out2 with an optimum gainand the time shift; the results are shown in Fig. 2(c). Asreferences, in Fig. 2(d), we also show traces of ˆ x out1 andˆ x out2 for the case of vacuum input. We can see that thevariances of the residual fluctuations in Fig. 2(c) are com-parable to those of the vacuum input case in Fig. 2(d).The nice cancellation with a simple time shift means thatthe gate converts the instant input signals to the instantoutput signals without memory-like effects in this timescale. Without the added random signals, there is stillsome positive correlation independent of the input signal N o i s e po w e r [ d B ] ^ x out1 ^ x out2 ^ p out1 ^ p out2 FIG. 3. Power spectra of output quadratures when a random signal is added to ˆ x in1 . Black: shot noises. Red: results ofvacuum-state input. Magenta: results of coherent-state input. Green: cancellation of the random signal. in ˆ x out1 and ˆ x out2 . On the other hand, when we look atˆ p out1 and ˆ p out2 in Fig. 2(e), there is a negative correlation.Figures 2(d) and (e) show the quantum entanglementgenerated by the gate interaction.Next, in order to evaluate the cancellation more pre-cisely, we perform Fourier transform to the results, andthe resulting power spectra are shown in Fig. 3. Thespectra for the vacuum-state input, the coherent-stateinput, the cancellation, and the homodyne shot noise asa reference are colored in red, magenta, green, and black,respectively. In the case of an ideal QND interaction ofvacuum inputs with G = 1, ˆ x out1 and ˆ p out2 are kept atthe shot-noise level, while ˆ x out2 and ˆ p out1 are increased by3 dB from the shot-noise level, because a vacuum fluc-tuation of ˆ x in1 or ˆ p in2 is added. Our results are in goodagreement with this, though there are some excess noiseincreases due to finite squeezing of ancillary states. Whenthe input white signal is added to ˆ x in1 , the powers of ˆ x out1 and ˆ x out2 increase by the same amount, showing the unitygain of the QND interaction, while those of ˆ p out1 and ˆ p out2 do not increase, showing negligible crosstalk between x and p quadratures. Comparing the vacuum-input (red)trace and the signal-canceled (green) trace, we can seethat the cancellation is almost perfectly working for upto about 100 MHz. Further discussions of the cancella-tions by introducing response functions are included inthe SM [33].Finally, we evaluate the QND quantities T S , T P , and V S | P for both ˆ x and ˆ p quadratures. The success of QNDmeasurements is commonly verified by the criteria [32]1 < T S + T P , V S | P < . (3)The experimentally determined values are T S + T P =1 . ± . > V S | P = 0 . ± . < x quadratures, T S + T P = 1 . ± . > V S | P =0 . ± . < p quadratures. Therefore, wesucceeded in construction of a QND gate that enablesreal-time QND measurements for both conjugate quadra-tures with the bandwidth of about 100 MHz. For a moredetailed analysis, we show the QND quantities at eachfrequency in Fig. 4. All of T S , T P , and V S | P satisfy theQND criteria up to about 100 MHz. As for V S | P , becauseof the finite bandwidth of the ancillary squeezed vacua,the correlation degrades at higher frequencies, however, C ond i t i ona l v a r i an c e C ond i t i ona l v a r i an c e T r an s f e r c oe ff i c i en t T r an s f e r c oe ff i c i en t (a) (b)(c) (d) FIG. 4. Spectra of the QND quantities. (a), (b) Transfercoefficients when the random signal is added to ˆ x in1 or ˆ p in2 , re-spectively. Blue, green and red traces are T S , T P and T S + T P ,respectively. (c), (d) Conditional variances for vacuum in-puts, i.e., the variances of ˆ x out1 − g x ˆ x out2 and g p ˆ p out1 + ˆ p out2 atthe gain g x = 0 .
41 and g p = 0 .
39, respectively [33], normal-ized by the shot noise spectrum. Blue and green traces arefor the cases with and without the ancillary squeezed vacua,respectively. there are still sub-shot-noise correlations for up to about100 MHz. The two output modes are entangled, whichis described in the SM [33].
Conclusions .—We experimentally demonstrated anoptical two-mode QND interaction gate that enables real-time QND measurements on temporally fluctuating ran-dom signals. We also showed that the interaction workson a broad spectrum, namely up to about 100 MHz inthe frequency domain. The capability of the gate todeal with instantaneous signals is confirmed by the can-cellation of random signals. This scheme is applicableto any quantum states in wave-packet modes, includingnon-Gaussian states, and thus perfectly suitable for im-plementing non-Gaussian gates [5, 12], generating time-multiplexed cluster states [11], diagnosing the error syn-drome in quantum error correction [3–5], and implement-ing gate sequences in a loop-based architecture [30].
ACKNOWLEDGMENTS
This work was partly supported by CREST (JP-MJCR15N5) and PRESTO (JPMJPR1764) of JST, JSPS KAKENHI, APSA. P. v. L. was supported byQcom (BMBF). Y. S., S. Y., and T. S. acknowledge fi-nancial support from ALPS. [1] V. B. Braginsky and F. Y. Khalili,
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SI. DETAILED EXPERIMENTAL SETUP AND FREQUENCY SPECTRAA. Electric filters and response function
The homdyne signals for verification as well as the white signals for the QND input are stored by an oscilloscope(DPO7054, Tektronix) after a low-pass filter (LPF) and a high-pass filter (HPF). The LPF is a commercially availablefilter whose cut-off frequency is 100 MHz (Mini-Circuits, BLP-100+). We plot the frequency characteristics of theLPF in Fig. S5. The HPF is a homemade 1st-order filter with a cutoff frequency of 1 MHz, which is used in order toremove low-frequency noise around the laser carrier frequency. We plot the frequency characteristics of the HPF inFig. S6. The mode function g mode ( t ) is mainly determined by the LPF. The time-domain response function g filter ( t )calculated from the gain and phase in Fig. S5 is shown in Fig. S7. As mentioned in the main text, the time-reversalresponse function with time shifts is the effective mode function g mode ( t ) for the QND measurements. G a i n [ d B ] −180−120−60060120180 P ha s e [ deg r ee ] LPF-1 G a i n [ d B ] −180−120−60060120180 P ha s e [ deg r ee ] LPF-2 G a i n [ d B ] −180−120−60060120180 P ha s e [ deg r ee ] LPF-3
FIG. S5. Frequency characteristics of the LPFs, obtained by a network analyzer (Keysight, E5061B). Blue: gain. Green: phase. -4 -3 -2 -1 Frequency [MHz]−100−80−60−40−20020 G a i n [ d B ] −180−120−60060120180 P ha s e [ deg r ee ] HPF-1 -4 -3 -2 -1 Frequency [MHz]−100−80−60−40−20020 G a i n [ d B ] −180−120−60060120180 P ha s e [ deg r ee ] HPF-2 -4 -3 -2 -1 Frequency [MHz]−100−80−60−40−20020 G a i n [ d B ] −180−120−60060120180 P ha s e [ deg r ee ] HPF-3
FIG. S6. Frequency characteristics of the HPFs, obtained by a network analyzer (Keysight, E5061B). Blue: gain. Green:phase. A m p li t ude [ a . u .] LPF-1LPF-2LPF-3
FIG. S7. Obtained response function of the LPF. N o r m a li z ed po w e r [ d B ] FIG. S8. Power spectrum of the input white signal. Green: unfiltered signal. Blue: filtered signal. Red: oscilloscope noisefloor.
B. White signal source
Figure S8 shows the power spectrum of the white signal used for the input of the QND gate. The white signalis amplified thermal noises of resistors and operational amplifiers (OPA847, Texas Instruments). The trace in greenrepresents unfiltered signals, while the trace in blue represents filtered signals which corresponds to the signals storedby the oscilloscope in the actual QND experiment.
C. Homodyne detectors
Figure S9 shows the optical shot-noise spectra with a local oscillator (LO) power of 10 mW, together with thedetector dark noise spectra, of the four homodyne detectors (two for feed-forward operations and two for QNDmeasurements). We show both the filtered and unfiltered cases. The shot-noise spectra are flat up to about 100 MHzfor all of the four detectors. The clearance between the shot noise and the dark noise is more than 10 dB even at100 MHz.
D. Ancillary squeezed vacua
Figure S10 shows the power spectra of the squeezed and anti-squeezed quadratures of the ancillary squeezed vacuanormalized by the shot noise spectrum. The power of the pump beam is 85 mW. Both of the two squeezed vacua N o r m a li z ed po w e r [ d B ] HD-A
HD-B
HD-1
HD-2
FIG. S9. Noise power spectra of four homodyne detectors. Blue: optical shot noise spectra with the LPF. Green: optical shotnoise spectra without the LPF. Red: detector dark noise spectra with the LPF. Cyan: detector dark noise spectra without theLPF. Magenta: oscilloscope noise floor. N o r m a li z ed po w e r [ d B ] FIG. S10. Noise power spectra of ancillary squeezed vacua, normalized by the shot-noise spectrum. Green: anti-squeezedquadrature. Blue: squeezed quadrature. show about − − Beam-1EOM Beam-2 Beam-3AOMAOM
QND gate
Random signal genration in Input-1Random signal genration in Input-2The same setupDetectorDetectorfor Input-2 Fluctuatingsignals
FIG. S11. Experimental setup for input signal preparation. Beam-2 is sent to either Input-1 or Input-2.
E. Control of optical systems
In order to lock interference phases in the QND gate, we use weak laser beams as phase references for each opticalpaths. These reference beams are temporally turned on and off by switching a pair of acousto-optic modulators(AOMs). We control the optical systems by feedback when the reference beams are on, while the system is held andthe QND gate is tested when they are off. The duration of ON time is 1400 µ s and that of OFF time is 600 µ s. TheQND measurement data are acquired within 10 µ s in the OFF time, during which the drift of the optical system isnegligible.However, there are some beams which cannot be turned off. Some are carrier beams to generate input randomsignals by modulations, and others are carrier beams for feed-forward operations in the squeezing gates. The lasernoises of these beams disturb the homodyne signals. The noises by the input carrier beams are more significant thanthose by the feed-forward carrier beams because of the differences in optical path lengths. Since the input beams passthrough the optical delay line before interference with the LOs, the phase noises look larger in the output homodynesignals. These noises are filtered out by the HPF in Fig. S6 and thus they are not so significant problems, however,in order to further remove them, we employ procedures as follows.The optical setup for the input random signal is shown in Fig. S11. For each of Input-1 and Input-2, three beams areused. Beam-1 is the phase reference for the QND gate. Beam-2 is the carrier beam to convert the random electronicsignals to optical signals by phase modulation. While Beam-1 is temporally turned off during data acquisition, Beam-2is always on. We use Beam-3, which is always on, for canceling the carrier component of Beam-2, leaving only themodulation sideband. Note that only Beam-1 is used when we use vacuum states as input states. The quadratureto add the random signal is selected via the relative phase between Beam-1 and Beam-2. For example, for Input-1,when Beam-1 and Beam-2 are locked in phase, the random signal is added to ˆ x in1 . On the other hand, when Beam-1and Beam-2 are locked 90-degree out of phase, the random signal is added to p in1 . Beam-3 is always locked to theopposite phase with Beam-2, removing the carrier component of Beam-2. F. Feed-forward operation
The feed-forward operations in the squeezing gates cancel the anti-squeezed noises of the ancillary squeezed vacua.Here we explain this by using equations. In the squeezing gate, first the input state (quadrature operators ˆ x in andˆ p in ) is coupled with an ancillary squeezed state (quadrature operators ˆ x (0) e − r and ˆ p (0) e r with a squeezing parameter r ) by a beam splitter with a reflectivity R .ˆ x int-1 = √ R ˆ x in + √ − R ˆ x (0) e − r , ˆ p int-1 = √ R ˆ p in + √ − R ˆ p (0) e r , (S4a)ˆ x int-2 = √ − R ˆ x in − √ R ˆ x (0) e − r , ˆ p int-2 = √ − R ˆ p in − √ R ˆ p (0) e r . (S4b)Next, as a feed-forward operation, the anti-squeezed quadrature of a beam-splitter output ˆ p int-2 is measured and usedfor cancellation of the anti-squeezed noise ˆ p (0) e r in the other output quadrature ˆ p int-1 ,ˆ x out = ˆ x int-1 = √ R ˆ x in + √ − R ˆ x (0) e − r , ˆ p out = ˆ p int-1 + (cid:114) − RR ˆ p int-2 = 1 √ R ˆ p in . (S5)In the ideal limit of r → ∞ , the excess noise term ˆ x (0) e − r vanishes, and Eq. (S5) approaches the ideal squeezingtransformation where the squeezing degree is determined by the reflectivity R .For the cancellation of the anti-squeezed noises, unlike the previous narrowband experiments [16, 17], the electronicsignal for the feed-forward must be synchronized with the optical signal, in other words, the phase lags must bematched at all the frequencies. For this purpose, we use high-speed homodyne detectors and amplifiers with a flatgain and a linear dispersion, and the optical delay line for the compensation of the electronic delay. We confirmedthe broadband cancellation by using a network analyzer (MS4630B, ANRITSU), which is shown in Figs. S12 andS13. Modulation signals are added by an EOM before the OPOs to the ancillary quadratures to be anti-squeezed,and they are canceled by the feed-forward. Figures S12(a), S12(b), S13(a), and S13(b) are the gains and phases ofthe modulated reference beams through the optical delay line. The gains decrease at higher frequencies due to thebandwidth of the OPO cavities. They are used for calibration of the traces in the other figures in Figs. S12 and S13.0 −25−20−15−10−5 G a i n [ d B ] −180−120−60 P ha s e [ deg r ee ] −25−20−15−10−5 G a i n [ d B ] P ha s e [ deg r ee ] −70−60−50−40−30−20−10 G a i n [ d B ] −180−120−60 P ha s e [ deg r ee ] (a)(b) (c)(d) (e)(f) FIG. S12. Cancellation of the modulated signal by the feed-forward in the squeezing gate-A. (a), (b) Gain and phase of thereference beam in the squeezing gate-A, used for calibration of the other traces. (c), (d) Gain and phase of the feed-forwardbeam in the squeezing gate-A. (e), (f) The results of the cancellation of the modulated signal.
Figures S12(c), S12(d), S13(c), and S13(d) are the gains and phases through the feed-forward electronic paths. Thegains are flat and the phases are opposite (180 ◦ ) for up to 100 MHz. Figures S12(e), S12(f), S13(e), and S13(f) arethe residual modulation signals after the cancellation. The extinction ratios of the modulated signals are more than20 dB for up to 100 MHz. SII. RESPONSE OF THE QND GATEA. General theory of response functions and cancellation
We consider a linear and static system y ( t ) = (cid:90) f ( t − τ ) w ( τ ) dτ + v ( t ) , (S6)where f ( t ) is a response function, w ( t ) is an input signal, y ( t ) is an output signal, and v ( t ) is an excess noise whichis independent of w ( t ), i.e., the cross-correlation vanishes, R wv ( t ) = (cid:104) w ( τ ) v ( τ + t ) (cid:105) = (cid:90) w ( τ ) v ( τ + t ) dτ = 0 . (S7)The response function f ( t ) is obtained by deconvolution from the input-output cross-correlation. The autocorrelation R ww ( t ) and the cross-correlation R wy ( t ) are, R ww ( t ) = (cid:90) w ( τ ) w ( τ + t ) dτ, (S8a) R wy ( t ) = (cid:90) w ( τ ) y ( τ + t ) dτ, = (cid:90) (cid:90) w ( τ ) w ( τ (cid:48) ) f ( τ − τ (cid:48) + t ) dτ dτ (cid:48) , (S8b)1 −25−20−15−10−5 G a i n [ d B ] −180−120−60 P ha s e [ deg r ee ] −25−20−15−10−5 G a i n [ d B ] P ha s e [ deg r ee ] −70−60−50−40−30−20−10 G a i n [ d B ] −180−120−60 P ha s e [ deg r ee ] (a)(b) (c)(d) (e)(f) FIG. S13. Cancellation of the modulated signal by the feed-forward in the squeezing gate-B. (a), (b) Gain and phase of thereference beam in the squeezing gate-B, used for calibration of the other traces. (c), (d) Gain and phase of the feed-forwardbeam in the squeezing gate-B. (e), (f) The results of the cancellation of the modulated signal. or in the frequency domain, S ww ( ω ) = | W ( ω ) | , (S9a) S wy ( ω ) = | W ( ω ) | F ( ω ) . (S9b)Therefore, the response function is obtained in the frequency domain by F ( ω ) = S wy ( ω ) S ww ( ω ) . (S10)The obtained response function f ( t ) gives the optimal cancellation of the input signal, i.e., (cid:42)(cid:20) y ( t ) − (cid:90) h ( t − τ ) w ( τ ) dτ (cid:21) (cid:43) = (cid:10) v ( t ) (cid:11) + (cid:42)(cid:26)(cid:90) [ f ( t − τ ) − h ( t − τ )] w ( τ ) dτ (cid:27) (cid:43) , (S11)which is minimized when h ( t ) = f ( t ). Note that the cross terms vanish by using Eq. (S7). B. Experimental response functions
If the QND gate is not working instantaneously, the QND gate transformations in the time domain are generallyin the form of ˆ x out1 ( t ) = (cid:90) f x → ( t − τ )ˆ x in1 ( τ ) dτ + (other noise terms) , (S12a)ˆ x out2 ( t ) = (cid:90) f x → ( t − τ )ˆ x in1 ( τ ) dτ + (cid:90) f x → ( t − τ )ˆ x in2 ( τ ) dτ + (other noise terms) , (S12b)ˆ p out1 ( t ) = (cid:90) f p → ( t − τ )ˆ p in1 ( τ ) dτ − (cid:90) f p → ( t − τ )ˆ p in2 ( τ ) dτ + (other noise terms) , (S12c)ˆ p out2 ( t ) = (cid:90) f p → ( t − τ )ˆ p in2 ( τ ) dτ + (other noise terms) . (S12d)2 −150−100 −50 0 50 100 150Time [ns]−0.20.00.20.40.60.8 A m p li t ude [ a . u .] R ww ( t ) A m p li t ude [ d B ] j S ww ( ! ) j −150−100 −50 0 50 100 150Time [ns]−0.20.00.20.40.60.8 A m p li t ude [ a . u .] R wy ( t ) A m p li t ude [ d B ] j S wy ( ! ) j −150−100 −50 0 50 100 150Time [ns]−0.050.000.050.100.150.200.25 A m p li t ude [ a . u .] f ( t ) A m p li t ude [ d B ] j F ( ! ) j FIG. S14. Experimental input autocorrelation, input-output cross-correlation, and response function ( f in → ∗ f x → ∗ f → out )( t )in the time domain (top panels) and in the frequency domain (bottom panels). −50 −25 0 25 50 75 100 125 150Time [ns]−0.2−0.10.00.10.20.30.40.50.6 A m p li t ude [ a . u .] (i)(ii)(iii)(iv)(v)(vi)(vii) FIG. S15. Estimated response functions, used for the cancellation in Fig. S16. (i) ( f in → ∗ f x → ∗ f → out )( t ). (ii) ( f in → ∗ f x → ∗ f → out )( t ). (iii) ( f in → ∗ f x → ∗ f → out )( t ). (iv) ( f in → ∗ f p → ∗ f → out )( t ). (v) ( f in → ∗ f p → ∗ f → out )( t ). (vi) ( f in → ∗ f p → ∗ f → out )( t ). (vii) ( f in → ∗ f → out )( t ). We want to apply the theory in Sec. SII A to this QND system. For the estimation of the response functions, therandom signals are used. As an example, we consider the case where a random signal α ( t ) is added to the vacuumfluctuation ˆ x (0)1 ( t ) as ˆ x in1 ( t ) = ˆ x (0)1 ( t ) + α ( t ) , (S13)and the other three quadratures are kept to vacuum levels. In this case, in theory, by examining the transfer of therandom signal α ( t ) to the two output quadratures x out1 ( t ) and x out2 ( t ), response functions f x → ( t ) and f x → ( t ) areobtained, respectively. Note that the vacuum fluctuations, though they are white and random, cannot be used for theestimation of the response functions. As discussed in Sec. SII A, the important thing is that we know the input signalin order to obtain the cross correlation. In reality, we cannot obtain the response functions with the procedures inSec. SII A. The actual response functions obtained experimentally are ( f in → k ∗ f x,pk → l ∗ f l → out )( t ), where f in → k ( t ) is aresponse function of a conversion from an electronic signal to an optical signal, f l → out ( t ) is a response function of aconversion from an optical signal to an electronic signal, and ∗ denotes a convolution.As an example, we show the autocorrelation, the cross-correlation, and the obtained response function ( f in → ∗ f x → ∗ f → out )( t ) in Fig. S14. All the other experimentally estimated response functions from the input electronic3signals to output the electronic signals are shown as traces (i)–(vi) in Fig. S15. All the response functions have thesame shape. Note that, although we use LPFs and HPFs for output homodyne signals, the same filters are appliedbefore the storage of the input signal as shown in Fig. 1(b) in the main text and thus the effect of the filters arecanceled in the response functions. These response functions improve the cancellation in Fig. 3 in the main text.Figure S16 shows all the power spectra of cancellation with and without the response functions when the randomsignal is added to one of the four input quadratures ˆ x in1 , ˆ x in2 , ˆ p in1 , and ˆ p in2 . Black, red, magenta, blue, and green tracesare the spectra for the shot noises as references, the QND outputs with vacuum inputs, those with the random signalinput, the cancellation with the response functions, and the cancellation without them, respectively. The signals areperfectly cancelled when the response functions are used, which means that the evolution of the signals through theQND gate is completely predictable.However, we note that over-150-MHz components of the response functions do not actually represent the responseof the QND gate but are determined by other reasons. For the frequencies higher than 150 MHz, the homodynesignals are highly attenuated by the LPF and thus electronic noises are dominant. While these electronic noises havea negligible cross-correlation between channels, they contribute to the autocorrelation. Even though we subtractedthe background electronic noises obtained without the optical LOs, there were still some residual noises, by which thedenominator becomes much larger than the numerator in Eq. (S10) over 150 MHz. As a result, the response functionslook as if they have a limited bandwidth of less than 150 MHz. The dull shape of the response functions shown inFig. S15 are because of these situations.In order to estimate the response function of the QND gate itself f x,pk → l ( t ), we conducted the following experiment.As references, we estimated the response functions ( f in → ∗ f → out )( t ) for conversion of electric signals to opticalsignals and vice versa without the QND gate, and obtained a trace (vii) in Fig. S15. Here, we assumed negligibledifferences among peripheral response functions ( f in → k ∗ f l → out )( t ). The trace (vii) has the same shape as those ofthe traces (i)-(vi) with a time difference of 11 ns. This time difference simply represents the difference of the positionsof the homodyne detectors, and does not directly represent the QND gate latency of about 13 ns corresponding tothe optical path length of about 3.8 m. The response functions of the QND gate are obtained by the deconvolutionof the traces (i)-(vi) by the trace (vii), and the results in the frequency domain are shown in Fig. S17. The obtainedspectra are flat for up to 100 MHz, and thus we conclude that the response functions of the QND gate are like a deltafunction in the considered time scale. Inner products of all the traces (i)–(vii) in Fig S15 with the time shift of 11 nsare summarized in Tab. S1. SIII. TRANSFER COEFFICIENTS AND CONDITIONAL VARIANCES
As discussed in Sec. SII B, the response of the QND gate is like a delta function in the considered time scale.Therefore, we can apply the conventional QND criteria [32] to the filtered quadrature values at each time, withoutconsidering a complicated mixing of quadratures at different times. Here, we summarize the QND criteria, especially,the connections between the QND quantities and the signal-to-noise ratios (SNRs).General linear conversions of a signal observable ˆ A S and a probe observable ˆ A P by a nonideal QND gate areˆ A outS = G S , S ˆ A inS + G S , P ˆ A inP + G S , NC ˆ N COM + ˆ N S , (S14a)ˆ A outP = G P , S ˆ A inS + G P , P ˆ A inP + G P , NC ˆ N COM + ˆ N P , (S14b)where ˆ N COM is a correlated component, and ˆ N S and ˆ N P are uncorrelated components, of excess noises of the gate. TABLE S1. Inner products between all the response functions (i)-(vii) in Fig. S15.Response function (i) (ii) (iii) (iv) (v) (vi) (vii)(i) 1 0.989 0.991 0.990 0.994 0.990 0.976(ii) - 1 0.989 0.990 0.988 0.995 0.978(iii) - - 1 0.985 0.986 0.990 0.972(iv) - - - 1 0.992 0.991 0.982(v) - - - - 1 0.989 0.982(vi) - - - - - 1 0.981(vii) - - - - - - 1 −10−5 N o i s e po w e r [ d B ] ^ x in1 −10−5 ^ x out1 −10−5 ^ x out2 −10−5 ^ p out1 −10−5 ^ p out2 −10−5 N o i s e po w e r [ d B ] ^ x in2 −10−5 ^ x out1 −10−5 ^ x out2 −10−5 ^ p out1 −10−5 ^ p out2 −10−5 N o i s e po w e r [ d B ] ^ p in1 −10−5 ^ x out1 −10−5 ^ x out2 −10−5 ^ p out1 −10−5 ^ p out2 −10−5 N o i s e po w e r [ d B ] ^ p in2 −10−5 ^ x out1 −10−5 ^ x out2 −10−5 ^ p out1 −10−5 ^ p out2 FIG. S16. Power spectra of all the cases where a random signal is added to one of the input quadratures ˆ x in1 , ˆ x in2 , ˆ p in1 and ˆ p in2 .Black: shot noises. Red: the QND outputs with vacuum-state inputs. Cyan: optical random signal at the input. Magenta: theQND outputs with the random signal input. Green: cancellation of the random signal without the response functions. Blue:cancellation of the random signal with the response functions. A m p li t ude [ d B ] j F x ! ( ! ) j A m p li t ude [ d B ] j F x ! ( ! ) j A m p li t ude [ d B ] j F x ! ( ! ) j A m p li t ude [ d B ] j F p ! ( ! ) j A m p li t ude [ d B ] j F p ! ( ! ) j A m p li t ude [ d B ] j F p ! ( ! ) j FIG. S17. Response functions of the QND gate in the frequency domain. Blue: response function | F in → ( ω ) F → out ( ω ) | . Green:response functions | F in → ( ω ) F x,pk → l ( ω ) F → out ( ω ) | . Red: response functions | F x,pk → l ( ω ) | . The success criteria of the QND measurements are [32],1 < T S + T P , V S | P < . (S15)The transfer coefficients T S , T P and the conditional variance V S | P are defined as T S = C A inS ˆ A outS = | (cid:104) ˆ A inS ˆ A outS (cid:105) − (cid:104) ˆ A inS (cid:105) (cid:104) ˆ A outS (cid:105) | V ˆ A inS V ˆ A outS , (S16a) T P = C A inS ˆ A outP = | (cid:104) ˆ A inS ˆ A outP (cid:105) − (cid:104) ˆ A inS (cid:105) (cid:104) ˆ A outP (cid:105) | V ˆ A inS V ˆ A outP , (S16b) V S | P = V ˆ A outS (1 − C A outS ˆ A outP )= V ˆ A outS (cid:32) − V ˆ A outS ˆ A outP V ˆ A outS V ˆ A outP (cid:33) = V ˆ A outS (cid:32) − | (cid:104) ˆ A outS ˆ A outP (cid:105) − (cid:104) ˆ A outS (cid:105) (cid:104) ˆ A outP (cid:105) | V ˆ A outS V ˆ A outP (cid:33) , (S16c)where V ˆ X ˆ Y , V ˆ X , and C ˆ X ˆ Y are a covariance, a variance, and a correlation, respectively, V ˆ X ˆ Y = (cid:104) ˆ X ˆ Y (cid:105) − (cid:104) ˆ X (cid:105) (cid:104) ˆ Y (cid:105) , (S17a) V ˆ X = V ˆ X ˆ X , (S17b) C ˆ X ˆ Y = V ˆ X ˆ Y (cid:112) V ˆ X V ˆ Y , (S17c)and the signal input state is assumed to be a coherent state, V ˆ A inS = 1, i.e., the latter part of Eq. (S15) means that thesignal observable is squeezed by the QND measurement. Note that the transfer coefficients and the conditional varianceare T S = 1, T P = G/ (1 + G ), and V S | P = 1 / (1 + G ), for the ideal QND interaction, ˆ A outS = ˆ A inS , ˆ A outP = G ˆ A inS + ˆ A inP ,with a coherent-state probe input V ˆ A inP = 1. The excess noises of the gate decrease the transfer coefficients and6increase the conditional variance. With the general linear conversions in Eq. (S14), the transfer coefficients are T S = G , S V ˆ A inS G , S V ˆ A inS + G , P V ˆ A inP + G , NC V ˆ N COM + V ˆ N S , (S18a) T P = G , S V ˆ A inS G , S V ˆ A inS + G , P V ˆ A inP + G , NC V ˆ N COM + V ˆ N P , (S18b)and the conditional variance is discussed later.The transfer coefficients T S and T P are experimentally obtained by examining the transfer of the SNRs. For thispurpose, we add a signal α to the signal input ˆ A inS = δ ˆ A inS + α , where δ ˆ A inS is a vacuum noise fluctuation (cid:104) δ ˆ A inS (cid:105) = 0, V δ ˆ A inS = 1, and the power is compared with that of the case without the input signal ˆ A inS = δ ˆ A inS . The SNR at thesignal input is SNR inS = α V ˆ A inS = (cid:104) ( δ ˆ A inS + α ) (cid:105) − (cid:104) ( δ ˆ A inS ) (cid:105)(cid:104) ( δ ˆ A inS ) (cid:105) , (S19)and thus obtained experimentally from the powers of the two cases (cid:104) ( δ ˆ A inS + α ) (cid:105) and (cid:104) ( δ ˆ A inS ) (cid:105) . On the other hand,the output signal and probe observables become ˆ A outS = δ ˆ A outS + G S,S α and ˆ A outP = δ ˆ A outP + G P,S α , where δ ˆ A outS and δ ˆ A outP are noise fluctuations without the input signal α . We assume (cid:104) δ ˆ A outS (cid:105) = (cid:104) δ ˆ A outP (cid:105) = 0 without loss of generality.The SNRs at the signal and probe outputs are,SNR outS = G α V ˆ A outS = (cid:104) ( δ ˆ A outS + G S,S α ) (cid:105) − (cid:104) ( δ ˆ A outS ) (cid:105)(cid:104) ( δ ˆ A outS ) (cid:105) = G , S α G , S V ˆ A inS + G , P V ˆ A inP + G , NC V ˆ N COM + V ˆ N S , (S20a)SNR outP = G α V ˆ A outP = (cid:104) ( δ ˆ A outP + G P,S α ) (cid:105) − (cid:104) ( δ ˆ A outP ) (cid:105)(cid:104) ( δ ˆ A outP ) (cid:105) = G , S α G , S V ˆ A inS + G , P V ˆ A inP + G , NC V ˆ N COM + V ˆ N S . (S20b)and thus obtained experimentally from the powers of the two cases (cid:104) ( δ ˆ A outS + G S,S α ) (cid:105) , (cid:104) ( δ ˆ A outP + G P,S α ) (cid:105) , and (cid:104) ( δ ˆ A outS ) (cid:105) , (cid:104) ( δ ˆ A outP ) (cid:105) . By using Eqs. (S18)–(S20), we obtain, T S = SNR outS SNR inS , T P = SNR outP SNR inS . (S21)Therefore, T S and T P represent the degradation of the SNR when the signal input α is transferred to the signal andprobe outputs, respectively.The conditional variance V S | P corresponds to the minimum variance of ˆ A outS − g ˆ A outP where the subtraction gain g is an optimization parameter. The variance of ˆ A outS − g ˆ A outP is a quadratic polynomial in g , V ˆ A outS − g ˆ A outP = (cid:104) ( δ ˆ A outS − gδ ˆ A outP ) (cid:105) = V ˆ A outP g − V ˆ A outS ˆ A outP g + V ˆ A outS = V ˆ A outP (cid:32) g − V ˆ A outS ˆ A outP V ˆ A outP (cid:33) + V S | P , (S22)which is minimized at g = V ˆ A outS ˆ A outP /V ˆ A outP .The experimental values are summarized in Tab. S2. The variances (cid:104) (ˆ x out1 − g x ˆ x out2 ) (cid:105) and (cid:104) ( g p ˆ p out1 + ˆ p out2 ) (cid:105) forvarious subtraction and addition gains are plotted in Fig. S18.7 TABLE S2. Verification of transfer coefficients in the QND gate. The coherent state amplitude is injected either to x in1 or p in2 .Coherent state input x in1 p in2 T S ± ± T P ± ± T S + T P ± ± V S | P ± ± g x P o w e r g p P o w e r (a) (b) FIG. S18. Variances of (a) ˆ x out1 − g x ˆ x out2 and (b) g p ˆ p out1 + ˆ p out2 . Red markers: experimental results when ancillary squeezedvacua are used. Cyan markers: experimental results when ancillary squeezed vacua are not used. Magenta curves: theoreticalvariance when ancillary squeezed vacua with − . G = 1. Gray lines: entangledcriterion. SIV. QUANTUM ENTANGLEMENT
The sub-shot-noise conditional variances V S | P < x and ˆ p quadratures are not a sufficient conditionfor entanglement. A sufficient condition based on the Duan-Simon criterion is [16, 34, 35], ∃ g, (cid:104) (ˆ x out1 − g ˆ x out2 ) (cid:105) + (cid:104) ( g ˆ p out1 + g ˆ p out2 ) (cid:105) < | g | . (S23)In Fig. S18, there are gray lines (cid:104) (ˆ x out1 − g ˆ x out2 ) (cid:105) = 2 | g | and (cid:104) ( g ˆ p out1 + ˆ p out2 ) (cid:105) = 2 | g | , and there is a region of gg