## Introduction

Quantum sensing and measurement-based quantum computing utilise quantum correlated states1,2. The most mature technology for these applications is based on quantum states of the electro-magnetic field. Laser interferometers that are used as telescopes for gravitational-wave astronomy achieve unprecedented sensitivities based on states having a squeezed photon counting statistic3,4,5,6. It was proposed to further improve them by bipartite Gaussian entangled states7,8,9,10. Optical measurement-based quantum computing based on multi-partite entangled cluster states11,12 was pushed forward recently13.

Measurements in the regime of Gaussian quantum statistics concern two non-commuting observables. In terms of dimensionless operators that are normalised to the variance of the ground state, they are often named $$\hat{X}$$ and $$\hat{Y}$$. They need to be measured both in order to determine the full energy of a quantised harmonic oscillator, similar to position and momentum. The Heisenberg uncertainty relation14,15,16,17 is a useful reference to distinguish between semi-classical measurements18 and those that exploit entanglement19. Experimentally achievable Gaussian entanglement has been characterised by co-variances derived from ensemble measurements in stationary settings8,20,21,22. So far, Gaussian entanglement was not used to improve measurements of phase space displacements that changed after a single measurement window. In such a time-dependent setting, averaging would not constitute a suitable approach for improving the signal-to-noise ratio.

Here, we present the monitoring of a dynamical phase space trajectory α(t) through the simultaneous measurements of two non-commuting observables (X(ti); Y(ti)) at subsequent times ti with an imprecision much lower than the reference limit as given by the Heisenberg uncertainty relation. The non-classical improvement provides the same benefit as ten times averaging, which, however, is possible only in a stationary setting. The experimental achievements presented here are entirely based on individually sampled two-dimensional data points.

## Results

### Uncertainty relations

Our experiment uses quantities that are also used in optical communication and optical quantum computing13, namely phase and amplitude modulation depths carried by quasi-monochromatic laser light of optical frequency ν, see the figure in Supplementary Information. The depth of the amplitude modulation (amplitude modulation index) in the frequency band f ± Δf, with νf > Δf, is quantified by the dimensionless operator $${\hat{X}}_{f,{{\Delta }}f}$$23. This operator is also known as ‘amplitude quadrature amplitude’. The corresponding depth of phase modulations is (in the limit of weak phase modulations) quantified by the operator $${\hat{Y}}_{f,{{\Delta }}f}$$. This operator is also known as ‘phase quadrature amplitude’ 24. $${\hat{X}}_{f,{{\Delta }}f}$$ and $${\hat{Y}}_{f,{{\Delta }}f}$$ do not commute. In the following, we skip the indices and normalise the commutator to $$[\hat{X},\hat{Y}]=2{{{{{\rm{i}}}}}}$$, which results in the Heisenberg uncertainty relation

$${{\Delta }}\hat{X}{{\Delta }}\hat{Y}\ge 1,$$
(1)

where Δ denotes the standard deviation of the measured eigenvalues of the respective operator. $$\hat{X}$$ and $$\hat{Y}$$ span a phase space, in which the uncertainty area is bounded from below accordingly. The lower bound in Eq. (1) refers to ‘ideal’ measurements performed with semi-classical devices, which do not use quantum correlations. ‘Ideal’ means that $$\hat{X}$$ is measured on a first copy and $$\hat{Y}$$ is measured on a second copy of the system. Ideal measurements are only possible in (quasi-)stationary settings. If the quantities $$\hat{X}$$ and $$\hat{Y}$$ change on time scales that are not much longer than 1/Δf, they need to be measured simultaneously. In this case, splitting the system into two independently measured subsystems is required (beam splitter in Fig. 1, and BS3 in Fig. 2). Furthermore, averaging is not possible unless the trajectory repeats after some time. The splitting reduces the signal-to-noise ratio in comparison with ideal measurements. The splitting can be described as opening a new port through which vacuum uncertainty couples to the measurement (port B in Fig. 1, left), if not an entangled reference system is superimposed via this port, as shown in in Figs. 1b and 2. In the absence of quantum correlations, simultaneous measurements at times ti need to cope with at least doubled minimal quantum uncertainty, which increases standard deviations by at least the factor $$\sqrt{2}$$18, yielding

$${{\Delta }}\left(\hat{X}({t}_{i})\right){{\Delta }}\left(\hat{Y}({t}_{i})\right)\ge 2.$$
(2)

The above inequality represents the fundamental precision limit when two conjugate observables are measured simultaneously on a single system with respect to reference values of a semi-classical measurement device. Note that inequality (2) relates to a Gaussian state in the Husimi Q representation25, whereas inequality (1) relates to the Wigner representation26.

### Measurements with respect to quantum entangled references

Let $$[\hat{X},\hat{Y}]\,=\,2{{{{{\rm{i}}}}}}$$ describe a quantum system of interest and $$[{\hat{X}}_{0},{\hat{Y}}_{0}]\,=\,2{{{{{\rm{i}}}}}}$$ another quantum system. A short calculation leads to the zero-commutator $$[\hat{X}\,\pm \,{\hat{X}}_{0},\hat{Y}\,\mp \,{\hat{Y}}_{0}]\,=\,0$$, which results in

$${{\Delta }}(\hat{X}\pm {\hat{X}}_{0})\ {{\Delta }}(\hat{Y}\mp {\hat{Y}}_{0})\ge 0.$$
(3)

This inequality describes the fact that X and Y of a system can be measured simultaneously with arbitrary precision with respect to the corresponding quantities X0 and Y0 of a reference system. Usually, a reference system has its own quantum uncertainty. If, however, an ensemble of quantum systems is available that are all entangled with a reference system at hand, the measurement of a phase space trajectory $$(\langle \hat{X}\rangle ;\,\langle \hat{Y}\rangle )(t)$$ with a sub-Heisenberg imprecision is possible.

### Experimental setup

Figure 2 shows the schematics of our experiment. A commercial erbium-doped fibre laser generated 1 W of quasi-monochromatic light at the wavelength of 1550 nm. About half of the light was frequency doubled to provide the pump light for two squeezed-light resonators. The latter used resonator-enhanced degenerate type 0 optical-parametric amplification in periodically poled potassium titanyl phosphate. The two output fields carried modulation spectra around 5 MHz in squeezed vacuum states and were overlapped at balanced beam splitter BS1. The results were two fields whose modulations were strongly EPR entangled, which was characterised in a precursor experiment20. Here, we recombined the entangled beams on a second balanced splitter (BS3). The optical path length difference was controlled to convert them back to two squeezed beams. Due to necessarily imperfect interference contrasts at the two beam splitters, the final squeeze factors could only be lower than the initial squeeze factors of the input modes (subscripts 1 and 2). The BHDs used optical local oscillators (LOs) of about 10 mW from the joint fibre laser. The phase differences between the LOs and the squeezed fields were stably controlled to 0° and 90°, respectively. BHD1 at 0° sampled values of a squeezed amplitude quadrature amplitude and BHD2 at 90° sampled values of a squeezed phase quadrature amplitude, both with a sampling frequency of 200 MHz. To avoid aliasing, we applied an analogue lowpass filter with a corner frequency of 50 MHz to each channel. Post processing was done with a self-written Python script, which was used to digitally demodulate the data at f = 5 MHz and subsequent finite impulse response-lowpass filtering with a cut-off frequency of Δf/2 = 10 kHz. Figure 3 represents the entanglement quality of our setup in terms of variances.

The time-dependent displacement α(t) in our setup corresponded to the time-dependent modulation at f = 5 MHz of a coherent carrier field that was transmitted through the high reflectivity mirror BS2 (R = 99.99%). The high reflectivity minimised decoherence, i.e. optical loss to the entanglement. Changing the peak voltage to the electro-optical modulator (EOM, as shown in Fig. 2), changed the absolute value of the displacement α. Changing the DC voltage to the piezo-actuated phase-shifter (U1) changed the differential excitation in $$\langle \hat{X}\rangle$$ and $$\langle \hat{Y}\rangle$$. The time series produced at BHD1 and BHD2 represented simultaneous measurements of the system’s conjugate displacement components with respect to the corresponding values of the (entangled) reference. Since $$\langle {\hat{X}}_{0}\rangle \,=\,\langle {\hat{Y}}_{0}\rangle \,=\,0$$, the data serve for monitoring the trajectory $$(\langle \hat{X}\rangle ;\,\langle \hat{Y}\rangle )(t)$$.

### Experimental results

Figure 4 shows two phase space trajectories $$(\langle \hat{X}\rangle ;\,\langle \hat{Y}\rangle )(t)$$ (solid lines) measured with precision suspending inequalities (1) and (2). Added are individual data points from simultaneous measurements of (X − X0)(ti) and (Y + Y0)(ti), when the interrogated system was entangled with the reference system. Also shown are individual data points from simultaneous measurements of (X − X0)(ti) and (Y + Y0)(ti), when the entanglement source was switched off and the modulations $$\langle \hat{X}\rangle$$ and $$\langle \hat{Y}\rangle$$ set to 0. These data points accumulated around the phase space origin and were used to derive the factor by which the inequalities (1) and (2) were surpassed. The standard deviations in (X − X0)(ti) and (Y + Y0)(ti) around the actual phase space trajectories $$(\langle \hat{X}\rangle ;\,\langle \hat{Y}\rangle )(t)$$ (solid line) were reduced by more than $$\sqrt{10}$$. This factor is highlighted by the different radii of the small circles. The phase space trajectories were thus tracked with an uncertainty product that violated inequality (2) by slightly more than a factor of 10. As expected, the factor by which Heisenberg’s uncertainty limit was surpassed directly corresponded to the strength of the entanglement. Increasing the entanglement strength requires further reduction of optical loss, including further increase of photo-electric detection efficiency27.

Figure 4a represents a constant modulation depth, while the kind of modulation was continuously changed. The system had a pure amplitude modulation when $$\langle \hat{Y}\rangle (t)=0$$ and a pure phase quadrature modulation when $$\langle \hat{X}\rangle (t)=0$$. The amplitude of the AC voltage at the EOM (U2) was constant and just the DC voltage at the piezo actuator (U1) continuously changed. The trajectory started at about $$(\langle \hat{X}\rangle ;\,\langle \hat{Y}\rangle )=(-3.7\sqrt{2};5.8\sqrt{2})$$, completed almost a full cycle, returned and stopped at about $$(-5.3\sqrt{2};-4.3\sqrt{2})$$. The bottom panel shows another example trajectory whose modulation depth also changed, resulting in a phase and amplitude dependent trajectory.

## Discussion

Our experiment demonstrates that any individual measurement of two non-commuting observables at the same time improved when entanglement is exploited. The observables considered are the real and imaginary parts of phase space displacement, which describe the depths of the amplitude modulation and the phase quadrature around a selected radio frequency of a continuous-wave quasi-monochromatic laser beam. The temporal change of the modulations depths corresponds to a trajectory in phase space. Our experiment shows that the trajectory can be monitored by a large number of individual measurement pairs with a tenfold reduced imprecision in each of the observables compared with the variances of measurements on the ground state (without entanglement). The reduction factors correspond to the squeeze factor in both observables (10 dB) realised by the entanglement resource. We conclude that the time-varying displacement can in principle be monitored with arbitrary precision without averaging, i.e. even for random walks. We thus further conclude that the often-quoted interpretation of Heisenberg’s uncertainty relation ‘two non-commuting observables of a quantum system cannot be measured simultaneously with arbitrary precision’ is incorrect. In light of our experiment, the statement becomes correct, if completed by ‘...with respect to a reference system that was or has been coupled to a thermal environment’, since in this case the reference system cannot be quantum correlated.

The phase space displacement in our experiment is overlapped with the entangled fields via a high reflectivity beam splitter. In principle, the displacement can also be produced directly in the beam path of one of the entangled states by a combination of an amplitude and a phase modulator, because the entangled states are carried by accompanying monochromatic fields. An important issue is to keep the entanglement decoupled from the environment before the measurement. The reduction factor in the quantum imprecision achieved is of practical significance and supports the emergent field of quantum sensing. In gravitational-wave observatories, entangled light provides additional sensitivity improvements compared to squeezed light3,4,5,6 by mitigating disturbances from back-scattered light8 and from quantum radiation pressure7,9,10. The setup realised constitutes state of the art quantum optics technology suitable for the generation and detection of Gaussian cluster states for measurement-based quantum computing11,12,13.