Experimental test of error-disturbance uncertainty relation with continuous variables

TL;DR

This paper experimentally tests the error-disturbance uncertainty relation for continuous variables in quantum mechanics, verifying the violation of Heisenberg's relation while confirming Ozawa's and Branciard's formulations using optical Gaussian states.

Contribution

First experimental verification of error-disturbance uncertainty relations for continuous variables in quantum optics, demonstrating violations of Heisenberg's relation and validation of alternative formulations.

Findings

Heisenberg's EDR is violated in the experiment.

Ozawa's and Branciard's EDR are validated for continuous variables.

Experimental setup uses heterodyne measurement on optical Gaussian states.

Abstract

Uncertainty relation is one of the fundamental principle in quantum mechanics and plays an important role in quantum information science. We experimentally test the error-disturbance uncertainty relation (EDR) with continuous variables for Gaussian states. Two conjugate continuous-variable observables, amplitude and phase quadratures of an optical mode, are measured simultaneously by using a heterodyne measurement system. The EDR with continuous variables for a coherent state, a squeezed state and a thermal state are verified experimentally. Our experimental results demonstrate that Heisenberg's EDR with continuous variables is violated, yet Ozawa's and Branciard's EDR with continuous variables are validated.

Equations

$$\epsilon (A)\eta (B)\ge C_{AB},$$

$$\epsilon (A)\eta (B)+\epsilon (A)\sigma (B)+\sigma (A)\eta (B)⩾C_{AB}.$$

$$\displaystyle\big{[}\varepsilon^{2}(A)\sigma^{2}(B)+\sigma^{2}(A)\eta^{2}(B)$$

$$\displaystyle+2\varepsilon(A)\eta(B)\sqrt{\sigma^{2}(A)\sigma^{2}(B)-C_{AB}^{2}}\big{]}^{1/2}\geqslant C_{AB},$$

$${\epsilon }^2(A)=⟨(C-A{)}^2⟩$$

$${\eta }^2(B)=⟨(D-B{)}^2⟩$$

Full text

I Introduction

As one of the cornerstones of quantum mechanics, uncertainty relation describes the measurement limitation on two incompatible observables. Uncertainty relation has a huge impact on quantum information technology, such as entanglement verification Buscemi , quantum key distribution Furrer , quantum dense coding Bennett and security of quantum cryptography Gisin . Heisenberg’s original uncertainty relation is related to measurement effect, which states that we cannot acquire perfect knowledge of a state without disturbing it Heisenberg .

There are two kinds of uncertainty relations, which are the preparation uncertainty relation and the measurement uncertainty relation, depending on whether you are talking about average measurement or one-shot measurement in the understanding of Heisenberg’s spirit. The preparation uncertainty which studied the minimal dispersion of two quantum observables before measurement Kennard ; Weyl ; Rob29 . The Robertson uncertainty relation Rob29 , reads as $\sigma(x)\sigma(p)\geq\hbar/2$, is a typical example in this sense, where $\sigma(x)$ and $\sigma(p)$ are the standard deviations of position and momentum. For such uncertainty relation, the measurements of $x$ and $p$ are performed on an ensemble of identically prepared quantum systems. The measurement uncertainty relation thinks the Heisenberg’s uncertainty principle should be based on the observer’s effect, which means that measurements of certain system cannot be made without affecting the system. This kind of uncertainty relation which studies to what extent the accuracy of a position measurement is related to the disturbance of the particle’s momentum, so is also called the error-disturbance relation (EDR) Ozawa03 ; Hall04 .

The Heisenberg’s EDR is generally expressed as

[TABLE]

where $C_{AB}=\left|\left\langle[A,B]\right\rangle\right|/2$, $[A,B]=AB-BA$, $\varepsilon(A)=\langle(C-A)^{2}\rangle^{1/2}$ and$\ \eta(B)=\langle(D-B)^{2}\rangle^{1/2}$ represent the root-mean-squared (RMS) difference between the initial value of $A$ and $B$ and the outcome value of a measurement of $C$ and $D$, respectively. However, it has been shown that Heisenberg’s EDR may be violated in some cases Balllentine . After that heated debates on EDR have taken place and new formulated of EDRs have been put forward Ozawa03 ; Ozawa04 ; Branciard ; Hall04 ; Werner1 ; Werner2 ; PhysRevA022106 ; PhysRevA032 ; PhysRevLett050401 ; lu ; Barchielli2017 ; Barchielli2018 . Ozawa proposed the EDR as

[TABLE]

After that Branciard improved the Ozawa’s EDR as Branciard

[TABLE]

which is tighter than Ozawa’s EDR. The experimental tests of the uncertainty relations have been demonstrated in photonic EXPphotons1 ; EXPphotons2 ; EXPphotons3 ; EXPphotons4 ; EXPphotons5 ; EXPphotons6 , spin-1/2 EXPpolarizedneutrons1 ; EXPpolarizedneutrons2 ; EXPphotons7 ; EXPphotons8 , nuclear spin EXPW1 , and ion trap EXPW2 ; EXPW3 systems. All of these experiments are in discrete-variable systems. Until recently, the test of the error-tradeoff uncertainty relation with continuous variables is experimentally demonstrated by using an Einstein-Podolsky-Rosen (EPR) entangled state Yang .

In this paper, we report the experimental test of EDR with continuous variables by using a heterodyne measurement system. In our experiment, we test the EDR for three different Gaussian states, which are coherent state, squeezed state and thermal state, respectively. A vacuum mode is used as meter mode in the measurement system. Our experimental results demonstrate that Heisenberg’s EDR with continuous variables is violated, yet Ozawa’s and Branciard’s EDR with continuous variables are validated.

II The principle and experimental setup

The amplitude and phase quadratures of an optical mode are incompatible continuous-variable observables and cannot be measured simultaneously. A heterodyne measurement system, which is a joint measurement apparatus, can be used to measure the approximation of $A$ and $B$ with the compatible observables $C$ and $D$ as shown in Fig. 1(a). The signal mode $\rho$ with incompatible observables $A=\hat{x}_{\rho}$ and $B=\hat{p}_{\rho}$ is coupled with a meter mode via a beam-splitter (BS), where $\hat{x}=\hat{a}+\hat{a}^{{\dagger}}$ and $\hat{p}=(\hat{a}-\hat{a}^{{\dagger}})/i$ denote the amplitude and phase quadrature of an optical mode, respectively. The signal mode are prepared as coherent state, squeezed state and thermal state, respectively, and a vacuum state $\nu$ is used as the meter mode in our experiment. The amplitude quadrature $C=\hat{x}_{c}=\sqrt{T}\hat{x}_{\rho}-\sqrt{R}\hat{x}_{\nu}$ and phase quadrature $D=\hat{p}_{d}=\sqrt{R}\hat{p}_{\rho}+\sqrt{T}\hat{p}_{\nu}$ of two output modes $c$ and $d$ of BS are measured by two homodyne detectors simultaneously, which are used to approximate $A$ and $B$, respectively, where $T$ is the transmission efficiency of the BS, and $R=1-T$. The root-mean-square error and disturbance are expressed as

[TABLE]

[TABLE]

The experimental setup for test of EDR is illustrated in Fig. 1(b). A laser generates both 1080 nm and 540 nm optical fields simultaneously. The 1080 nm optical field is used as the injected signal of a nondegenerate optical parametric amplifier (NOPA) and the local oscillator fields of homodyne detectors. The 540 nm optical field serves as the pump field of the NOPA. A half-waveplate (HWP) and a polarization beam-splitter (PBS), which are placed after the NOPA, are used to obtain different signal modes. The measurement apparatus is composed by a BS and two homodyne detectors. The AC output signals from HD1 and HD2 are mixed with a local reference signal of 3 MHz, and then filtered by low-pass filter with a bandwidth of 30 kHz and amplified 1000 times (Low noise preamplifier, SRS, SR560), respectively. And then the two signals from the outputs of the preamplifiers are recorded by a digital storage oscilloscope simultaneously. A sample size of 5$\times 10^{5\text{ }}$data points is used for all quadrature measurements. The interference efficiency between signal and local oscillator fields is 99% and the quantum efficiency of photodiodes are 99.6%.

III Results

A coherent state is prepared when the pump field of NOPA is blocked and only the injected field passes through the NOPA. The variances of amplitude and phase quadratures of the coherent state and the vacuum state (meter mode) are $\sigma^{2}(\hat{x}_{\rho})=\sigma^{2}(\hat{p}_{\rho})=1$, and $\sigma^{2}(\hat{x}_{\nu})=\sigma^{2}(\hat{p}_{\nu})=1$, respectively. When the NOPA is operated at the parametric deamplification situation and the half-waveplate after the NOPA is set to $22.5^{\circ}$, an x-squeezed and a p-squeezed states are prepared. The x-squeezed state is used as the signal mode in the test of EDR for squeezed state. The variances of the amplitude and phase quadratures of the x-squeezed state are $\sigma^{2}(\hat{x}_{\rho})=e^{-2r},\sigma^{2}(\hat{p}_{\rho})=e^{2r}$, respectively, where $r$ is the squeezing parameter EPRSU . In the experiment, the squeezed state with -2.9 dB squeezing and 3.9 dB antisqueezing is generated by NOPA. When the half-waveplate after the NOPA is set to $0^{\circ}$, the Einstein-Podolsky-Rosen entangled state is generated. Each mode of the entangled state is a thermal state, and one of them is used for the test of EDR for thermal state. The variances of the amplitude and phase quadratures of the thermal state are $\sigma^{2}(\hat{x}_{\rho})=\sigma^{2}(\hat{p}_{\rho})=(e^{-2r}+e^{2r})/2$.

The dependence of error of the amplitude quadrature $\varepsilon(A)$ and disturbance of the phase quadrature $\eta(B)$ on the transmission efficiency of BS for three different Gaussian signal modes are shown in Fig. 2(a), 2(b) and 2(c), respectively. The error $\varepsilon(A)$ decreases with the increasing of the transmission efficiency of the BS, while the disturbance $\eta(B)$ increases with the increasing of the transmission efficiency for all of the three Gaussian states. When the error reaches minimum value, the maximum disturbance is caused. The reduction of the disturbance can be realized by introducing error on the other observable. When a x-squeezed state serves as signal mode, the maximum error is less than the case that coherent state serves as signal field with the cost of the greater maximum disturbance for the anti-squeezing of the phase quadrature [Fig. 2 (b)]. When the signal mode is a thermal state, both the error and disturbance of the state are larger than that of coherent state at the same transmission efficiency of BS, as shown in Fig. 2(a) and 2(c).

The dependence of the left hand side of Ozawa’s (red curve), Branciard’s (blue curve) and Heisenberg’s (green curve) EDRs with continuous variables on the transmission efficiency of BS for three Gaussian states are shown in Fig. 2(d), 2(e) and 2(f), respectively. It is clear that the Ozawa’s and Branciard’s EDR with continuous variables are valid while the Heisenberg’s EDR with continuous variable is violated. Comparing the blue curve and red curve, we can see that the Branciard’s EDR is tighter than Ozawa’s EDR with continuous variables. When the transmission efficiency is 50%, the left hand side of Branciard’s EDR with continuous variables reaches its minimum value in case of coherent state and thermal state. In the case of x-squeezed state serves as signal mode, the Branciard’s inequality is minimized when the transmission efficiency is about 95% for the unsymmetrical of the variances of amplitude and phase quadratures of the squeezed state.

The comparison of the lower bounds of EDRs for three Gaussian states in the error-disturbance plot are shown in Fig. 3. The results for coherent state, squeezed state and thermal state are shown in Fig. 3(a), 3(b), and 3(c), respectively. All the experimental results demonstrate that Heisenberg’s EDR is violated, yet Ozawa’s and Branciard’s EDR with continuous variables are valid.

IV Conclusion

We experimentally test the Heisenberg’s, Ozawa’s and Branciard’s EDRs with continuous variables by using a heterodyne measurement system. Three different Gaussian states, i.e., coherent state, squeezed state and thermal state are used as signal mode to test the EDRs. All the experimental results demonstrate that Heisenberg’s EDR is violated, yet Ozawa’s and Branciard’s EDR are validated. Our work represents an important advance in understanding fundamentals of physical measurement and sheds light on the developing of quantum information technology.

ACKNOWLEDGMENTS

This research was supported by the NSFC (Grant No. 11834010), the program of Youth Sanjin Scholar, the National Key R&D Program of China (Grant No. 2016YFA0301402), and the Fund for Shanxi ”1331 Project” Key Subjects Construction.