Nonunity gain minimal-disturbance measurement

We propose and experimentally demonstrate an optimal nonunity gain Gaussian scheme for partial measurement of an unknown coherent state that causes minimal disturbance of the state. The information gain and the state disturbance are quantified by the noise added to the measurement outcomes and to the output state, respectively. We derive the optimal trade-off relation between the two noises and we show that the tradeoff is saturated by nonunity gain teleportation. Optimal partial measurement is demonstrated experimentally using a linear optics scheme with feedforward.

Figure 1

(Color online) The principles of a minimal-disturbance measurement of coherent states. The input state is drawn from an unknown distribution of coherent states, say ${|\alpha ⟩}_{\text{in}}={|x+ip⟩}_{\text{in}}$, and the task is to acquire information about the state (through a measurement) in such a way that the state is minimally disturbed according to quantum mechanics. This is the essence of a minimal-disturbance measurement. There are two outputs of the protocol: a classical one yielding information about the input state (in the form of two numbers, say ${\overline{x}}_{\text{cl}}$ and ${\overline{p}}_{\text{cl}}$) and a quantum one, namely, the postmeasurement state $\rho$. In this paper we consider only cases where the classical data as well as the disturbed quantum state are inflicted by additive phase-insensitive Gaussian noise. The gain $g$ of the protocol is defined by the ratio between the input and output mean values: $g=\text{Tr}\left(x\rho \right)∕⟨{\alpha }_{\text{in}}|x|{\alpha }_{\text{in}}⟩=\text{Tr}\left(p\rho \right)∕⟨{\alpha }_{\text{in}}|p|{\alpha }_{\text{in}}⟩$.

Figure 2

Optimal tradeoff between the output noise ${\nu }_{\text{out}}$ and the noise in measurement outcomes ${\nu }_{\text{cl}}$ for a single-mode Gaussian operation with optimal gain $g={\nu }_{\text{cl}}∕\sqrt{{\nu }_{\text{cl}}^2-1}$ (solid curve), amplifying operation with $g=2$ (dashed curve), unity gain operation with $g=1$ (dotted curve), and attenuating operation with $g=0.8$ (dash-dotted curve). See the text for details.

Figure 3

(Color online) The experimental scheme. AM: Amplitude modulator; PM: Phase modulator; $|\alpha ⟩$: The incoming coherent state to be measured; HWP: Half wave plate; PBS: Polarizing beam splitter cube; BS: Beam splitter; $D\left(x,p\right)$: Displacement operation; B: Auxiliary beam; BS(R,T): Variable beam splitter with reflectivity $R$ and transmissivity $T$ for intensity; A: Vacuum mode; G: electronic gains; $x_{\text{cl}},p_{\text{cl}}$: Classical measurement outcomes; $\rho$: Output state; LO: Local oscillator beam; ESA: Electronic spectrum analyzer.

Figure 4

Experimental results for MDM with optimized gain. Variances $⟨n_{\text{out},x}^2⟩$ (left figure) and $⟨n_{\text{out},p}^2⟩$ (middle figure) of the added noises in the amplitude and phase quadratures are plotted against the quantity $⟨n_{\text{cl},x}^2⟩$ and $⟨n_{\text{cl},p}^2⟩$ characterizing the noise added into the outcomes of simultaneous measurement of amplitude and phase quadratures. We also make use of Eq. (7) to plot ${\nu }_{\text{out}}$ against ${\nu }_{\text{cl}}$ (right figure). The solid line represents the theoretical relation ${\nu }_{\text{out}}=1∕{\nu }_{\text{cl}}$. The experimental data were obtained by taking into account the detection efficiency of $83\%$ at the homodyne detector. The error bars in the $x$-axis steam from the uncertainty in the measurement of the beam splitter transmission ($2\%$ deviation). The error bars in the $y$ axis are caused by 0.1 dB relative measurement accuracy of the electronic spectrum analyzer and 0.1 dB deviation of the homodyning efficiency.

Figure 5

Experimental results for MDM with fixed gains $g^2=0.5$ (a), $g^2=0.8$ (b), and $g^2=1.3$ (c). Variances $⟨n_{\text{out},x}^2⟩$ (left figure) and $⟨n_{\text{out},p}^2⟩$ (middle figure) of the added noises in the amplitude and phase quadratures are plotted against the $⟨n_{\text{cl},x}^2⟩$ and $⟨n_{\text{cl},p}^2⟩$ characterizing the noise added into the outcomes of simultaneous measurement of amplitude and phase quadratures. We also make use of Eq. (7) to plot ${\nu }_{\text{out}}$ against ${\nu }_{\text{cl}}$ (right figure). The optimality windows (gray shaded regions) are determined by Eqs. (14, 15). The solid line represents the theoretical tradeoff (13). The experimental data were obtained by taking into account the detection efficiency of $83\%$ at the homodyne detector. The error bars in the $x$ axis stem from the uncertainty in the measurement of the beam splitter transmission ($2\%$ deviation). The error bars in the $y$ axis are caused by 0.1 dB relative measurement accuracy of the electronic spectrum analyzer and 0.1 dB deviation of the homodyning efficiency.