Bell inequalities for falsifying mesoscopic local realism via amplification of quantum noise

Macroscopic realism (MR) per se specifies that a system which has two macroscopically distinct states available to it (such as a cat being dead or alive) is at all times predetermined to be in one or other of those two states. A minimal assumption of a macroscopic realistic theory therefore is the validity of a hidden variable ${\lambda }_M$ that predetermines the outcome (whether dead or alive) of a measurement $\widehat{M}$ distinguishing the two states. Proposals to test MR generally introduce a second premise to further qualify the meaning of MR. Thus, we consider a model, macroscopic local realism (MLR), where the second premise is that measurements at one location cannot cause an instantaneous macroscopic change $\delta$ to the results of measurements made on a second system at another location. To provide a practical test, we define the intermediate concept of $\delta$-scopic local realism ($\delta$-LR), where $\delta \ne 0$ can be quantified, but need not be macroscopic. By considering the amplification of quantum fluctuations, we show how negation of $\delta$-LR is possible using fields violating a continuous variable Bell inequality. A modified Bell-Clauser-Horne-Shimony-Holt inequality is derived that tests $\delta$-LR, and a quantitative proposal given for experiments based on polarization entanglement. In the proposal, $\delta$ is the magnitude of the quantum noise scaled by an adjustable coherent amplitude $\alpha$ that can also be considered part of the measurement apparatus. Thus, $\delta$ is large in an absolute sense, but scales inversely with the square root of the system size, which is proportional to ${\left|\alpha \right|}^2$. We discuss how the proposed experiment gives a realization of a type of Schrödinger-cat experiment without problems of decoherence.

Figure 1

Testing $\delta$-scopic LR using a Bell inequality: Here, $P\left(n\right)$ is the probability of obtaining a result $n$ in a hypothetical experiment. $P_0$ is the probability of a result between $-\delta$ and $+\delta$. We consider experiments where the binned outcomes of a measurement ${\widehat{M}}^A$ made on system $A$ are distinct by an amount $2\delta$. Similarly, we consider that the outcomes of a measurement ${\widehat{M}}^B$ made on a second system $B$ (spatially separated from the first) are distinct by $2\delta$. This is described in the diagram with $P_0=0$. An example of such an experiment is illustrated in Fig. 2, which introduces two adjustable coherent fields with amplitude $\alpha$. The outcomes $n$ for measurements $J_{\theta }^A,J_{\varphi }^B$ defined in Fig. 2 are binned into one of the regions 1,0,2. As $\alpha \to \infty$, $P_0\to 0$ (for fixed $\delta )$. In that case, the two outcomes given by $n<-\delta$ and $n>\delta$ are said to be $\delta$-scopically distinct, and are referred to as “dead” and “alive” as $\delta \to \infty$. A method accounting for nonzero $P_0$ is given in Sec. 4.

Figure 2

Two entangled cat systems are created as part of a measurement process: The modes $a_1$ and $b_1$ are prepared in an microscopic entangled state $|\psi \rangle$. The modes $a_2$ and $b_2$ are independent intense coherent states $|\alpha \rangle$. The fields $a_1,a_2$ and $b_1,b_2$ are combined at each location $A$ and $B$ to create mode pairs $a_{\pm }$ and $b_{\pm }$. For $\alpha \to \infty$, these form two cat systems at $A$ and $B$. Final measurements of the number differences $n,m$ given by $J_{\theta }^A=(N_{+}-N_{-})/2$ and $J_{\varphi }^B=(N_{+}-N_{-})/2$ are made using a polarizer beam splitter at each detector. This gives an amplified readout of the quadrature phase amplitudes $x_{\theta }$ and $x_{\varphi }$ of the fields $a_1$ and $b_1$. Whether the cat systems are considered dead or alive is determined by the sign of $J_{\theta }^A$ and $J_{\varphi }^B$. For a fixed $\delta$, we define the outcomes $J_{\theta ,\varphi }^{A/B}$ to be dead if $J_{\theta ,\varphi }^{A/B}<-\delta$ and alive if $J_{\theta ,\varphi }^{A/B}>\delta$. As $\alpha \to \infty$, the probability $P_0$ of a result $-\delta \le J_{\theta ,\varphi }^{A/B}\le \delta$ becomes zero (Fig. 1). This is true for any large $\delta$, provided $\alpha \gg \delta$. Thus, the dead and alive outcomes become macroscopically distinguishable as $\alpha \to \infty$.

Figure 3

Signature of the cat state created by the apparatus of Fig. 2: The number differences $J_{\theta /\varphi }^{A/B}=(N_{+}-N_{-})/2$ at the sites $A$ and $B$ are denoted $n$ and $m$, respectively, and are binned to the values $S_{\theta ,\varphi }^{A/B}=1$ or $-1$ according to sign as described in the text. Left: The expectation values $E=〈S_{\theta }^A{S}_{\varphi }^B〉-〈S_{\theta }^A{S}_{{\varphi }^{\prime }}^B〉+〈S_{{\theta }^{\prime }}^A{S}_{\varphi }^B〉+〈S_{{\theta }^{\prime }}^A{S}_{{\varphi }^{\prime }}^B〉$ generated when modes $a_1$ and $b_1$ are prepared in the state $|\psi \rangle$ given by Eq. (3.7) will violate the Bell inequality (3.6) for all $\alpha \to \infty$. Right: A contour graph of the probability for joint outputs $n$ and $m$ (here, $\theta =\varphi$). The absolute values of the number difference outputs $n,m$ increase with $\alpha$.

Figure 4

Practical method for testing $\delta$-scopic LR using a Bell inequality: The outcomes of each measurement ${\widehat{J}}_{\theta }^A,{\widehat{J}}_{\varphi }^B$ indicated in Fig. 2 are binned into one of the regions 1,0,2. As $\alpha \to \infty$, $P_0\to 0$. A method for testing $\delta$-scopic LR where $P_0\ne 0$ is given in Sec. 4. This involves considering that the system is probabilistically in one of two states, that have overlapping outcomes. The first state is defined as producing an outcome in the combined regions 1 or 0; the second state is defined as producing an outcome in regions 0 or 2.