1. INTRODUCTION

The question of what is a macroscopic quantum state has received a lot of attention over the last decade [1–17]. The motivation is not to address a new question, as it dates back from the early days of quantum theory [18], but rather comes from the experimental progress, which now allows one to harness large systems while highlighting their quantum nature. Quantum optics experiments reporting on squeezing operations provide a nice example. They are obtained from a ${\chi }^2$ nonlinearity and can result in largely entangled states. The entanglement can be further detected with homodyne detections by means of the Duan–Simon criterion [19,20]. When the ${\chi }^2$ nonlinearity is seeded by coherent states and/or embedded in a high finesse cavity, entanglement in squeezed states can be demonstrated with a large number of photons [21–29]. This naturally raises the question of whether squeezed states have macroscopic quantum features, which is a question of deep relevance because, thus far, squeezed states have been combined with conditional detections [30–36] for exploring quantum effects in many photon states.

In the literature, there exist different criteria for quantifying the macroscopic quantumness [3–14]. Typically, this includes a definition that assigns a number to a quantum state, which is here called effective size (or simply size). These criteria can be grouped into two categories. The first addresses the question of whether a two-component superposition $|{\varphi }_0〉+|{\varphi }_1〉$ is macroscopic, i.e., whether $|{\varphi }_0〉$ and $|{\varphi }_1〉$ are macroscopically distinct. For example, the proposal of [8] states that two spin states are macroscopically distinct if they can be distinguished from a small number of their spins, as a dead cat and a live cat can be distinguished from a small number of their cells. We also can refer to the proposals of [13,37] defining two states as being macroscopically distinct if they can be distinguished with a coarse-grained measurement, as a dead cat and a live cat can be distinguished with a detector having a very limited resolution. The second category aims to identify quantum states that are able to show some kind of macroscopic quantum effect. This term characterizes experimental evidence that cannot be explained by an accumulated quantum effect originated at the microscopic level of the system. For pure states, a large variance with respect to given observables and Hamiltonians is a sufficient signature for quantum fluctuations that are persistent on a macroscopic level. For mixed states, one typically uses a convex function that reduces to the variance for pure states. For example, the proposal of [11] shows how the notion of macroscopicity can be linked to the so-called quantum Fisher information [38].

Not all of these measures are able to correctly handle two-mode squeezed states for different reasons. Thus far, only the contributions [7,10,39] explicitly treat two-mode squeezed vacuum. This is valuable, but these criteria also leave open questions, which are discussed in more detail in the course of this paper. In short, Cavalcanti and Reid [7,39] propose a bound to measure the size of a two-mode squeezed state, which, however, does not give satisfactory answers to nonsqueezed states. This drawback is compensated by Lee and Jeong [10], but the mode additivity of their measure remains problematic.

In this paper, we aim to reinforce arguments in favor of assigning a macroscopic quantum nature to two-mode squeezed states. For this, we extend existing measures for macroscopic quantum states [13,40] to be able to investigate two-mode, continuous variable (cv) quantum states. Apart from avoiding problems that appeared in previous attempts [7,10,39], this broadens the range of arguments supporting the macroscopic quantumness of a two-mode squeezed vacuum. Importantly, we prove that the effective size of two-mode squeezed vacuum states (with $N$ mean photons) is basically the same as superpositions of coherent states with opposite phases $|\alpha 〉+|-\alpha 〉$ and ${|\alpha |}^2\tanh {|\alpha |}^2=N$ but with the great advantage that they are much easier to create [41]. The tools we propose allow one to bound the size of states obtained experimentally as well as their usefulness for parameter estimation beyond the classical limit. Aside from a fundamental interest, our results ultimately have important applications for quantum metrology. Furthermore, we discuss the impact of noise and technical limitations that cumber the verification of macroscopic quantumness of two-mode squeezed states. Finally, we use data from some performed experiments [21–29] to lower-bound the effective size achieved thus far.

2. TWO-MODE VACUUM SQUEEZED STATES

As an example of two-mode squeezed states, let us consider the two-mode squeezed vacuum, which is obtained from a parametric process in which photons from a pump laser decay spontaneously into photon pairs—one in mode 1, its twin in mode 2—while preserving energy and momentum. The corresponding propagator $\overline{S}(g)=e^{g(a_1{a}_2-a_1^{†}{a}_2^{†})}$, with squeezing parameter $g$, applies straightforwardly on the vacuum if written in the normal order. This results in

3. MACROSCOPIC QUANTUMNESS OF TWO-MODE CV STATES

A. Macroscopic Distinctness for cv States

While several definitions have been proposed to identify states that are macroscopically distinct [5,8,9,13], here we focus on the proposal of [13] based on coarse-grained measurements. This choice is arbitrary to some extent. Note, however, that the extension that we propose below easily applies to the measure of [8]. The extension of measures of [9] to two-mode squeezed states is less obvious, as they primarily address spin systems, but the link between measures for spins and photons presented in [17] might be the way to proceed.

The basic principle of the measure of macroscopicity based on coarse-grained measurement is simple. It can be seen as a game where Alice chooses a state in the set $\{|{\varphi }_0〉,|{\varphi }_1〉\}$ with equal a priori probabilities and sends it to Bob. Bob has to guess which one has been sent using a coarse-grained measurement only. It can be any measurement, provided that its resolution is limited. The quantum superposition state $|{\varphi }_0〉+|{\varphi }_1〉$ is qualified macroscopic if Bob wins the game with a detector having no microscopic resolution. Concretely, if one focuses on a noisy photon-counting detector, for example, the size of $|{\varphi }_0〉+|{\varphi }_1〉$ is characterized by the noise that one can tolerate to distinguish $|{\varphi }_0〉$ and $|{\varphi }_1〉$.

To extend this measure to cv states, we can mimic its original idea by introducing a 50/50 binning of measurement outcomes. For a two-mode squeezed vacuum state in particular, Alice measures her mode with a given quadrature and bins the result with respect to its sign. As Alice’s measurement is assumed to be accurate, this binning corresponds to equiprobable projections onto two orthogonal subspaces of the measured state. Bob has to guess whether she received a positive or negative outcome by measuring his mode with a noisy measurement. The distinguishability of components that Bob receives is again given by the noise that can be tolerated to win the game. Note that the measurement of correlated quadratures maximizes the probability to correctly guess Alice’s outcome. Concretely, the probability that Alice obtains the result $x_1$ and Bob $x_2$ knowing that they measure the quadratures ${\overline{X}}_1^0$ and ${\overline{X}}_2^0$ is given by ${|p(x_1,x_2,\sigma )|}^2=\mathrm{tr}(|{\psi }_{\mathrm{tms}}〉〈{\psi }_{\mathrm{tms}}|\delta ({\overline{X}}_1^0-x_1)g_{\sigma }({\overline{X}}_2^0-x_2))$, where $g_{\sigma }$ stands for the noise of Bob’s measurement device. We assume that $g_{\sigma }$ is a Gaussian with spread $\sigma$ and zero mean. Hence, the probability that Bob correctly guesses the sign of Alice’s result is given by $P_{\sigma }^{\mathrm{guess}}={\int }_0^{+\infty }{|p(x_1,x_2,\sigma )|}^2\mathrm{d}{x}_1\mathrm{d}{x}_2+{\int }_{-\infty }^0{|p(x_1,x_2,\sigma )|}^2\mathrm{d}{x}_1\mathrm{d}{x}_2$. We find

Let us now focus on practical considerations. The observation that Alice and Bob’s $x$ quadratures of the two-mode squeezed vacuum state are “macroscopically” correlated (correlated at a large scale, larger than the detector’s resolution) is at the heart of our generalization of the coarse-grained measure. These correlations can be revealed by measuring the joint probability distribution ${|p(x_1,x_2,0)|}^2$ with accurate quadrature measurements. (For simplicity, we introduce $p(x_1,x_2)=p(x_1,x_2,0)$, which stands for the probability amplitudes without noise.) Although this approach is sufficient to measure the size of a given state in theory, one also has to ensure that those correlations are truly quantum in practice. In mathematical terms, we can always write the state that is shared by Alice and Bob in the $x$ basis:

To do so, we consider the effect of imperfect coherences (decoherence) $f(x_1,{\overline{x}}_1,x_2,{\overline{x}}_2)\ne 1$ on the observed violation of the Duan–Simon witness. Note first that the variance $V({\overline{X}}_1^0-{\overline{X}}_2^0)$ can be directly obtained from ${|p(x_1,x_2)|}^2$. For the second term required in Eq. (3), we can show that the variance in the presence of decoherence (see Appendix A),

B. General Measures for Multimode cv States

Beside measures for macroscopic distinguishability, there have been recent proposals that aim to go beyond the basic structure $|{\varphi }_0〉+|{\varphi }_1〉$ [4,6,10,11]. While the measures of [4,6,11] were originally defined for spin systems, the definition of [10] is directly suitable for cv photonic states. For pure states, these three proposals are comparable because a state $|\psi 〉$ is called macroscopically quantum if it shows a large variance with respect to a restricted class of operators. In the spin case, the proposals [4,6,11] focus on sums of local operators (henceforth simply called “local operators”), whereas Lee and Jeong [10] define their measure for pure states proportional to $V({\overline{X}}^0)+V({\overline{X}}^{\pi /2})$. In [17], it was argued that local operators in the spin case play to some extent the same role as quadrature operators in mono-mode photonic systems.

The common feature of the proposals for mixed states is that the measures [4,6,10,11] are convex in the state, which is an important and natural feature for the present purpose. There are no clear arguments in favor of one of the proposals. Nevertheless, we focus here on the quantum Fisher information (QFI) [38], which is denoted as ${\mathcal{F}}_{\rho }(\overline{X})$ for the state $\rho$ and the operator $\overline{X}$. Importantly, the QFI is the convex roof of the variance [43,44] (up to a factor four); that is, it is the largest convex function that reduces to the variance for pure states. For experiments, it is interesting to note that there exist lower bounds on the QFI based on measurable quantities [45].

The extension to photonic states with $n>1$ modes is not straightforward. Indeed, a multimode version for the measure of Lee and Jeong was proposed [10]. However, it is additive and, hence, a bunch of “kitten states” ${|{\psi }_{\alpha }〉}^{\otimes n}\propto {(|\alpha 〉+|-\alpha 〉)}^{\otimes n}$ (with potentially small $\alpha$ but large $n$) is as macroscopically quantum as a “big” single cat state $|{\psi }_{\sqrt{n}\alpha }〉\propto |\sqrt{n}\alpha 〉+|-\sqrt{n}\alpha 〉$. Here, we propose instead to use a similar account that has been successfully applied in the spin case [4,6,11]. The idea is that the effective size of a product state is the average value of its components, while entangled states should be able to profit from quantum correlations between the modes. Both requirements are achieved by defining the effective size for $\rho$ as

We now come to the evaluation of the effective size for the two-mode squeezed vacuum state. It is simple to see that the variance is largest for the quadratures that are maximally correlated. For the state in Eq. (1), these are the operators ${\overline{X}}_1^0+{\overline{X}}_2^0$ and ${\overline{X}}_1^{\pi /2}-{\overline{X}}_2^{\pi /2}$. The effective size for each of these choices reads $N_{\mathrm{eff}}({\psi }_{\mathrm{tms}})=V({\overline{X}}_1^0+{\overline{X}}_2^0)=e^{2g}\approx 2N$, which is approximately half of the value as for the cat state with the same photon number, $N_{\mathrm{eff}}(|{\psi }_{\alpha }〉)\approx 4N$. Again, we conclude that two-mode squeezed states are compatible with photonic cat states.

In principle, the effective size of a pure state could be determined by witnessing a large variance for sums of quadrature operators. However, for mixed states, a large variance is not sufficient. Instead, one has to verify a large value of a convex function like the QFI. Since this quantity is typically only accessible through a full state tomography, one has to find other means to estimate it. Recently, a general lower bound on the QFI has been found [45]. It was shown that, for any quantum state $\rho$ and any pair of operators $A$, $B$, it holds that $V_{\rho }(A){\mathcal{F}}_{\rho }(B)\ge {〈i[A,B]〉}_{\rho }^2$, which is a tighter version of the Heisenberg uncertainty relation. Here, we use this inequality to bound the QFI from below. For $B={\overline{X}}_1^0+{\overline{X}}_2^0$, we set $A={\overline{X}}_1^{\pi /2}+{\overline{X}}_2^{\pi /2}$ and find $i[A,B]=-2$. Hence, one has

Note that Eq. (10) [as well as Eq. (8)] resembles the ideas of [7,39] for a generalized notion of macroscopic quantum coherences. However, in these works, solely the bound in Eq. (10) is used, which restricts one to investigate squeezed states. In contrast, we use this expression only to give a bound on the definition in Eq. (9), which also can deal with nonsqueezed states. Hence, our measure is general enough to compare two-mode squeezed states with other states such as cat states.

4. ON THE DIFFICULTY TO CERTIFY THE QUANTUM NATURE OF TWO-MODE SQUEEZED STATES

The common feature of measures for macroscopicity presented before is the requirement to reveal narrow variances, especially when dealing with large size states. How difficult is it in practice? To answer this question, we consider the effect of various experimental imperfections on the observed variance $V_{{\psi }_{\mathrm{tms}}}({\overline{X}}_1^{\pi /2}+{\overline{X}}_2^{\pi /2})$.

An experimental issue for the detection of highly squeezed states is the size of the local oscillator $|\alpha 〉$ (LO) used for homodyne detection. On the one hand, the mean total photon number $N+{|\alpha |}^2$ has to be smaller than some value $D_{\mathrm{th}}$ in such a way that the detectors are not saturated. On the other hand, $\alpha$ has to be large enough in order to be a good phase reference because a limited size of the LO is similar to a phase noise. Let us show this quantitatively. In a homodyne measurement, the system is mixed with the LO on a balanced beam splitter with the phase $\theta$. The quadrature is accessed via the difference in the mean photon number of the two outputs of the beam splitter $\frac{I_1-I_2}{|\alpha |}$. While for the mean of any quadrature, this is exactly given by $〈\frac{I_1-I_2}{|\alpha |}〉=〈X_{\theta }〉$, for the second moment $〈\frac{{(I_1-I_2)}^2}{{|\alpha |}^2}〉=〈X_{\theta }^2〉+\frac{〈a^{†}a〉}{{|\alpha |}^2}$, this correspondence with the measured result is not exact. Accordingly, for a finite size of the local oscillator, the observed variance is bounded by $V({\overline{X}}_1^{\pi /2}+{\overline{X}}_2^{\pi /2})\ge \frac{4{\sinh }^2(g)}{{|\alpha |}^2}$. Because the size of the LO is limited by the detector saturation threshold ${|\alpha |}^2\le D_{\mathrm{th}}-2{\sinh }^2(g)$, one has $N_{\mathrm{eff}}^{\max }({\psi }_{\mathrm{tms}})=\frac{D_{\mathrm{th}}-2{\sinh }^2(g)}{4{\sinh }^2(g)}$.

In each case, we clearly see that it becomes increasingly difficult to observe narrow variances with two-mode squeezed states as their size increases. This is in agreement with recent results [47–50] stating that it is difficult to observe the quantum nature of macroscopic states.

This naturally raises the question of the size of states that can be observed in practice. Note that Eqs. (8) and (10) are general, i.e., the variance along the conjugate of quadratures that are maximally correlated gives a bound on the size of the measured state. We use experimental data obtained in various setups in which the ${\chi }^2$ nonlinearity is either seeded or embedded in a cavity [21–24,27,28] to bound the effective size of the produced states (see Fig. 1). All these experiments have in common that the Duan–Simon criterion is used to reveal entanglement and the photon number is large. We clearly see that their effective size cannot be compared with their mean photon number. In the seeded case, the reason is that the seed increases the photon number but does not change the variance. Similarly with a cavity, the photon number can be large even if the gain slightly dominates the loss provided that the cavity finesse is large but the variance of interest is limited by the ratio between the gain and the loss only (see Appendix B). Interestingly, the results presented in Fig. 1 can be used directly to quantitatively estimate the metrological usefulness of states realized experimentally, as the size $N_{\mathrm{eff}}$ gives the QFI through the formula in Eq. (9).

 

Fig. 1. Bounds on the effective size $N_{\mathrm{eff}}$ (blue squares) of two-mode squeezed states obtained from experimental data reported in [21–29] using the inequality in Eq. (10). The red triangles indicate the minimal photon number $N$ necessary for a cat state $|\alpha 〉+|-\alpha 〉$ to have the same effective size according to Eq. (9). For example, the state reported in [21] has a size $N_{\mathrm{eff}}\ge 1.2$ for which one needs at least a cat state with $N\approx 0.2$ for the same size.

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5. CONCLUSION

We propose to consider a two-mode squeezed vacuum as being macroscopically quantum. Our conclusion is based on the results from two measures for macroscopic quantum states [11,13], which have been extended for the purpose of studying two-mode cv states. It is straightforward to generalize our argument to general two-mode squeezed states. This study is in line with former works [7,10,39], but it overcomes some of their problems. On a quantitative level, we see that two-mode squeezed vacuums are comparable with superpositions of coherent states, which are generally considered as archetypal Schrödinger cat states. However, one should acknowledge two advantages of the former compared with the latter: Two-mode squeezed states are not only relatively easy to generate. The experimental verification of macroscopic quantumness can be simply achieved by demonstrating strong (anti-)correlations between the quadratures of the two modes. Nevertheless, in agreement with previous findings [48,50], the precise control of the entire experiment, especially of the measurement, is imperative for showing large quantumness.