Quantifying the mesoscopic nature of the Einstein-Podolsky-Rosen nonlocality

TL;DR

This paper investigates mesoscopic EPR steering nonlocality in photonic and BEC systems, quantifying nonlocal effects involving large numbers of particles, and provides methods to estimate particle counts in entangled states.

Contribution

It introduces a way to quantify mesoscopic EPR nonlocality and entanglement in large-particle systems, extending nonlocality analysis beyond microscopic scales.

Findings

EPR nonlocality can be observed with up to 10^5 photons or tens of atoms.

Variance criteria can estimate the minimum number of particles in entangled states.

Methods are applicable to experimental data for large-scale quantum systems.

Abstract

Evidence for Bell's nonlocality is so far mainly restricted to microscopic systems, where the elements of reality that are negated predetermine results of measurements to within one spin unit. Any observed nonlocal effect (or lack of classical predetermination) is then limited to no more than the difference of a single photon or electron being detected or not (at a given detector). In this paper, we analyze experiments that report Einstein-Podolsky-Rosen (EPR) steering form of nonlocality for mesoscopic photonic or Bose-Einstein condensate (BEC) systems. Using an EPR steering parameter, we show how the EPR nonlocalities involved can be quantified for four-mode states, to give evidence of nonlocal effects corresponding to a two-mode number difference of $10^{5}$ photons, or of several tens of atoms (at a given site). We also show how the variance criterion of Duan-Giedke-Cirac and Zoller for EPR entanglement can be used to determine a lower bound on the number of particles in a pure two-mode EPR entangled or steerable state, and apply to experiments.

Equations

$$\epsilon \equiv {\Delta }_{inf}{X}_B{\Delta }_{inf}{P}_B<1$$

$$({\Delta }_{inf,{\delta }_X}{X}_B{)}^2$$

$${\epsilon }_{\delta }\equiv {\Delta }_{inf,{\delta }_X}{X}_B{\Delta }_{inf,{\delta }_P}{P}_B<1$$

$${\Delta }_{inf,{\delta }_J}(J_B^Z){\Delta }_{inf,{\delta }_J}(J_B^Y)<|⟨J_B^X⟩|/2,$$

$$\displaystyle D\equiv\frac{[\Delta(J_{A}^{Z}+J_{B}^{Z})]+[\Delta(J_{A}^{Y}+J_{B}^{Y})]^{2}}{\bigl{|}\langle J_{A}^{X}\rangle\bigr{|}+\bigl{|}\langle J_{B}^{X}\rangle\bigr{|}}$$

Full text

Einstein-Podolsky-Rosen (EPR) presented in 1935 a seemingly compelling argument that quantum mechanics was incomplete epr . In their gedanken experiment, properties of a system $B$ can be predicted ultra-precisely, by the measurements of a distant observer, popularly called Alice. EPR assumed no “spooky action-at-a-distance”, to argue that Alice’s measurement is noninvasive, and therefiore that Alice’s prediction represents a predetermined property (an “element of reality”) of system $B$. Further, they showed that the set of all such predetermined properties could not be consistent with any local quantum state description for $B$, and thus concluded that quantum mechanics was an incomplete theory. The assumptions made in EPR’s argument are collectively known as local realism (LR). Bell’s theorem negated these premises, by showing LR could be falsified Bell .

Understanding whether and how local realism fails macroscopically remains an open question in physics. Loophole-free experiments confirming Bell’s theorem are so far limited to microscopic systems e.g. system $B$ is a single photon or electron bell-ep ; loophole . In these cases, the predetermined properties that EPR called elements of reality give predictions to within a single spin unit. The failure of LR that is inferred from the experiments is therefore a microscopic effect only, in the sense that this pertains only to predictions specified to an accuracy of one spin unit, for a microscopic particle. Similar accuracies are required in almost all of the experiments predicted to violate LR for multi-particle systems higher-n-bell ; cv-bell .

By contrast, EPR’s experiment (called an “EPR-steering” experiment schosteer ; hw ; steereric ; eprr ; saunexp ) has been investigated experimentally for mesoscopic optical fields ou ; rrmp ; spinpolepr ; Aiko_2010_PRA_exp(aa) ; Tobias2011_cc ; Walborn2011 ; Aoki_2010_(a) ; Hannover2012best_(d) ; Hannover2013best ; Howell2013 ; pengbest_(b) ; recentD_(c) ; photon-pair-2016 , atomic ensembles eprnaturecommun ; SteerAt-obert ; treu-matteo ; Plozik ; Polzik2011EPR_macro_(e) ; muschik ; muschik2 ; Optica2017 , and, recently, for mesoscopic mechanical oscillators paulo ; eprentosc ; polamaki ; roman-schnabel-ent-epr-objects ; recent-ent-osc ; recnet-osci-2 ; HeReid2013 ; Simon2014 ; bell-osc . In many of these experiments, not only are the systems sizeable, but the outcomes are over a larger range, corresponding to several or many spin units. There is thus the possibility to test for a mesoscopic EPR nonlocal effect, where the predetermined “elements of reality” that are falsified give predictions with an indeterminacy of several spin units. One may then ask how much “spooky action-at-a-distance” is occurring in terms of spin units? How to do the quantification is not obvious, however. It is not simply the size of the entangled system, nor the range of outcomes. Previous measures inform us how many atoms are mutually entangled sorsolm-1 ; multi , or what fraction of particle pairs behave locally versus nonlocally other-measures , but these need not imply large differences in the actual outcomes of observables due to nonlocal effects.

The situation is clear if the physical quantities measured by observers Bob and Alice (at different locations) have two mesoscopically-distinct outcomes $+$ and $-$, e.g. $N$ particles in an up position versus $N$ particles in a down position ($N\gg 1$) Schrodinger ; LG . One may then extend EPR’s premises, to define $\delta$-scopic local realism ($\delta$LR), which asserts mlr ; recent-mlr : (1) any measurement by Alice cannot instantly induce a change $\delta=2N$ to the outcome of measurement at Bob’s location; and (2) if the outcome $+$ or $-$ for Bob’s system can be predicted with certainty by Alice, then Bob’s system is always predetermined to be in a state that gives either the result $+$ or $-$. The failure of such premises then implies a “spooky action” effect of size $\delta=2N$. However, examples of EPR systems with just two outcomes $\pm$ separated by $\delta$ are limited.

In this paper, we present a practical approach to test EPR premises based on $\delta$-scopic local realism ($\delta$LR). We consider a version of EPR’s argument based on the premise of $\delta$LR, where the separation of outcomes $+$ and $-$ is quantified by $\delta$, but where there is a continuous range of outcomes. Our analysis leads to a criterion sufficient to demonstrate EPR $\delta$-scopic nonlocality, that there is inconsistency between the completeness of quantum mechanics and $\delta$LR. The size of $\delta$ quantifies in the $\delta$LR assumption the upper bound on the amount of change that can occur to Bob’s system, due to Alice’s measurements. Failure of $\delta$LR where $\delta$ is large implies large nonlocal effects. We analyze EPR experiments which have a mesoscopic, continuous range of outcomes for Alice and Bob’s measurements, to present preliminary evidence for quantifiable mesoscopic EPR nonlocalities.

Quantifying the EPR Paradox: The EPR argument can be generalised to pairs of measurements $\{X_{A},P_{A}\}$ and $\{X_{B},P_{B}\}$ on two spatially separated systems $A$ and $B$. We consider $X_{B}$, $P_{B}$ to be scaled non-commuting observables which satisfy $[X_{B},P_{B}]=2$, so that the Heisenberg uncertainty relation is $\Delta X_{B}\Delta P_{B}\geq 1$. To demonstrate the paradox, one measures the variances $V_{B}(X_{B}|X_{A})$and $V_{B}(P_{B}|P_{A})$ of the respective conditional distributions $P(X_{B}|X_{A})$ and $P(P_{B}|P_{A})$ eprr . Here $P(X_{B}|X_{A})$ is the probability for result $X_{B}$, given a measurement $X_{A}$. The average conditional variance $(\Delta_{inf}X_{B})^{2}=\sum_{X_{A}}P(X_{A})V_{B}(X_{B}|X_{A})$ determines the accuracy of inference of the results for $X_{B}$, based on the measurements at $A$. The $\Delta_{inf}P_{B}$ is defined similarly. Using EPR’s logic, these inference variances define the average indeterminacy of the two respective elements of reality, for $X_{B}$ and $P_{B}$. If

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then an EPR paradox arises, since the simultaneous predetermination for $X_{B}$ and $P_{B}$ is more accurate than allowed by the uncertainty principle rrmp ; eprr . The condition (1) is a condition for “EPR steering of system $B$” hw ; steereric .

We now construct a quantified version of the EPR argument, by relaxing EPR’s premises. The assumptions of $\delta_{X}$-scopic local realism ($\delta_{X}$LR) are: (1) A measurement made at $A$ might disturb the system $B$, so that the outcome for a simultaneous measurement $X$ on $B$ can be altered, but the change to the outcome cannot be greater (in magnitude) than $\delta_{X}$. (2) If the value for a physical quantity $X$ is predictable, without disturbing the system by more than $\delta_{X}$, then the value of that physical quantity is a predetermined property of the system (the “element of reality” for $X$), the predetermined value being given to within $\pm\delta_{X}$ of the predicted value. We will refer to $\delta_{X}$ as the degree of “nonlocal indeterminacy”, with respect to the EPR observable $X$.

The assumption of $\delta$LR changes the condition for an EPR paradox, making it more difficult to demonstrate the paradox. Applying $\delta_{X}$LR, the indeterminacy in the predictions for $X_{B}$ associated with the element of reality has increased, but by a limited amount only. We show in the Supplemental materials that the maximum value of this indeterminacy becomes supp

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where $P(X_{A},X_{B})$ is the joint probability. Defining $\Delta_{inf,\delta_{P}}P_{B}$ in a similar manner, the experimental realisation of

[TABLE]

will therefore imply an inconsistency between the premise of $\delta$LR and the completeness of quantum mechanics. The calculation of $\varepsilon_{\delta}$ is straightforward, once the distributions $P(X_{A},X_{B})$ and $P(P_{A},P_{B})$ are known. When $\delta=0$, Eq. (3) reduces to the standard EPR condition (1). The inequality is progressively more difficult to satisfy, as $\delta$ increases.

Gaussian $\delta$-scopic EPR nonlocality: We consider EPR experiments based on field modes at locations $A$, $B$. $X_{A/B}$, $P_{A/B}$ are defined according to $a=(X_{A}+iP_{A})/2$ and $b=(X_{B}+iP_{B})/2$, where $a$, $b$ are the annihilation operators of each mode. The $\delta$-scopic EPR inequality reduces to Eq. (3). A widely-used source of EPR-correlated fields is the parametric amplifier, the ideal output of which is the two-mode squeezed state rrmp ; eprr . Here, the conditionals $P(X_{B}|X_{A})$ and $P(P_{B}|P_{A})$ are Gaussian. Moreover, a Gaussian profile is maintained in non-ideal situations where losses and thermal noise are present rrmp ; cv-review ; gauscv .

Assuming Gaussianity, the prediction of $\varepsilon_{\delta}$ given measured values of $\Delta_{inf}X_{B}$ and $\Delta_{inf}P_{B}$ is straightforward. Using Eq. (LABEL:eq:neweprdeltaline) and that for a Gaussian distribution $\langle|X_{B}-\mu_{X}|\rangle=\Delta_{inf}X_{B}\sqrt{2/\pi}$ (where $\mu_{X}$ is the mean of $P(X_{B}|X_{A})$), we find $\varepsilon_{\delta}=\sigma^{2}+\delta^{2}+2\delta\sigma\sqrt{2/\pi}$ supp . For the sake of simplicity, we have taken $\sigma=\Delta_{inf}X_{B}=\Delta_{inf}P_{B}$ and $\delta=\delta_{X}=\delta_{P}$. We see that \varepsilon{\color[rgb]{0,0,0}<\left[-\delta\sqrt{2/\pi}+\sqrt{2\delta^{2}/\pi-(\delta^{2}-1)}\right]^{2}} will be sufficient to imply the $\delta$-scopic EPR nonlocality.

Extensive data has been reported for continuous variable EPR experiments ou ; rrmp ; spinpolepr ; Aiko_2010_PRA_exp(aa) ; Tobias2011_cc ; Walborn2011 ; Aoki_2010_(a) ; Hannover2012best_(d) ; Hannover2013best ; Howell2013 ; eprnaturecommun ; SteerAt-obert ; treu-matteo ; recentD_(c) ; pengbest_(b) ; photon-pair-2016 (see Fig. 1). Gaussian distributions are predicted in almost all cases plotted (including the data indicated by (g)) as has been verified experimentally gauscv ; rrmp . For rigorous testing, a full construction of the distributions with space-like separated measurement events is required Bell ; loophole . With this proviso, we note that the recently achieved values of the EPR parameter $\varepsilon\sim 0.176$ Hannover2013best will imply a $\delta$-scopic EPR nonlocality, with $\delta\sim 0.633$.

To determine the* *significance of the value of $\delta$, one needs to resort to the details of the individual experiments. The nonlocal indeterminacy $\delta$ is given relative to the quantum noise level, which for the optical experiments is usually considered microscopic. On the other hand, entanglement has now been detected between two mechanical oscillators paulo ; eprentosc ; recnet-osci-2 , and between an oscillator and a field polamaki . Entanglement however does not imply the EPR steering condition (1). It has been proposed to detect the EPR condition (1) for these cases paulo ; roman-schnabel-ent-epr-objects ; HeReid2013 ; Simon2014 , where $X_{B}$, $P_{B}$ refer to the quadratures of the phonon modes of the oscillator. Eq. (3) enables a quantification of the EPR nonlocality that would be observed in such an experiment. Here, $\delta$ is quantifiable at the Planck scale probeplanck , and corresponds to a nonlocal indeterminacy with respect to mechanical motion.

EPR nonlocality using Schwinger spins: For some experiments, the quantum noise level and hence $\delta$ may correspond to a large number of photons. This is understood by considering the Heisenberg relation \Delta J^{Z}\Delta J^{Y}\geq\bigl{|}\langle J^{X}\rangle\bigr{|}/2 for spin systems, where measurements are made of the spin components $J^{X,Y,Z}$. For high spins, \bigl{|}\langle J^{X}\rangle\bigr{|} can be large.

Indeed, EPR states exist for which \bigl{|}\langle J^{X}\rangle\bigr{|} is a scalable large number. In these cases, the EPR observables are two-mode Schwinger spins, defined as $J_{A}^{X}=(a_{+}^{\dagger}a_{-}+a_{+}a_{-}^{\dagger})/2$, $J_{A}^{Y}=(a_{+}^{\dagger}a_{-}-a_{+}a_{-}^{\dagger})/2i$, $J_{A}^{Z}=(a_{+}^{\dagger}a_{+}-a_{-}^{\dagger}a_{-})/2$, and $J_{B}^{X}=(b_{+}^{\dagger}b_{-}+b_{+}b_{-}^{\dagger})/2$, $J_{B}^{Y}=(b_{+}^{\dagger}b_{-}-b_{+}b_{-}^{\dagger})/2i$, $J_{B}^{Z}=(b_{+}^{\dagger}b_{+}-b_{-}^{\dagger}b_{-})/2$, where $a_{\pm}$, $b_{\pm}$ are annihilation operators for four modes mlr . The four modes are created from spatially separated modes $a$, $b$ prepared in an EPR state $|\psi\rangle$. Each mode $a$ $b$ interferes (via a beam splitter) with an intense “local oscillator” field (denoted by annihilation mode operators $b_{LO}$, $a_{LO}$). This creates a macroscopic photonic state $|\psi\rangle_{M}$ involving four fields $a_{\pm}=(a\pm a_{LO})/\sqrt{2}$, $b_{\pm}=(b\pm b_{LO})/\sqrt{2}$ at sites $A$ and $B$ respectively. The fields at each site pass through second polarising beam splitters set at respective angles $\theta_{A}$ and $\theta_{B}$. The number of particles in each arm is detected, as a large number, and the difference gives a measure of $J^{Z}$, $J^{Y}$ or $J^{Z}$ depending on the choice of $\theta_{A},\theta_{B}$. Based on the Heisenberg uncertainty relation, the EPR criterion is

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which normalises to Eq. (3) on defining $X_{B}/P_{B}=J_{B}^{Z/Y}/\sqrt{|\langle J_{B}^{X}\rangle|/2}$ and $\delta=\delta_{J}/\sqrt{|\langle J_{B}^{X}\rangle|/2}$. Here, $|\langle J_{B}^{X}\rangle|=|\langle b_{LO}^{\dagger}b_{LO}-b^{\dagger}b\rangle/2|$ which becomes $\langle b_{LO}^{\dagger}b_{LO}\rangle/2$ since $\langle b^{\dagger}b\rangle/\langle b_{LO}^{\dagger}b_{LO}\rangle$ is small. The intensity of the local oscillator is macroscopic and $\delta_{J}\sim\delta\sqrt{\langle b_{LO}^{\dagger}b_{LO}\rangle/4}$ (the nonlocal indeterminacy in the values of $J_{X}^{B}$, $J_{Y}^{B}$) can therefore also be large.

EPR nonlocality for Schwinger spins has been realised in the experiments of Bowen et al spinpolepr , where $a_{\pm}$ ($b_{\pm}$) correspond to two orthogonal horizontally (“H” or “$x$”) and vertically (“V” or “$y$”) polarised field modes at $A$ ($B$). From the description of their experiment spinpolepr ; supp , $|\langle J_{B}^{X}\rangle|\sim 10^{11}$ photons, implying a $\delta_{J}$ of order $10^{5}$ photons. The relative value $\delta_{J}/|\langle J_{B}^{X}\rangle|$ is however small.

**Analogy to Schrödinger cat: The Schwinger-spin experiment provides a simple parallel to Schrödinger’s cat gedanken experiment Schrodinger ; mlr . In the original cat paradox, a macroscopic superposition is created by the process of measurement, which couples the microscopic system (prepared in a superposition state) to a measurement apparatus. In the experiment, the microscopic EPR state $|\psi\rangle$ is indeed coupled to a macroscopic system (the local oscillator fields) at each site, and a four-mode amplified state $|\psi\rangle_{M}$ is produced that enables a macroscopic readout of $X_{A/B}$ and $P_{A/B}$ of the original fields.

The many-particle state $|\psi\rangle_{M}$ is created prior to the measurements $J_{A}^{\theta_{A}}$, $J_{B}^{\theta_{B}}$ and it is this feature that enables the demonstration of mesoscopic nonlocality. The $|\psi\rangle_{M}$ is a superposition of states with definite outcomes for $J_{B}^{\theta_{B}}$, where those outcomes are given by $J_{B}^{\theta_{B}}=EX_{B}/2$, or $EP_{B}/2$ (here $E^{2}=\langle b_{LO}^{\dagger}b_{LO}\rangle$) depending on $\theta_{B}$. The superposition $|\psi\rangle_{M}$ comprises many states that have a large range of continuous outcomes for $J_{B}^{\theta_{B}}$, rather than just two distinct states as in Schrodinger’s case. In the Supplemental Materials we prove that the observation of the $\delta_{J}$-scopic EPR nonlocality can only arise if $|\psi\rangle_{M}$ comprises at least two states that differ in outcome for $J_{B}^{\theta_{B}}$ by at least $\delta_{J}$ supp . Such states (when $\delta_{J}$ is large) give a nonzero mesoscopic quantum coherence, and signify a generalised Schrodinger-cat paradox (refer Refs. cat-sup-in-cv ; cavalr ; frowis ; flor-cat-rmp ). These states, for which EPR nonlocality is also demonstrated, are well nested within the overall superposition state however. For an experimental value $\epsilon\sim 0.42$ (where $\delta\sim 0.4$), their separation is typically $\delta_{J}=0.4E/2$, whereas the state $|\psi\rangle_{M}$ predicts a Gaussian distribution for $J_{B}^{\theta_{B}}$, with $\Delta J_{B}^{\theta_{B}}\sim 1.3E/2$.

The $\delta$-scopic EPR nonlocality manifests without the significant decoherence that normally prevents formation of Schrödinger cat states, because the separation $\delta_{J}$ between states of the superposition is not amplified relative to the quantum noise level. The EPR steering parameter $\varepsilon_{\delta}$ is unchanged, consistent with the requirement that entanglement cannot be created by local entangling transformations, such as produced by beam splitters ent-measure .

EPR nonlocality between distinct atom groups:* *The experiments of Refs. Plozik ; Polzik2011EPR_macro_(e) ; muschik ; muschik2 investigate EPR entanglement between two spatially separated macroscopic atomic ensembles, $A$ and $B$. In their experiment, $J_{A/B}^{X,Y,Z}$ are the collective spins of each ensemble, defined relative to two atomic levels. The observation of the condition $D\equiv\{[\Delta(X_{A}-X_{B})]^{2}+\Delta[(P_{A}+P_{B})]^{2}\}/4<1$ implies entanglement between subsystems $A$ and $B$ duan . For spins, this entanglement condition becomes raymer

[TABLE]

Measurements give $D\sim 0.8$ for thermal atomic ensembles Plozik ; Polzik2011EPR_macro_(e) . The value of $D\sim 0.5$ would imply an EPR steering nonlocality (4) rrmp ; beceprolsen ; atoms-njp . For a rigorous demonstration of EPR nonlocality, it is however necessary to measure the EPR observables independently and locally, so that information is gained simultaneously about each of $J_{A}^{\theta_{A}}$ and $J_{B}^{\theta_{B}}$.

EPR steering correlations meeting the condition (1) have however recently been observed for matter-wave fields created with Bose-Einstein condensates (BEC) eprnaturecommun ; EntAtoms ; SteerAt-obert ; treu-matteo ; heidelepr . In the experiments eprnaturecommun ; heidelepr ; SteerAt-obert , twin atom-beam states are generated by a parametric interaction heidelepr . Atoms are created in pairs, one into each group $A$ and $B$ that correspond to different spins. The atom field quadratures $X_{A/B}$, $P_{A/B}$ are measured by an atomic homodyne method, where the local oscillators are a different, larger group consisting of $E=\langle a_{LO}^{\dagger}a_{LO}\rangle\sim\langle b_{LO}^{\dagger}b_{LO}\rangle>10^{2}$ atoms heidelepr . Experiments of Piese et al observe EPR correlations (1) between mesoscopic atomic groups $A$ and $B$ with $\varepsilon\sim 0.85$ (with no spatial separation) eprnaturecommun . Recently, Kunkel et al observe an EPR steering with $\varepsilon\sim 0.71$ between atom groups spatially separated by $\sim 10\mu m$ SteerAt-obert . Assuming results are unchanged if the experiment were reconfigured along the lines of the Schwinger spin experiments, and that the distributions are approximately gaussian, this suggests nonlocalities with $\delta_{J}>E\delta>20$ atoms.

Using a different method of generation, Fadel et al observe EPR steering correlations ($\varepsilon\sim 0.74$) between the Schwinger spins $J_{A}^{Z}$, $J_{B}^{Z}$ (and $J_{A}^{Y}$, $J_{B}^{Y}$) of two atomic groups separated by \sim 4$$\mu m treu-matteo . Here, group $A$ ($B$) consists of two BEC components, $a_{\pm}$ ($b_{\pm}$). Their measurement of spins ($J^{X}$ or $J^{Y}$) is achieved with a variable pulse rotation $\theta$, in analogy to the polarising beam splitter of Bowen et al. (although without independent selection of the two measurement angles). The BEC experiments thus reveal quantifiable nonlocal indeterminacies in the atom number differences $J_{B}^{\theta_{B}}$ at the given site. The nonlocality being tested here is whether an action at the site $A$ can create a change in the number of atoms between the groups $b_{+}$ and $b_{-}$, at site $B$.

**Quantification of number of bosons in the two-mode entangled state: The large values of $\delta_{J}$ arise for four-mode states. However, the two-mode EPR state $|\psi\rangle$ can itself be constrained to have a certain degree of “largeness”. For any entangled pure state $|\psi\rangle$, the mean total number of bosons is $\bar{n}=\langle\psi|a^{\dagger}a+b^{\dagger}b|\psi\rangle$. The value of $D$ places a lower bound on $\bar{n}$. Using the identity $|\langle ab\rangle|\leq\sqrt{(\langle a^{\dagger}a\rangle+1)\langle b^{\dagger}b\rangle}$ hillezub , we find $D\geq D_{\overline{n}}^{(l)}$ supp , where $D_{\bar{n}}^{(l)}=1+\bar{n}-\bar{n}\sqrt{1+2/\bar{n}}$ decreases with $\bar{n}$, and is achieved for the two-mode squeezed state for which $D=(1-x)/(1+x)$. In an experiment, the two-mode system is generally not a pure state. The measured $\langle a^{\dagger}a+b^{\dagger}b\rangle$ does not then reflect the mean number of bosons in an entangled state, because there may be components of the mixture that are not entangled. However, the observation of $D<D_{\bar{n}}^{(l)}$ certifies that a pure entangled state $|\psi\rangle$ with $\langle a^{\dagger}a+b^{\dagger}b\rangle>\bar{n}$ must be a component of the mixed state (refer supp ). Similarly, by expanding all pure states in the basis of number states $|i\rangle_{a}|j\rangle_{b}$, we prove in the Supplemental Materials that the value $D$ places a lower bound on the minimum number of bosons $n_{0}=i+j$ contributing a nonzero term to the expansion. If the bosons are atoms, the state $|i\rangle_{a}|j\rangle_{b}$ has an entanglement depth of $n_{0}=i+j$, meaning all $n_{0}$ atoms are mutually entangled sorsolm-1 ; multi ; supp .

Experimental values of $D$ are plotted in Fig. 2. The values {\color[rgb]{1,0,0}{\color[rgb]{0,0,0}D<0.2228}} confirm two-mode optical EPR states $|\psi\rangle$ involving more than $10$ photons ($n_{0}>10$). These states are different to states constructed from photon pairs, for which $n_{0}\leq 2$. Where $D<0.5,$ the two modes of the state $|\psi\rangle$ are both EPR steerable qi-discord-two-way-steer . Measurements by Piese et al eprnaturecommun observe $D<0.43$, implying* two-way *EPR steerable states $|\psi\rangle$ with more than $3$ atoms (if spatial separation could be achieved) supp .

Conclusion: We have given evidence for a mesoscopic EPR nonlocality that “delocalises” $\delta_{J}\sim 10^{5}$ photons between two polarisation modes at a given site. This represents a tenth of the full range of outcomes (defined as that within $3$ standard deviations of the mean) for the polarisation photon-number difference at the site. Recent experiments with Bose-Einstein condensates show similar EPR nonlocalities involving four atomic modes. This motivates new experiments where it may be feasible to demonstrate an EPR nonlocality “delocalising” $\delta_{J}\sim 10$ atoms across two highly-occupied atomic modes at a given site. The criteria presented in this paper may also have a practical application. The curves of Fig. 1 can be used to detect a genuine EPR effect, even when a causal effect is present. If the maximum disturbance due to the causal effect can be quantified (to be $\delta_{C}$ say), then an EPR nonlocality can be deduced if $\epsilon_{\delta}<1$ where $\delta>\delta_{C}$. An example of such a causal effect (“cross-talk”) is given in Ref. treu-matteo .