An Einstein-Podolsky-Rosen argument based on weak forms of local realism not falsifiable by GHZ or Bell experiments

The premises presented in the original 1935 argument given by EPR are based on the assumptions of local realism. The premises, which we refer to as EPR’s local realism (LR), are summarized as two Assertions for space-like separated systems, $A$ and $B$.

The assertions defining macroscopic local realism (MLR) are identical to those of EPR’s local realism, except that the assertions are weaker, being restricted to apply only to the subset of systems where the outcomes for all relevant measurements, $S_{\theta}^{A}$ and $S_{\phi}^{B}$, correspond to macroscopically distinct states of the system. This means that the systems upon which those measurements are made can be viewed as having two (or more) macroscopically distinct states available to them, so that the Leggett-Garg definition of macroscopic realism legggarg-1 can be applied, to separately posit deterministic macroscopic realism (dMR), as below. It would be argued that the assumption of realism is more robustly justified for macroscopically distinct states s-cat-1935 .

Weak macroscopic realism (wMR) involves weaker (i.e. less restrictive) assumptions than macroscopic local realism. Macroscopic local realism implies wMR, but the converse is not true. The assertions for wMR are modified so that EPR’s local realism applies to the systems after the selection of the measurement settings, at time $t_{f}$ in the Figure 2.

From (4), it is clear that the outcomes of spin-$z$ measurements on each particle are anticorrelated. Similarly, we may measure the component $\sigma_{y}$ of each particle. To predict the outcomes, we transform the state into the $y$ basis, noting the transformation

	$\displaystyle|\uparrow\rangle_{y}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}}(|\uparrow\rangle_{z}+i|\downarrow\rangle_{z})$
	$\displaystyle|\downarrow\rangle_{y}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}}(|\uparrow\rangle_{z}-i|\downarrow\rangle_{z})$		(5)

where $|\uparrow\rangle_{y}$ and $|\downarrow\rangle_{y}$ are the eigenstates of $\sigma_{y}$, with respective eigenvalues $+1$ and $-1$. Hence we find $|\uparrow\rangle_{z}=(|\uparrow\rangle_{y}+|\downarrow\rangle_{y})/\sqrt{2}$ and $|\downarrow\rangle_{z}=-i(|\uparrow\rangle_{y}-|\downarrow\rangle_{y})/\sqrt{2}$. The state becomes in the new basis

$\displaystyle|\psi_{B}\rangle$

$\displaystyle=$

$\displaystyle\frac{i}{\sqrt{2}}(|\uparrow\rangle_{y}|\downarrow\rangle_{y}-|\downarrow\rangle_{y}|\uparrow\rangle_{y}).$

(6)

Denoting the respective measurements at each site by $\sigma_{y}^{A}$ and $\sigma_{y}^{B}$, we see that the spin-$y$ outcomes at $A$ and $B$ are also anticorrelated.

An EPR-Bohm paradox follows from the following argument (Figure 1). By making a measurement of $\sigma_{z}^{B}$ on particle $B$, the outcome for the measurement $\sigma_{z}^{A}$ on particle $A$ is known with certainty. EPR present their Assertions of local realism (LR) epr , summarized in Section II.A.1. Invoking EPR Assertion LR (2) that there is no disturbance to system $A$ due to the measurement at $B$, EPR’s Assertion LR (1) therefore implies system $A$ can be ascribed a hidden variable $\lambda_{z}^{A}$ mermin-ghz , which predetermines the outcome for the measurement $\sigma_{z}^{A}$ should that measurement be performed. However, the outcomes at $A$ and $B$ are also anticorrelated for measurements of $\sigma_{y}$ at both sites. The assumption of EPR’s premises therefore ascribes two hidden variables $\lambda_{z}^{A}$ and $\lambda_{y}^{A}$ to the system $A$, which simultaneously predetermine the outcome of either $\sigma_{z}$ or $\sigma_{y}$ at $A$, should either measurement be performed. This description is not compatible with any quantum wavefunction $|\psi\rangle$ for the spin $1/2$ system $A$. The conclusion is that if EPR’s local realism is valid, then quantum mechanics gives an incomplete description of physical reality.

The above conclusions draw on the assumption that the subsystem $A$ is described quantum mechanically as a spin $1/2$ system. For such a system, the Pauli spin variances defined by $(\Delta\sigma_{i})^{2}=\langle\sigma_{i}^{2}\rangle-\langle\sigma_{i}\rangle^{2}$ satisfy the uncertainty relation $(\Delta\sigma_{x})^{2}+(\Delta\sigma_{y})^{2}+(\Delta\sigma_{z})^{2}\geq 2$ hofmann-take . Since $(\Delta\sigma_{z})^{2}\leq 1$, this implies hofmann-take

$$(\Delta\sigma_{y})^{2}+(\Delta\sigma_{z})^{2}\geq 1.$$

(7)

For a quantum state description of the system $A$, the values of $\sigma_{y}$ and $\sigma_{z}$ cannot be simultaneously precisely defined. A realisation has been given for two spin $1/2$ particles (photons) which showed near-perfect correlation for both of two orthogonal spins (orthogonal linear polarizations), for a subensemble where both photons are detected aspect-Bohm .

A stricter argument not dependent on the assumption of a spin $1/2$ system is possible, if the experimentalist can measure the correlation of all three spin components Bohm ; epr-rmp ; bohm-test-uncertainty . Consider the spin-$x$ measurements $\sigma_{x}^{A}$ and $\sigma_{x}^{B}$. The eigenstates of $\sigma_{x}$ are

	$\displaystyle|\uparrow\rangle_{x}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}}(|\uparrow\rangle_{z}+|\downarrow\rangle_{z})$
	$\displaystyle|\downarrow\rangle_{x}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2}}(|\uparrow\rangle_{z}-|\downarrow\rangle_{z}).$		(8)

The state (4) becomes in the spin-$x$ basis

$\displaystyle|\psi_{B}\rangle$

$\displaystyle=$

$\displaystyle\frac{1}{\sqrt{2}}(|\downarrow\rangle_{x}|\uparrow\rangle_{x}-|\uparrow\rangle_{x}|\downarrow\rangle_{x}).$

(9)

The spin-$x$ outcomes at $A$ and $B$ are also anticorrelated. According to the EPR premises, it is therefore possible to assign a hidden variable $\lambda_{x}^{A}$ to the subsystem $A$ that predetermines the outcome of the measurement $\sigma_{x}^{A}$. Hence, local realism implies that the system $A$ would at any time be described by three precise values, $\lambda_{x}^{A}$, $\lambda_{y}^{A}$ and $\lambda_{z}^{A}$, which predetermine the outcomes of measurements $\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ respectively. Each of $\lambda_{x}$, $\lambda_{y}$ and $\lambda_{z}$ has the value $+1$ or $-1$ . Since always $|\lambda_{z}^{A}|=1$, such a hidden variable description cannot be given by a local quantum state $|\psi\rangle$ of $A$, as this would be a violation of the quantum uncertainty relation

$$\Delta\sigma_{x}\Delta\sigma_{y}\geq|\langle\sigma_{z}\rangle|,$$

(10)

which applies to all quantum states. Hence, we arrive at an EPR paradox, where local realism implies an inconsistency with the completeness of quantum mechanics.

We consider the system to be prepared at time $t_{1}$ in the entangled cat state cat-bell-wang-1 ; cat-det-map

$$|\psi_{Bell}\rangle=\mathcal{N}(|\alpha\rangle|-\beta\rangle-|-\alpha\rangle|\beta\rangle).$$

(15)

Here $|\alpha\rangle$ and $|\beta\rangle$ are coherent states for single-mode fields $A$ and $B$, and we take $\alpha$ and $\beta$ to be real, positive and large. $\mathcal{N}=\frac{1}{\sqrt{2}}\{1-\exp(-2\left|\alpha\right|^{2}-2\left|\beta\right|^{2})\}^{-1/2}$ is the normalization constant. The phase of the coherent amplitudes $\alpha$ and $\beta$ are defined as real relative to an fixed axis, which is usually defined by a phase specified in the preparation process. For example, this is may be fixed by the phase of a pump field, as in the coherent state superpositions generated by nonlinear dispersion yurke-stoler-1 .

For each system $A$ and $B$, one may measure the field quadrature phase amplitudes $\hat{X}_{A}={\color[rgb]{1,0,0}{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\frac{1}{\sqrt{2}}}}}(\hat{a}+\hat{a}^{\dagger})$, $\hat{P}_{A}={\color[rgb]{1,0,0}{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\frac{1}{i\sqrt{2}}}}}(\hat{a}-\hat{a}^{\dagger})$, $\hat{X}_{B}=\frac{1}{\sqrt{2}}(\hat{b}+\hat{b}^{\dagger})$ and $\hat{P}_{A}={\color[rgb]{1,0,0}{\color[rgb]{0,0,1}{\color[rgb]{0,0,0}\frac{1}{i\sqrt{2}}}}}(\hat{a}-\hat{a}^{\dagger})$, which are defined in a rotating frame yurke-stoler-1 . The boson destruction mode operators for modes $A$ and $B$ are denoted by $\hat{a}$ and $\hat{b}$. As $\alpha$ $\rightarrow\infty$, the probability distribution $P(X_{A})$ for the outcome $X_{A}$ of the measurement $\hat{X}_{A}$ consists of two well-separated Gaussians which can be associated with the distributions for the coherent states $|\alpha\rangle$ and $|-\alpha\rangle$. (Any central component due to interference vanishes for large $\alpha$, $\beta$). Hence, the outcome $X_{A}$ distinguishes between the states $|\alpha\rangle$ and $|-\alpha\rangle$. Similarly, $\hat{X}_{B}$ distinguishes between the states $|\beta\rangle$ and $|-\beta\rangle$.

We define the outcome of the “spin” measurement $\hat{S}^{A}$ to be $S^{A}=+1$ if $X_{A}\geq 0$, and $-1$ otherwise. Similarly, the outcome of the measurement $\hat{S}^{B}$ is $S^{B}=+1$ if $X_{B}\geq 0$, and $-1$ otherwise. The result is identified as the spin of the system i.e. the qubit value. For each system, the coherent states become orthogonal in the limit of large $\alpha$ and $\beta$, in which case the superposition maps to the two-qubit Bell state $|\psi_{Bell}\rangle=\frac{1}{\sqrt{2}}(|+\rangle_{a}|-\rangle_{b}-|-\rangle_{a}|+\rangle_{b})$, given by (4). At time $t_{1}$, the outcomes $S^{A}$ and $S^{B}$ are anticorrelated.

In order to realise the EPR-Bohm paradox, it is necessary to identify the noncommuting spin observables and the appropriate unitary rotations $U$ at each site required to measure these. For this purpose, we examine the systems $A$ and $B$ as they evolve independently according to local transformations $U_{A}(t_{a})$ and $U_{B}(t_{b})$, defined as

$$U_{A}(t_{a})=e^{-iH_{NL}^{A}t_{a}/\hbar},\ \ U_{B}(t_{b})=e^{-iH_{NL}^{B}t_{b}/\hbar}$$

(16)

where

$$H_{NL}^{A}=\Omega\hat{n}_{a}^{k},\ \ H_{NL}^{B}=\Omega\hat{n}_{b}^{k}.$$

(17)

Here, $t_{a}$ and $t_{b}$ are the times of evolution at each site, $\hat{n}_{a}=\hat{a}^{\dagger}\hat{a}$ and $\hat{n}_{b}=\hat{b}^{\dagger}\hat{b}$, and $\Omega$ is a constant. We consider $k=2$, noting that $k=4$ allows a Bell test manushan-bell-cat-lg . The dynamics of this evolution is well known yurke-stoler-1 ; collapse-revival-bec-2 ; collapse-revival-super-circuit-1 ; wright-walls-gar-1 . If the system $A$ is prepared in a coherent state $|\alpha\rangle$, then after a time $t_{a}=\pi/2\Omega$ the state of the system $A$ becomes manushan-cat-lg ; manushan-bell-cat-lg ; macro-bell-lg ; yurke-stoler-1 ; cat-states-super-cond ; cat-states-mirrahimi

$\displaystyle U_{\pi/4}^{A}|\alpha\rangle$

$\displaystyle=$

$\displaystyle e^{-i\pi/4}(|\alpha\rangle+i|-\alpha\rangle)/\sqrt{2}.$

(18)

Here we define $U_{\pi/4}^{A}=U_{A}(\pi/2\Omega)$. A similar transformation $U_{\pi/4}^{B}$ is defined at $B$ for $t_{b}=\pi/2\Omega$. This state is a superposition of two macroscopically distinct states, and is referred to as a cat state after Schrödinger’s paradox s-cat-1935 ; scat-rmp-frowis . Further interaction for the whole period $t_{a}=2\pi/\Omega$ returns the system to the coherent state $|\alpha\rangle$.

The macroscopic version of the EPR-Bohm paradox is depicted in Figure 3. We consider the spin-$1/2$ observables $\hat{S}_{z}=|+\rangle\langle+|-|-\rangle\langle-|$, $\hat{S}_{x}=|+\rangle\langle-|+|-\rangle\langle+|$ and $\hat{S}_{y}=\frac{1}{i}(|+\rangle\langle-|-|-\rangle\langle+|)$, defined for orthogonal states $|\pm\rangle$ of a two-level system, which we also denote by $|\uparrow\rangle_{z}$ and $|\downarrow\rangle_{z}$. Here, we identify the eigenstates $|\pm\rangle$ of $\hat{S}_{z}^{A}$ ($\hat{S}_{z}^{B}$) as the coherent states $|\pm\alpha\rangle$ ($|\pm\beta\rangle$) respectively, with $\alpha$ and $\beta$ real, and in the limit of large $\alpha$ and $\beta$ where orthogonality is justified. In this limit, we define

	$\displaystyle\hat{S}_{z}^{A}$	$\displaystyle=$	$\displaystyle|\alpha\rangle\langle\alpha|-|-\alpha\rangle\langle-\alpha|$
	$\displaystyle\hat{S}_{x}^{A}$	$\displaystyle=$	$\displaystyle|\alpha\rangle\langle-\alpha|+|-\alpha\rangle\langle\alpha|$
	$\displaystyle\hat{S}_{y}^{A}$	$\displaystyle=$	$\displaystyle\frac{1}{i}(|\alpha\rangle\langle-\alpha|-|-\alpha\rangle\langle\alpha|)$		(19)

for system $A$. The scaling corresponds to Pauli spins $\overrightarrow{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$. The spins $\hat{S}_{z}^{B}$, $\hat{S}_{x}^{B}$ and $\hat{S}_{y}^{B}$ for system $B$ are defined in identical fashion on replacing $\alpha$ with $\beta$. We omit operator “hats”, where the meaning is clear.

The EPR-Bohm paradox requires measurement of $S_{z}^{A}$ and $S_{z}^{B}$. The system in the state (15) is prepared for the pointer stage of the measurements of $S_{z}$. This is because for this system, a measurement of (the sign of) $\hat{X}_{A}$ and $\hat{X}_{B}$ is all that is required to complete the $S_{z}^{A}$ and $S_{z}^{B}$ measurement. The local measurement constitutes an optical homodyne, in which the fields are combined with a strong field across a beamsplitter with a relative phase shift $\vartheta$, followed by direct detection in the arms of the beam splitter yurke-stoler-1 . Here, $\vartheta$ is chosen to measure $\hat{X}_{A}$ ($\hat{X}_{B}$), the axis so that $\alpha$ ($\beta$) as real. The $\vartheta$ is defined by the preparation process, usually involving a pump field.

The EPR-Bohm argument also requires measurements of $S_{y}^{A}$ and $S_{y}^{B}$ on the Bell state (15) prepared at time $t_{1}$ (Figure 3). Here, it is required to adjust the measurement-setting by applying a local unitary transformation $U_{y}$ at each site.

The eigenstates of $\hat{S}_{y}$ are often written in the form $|\uparrow\rangle_{y}=(|\uparrow\rangle_{z}+i|\downarrow\rangle_{z})/\sqrt{2}$ and $|\downarrow\rangle_{y}=(|\uparrow\rangle_{z}-i|\downarrow\rangle_{z})/\sqrt{2}$, but the normalization can vary by a phase factor. We can abbreviate as $|\pm\rangle_{y}=\frac{1}{\sqrt{2}}(|\pm\rangle+i|\mp\rangle)$, denoting $|\uparrow\rangle$ as $|+\rangle$, and $|\downarrow\rangle$ as $|-\rangle$, interchangeably. We choose

	$\displaystyle|\uparrow\rangle_{y}$	$\displaystyle=$	$\displaystyle\frac{e^{-i\pi/4}}{\sqrt{2}}(|\uparrow\rangle_{z}+i|\downarrow\rangle_{z})$
	$\displaystyle|\downarrow\rangle_{y}$	$\displaystyle=$	$\displaystyle\frac{e^{-i\pi/4}}{\sqrt{2}}(|\downarrow\rangle_{z}+i|\uparrow\rangle_{z})=\frac{e^{i\pi/4}}{\sqrt{2}}(|\uparrow\rangle_{z}-i|\downarrow\rangle_{z}),$

and will denote the eigenstates at different sites by a subscript. We have temporarily dropped for convenience the superscripts and subscripts indicating the $A$ and $B$, since the transformations are local and apply independently to both sites. It is readily verified that the $\hat{S}_{y}|\uparrow\rangle_{y}=|\uparrow\rangle_{y}$ and $\hat{S}_{y}|\downarrow\rangle_{y}=|\downarrow\rangle_{y}$ i.e. $\hat{S}_{y}^{A}|\pm\rangle_{y,A}=\pm|\pm\rangle_{y,A}$ and $\hat{S}_{y}^{B}|\pm\rangle_{y,B}=\pm|\pm\rangle_{y,B}$.

Now we consider how to perform the measurement of $\hat{S}_{y}$. As explained in Section II, the first stage of measurement involves a unitary operation $U_{y}$, giving a transformation to the measurement basis, so that the system is then prepared for the second stage of measurement, which is the “pointer [stage of] measurement” of $\hat{S}_{y}$. The pointer stage constitutes a measurement of the sign $\hat{S}$ of $\hat{X}$, which for large $\alpha$ ($\beta$) will (after $U_{y}$ has been applied) directly yield the outcome $\pm 1$ for the system prepared in $|\pm\rangle_{y}$. To establish $U_{y}$, following the procedure of Eqs. (4-6) and (11-13), any state

$$|\psi\rangle=c_{+}|\uparrow\rangle_{z}+c_{-}|\downarrow\rangle$$

(21)

written in the $z$ basis can be transformed into the $y$ basis, by substituting

	$\displaystyle|\uparrow\rangle_{z}$	$\displaystyle\rightarrow$	$\displaystyle(e^{i\pi/4}|\uparrow\rangle_{y}+e^{-i\pi/4}|\downarrow\rangle_{y})/\sqrt{2}$
	$\displaystyle|\downarrow\rangle_{z}$	$\displaystyle\rightarrow$	$\displaystyle-i(e^{i\pi/4}|\uparrow\rangle_{y}-e^{-i\pi/4}|\downarrow\rangle_{y})/\sqrt{2},$		(22)

This gives

$$|\psi\rangle=d_{+}|+\rangle_{y}+d_{-}|-\rangle_{y}$$

(23)

where $d_{\pm}=(c_{\pm}\mp ic_{-})e^{\pm i\pi/4}$. To obtain the transformed state (23), ready for the pointer stage of measurement of $S_{y}$, the system is thus evolved according to

$$U_{y}|\psi_{Bell}\rangle$$

(24)

where $U_{y}\equiv U_{\pi/4}^{-1}=U^{-1}(\pi/2\Omega)$. We explain this result further in the Appendix A for the purpose of clarity.

The $U_{y}$ is the inverse of the transformation $e^{-iH_{NL}t/\hbar}$ where $t=\pi/2\Omega$, given by (18). The $U_{y}$ is achieved by evolving the local system for a time $t=-\pi/2\Omega\equiv 3\pi/2\Omega$, since the solutions are periodic.

The EPR-Bohm argument is as follows (Figure 3). Consider the system prepared in the Bell state (15) at time $t_{1}$ ($\alpha$, $\beta\rightarrow\infty$). One first measures $S_{z}^{A}$ and $S_{z}^{B}$ for this state at time $t_{1}$. The anticorrelation of the Bell state means that the result for $S_{z}^{A}$ at $A$ can be predicted with certainty by the measurement at $B$.

The EPR-Bohm argument continues, by considering measurements of $S_{y}^{A}$ and $S_{y}^{B}$ on the Bell state (15) prepared at time $t_{1}$ (Figure 3). The measurement of $S_{y}^{A}$ ($S_{y}^{B}$) is thus made by applying the local unitary rotation $U_{y}^{A}$ ($U_{y}^{B}$) to the Bell state prepared at $t_{1}$, followed by a measurement of the sign of $X_{A}$ ($X_{B}$). The state of the system prepared after the unitary rotations $U_{y}^{A}$ and $U_{y}^{B}$ is also given as the Bell state (15), which we write as

$$|\psi_{Bell}\rangle_{y,y}=\frac{1}{\sqrt{2}}(|-\rangle_{y}|+\rangle_{y}-|+\rangle_{y}|-\rangle_{y})$$

(25)

where we have taken $\alpha$, $\beta$ large. As with the original paradox given in Section III.A.1, as seen by Eq. (6), the final measurements of $X_{A}$ and $X_{B}$ therefore reveal an anticorrelation between $S_{y}^{A}$ and $S_{y}^{B}$, and the result for $S_{y}^{A}$ can be revealed, with certainty, by measurement of $S_{y}^{B}$ at site $B$. Here, we note the Comment in the above section, that $|\pm\rangle_{y}$ are the macroscopically distinct coherent states $|\alpha\rangle$ and $|-\alpha\rangle$ (or $|\beta\rangle$ and $|-\beta\rangle$) that are realised at the time $t_{f}$ in Figure 3, which corresponds to the time after the transformation $U_{y}$ has been carried out at each location.

The Bohm-EPR argument continues. The correlation between the spins enables an experimentalist at $B$ to determine with certainty either $S_{z}^{A}$ or $S_{y}^{A}$, for the system prepared at $t_{1}$, by choosing the suitable measurement at the site $B$. Assuming EPR’s local realism, this implies that both the spins of system $A$ are predetermined with certainty. Following along the lines of the two-spin paradox of Section III.A.1, this constitutes Bohm’s EPR paradox, because it is not possible to define a local quantum state $\varphi$ for the spin $1/2$ system $A$ at the time $t_{1}$ with simultaneously specified values for both $\hat{S}_{z}^{A}$ and $\hat{S}_{y}^{A}$. The paradox is the inconsistency between macroscopic local realism and the completeness of quantum mechanics.

In the gedanken experiment, it is assumed that the system $A$ is described quantum mechanics as a spin $1/2$ system, which is valid as $\alpha\rightarrow\infty$, where the two coherent states $|\alpha\rangle$ and $|-\alpha\rangle$ are orthogonal. In the same limit, the spin outcomes for $\hat{S}_{z}^{A}$ and $\hat{S}_{z}^{B}$, and also for $\hat{S}_{y}^{A}$ and $\hat{S}_{y}^{B}$, are perfectly anticorrelated, so that this realisation of the EPR-Bohm paradox strictly follows in the macroscopic limit, where $\alpha\rightarrow\infty$. Proposals for finite $\alpha$ that also account for imperfect anticorrelation of the spins are presented in Appendices C and D.