A Commentary on Bohm's Discussion of Einstein-Rosen-Podolsky (EPR) Paper

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Bohm's 1957 paper “Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky” co-authored by Aharonov is about the nature of correlations that were the subject of the EPR paper.

They introduce a model to explain EPR correlations. According to that model the quantum mechanical (QM) description of many-body system breaks down when the systems are separated and the parties acquire a separable state which produces the same correlation on average whereas individual pairs are not always anti-correlated anymore. Then they remind an already done experiment that gives different observable results for this separable model and quantum entangled model. The experiment is in agreement with QM description.

Bohm uses an entanglement of discrete quantities, spin of spin-1/2 electrons or polarization of photons, to be able to have an experimental setup that is easier to implement than the original Gedanken experiment suggestion of EPR that involves entanglement of continuous DoF's such as position and momentum. Discrete version is mathematically simpler too.

Instead of

\[\Psi\left(x_{1},x_{2}\right)=\int_{-\infty}^{\infty}e^{i\left(x_{1}-x_{2}+x_{0}\right)p/\hbar}dp\]
Bohm proposes to use the singlet state
\[\psi=\frac{1}{\sqrt{2}}\left[\psi_{+}\left(1\right)\psi_{-}\left(2\right)-\psi_{-}\left(1\right)\psi_{+}\left(2\right)\right]\]
in our notation
\[\ket{\Psi}=\left(\ket{u+u-}-\ket{u-u+}\right)/2\]
where $u$ is an arbitrary direction. The reason of perfect correlation in momenta in the case of EPR state was the fact that it was a state of zero total momentum. In the singlet state the total spin angular momentum is zero. Hence the existence of correlation is independent of which direction is used for the spin measurement. As long as both parties are measured at the same direction the outcomes are perfectly anti-correlated, hence $u$ is arbitrary.

"Before the measurement has taken place (even while the atoms are still in flight) we are free to choose any direction as the one in which the spin of particle $A$ (and therefore of particle $B$) will become definite." This idea of changing the measurement direction on the flight is the basis of the Bell experiments that closed "communication loophole".


Bohm mentions and old and incorrect interpretation of the uncertainty principle as a representation of the disturbance effects of the measurement on incompatible quantities. “This point of view more generally implies that the definiteness of any desired component of the spin is (along with the indefiniteness of the other two components) a potentiality which can be realized with the aid of a suitably oriented spin-measuring apparatus” Bohm does not insist on using a concept such as element of reality (EoR) but simply uses the term “definiteness”. Definiteness is humbler than reality in scope and is a cleaner concept in definiton. According to this view wave-like and particle-like behaviors of an electron are different and mutually incompatible potentialities which are realized by applying some certain experimental setups.

Bohm says that this disturbance based explanation is not satisfactory in EPR experiment. Because of the lack of the direct physical interaction between the party-2 and the apparatus-1, why party-2 “realizes its potentiality for a definite spin is precisely the same direction as that of” party-1, cannot be explained.

At the time when the paper is written quantum optics and other means of manipulating individual quantum systems was not developed enough to directly implement the EPR Gedanken experiment or any other setup that will create EPR correlations. Hence Bohm feels free to develop alternative models to “yet experimentally unverified extrapolation of the many-body Schrödinger and Dirac equations to the case where the particle's wave functions do not overlap”. Einstein proposed the idea that “QM may break down when particles are far enough apart” in a private communication with Bohm.

All these detailed discussion and reasonable arguments against QM amazes me about QM's accuracy. The validity region of QM is so broad. Even though when it was put forward the experimental abilities were not enough to test all of its implications hence it basically consisted of speculations, those speculations turned out to be correct in the course of the history thanks to the deep intuitions of the founders.

The alternative model is proposed by Furry 1936. In that model, when particles “loose contact” wave function turns into a product state of zero total spin, as if entanglement is some kind of thread that snaps when pulled from both sides,

\[\psi=\psi_{+\theta,\varphi}\left(1\right)\psi_{-\theta,\varphi}\left(2\right)\]
\[\ket{\psi}=\ket{v+}\ket{v-}\]
where $v$ is an arbitrary unit vector. The down side of this state is that, now spin angular momentum is zero only for the $v$ direction. In other two orthogonal directions total spin fluctuates and is not zero for individual pairs.
\begin{eqnarray*}\sqrt{2}\sigma_{v}^{T}\ket{\psi} & = & \left(\sigma_{v}\otimes I+I\otimes\sigma_{v}\right)\ket{v+}\otimes\ket{v-}\\ & = & \sigma_{v}\ket{v+}\otimes I\ket{v-}+I\ket{v+}\otimes\sigma_{v}\ket{v-}\\ & = & +\ket{v+}\otimes I\ket{v-}-\ket{v+}\otimes\ket{v-}\\ & = & 0\ket{\psi}\end{eqnarray*}
\begin{eqnarray*}\sqrt{2}\sigma_{x}^{T}\ket{\psi} & = & \sigma_{x}\ket{z+}\otimes I\ket{z-}+I\ket{z+}\otimes\sigma_{x}\ket{z-}\\ & = & \ket{z-}\ket{z-}+\ket{z+}\ket{z+}\end{eqnarray*}

Half of the time they are both down and other half of the time both up so that on average total spin is zero too but not in individual cases. This is correct for only $x$ (or $y$) directions. For other directions the total spin will not be zero. So, the additional assumption is that, $v$ is a random direction with a uniform distribution in all solid angle.


In a future post I will try to clarify what the constraints of local reality imposes on the correlations between two spatially separated systems with two measurements both of which has two outcomes.

The only advantage of the this model is to avoid the so-called EPR paradox, the instantaneous interaction needed to explain the correlations.

But an experiment that refutes this model is shown hence the “paradox” cannot be resolved by breaking down QM for far away particles.

This is a moment of enlightenment about the nature of uncertainty principle: “we can no longer retain the notion that a definite value of a given variable is essentially realized through interaction with an appropriate apparatus, and that the indeterminacy principle represents only an uncontrollable fluctuation in complementary variables brought about by a disturbance originating in the apparatus.” Which means that the uncertainty principle is not an arbitrary lower limit on the amount of the disturbances due to making measurement, a limit on the means of gaining knowledge about systems, but it is an inherent constraint about the nature of the states that a physical system can have and it is independent of the measurement processes.

First view implies that the disturbance due to the mechanism of the apparatus a formerly definite quantity will have a value that is unknown until it is going to be measured, which will just give a probability distribution about the prior information about the system. This situation is different than a superposition, which is an inherent indefiniteness.

Bohm mentions Bohr's ideas to overcome the paradox. Once I tried to read Bohr's response to EPR paper, but it was so unintelligible. :-( What Bohr says, basically, according to Bohm, is that we are not allowed to ask questions that leads to the paradox, hence there is no paradox. Voila! “the observing apparatus plus what is observed form a single indivisible combined system not capable at the quantum-level of being analyzed correctly into separate and distinct parts” The apparatus and system at hand cannot be analyzed further, QM just lets us calculate the outcome probabilities using the “recipe”, and how those probabilities emerge is outside the realm of Bohr's QM and hence meaningless. It does not matter whether the system at hand is a single particle system or an entangled system of many particles or a system and an apparatus etc.

For sure, I do not like this approach. The combination of the system and the apparatus basically includes entanglement too, namely, the gauge of the apparatus is in one-to-one correlation with the quantity that is measured. And entanglement is basically the quantum version correlations. Though, the answer to the problem of how the superposition in that entangled combined system “collapses” to one of its terms, the infamous measurement problem, is still unknown.

Then Bohm mentions his own view that suggests a “deeper exploration of the quantum theory as a whole”. He agrees to treat the setup as a single combined system but says that single combined system may be subject to further analysis, according to his “casual interpretation of quantum theory” namely the Bohmian Mechanics (BM), where he introduces a hidden interaction via the so-called “quantum potential” that allows an interaction “even when their classical interaction potential is zero.” “It must be admitted, however, that this quantum potential seems rather artificial in form, besides being subject to the criticism ... that it implies instantaneous interactions between distant particles, so that it is not consistent with the theory of relativity”.

I am not knowledgeable in BM. But it looks like Bohm himself thinks that his framework is more like a mathematical data fitting than an actual physical theory because it is in conflict with special relativity. But I wonder whether it is really so. Does BM obeys no-signaling condition? If yes, then there is no contradiction.

Bohm also envisage a deeper theory underlying QM. “there has been developed a further new explanation of the quantum theory in terms of a deeper subquantum-mechanical level. The laws of this lower level are different from those of the quantum theory, but approach these latter laws as an approximation, much as the laws of atomic physics approach those of macroscopic physics when many atoms are involved”. Sound exciting!

Then Bohm describes the experiment that supports the existence of EPR-like correlations. Instead of working with spin-1/2 particles he proposes to use photons of which polarization DoF is effectively two dimensional too. The system at hand is “photons ... produced in the annihilation radiation of a positron-electron pair”. They have zero total momentum and their polarization is always orthogonal. He describes the math of bosons that uses creation and annihilations operators on the vacuum state to describe the state of the system that lives in Fock space, which is a bit more involved that the fermion version where number of particles is constant and the state lives in Hilbert space.


Most general state for a photon of wave number $k$ moving along $z$-axis is
\[\psi=rC_{k}^{x}\psi_{0}+sC_{k}^{y}\psi_{0}\]
\[\ket{\psi}=ra_{x}^{\dagger}\ket{\phi}+sa_{y}^{\dagger}\ket{\phi}\]
where $\ket{\phi}$ is the vacuum state with no photons, $a_{i}^{\dagger}$ are creation operators,$ $ $i$ indicates the direction of linear polarization. $r$ and $s$ are complex numbers. Then analogue of the singlet state will be
\begin{eqnarray*}\ket{\Psi} & = & \frac{1}{\sqrt{2}}\left(a_{x}^{\dagger}b_{y}^{\dagger}-a_{y}^{\dagger}b_{x}^{\dagger}\right)\ket{\phi}\\ & = & \frac{1}{\sqrt{2}}\left(a_{x'}^{\dagger}b_{y'}^{\dagger}-a_{y'}^{\dagger}b_{x'}^{\dagger}\right)\ket{\phi}\\ & \equiv & \frac{1}{\sqrt{2}}\left(\ket H\ket V-\ket V\ket H\right)\end{eqnarray*}
where $a$ is for left going photon and $b$ for right going. Primed coordinates are rotated coordinate around the $z$-axis. Last line shows the mathematical equivalency of this polarization state to the spin singlet state.

Because at that time it was experimentally not feasible to measure whether each and every pair of photons produced by the annihilation have orthogonal polarizations, Bohm applies to another experiment that has been done by Wu in 1950 that tests the existence of EPR-correlations in a more indirect way. ``This experiment consists in measuring the relative rate, $R$, of coincidences in the scattering of the two photons through some angle, $\theta$'' It is involved with the scattering theory of photons in a $e^{-}e^{+}\rightarrow\gamma\gamma$ scattering. I have shamefully no idea about the physics of this process. :-(

They calculate $R$ using 3 different assumptions:
A) QM is correct and particles are in the singlet state $\frac{1}{\sqrt{2}}\left(\ket H\ket V-\ket V\ket H\right)$

B1) Entanglement breaks down, particles are circularly polarized in opposite directions. $\left(\ket H+i\ket V\right)\left(\ket H-i\ket V\right)$.
B2) Entanglement breaks down, particles are linearly polarized in uniformly distribute random opposite directions $u$, $\ket{H_{u}}\ket{V_{u}}$.


B is the case where combined angular momentum is conserved only for the average of all pairs but not for individual pairs.

They call the plane that is spanned by the initial direction of motion of photon and direction after scattering $\pi$.

To do the calculations they use the scattering cross section of a single photon from an electron, according to the Klein-Nishina formula it is
\begin{eqnarray*}d\sigma_{1} & = & \frac{1}{2}r_{0}d\Omega\left(k^{2}/k_{0}^{2}\right)\left(\gamma-2\sin^{2}\theta\right)\\d\sigma_{2} & = & \frac{1}{2}r_{0}d\Omega\left(k^{2}/k_{0}^{2}\right)\gamma\end{eqnarray*}
where $\gamma=k_{0}/k+k/k_{0}$, $k_{0}$ is the wave number of the incident photon, $k$ that of the final photon, $r_{0}$ is the classical electronic radius, and $d\Omega$ is the element of solid angle. 1 is the case where initial direction of polarization is parallel to the plane $\pi$ and 2 is the case where it is perpendicular.

A general initial polarization state will be a linear combination of $\ket{\psi_{i}}=\left\{ \ket{HH},\ket{HV},\ket{VH},\ket{VV}\right\} $, $\ket{\Psi}=\sum_{i}c_{i}\ket{\psi_{i}}$. And in the most general case the calculation of cross-section contains terms $c_{i}c_{j}$ where $i\neq j$. But for the special case where $\pi_{1}$ and $\pi_{2}$ of two photons are parallel or perpendicular, those terms ``drop out from the expression for the probability of scattering''. So, if both $\pi_{i}$'s are placed on $zx$-plane, or, one of them on $zx$-and the other one on $zy$-plane, then the lack of crossing terms allows one to calculate the probabilities for individual $\ket{\psi_{i}}$ and then sum their contributions. And, ``for each case, $\ket{\psi_{i}}$, ... this probability reduces to just the product of the probabilities of scattering of the single photons''. Which simplifies the calculations!


Calculated value of $R$ (average for experimental solid angle) for the case A is $2.00$, case B1 $1.00$, B2 $1.5$. The experimental value is $2.04\pm0.08$.









This is a strong argument for the existence of singlet state for distant particles and against the “hypotheses implying a breakdown of the quantum theory that could avoid paradox of ERP”.

I think this article is a good example of clarifying philosophical concepts and carrying them into the realms of physics by quantifying them. It supports the existence of the controversial singlet state by finding experimental discrepancies between the observational fact and alternative models. It demonstrates a usage of scattering experiments in a “fundamentals experiment” which can be used to test the existence of entanglement in a two photon system. I wonder whether further applications can be found for the same physical setup.

One other unsolved mystery that was left is why Bohm changed the order of initials from EPR to ERP?

A Commentary on Einstein-Podolsky-Rosen (EPR) Paper

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EPR makes an argument about incompleteness of quantum mechanics (QM). Instead of accepting the randomness (an unavoidable outcome of uncertainty principle) as an ontological fact about the nature, they think of probability distributions an epistemological weakness of quantum theory.

According to them, there are some elements which exist in physical reality but does not have corresponding parts in the quantum theory. Therefore, even though QM is correct in their predictions, because it does not involve all information that can be gained about the physical reality, and because it works with partial information, its predictions are probabilistic.

They want to explain the superposition states and randomness due to superpositions via missing parts in the theory. That explanation solves the measurement problem too. It reduces the collapse of the wave function to an update of prior information about the system in Bayesian sense!

Their definition of completeness is as such: "every element of the physical reality must have a counterpart in the physical theory''. It sounds a bit vague until the "element of reality'' (EoR) is defined. Here is their definition: "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.'' They will claim that there are EoRs out there that are not represented in QM.

This is a definition of a hard to explain but intuitive notion and it is deeply soaked in the classical physical paradigm. A simpler way to put it might be: "any quantity that is assigned a real number is an element of reality''. "Disturbing the system while making the measurement'' and "amount of certainty'' etc. makes the definition bulky. The basic idea is that in a classical system each quantity is assigned a real number at any time.

Classically it is possible to measure anything without disturbing the system. For example we want to measure the temperature of a pool. Immerse a thermometer into the pool and read the mercury level. But, wait! In order to measure the temperature, to move the level of the mercury column to the final position, there has to be a heat exchange between two systems, hence there has to be a disturbance, right?

In principle it is possible to take the limit of this disturbance to zero. One can imagine to scale down the thermometer and to make it more and more sensitive so that the disturbance can be made arbitrarily small.

Or the amount of disturbance can be calculated later and be subtracted from the final result. If one knows the masses, heat capacities of the two interacting systems and the initial temperature of the thermometer, it is possible to figure out the initial temperature of the pool before the measurement.

In the classical realm where matter is continuous and is able to be prepared in any proportions it is possible to produce information about it (which means to encode the information about its state in another physical system) without altering its state. That is not the case in quantum realm. It is possible to copy bits in a computer memory without changing them. But, in general, it is impossible to copy quantum bits if its information content is not known beforehand, due to no-cloning theorem.

A more modern definition of EoR can be "any quantity that is assigned a ray in a Hilbert space is an element of reality''. (Actually an interesting question might be "which quantities can be in a superposition?'' Degrees of Freedoms (DoF) for sure. But AFAIK quantities such as the mass of an electron, or energy levels of hydrogen atom can't be in a superposition.)

EPR applies their definition of EoR -let's call their definition EPRR- to QM. According to EPRR definition, when a system is in the state $\ket a$, observables for which $\ket a$ is one of their eigenfunction are elements of reality. The value of EPRR is the corresponding eigenvalue.

\[ A\ket a=a\ket a \]

If $\ket a$ is not an eigenfunction of an observable, which is the case of a $B$ for which $\left[A,B\right]\neq0$, then that quantity does not have an EPRR, for that state.

\[B\ket a\neq b\ket a\]

This approach that associates definiteness (being determined) with EoR creates a strange sense of reality where the quantities' being an element of reality is state dependent. Through a time evolution, or after a measurement, a quantity can become a part of the reality or quit being part of the reality.

For EPR the weirdness of this situation is the reason to blame QM for being incomplete, whereas I believe that it is an indication of ill-posed-ness of the definition. Instead we should reconcile with the inherent indeterminacy of the nature and should not associate the EoR with $p=1$ certainty in predictions.

From this point of view the correct problem is "how the real number approximation emerges from this new notion of reality with rays in Hilbert space''. An explanation that is similar to the one that explains how Newtonian mechanics emerge from special relativistic space-time etc. is needed. It is a fancier way of asking how classical mechanics emerges from underlying quantum mechanics.

A way to give $B$ an EoR is to measure it. After the measurement the system will be in one of the eigenstates.
\[\ket a\xrightarrow{B}\ket b\]

\[B\ket b=b\ket b\]

But this disturbance changes the EPRR of $A$. Because $A\ket b\neq a\ket b$ $A$ lost being an EPRR. "in quantum mechanics ... when the momentum of a particle is known, its coordinate has no physical reality''

According to EPR the incompleteness imply the existence of EPRR for all quantities. But that cannot be the case. A $B$ measurement does not only alter the epistemological state in a way that destroys the knowledge about $A$, but it alters the EPRR of $A$ too. A successive measurements of $B$ and $A$ may put the system into a state which is different than the initial one

\[\ket{\chi}\xrightarrow{A}\ket a\xrightarrow{B}\ket b\xrightarrow{A}\ket{a'}\]

This by itself should indicate that the quantum state is more than a representation of knowledge about the system.

The main point of the article is to force us to choose one of these two: "(1) the quantum mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality'' So, either 2) QM is complete and non-commuting observables cannot have simultaneous reality, OR, 1) QM is incomplete, non-commuting quantities have simultaneous reality.

Then they try to prove that non-commuting quantities can have EPRR and hence QM is incomplete.

Their Gedanken experiment goes like this:

• Have a system made of two parties.
• Let them interact so that they have an entangled state that will give perfect correlations for local measurements of some arbitrary quantities at two parties, i.e. the outcomes of $A$ and $B$ measurements at party-1 are perfectly correlated with the outcomes of $P$ and $Q$ measurements at party-2.

 A state that has this property is the singlet state:

\[\ket{\Psi}=\left(\ket{u+u-}-\ket{u-u+}\right)/\sqrt{2}\]

 Local measurements of the same quantities at two parties will give opposite outcomes. $\ket{u+u-}-\ket{u-u+}$$=\ket{x+x-}-\ket{x-x+}$$=\ket{y+y-}-\ket{y-y+}$. For $A=P=\sigma_{x}$ and $B=Q=\sigma_{y}$, $A=-P$ and $B=-Q$ holds always. $A$ and $P$, and, $B$ and $Q$ are perfectly anti-correlated.

• Separate the parties spatially so that at the moment of measurement they do not directly interact. (They were mistaken in judging the nature of influence between the parties due to measurements.) 
• Measure $A$ on the first party. Due to the perfect correlation the outcome of party-1 allows us to predict the value of $P$ at the party-2 with certainty.
• Or measure a quantity $B$, that does not commute with $A$, on the first party. Similarly the outcome of $B$ at party-1 determines the value of $Q$ at party-2.
• "as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions''

\begin{eqnarray*}\ket{\Psi} & \xrightarrow{A} & \ket a\otimes\ket p\\\ket{\Psi} & \xrightarrow{B} & \ket b\otimes\ket q\end{eqnarray*}

• If $\left[P,Q\right]\neq0$ this setup shows that "since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system''. "it is possible to assign two different wave functions, $\ket p$ and $\ket q$, to the same reality'' 

There are some incorrect points in this reasoning due to the classical point of view. They tend to say that the two parties do not interact, hence measurement on party-1 cannot affect party-2. Even though they see that the choice of observable to measure at party-1 changes the set of possible states of party-2 after the measurement, they do not accept this change as a real influence. Relativistic locality is not

as restrictive as they think.

According to quantum point of view two parties non-locally interact, but the interaction is not a deterministic one. The measurement outcome at party-1 is random, hence the reduced state at party-2 is random too. Randomness is needed to prevent the violation of relativistic causality in the presence of non-local influences.

Seeing the effect of party-1 measurements on party-2 is difficult because it does not affect the distributions of party-2, the $p\left(Q=q\right)$ and $p\left(P=p\right)$. The statistical averages of quantities at party-2 is independent of what was measured at party-1.

\[p\left(Q=q\right)=\sum_{a}p\left(Q=q,A=a\right),\quad \left(Q=q\right)=\sum_{b}p\left(Q=q,B=b\right) \]

\[p\left(P=p\right)=\sum_{a}p\left(P=p,A=a\right),\quad \left(P=p\right)=\sum_{b}p\left(P=p,B=b\right)\]

One gets same distributions at party-2, whether $A$ is measured at party-1 or $B$. This is called the "no-signaling condition''.

But this does not mean that two different states at party-2 due to two different measurements at party-1, $\ket q$ and $\ket p$, do represent the same reality. The distributions of quantities of party-2 are statistical phenomena. There may be different joint distributions of party-1 and -2 with the same marginals for party-2. For a fixed quantity at party-2, say $Q$. Different quantities at party-1, say $A$ and $B$ gives different correlations, $\left\langle AQ\right\rangle \neq\left\langle BQ\right\rangle $, in general.

This is expected in classical realm too. But the amount of allowed differences in these correlations by LR and QM makes a distinction between them. Namely, the maximum value for 3 positive and 1 negative correlations

\[ \left\langle AQ\right\rangle +\left\langle AP\right\rangle +\left\langle BQ\right\rangle -\left\langle BP\right\rangle\]

is different in LR and QM. The outcome statistics of quantum systems cannot be explained using classical probability distributions of some missing information. It took $1969-1935=34$ years to quantify this distinction.

They mention another possible objection to their argumentation by themselves: "One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive.'' "Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but no both simultaneously, of the quantities $P$ and $Q$ can be predicted, they are not simultaneously real. This makes the reality of $P$ and $Q$ depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.''

Here for the sake of keeping EPRR state-independent they let its definition loose. But actually just as $A$ and $B$ do not have simultaneous realities, $Q$ and $P$ do not too. The entangled state do not improve the situation with the lack of simultaneous reality but just spread the uncertainty at one party to the other party via perfect correlations.

EPR accept that the state of party-2 depends on the measurement at party-1. But, at the same time, they assume that distant local measurements at party-1 cannot affect the EPRR at party-2, hence that change in the state does not reflect an alteration in reality.

They try to show the existence of EPRR of a system by $p=1$ predictions after random outcomes of a measurement on a party that is maximally entangled with the former one. Then, what about the reality condition of this helper system, party-1? Should it be maximally entangled to another system so that we can assign EPRR to quantities that belong to it too? In order to apply the same method for EPRR of party-1 we need a third party.

First of all, using $N>2$ parties, that is not possible. Mutual entanglement is exclusive. In a 3 party system, $A$, $B$, $C$. $A$ cannot be maximally entangled with $B$ and $C$ at the same time. (Similarly, $B$ cannot be with $A$ and $C$ etc.) There are two types of maximally entangled states for a 3 party system

1.  $\ket{\mbox{GHZ}}=\left(\ket{000}-\ket{111}\right)/\sqrt{2}$
2. $\ket{\mbox{W}}=\left(\ket{100}+\ket{010}+\ket{001}\right)/\sqrt{3}$

For a $\ket{\mbox{GHZ}}$ state measurement on a party gives the same measurement outcomes of other two parties with certainty but the first party cannot be assigned an EPRR just like the $N=2$ case. $\ket{\mbox{GHZ}}$ is a genuine 3 party entanglement which does not include any 2 party entanglement.

A $\ket{\mbox{W}}$ state, despite having 3 parties, has 3 2-party entanglement but they are not maximal, hence the correlations are not perfect, $p<1$.

The conclusion is that using their scheme it is impossible to assign EPRR to all parties.

Appendix

Show that EPR's choice of the continuous state allows maximum correlation for non-commuting observables.

Correlation of momenta:

\begin{eqnarray*}\Psi\left(x_{1},x_{2}\right) & = & \int_{-\infty}^{\infty}e^{i\left(x_{1}-x_{2}+x_{0}\right)p/\hbar}dp\\& = & \int_{-\infty}^{\infty}e^{i\left(-x_{2}+x_{0}\right)p/\hbar}e^{ix_{1}p/\hbar}dp\\& = & \int_{-\infty}^{\infty}\psi_{-p}\left(x_{2}\right)u_{p}\left(x_{1}\right)dp\end{eqnarray*}

Both $\psi_{-p}\left(x_{2}\right)$ and $u_{p}\left(x_{1}\right)$

are eigenfunction of momentum operators $P=i\hbar\frac{\partial}{\partial x_{2}}$,

$A=i\hbar\frac{\partial}{\partial x_{1}}$.

After a measurement

\[\int_{-\infty}^{\infty}\psi_{p'}\left(x_{2}\right)u_{p'}\left(x_{1}\right)dp'\xrightarrow{i\hbar\frac{\partial}{\partial x_{1}}}\psi_{-p}\left(x_{2}\right)u_{p}\left(x_{1}\right)\]

When $p_{1}=p$, $p_{2}=-p$ with certainty.

Correlation of positions:

\begin{eqnarray*}\Psi\left(x_{1},x_{2}\right) & = & \int_{-\infty}^{\infty}dp\, e^{i\left(x_{1}-x_{2}+x_{0}\right)p/\hbar}\\& = & \int_{-\infty}^{\infty}dp\, e^{i\left(-x_{2}+x_{0}\right)p/\hbar}e^{ix_{1}p/\hbar}\\& = & \int_{-\infty}^{\infty}dp\, e^{i\left(-x_{2}+x_{0}\right)p/\hbar}\int_{-\infty}^{\infty}dp'\,\delta\left(p-p'\right)e^{ix_{1}p'/\hbar}\\& = & \int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dp'\, e^{i\left(-x_{2}+x_{0}\right)p/\hbar}\hbar\int_{-\infty}^{\infty}dx\, e^{i\left(p-p'\right)x/\hbar}e^{ix_{1}p'/\hbar}\\& = & \hbar\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dp\int_{-\infty}^{\infty}dp'\, e^{i\left(-x_{2}+x_{0}\right)p/\hbar}e^{ipx/\hbar}e^{-ip'x/\hbar}e^{ix_{1}p'/\hbar}\\& = & \hbar\int_{-\infty}^{\infty}dx\,\int_{-\infty}^{\infty}dp\, e^{i\left(x-x_{2}+x_{0}\right)p/\hbar}\int_{-\infty}^{\infty}dp'\: e^{-ip'\left(x_{1}-x\right)/\hbar}\\& = & \int_{-\infty}^{\infty}dx\,\hbar\delta\left(x-x_{2}+x_{0}\right)\delta\left(x_{1}-x\right)\\& = & \int_{-\infty}^{\infty}dx\,\varphi_{x}\left(x_{2}\right)v_{x}\left(x_{1}\right)\end{eqnarray*}

Both $\varphi_{x}\left(x_{2}\right)$ and $v_{x}\left(x_{1}\right)$

are eigenfunction of position operators $Q=x_{2}$, $B=x_{1}$, with the eigenvalues $x=x_{2}-x_{0}$ and $x=x_{1}$, which for all measurement. Therefore

\begin{eqnarray*}x-x & = & x_{2}-x_{0}-x_{1}\\0 & = & x_{2}-x_{1}-x_{0}\end{eqnarray*}

hence

\[x_{2}-x_{1}=x_{0}\]

and constant. When $x_{1}=x$, then $x_{2}=x+x_{0}$.

It is possible to prepare this state, $\Psi$, with its perfect correlations

$x_{2}-x_{1}=x_{0}$ and $p_{1}+p_{2}=0$, despite the fact that$\left[P,Q\right]=\left[x_{2},p_{2}\right]=i\hbar$, because $\left[P-A,Q+B\right]=\left[x_{2}-x_{1},p_{1}+p_{2}\right]=0$.

An Example of that the Projection Operator is Independent of the Chosen Basis of the Subspace

$\newcommand{\tr}[1]{\text{Tr}\left\{#1\right\}}$ $\newcommand{\ket}[1]{\left|#1\right\rangle}$ $\newcommand{\bra}[1]{\left\langle#1\right|}$ $\newcommand{\braket}[2]{\left\langle#1\right| \left.#2\right\rangle}$ $\newcommand{\sandwich}[3]{\left\langle#1\right|#2 \left|#3\right\rangle}$ $\newcommand{\span}[1]{\text{Span}\left\{#1\right\}}$ $\newcommand{\proj}[2]{\text{Proj}_{#1}\left(#2\right )}$

Say we want to find the projection operator onto the $\mathcal{V}$ $= xy$-plane. If one uses the simplest orthonormal basis as $B = \left\{ \vec{x}, \vec{y} \right\}$. $\vec{x} = \begin{pmatrix}1\\0\\0\end{pmatrix}$, $\vec{y} = \begin{pmatrix}0\\1\\0\end{pmatrix}$, the projection operator becomes $\proj{\mathcal{V}}{\vec{v}}$ $\equiv M_\mathcal{V}\vec{v}$.  $$\begin{align}  M_\mathcal{V} &= \ket{x}\bra{x} + \ket{y}\bra{y} \\
&= \vec{x}\vec{x}+\vec{y}\vec{y} \\
&= \begin{pmatrix}1\\0\\0\end{pmatrix}\begin{pmatrix}1&0&0\end{pmatrix}+
\begin{pmatrix}0\\1\\0\end{pmatrix}\begin{pmatrix}0&1&0\end{pmatrix}\\
&= \begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}
+\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix} \\
&= \begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}
\end{align}$$
Lets pick another basis $B^\prime = \left\{ \vec{u}_1, \vec{u}_2 \right\}$. $\vec{u}_1 = \begin{pmatrix}\cos{\theta}\\ \sin{\theta}\\0\end{pmatrix}$, $\vec{u}_2 = \begin{pmatrix}-\sin{\theta}\\ \cos{\theta}\\0\end{pmatrix}$. $$\begin{align}  M_\mathcal{V} &= \ket{u_1}\bra{ u_1 } + \ket{ u_2}\bra{ u_2} \\
&= \vec{u}_1\vec{u}_1+\vec{u}_2\vec{u}_2 \\

&= \begin{pmatrix}\cos{\theta}\\ \sin{\theta}\\0\end{pmatrix} \begin{pmatrix}\cos{\theta}& \sin{\theta}&0\end{pmatrix} +
\begin{pmatrix}-\sin{\theta}\\ \cos{\theta}\\0\end{pmatrix} \begin{pmatrix}-\sin{\theta}& \cos{\theta}&0\end{pmatrix}\\
&= \begin{pmatrix} \cos{\theta}^2 & \cos{\theta}\sin{\theta} &0\\ \cos{\theta}\sin{\theta} & \cos{\theta}^2 &0\\0&0&0\end{pmatrix}
+\begin{pmatrix} \sin{\theta}^2 & -\cos{\theta}\sin{\theta} &0\\ -\cos{\theta}\sin{\theta} & \sin{\theta}^2 &0\\0&0&0\end{pmatrix} \\
&= \begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}
\end{align}$$
As we see, the answer is independent of $\theta$ hence we get the same projection operator for all possible orthonormal basis that span the $xy$-plane.

In QM, to get the probabilities we calculate the square of the norm of a vector, $\| \vec{v}_\mathcal{V} \|^2$. It can either be found by first finding the projection operator, then getting the projected component and then getting its norm etc. The end result is sandwiching the projection operator with the state (expectation value of the projection) $\sandwich{\psi}{M_\mathcal{V}}{\psi}$. Or by finding the projected vector's components on the orthonormal basis first by taking the inner product of the vector with each element of the basis.

Say $\vec{r}_\mathcal{V}$ is the projection of a vector $\vec{r}$ onto $xy$-plane. Let us express it in two different basis (corresponding to two different $\theta$ values) that span onto the $xy$-plane. $\vec{r}_{\mathcal{V},u_1}$ $= \braket{u_1}{r}$ $=\vec{u}_1\cdot \vec{r}$. $\vec{r}_{\mathcal{V},u_2}$ $= \braket{u_2}{r}$ $=\vec{u}_2\cdot \vec{r}$. And same relations for $\vec{w}_i$s. The norm square is $$\begin{align} \| \vec{r}_\mathcal{V} \|^2 & = \braket{r_\mathcal{V}}{r_\mathcal{V}} \\
&= \| \vec{r}_{\mathcal{V},u_1} \|^2 + \| \vec{r}_{\mathcal{V},u_2} \|^2 \\
&= \left| \braket{u_1}{r} \right|^2 + \left| \braket{u_2}{r} \right|^2 \\
&= \braket{r}{u_1}\braket{u_1}{r} + \braket{r}{u_2}\braket{u_2}{r} \\
&= \bra{r} \left( \ket{u_1}\bra{u_1} + \ket{u_2}\bra{u_2} \right) \ket{r} \\
&\equiv \sandwich{r}{M_\mathcal{V}}{r}
\end{align}$$
Pythagorean law applies here. For any basis the square of the sides will be equal to the square of the hypotenuse, which is the square norm that we are looking. Therefore these two methods of probability calculation are equivalent. $\left| \braket{u_1}{\psi} \right|^2 + \left| \braket{u_2}{\psi} \right|^2$ $\equiv \sandwich{\psi}{M_\mathcal{V}}{\psi}$. According to Euclidean geometry this is correct for any dimensional subspaces. $\sum_i^K \left| \braket{u_i}{\psi} \right|^2 = \sandwich{\psi}{M_\mathcal{V}}{\psi}$.