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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.
\[\ket a\xrightarrow{B}\ket b\]
\[\ket a\xrightarrow{B}\ket b\]
\[B\ket b=b\ket b\]
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.
\[\ket{\Psi}=\left(\ket{u+u-}-\ket{u-u+}\right)/\sqrt{2}\]
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
- $\ket{\mbox{GHZ}}=\left(\ket{000}-\ket{111}\right)/\sqrt{2}$
- $\ket{\mbox{W}}=\left(\ket{100}+\ket{010}+\ket{001}\right)/\sqrt{3}$
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 $\left[P-A,Q+B\right]=\left[x_{2}-x_{1},p_{1}+p_{2}\right]=0$.
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