Hello everyone, apologies for the delay!
In this post I will be talking about EPR, and in particular the way in which it is structured as an argument, which is something I wanted to understand as clearly as possible. There are a few things I feel are problematic, so I wanted to flag them for discussion.
As I've understood it, the argument in EPR works like this...
Let's call P the proposition that "QM is incomplete" (or, more accurately, "the quantum-mechanical description of physical reality given by wave functions is not complete"), and let Q be the proposition "Incompatible quantities cannot have simultaneous realities". "Incompatible quantities" here refers to quantities described by non-commuting operators.
EPR gives the following necessary condition for completeness: If every element of the physical reality has a counterpart in the physical theory, then the physical theory is complete. Furthermore, the sufficient condition for "having a counterpart in the physical theory" is: if without disturbing a system we can predict with certainty the value of a physical quantity, then there exists an element of reality with a counterpart in the physical theory. Let's call this the Criterion of Reality (COR).
The argument itself goes like this.
i) P ∨ Q ...(Premise 1)
ii) ¬P → ¬Q ...(Premise 2)
iii) P ∨ ¬Q ...(Equivalence from premise 2)
iv) P ∨ (Q ∧ ¬Q) ...(Distribution of ∨ over ∧ from i and iii)
v) P ...(Conclusion)
Argument for premise 1: "Either QM is incomplete, or incompatible quantities cannot have simultaneous realities"
Suppose ¬Q, i.e., incompatible quantities could have simultaneous realities. So, in accordance with COR, without disturbing a system we can predict with certainty the value of two physical quantities (position and momentum). However, by the uncertainty principle, the wave function of the system cannot describe both simultaneously. So there at least one element of physical reality with no counterpart in the physical theory. Therefore the theory does not fulfil the necessary criterion for completeness.
So either the theory is incomplete (P), or the supposition is false (so, Q).
So, P ∨ Q.
Argument for premise 2: "If QM is complete, then incompatible quantities could have simultaneous realities"
Suppose we have a case of quantum entanglement, where two systems I and II interact with each other in a way such that conservation of relative position and conservation of momentum hold. Let us also assume separability (when a measurement is made in I, there is some reality that pertains to II) and locality (no real change can take place in II as a consequence of a measurement in I).
Suppose now we make a position measurement in I. Because of the way the systems are entangled, we can make a prediction about the position in II, and since this prediction only depends on a measurement in I, according to locality, no real change takes place in II. Therefore, we can predict the position in II with certainty without disturbing the system II, and therefore, according to COR, there exists an element of reality corresponding to the position value in II (and the same can be imagined of momentum). We can then construct the two conditionals: (1) If a position measurement is made in I, then II has a real position value, and (2) if a momentum measurement is made in I, then II has a real momentum value.
Now, what if we don't make any measurement in I at all? By locality, the state of II has the same reality even if there is no measurement made in I. So locality suggests that the sufficient condition for II to have a real value for position or momentum is that (1) or (2) hold, respectively. But a conditional always holds when its antecedent is false (i.e., if no measurement is made). So given these assumptions, (1) and (2) hold simultaneously, so II has real values for both momentum and position simultaneously, so these incompatible quantities have simultaneous realities (AKA, ¬Q).
As far as I’ve understood, this is the structural gist of EPR. The main problem for me is that it makes use of several fuzzily grounded assumptions and vaguely defined concepts. However, formulated as above, it seems to make sense. I think ultimately the problem lies in the assumption of locality, which the Q theorist could simply deny.
The definition given for completeness is intuitively plausible, but one thing I think I’m still not getting is the COR. We can say there is a physical reality corresponding to and element of the physical theory if it can be predicted without disturbing the system. But isn’t the whole point that it does disturb the system? If the two are entangled in the relevant way, then a measurement on I disturbs the system I, and consequently disturbs II, doesn’t it? So if we can only get the position value of I by disturbing I, we can make a prediction on II but only by disturbing II, meaning that the value is not an element of the physical reality, which still satisfies the universal quantifier in the condition for completeness. So it seems to me, but I may very well have made an error here, that the COR presupposes the incompleteness of QM, and therefore cannot be legitimately used as a condition for the completeness of QM.
I’ll leave it here for now. All in all I found this topic quite challenging, so I welcome any corrections in my understanding of it!
EPR's derivation of the premise that either quantum mechanics is incomplete or non-commuting observables cannot be simultaneously real is quite confusing and, I think, is not helped by their presentation of it.
ReplyDeleteThe key problem for me was that it seems, until you look carefully, that they're using their criterion of reality to establish this premise whilst in fact they're not (if they did it would be invalid because their criterion is merely sufficient for reality).
They couch their discussion of non-commuting operators in terms of knowledge, ie:
``More generally, it is shown in quantum mechanics that, if two physical operators, say $A$ and $B$, do not commute, that is, if $AB\neq BA$, then the precise knowledge of one of them precludes such a knowledge of the other. Furthermore, any attempt to determine the latter experimentally will alter the state of the system in such a way as to destroy the knowledge of the first''.
The reference to knowledge primes you to think that their argument will use their criterion of reality (which is in terms of predictability) along the lines of: one cannot make predictions of non-commuting observables without disturbing the system; hence (using the criterion of reality) non-commuting observables aren't real. This (pseudo) argument, however, uses the criterion of reality the wrong way round.
In fact the argument relies on the definition of completeness and a notion of reality as entailing definite-valuedness:
If quantum mechanics is complete and quantities represented by non-commuting operators are simultaneously real;
then the quantum theory would have elements corresponding to those quantities (from definition of completeness);
then those elements of the theory would `contain' definite values for those quantities;
but quantum mechanics has no such elements;
hence the assumption is false;
hence either quantum mechanics is incomplete or quantities represented by non-commuting operators aren't simultaneously real.
Thank you Luc. I just wanted to say that I think you have made a mistake in your argument for the first premise. You said:
ReplyDelete"Suppose ¬Q, i.e., incompatible quantities could have simultaneous realities. So, in accordance with COR, without disturbing a system we can predict with certainty the value of two physical quantities (position and momentum)."
In this paragraph you forgot that COR is just about the sufficient condition for the reality of a quantity and not the necessary condition. That's why you said that according to COR if a quantity is real then we can predict its value.
I think EPR don't intend to use the criterion of reality for deriving that either the theory is incomplete or two quantities cannot have simultaneous reality.
ReplyDeleteIndeed, then one would have to use its form of implication backwards. But (which is probably what Jon pointed out), this can be concluded from the necessary condition for complete theories together with what looks like the uncertainty principle.
This part could roughly be reconstructed like this:
(1) A theory is complete if it describes a state, whenever that state is real. (p->(q->r))
(2) States described by non-commuting operators are not simultaneously described in quantum theory. (not-r)
(3) Therefore, quantum theory is incomplete or these states are not simultaneously real. (not-p or not-q)
Then the second part makes use of the reality criterion. It more or less says:
(1) Suppose quantum theory is complete.
(2) In the case of entangled states we can predict the state of both system with certainty and without disturbing them, by measuring the other system's state.
(4) Criterion of reality.
(5) Therefore both predictable states of B are simultaneously real.
(6) Therefore if quantum theory is complete, then two states described by non-commuting operators can have simultaneous reality.
What I found quite confusing was this splitting of the argument into two parts and what seems to be two different descriptions of two states where in one case states cannot and in the other case can be predicted.
ReplyDeleteI found the description in Norton's book much easier to follow, which focused only on this entanglement issue and apparently better reflects Einstein's view.
The argument there seems to be the following:
(1) In the case of two interacting and then separated systems, we can predict the state of system B by measuring that of A.
(2) Criterion of reality.
(3) Therefore, the state of B is real.
(4) But since measurement of a space-like distant system cannot change its state, B's state must have been real before the measurement of A.
(5) Before this measurement, however, the wave-function for B describes its state as dependent on A's state.
(6) If quantum theory was complete it would describe the unique state of B before the measurement independent from A. (some version of the necessary condition for complete theories)
(7) Therefore, quantum theory is not complete.
This reconstruction makes use of the premise that a measurement cannot cause or change distant states (in (4)) and assumes that such systems are real independently of each other (probably also in (4)), both of which according to Norton Einstein was not willing to give up.
So I’d like to try and clarify something that’s been troubling me with this topic. Basically it’s about when particular properties are acquired. I’ll use the example from our discussion last week about 2 coloured balls (one red and one blue) in two sealed boxes in separate parts of space. By opening one box and observing the colour of the ball inside you also learn the colour of the ball in the other box (I know there are some more fundamental differences between this system and a quantum one but it works for my question). My understanding currently is that this would be how Einstein and co would see it. The property of the balls (in this case there colour) was a definite physical reality independent of our action of observing the other ball. Say if the unobserved ball was blue; it was blue before the red ball was observed and would have continued to be blue even if no measurement of the red balls colour had been made. Is this a fair understanding (according to Einstein)? The alternative view would therefore be that neither ball had a set colour prior to the observation and only collapsed to this state after the observation of the first ball.
ReplyDeleteI just wanted to be sure I’m getting this right!