#### Coleman on GHZS

My background is the talk "Quantum Mechanics In Your Face" by Sidney Coleman which I consider as the best argument why quantum mechanics cannot be described by a local and realistic theory (from which I would conclude it is not realistic). In a nutshell, the argument goes like this: Consider the three qubit state state

$$\Psi=\frac 1{\sqrt 2}(\uparrow\uparrow\uparrow-\downarrow\downarrow\downarrow)$$

which is both an eigenstate of eigenvalue -1 for $\sigma_z\otimes\sigma_z\otimes\sigma_z$ and an eigenstate of eigenvalue +1 for $\sigma_x\otimes\sigma_x\otimes\sigma_z$ or any permutation. This means that, given that the individual outcomes of measuring a $\sigma$-matrix on a qubit is $\pm 1$, when measuring all in the z-direction there will be an odd number of -1 results but if two spins are measured in x-direction and one in z-direction there is an even number of -1's.

The latter tells us that the outcome of one z-measurement is the product of the two x-measurements on the other two spins. But multiplying this for all three spins we get that in shorthand $ZZZ=(XXX)^2=+1$ in contradiction to the -1 eigenvalue for all z-measurments.

The conclusion is (unless you assume some non-local conspiracy between the spins) that one has to take serious the fact that on a given spin I cannot measure both $\sigma_x$ and $\sigma_z$ and thus when actually measuring the latter I must not even assume that $X$ has some (although unknown) value $\pm 1$ as it leads to the contradiction. Stuff that I cannot measure does not have a value (that is also my understanding of what "not realistic" means).

#### Fruchtiger and Renner

Now to the recent Nature paper. In short, they are dealing with two qubits (by which I only mean two state systems). The first is in a box L' (I will try to use the somewhat unfortunate nomenclature from the paper) and the second in in a box L (L stands for lab). For L, we use the usual z-basis of $\uparrow$ and $\downarrow$ as well as the x-basis $\leftarrow = \frac 1{\sqrt 2}(\downarrow - \uparrow)$ and $\rightarrow = \frac 1{\sqrt 2}(\downarrow + \uparrow)$ . Similarly, for L' we use the basis $h$ and $t$ (heads and tails as it refers to a coin) as well as $o = \frac 1{\sqrt 2}(h - t)$ and $f = \frac 1{\sqrt 2}(h+f)$. The two qubits are prepared in the state

$$\Phi = \frac{h\otimes\downarrow + \sqrt 2 t\otimes \rightarrow}{\sqrt 3}$$.

Clearly, a measurement of $t$ in box L' implies that box L has to contain the state $\rightarrow$. Call this observation A.

Let's re-express $\rightarrow$ in the x-basis:

$$\Phi =\frac {h\otimes \downarrow + t\otimes \downarrow + t\otimes\uparrow}{\sqrt 3}$$

From which one concludes that an observer inside box L that measures $\uparrow$ concludes that the qubit in box L' is in state $t$. Call this observation B.

Similarly, we can express the same state in the x-basis for L':

$$\Phi = \frac{4 f\otimes \downarrow+ f\otimes \uparrow - o\otimes \uparrow}{\sqrt 3}$$

From this once can conclude that measuring $o$ for the state of L' one can conclude that L is in the state $\uparrow$. Call this observation C.

Using now C, B and A one is tempted to conclude that observing L' to be in state $o$ implies that L is in state $\rightarrow$. When we express the state in the $ht\leftarrow\rightarrow$-basis, however, we get

$$\Phi = \frac{f\otimes\leftarrow+ 3f\otimes \rightarrow + o\otimes\leftarrow - o\otimes \rightarrow}{\sqrt{12}}.$$

so with probability 1/12 we find both $o$ and $\leftarrow$. Again, we hit a contradiction.

One is tempted to use the same way out as above in the three qubit case and say one should not argue about contrafactual measurements that are incompatible with measurements that were actually performed. But Frauchiger and Renner found a set-up which seems to avoid that.

They have observers F and F' ("friends") inside the boxes that do the measurements in the $ht$ and $\uparrow\downarrow$ basis whereas later observers W and W' measure the state of the boxes including the observer F and F' in the $of$ and $\leftarrow\rightarrow$ basis. So, at each stage of A,B,C the corresponding measurement has actually taken place and is not contrafactual!

#### Interference and it did not happen

I believe the way out is to realise that at least from a retrospective perspective, this analysis stretches the language and in particular the word "measurement" to the extreme. In order for W' to measure the state of L' in the $of$-basis, he has to interfere the contents including F' coherently such that there is no leftover of information from F''s measurement of $ht$ remaining. Thus, when W''s measurement is performed one should not really say that F''s measurement has in any real sense happened as no possible information is left over. So it is in any practical sense contrafactual.

To see the alternative, consider a variant of the experiment where a tiny bit of information (maybe the position of one air molecule or the excitation of one of F''s neutrons) escapes the interference. Let's call the two possible states of that qubit of information $H$ and $T$ (not necessarily orthogonal) and consider instead the state where that neutron is also entangled with the first qubit:

$$\tilde \Phi = \frac{h\otimes\downarrow\otimes H + \sqrt 2 t\otimes \rightarrow\otimes T}{\sqrt 3}$$.

Then, the result of step C becomes

$$\tilde\Phi = \frac{f\otimes \downarrow\otimes H+ o\otimes \downarrow\otimes H+f\otimes \downarrow\otimes T-o\otimes\downarrow\otimes T + f\otimes \uparrow\otimes T-o \otimes\uparrow\times T}{\sqrt 6}.$$

We see that now there is a term containing $o\otimes\downarrow\otimes(H-T)$. Thus, as long as the two possible states of the air molecule/neuron are actually different, observation C is no longer valid and the whole contradiction goes away.

This makes it clear that the whole argument relies of the fact that when W' is doing his measurement any remnant of the measurement by his friend F' is eliminated and thus one should view the measurement of F' as if it never happened. Measuring L' in the $of$-basis really erases the measurement of F' in the complementary $ht$-basis.

## 3 comments:

As you pointed out, the experiment depends on /F's "lab" being measured such that its two internal states are coherent and interfere as seen by /W. The idealised lab is perfectly sealed, but somehow /W can measure it from outside, rather like a Schrodinger who can check some aspect of his cat without opening his box.

It's not just /W and W who must perform this magic. When the qubit "S" is passed from lab /L to lab L, the same thing must happen.

This may seem monstrously difficult to achieve, but it's not impossible. We could implement the measurements by having the F agents emit single polarised photons. If the photons were generated before they made their measurements, then polarised in a way dependent on the measurements, the decoherence of the agent (and the rest of the lab contents) would not matter.

And although we cannot create perfectly sealed "labs" to contain human beings, we have equivalent devices in quantum computers as well as many, many quantum experiments.

My point is; one day, we may get this experiment up and running, in some form. So, can we still confidently say that /F's measurement "didn't happen", i.e. is contrafactual, when it becomes possible one day to "open her lab" and see?

I've written what I consider an accessible and readable analysis of Renner's "paradox" and why it's not real. Hopefully it will help someone.

http://byrden.com/quantum/fr.html

I'm back again, because I wrote an even simpler explanation of the "paradox" after thinking about it from all angles. Hopefully this will be accessible to more people;

http://byrden.com/quantum/Renner-explained.html

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