Why Vanishing Commutators Imply There's No Activity At A Distance

I believe that an extremely similar spider web log ship service has been written inwards the yesteryear but I can't observe it or what I tin observe isn't quite the same together with then that's why I decided to write this i again.

Lots of people say that at that topographic point must endure non-local influences or or together with then activity at a distance inwards the existent world, together with this claim is implied yesteryear Bell's inequalities or something similar that. This declaration is completely wrong. Since the 1905 especial theory of relativity, nosotros accept known that the non-local or superluminal influences would endure equivalent – yesteryear the Lorentz transformation – to the influencing changing one's past, together with those are logically inconsistent.

So why don't entanglement experiments imply whatsoever activity at a distance?

In quantum mechanics, events are predicted probabilistically. Unless all the probabilities are calculated to endure 100% or 0%, together with they're commonly inwards the middle, nosotros can't say that the effect volition endure something or something else alongside certainty. We tin exclusively say that the effect volition endure something alongside or together with then probability; together with something else alongside or together with then other probability.

In this setup, the activity at a distance apparently means that the willful activity at i house which nosotros volition telephone weep upwardly Alaska (A) volition modify the probabilities of or together with then properties of outcomes of measurements at or together with then other house which nosotros volition telephone weep upwardly Boston (B). OK, let's imagine nosotros accept an entangled span of particles or other physical objects that were created as entangled somewhere inwards Texas, to locomote inwards general, but the subsystems accept propagated to Alaska together with Boston, respectively.

I chose Alaska together with Boston for them to endure on the left together with on the right. Alabama's seat didn't await convenient enough.




OK, nosotros desire to follow the probability that or together with then measuring inwards Boston volition accept or together with then property. It may endure whatsoever holding of whatsoever measuring of whatsoever quantity that is geometrically confined to Boston. Therefore, the corresponding quantity or operators could deportment or together with then extra indices that quest that at that topographic point are many options. But without a loss of generality, nosotros volition follow i detail holding together with it volition endure expressed yesteryear the projection operators \(P_B\). It's or together with then projection operator constructed from local fields inwards the Greater Boston area, if you lot utilisation the framework of quantum land theory.




The initial (entangled) density matrix describing both subsystems volition endure called but \(\rho\) together with nosotros assume \({\rm Tr}(\rho)=1\). I am using a density matrix inwards guild to endure actually general. But the give-and-take below may of course of pedagogy endure applied to the especial instance of the pure states when \(\rho=\ket\psi\bra\psi\), too.

Fine, together with then what is the probability that the holding inwards Boston volition hold? It volition endure simply\[

{\rm Tr} (P_B \rho P_B).

\] I wrote the projection operators on both sides, inwards guild to quest that \(P_B\) would human activity both on the ket vector \(\ket\psi\) together with on the bra vector \(\bra\psi\) inwards the pure Earth case. But you lot tin run into that due to the cyclicity of the trace, i of the \(P_B\) may endure moved "around the world" through the parentheses following to the other i together with nosotros may utilisation \(P_B^2 = P_B\), a defining holding of a projection operator. It follows that inwards the probability above, you lot could easily omit i of the 2 copies of \(P_B\).

In the pure Earth case, the probability given yesteryear the draw is reduced to \(|\bra\psi P_B \ket\psi|^2\), a classic illustration of Born's rule, but I volition no longer write the especial forms of the formulae for pure states below.

Fine. Note that inwards the instance of a spin-1/2 particle, the projection operator proverb that "the spin is upwardly alongside honor to the axis \(\vec n\)" where \(|\vec n|=1\) is simply\[

P_B = \frac{ 1 + \vec \sigma \cdot \vec n}{2}, \quad \vec\sigma \cdot \frac\hbar 2 \equiv \vec S.

\] But again, the residual of our give-and-take holds non exclusively for spins of the electrons but for whatsoever systems, together with non exclusively for the maximally entangled states, but for absolutely whatsoever states of whatsoever composite systems.

Now, the inquiry is whether or together with then willful activity done on the other side of America, inwards Alaska, may modify the probability\[

{\rm Tr} (P_B \rho P_B)

\] to something else. What are the willful actions? Well, at that topographic point are people inwards Alaska. And i of them, Sarah Palin – the exclusively cry inwards Alaska I was able to memorize – may press buttons inwards her laboratory which determines what holding of her subsystem is going to measured. For example, if the subsystem inwards Alaska is or together with then other spin-1/2 particle, Sarah Palin may press the push "x" or "y" or "z" which determines which polarization of the spin of her electron is gonna endure measured.

The determination is somewhat random, generally confined to her brain. It may endure chaotic, it may depend on or together with then objects inwards Russian Federation that Sarah Palin has seen from her house, together with and then on. I mentioned the spin but whatsoever determination that a human makes may endure parameterized yesteryear similar information as the selection of "x" or "y" or "z". The exclusively inherently quantum impact of this determination is the selection of the footing of possible "after the measuring states" of her subsystem, e.g. the Alaskan spin-1/2 particle.

Again, such decisions may endure arbitrary, influence whatsoever observable of the Alaskan system, together with then the operator \(P_{A,i}\) could accept extra indices. But again, our declaration would piece of occupation for all values of these extra indices, together with then without the loss of generality, nosotros volition omit these indices. So Sarah Palin decides to press a button, e.g. "y", together with this selection singles out the projection operators\[

P_{A,i}

\] that may human activity on her pure Earth or the density matrix. Here, \(i\) is the index indicating the possible results. For example, inwards the instance I mentioned, \(i=1,2\) agency \(\sigma_y=\pm 1/2\), respectively. We apparently have\[

\sum_i P_{A,i}=1.

\] Sarah Palin's measurement, when it's completed, induces a collapse of the moving ridge business office or density matrix, together with it's this collapse that is said to "influence the province of affairs inwards Boston" yesteryear the confused champions of non-locality. OK, did Sarah Palin influence Boston when she pressed a push together with extracted a resultant of the measurement? Let's await carefully.

The probability that Sarah Palin gets the \(i\)-th resultant is\[

{\rm Tr} (P_{A,i} \rho P_{A,i})

\] together with if she gets this \(i\)-th result, the density matrix for the composite organization changes from \(\rho\) to\[

\rho \to \frac{ P_{A,i} \rho P_{A,i} }{ {\rm Tr} (P_{A,i} \rho P_{A,i}) }.

\] I accept divided the density matrix yesteryear its draw inwards guild to choke on the draw of the ultimate density matrix equal to one. OK, at whatsoever rate, you lot tin run into that Sarah Palin has modified the density matrix – she has inserted her detail projection operator \(P_{A,i}\) on both sides from the density matrix. Let me cite that inwards the numerator, at that topographic point is no trace, together with then you lot cannot omit either re-create of the 2 projection operators (with the omission, the density matrix wouldn't fifty-fifty endure Hermitian). In the denominator, you lot tin omit i for the argue I accept already discussed dorsum inwards Boston.

OK, this looks similar a really dissimilar density matrix that depends on the \(i\)-th resultant of her Alaskan measurement. How does it impact Boston? How does it impact the probability that the Bostonian holding we're discussing volition endure measured as "Yes"? Well, it's simple. In the formula for the probability of the chosen Bostonian property\[

{\rm Tr} (P_B \rho P_B),

\] nosotros must but supersede \(\rho\) alongside the novel density matrix afterward the Alaskan measurement. It agency that the probability of the Bostonian holding volition modify to the conditional probability\[

\frac{{\rm Tr}(P_B P_{A,i} \rho P_{A,i} P_B) }{ {\rm Tr} (P_{A,i} \rho P_{A,i}) }

\] inwards the instance of the \(i\)-th resultant inwards Alaska. This looks complicated together with theme on or together with then materials inwards Alaska. So Sarah Palin has influenced Boston, hasn't she? Wait a minute.

No i inwards Boston could accept learned most the value of the index \(i\) indicating the effect of the Alaskan measuring yet. Also together with as importantly, Sarah Palin couldn't "order" the resultant to endure \(i\) – the results are randomly chosen yesteryear Mother Nature, non yesteryear Sarah Palin. So the probability that the Bostonian holding is obeyed must endure calculated as the probabilistic weighting of the conditional probabilities inwards a higher house multiplied yesteryear the probabilities of the \(i\)-th outcomes inwards Alaska.

So afterward Sarah Palin picked her measuring type together with measured the effect chosen yesteryear Mother Nature, but earlier Boston learned most that outcome, the probability of the chosen Bostonian holding is clearly\[

\sum_i \frac{{\rm Tr}(P_B P_{A,i} \rho P_{A,i} P_B) }{ {\rm Tr} (P_{A,i} \rho P_{A,i}) } {\rm Tr} (P_{A,i} \rho P_{A,i})=\dots

\] I accept copied together with pasted the previous complicated formula, added the nub over \(i\) on the left, together with added the probability that she measured the corresponding \(i\)-th outcome. But as you lot tin see, that probability is precisely the denominator that I previously inserted to normalize the density matrix, together with then those 2 cancel. The aspect inwards a higher house is hence equal to\[

\dots = \sum_i {\rm Tr}(P_B P_{A,i} \rho P_{A,i} P_B)=\dots

\] That looks much simpler. Now, if the operators inwards Alaska together with Boston commute alongside each other – together with they commute alongside each other if the regions are spacelike-separated, nosotros have\[

P_{A,i} P_B = P_B P_{A,i}.

\] But if that's so, nosotros tin switch the guild of the 2 side yesteryear side operators (in Alaska vs Boston) at both places, to get\[

\dots = \sum_i {\rm Tr}(P_{A,i} P_B \rho P_B P_{A,i})=\dots

\] Now it's simple. We utilisation the cyclic holding of the draw which allows us to motion i of the operators \(P_{A,i}\) following to the other one, to the contrary side of the parentheses. We utilisation the holding of the projection operator \(P_{A,i}^2 = P_{A,i}\) together with the aspect inwards a higher house is hence equal to\[

\dots = \sum_i {\rm Tr}(P_{A,i} P_B \rho P_B)=\dots

\] which differs yesteryear the erasure of i of the 2 copies of the Alaskan projection operator. But yesteryear linearity, nosotros may utilisation the fact that the nub of the Alaskan projection operators over \(i\) is but equal to the identity operator \(1\), together with then the aspect inwards a higher house is but equal to\[

\dots = {\rm Tr}(P_B \rho P_B).

\] That's the same probability of the holding inwards Boston that nosotros had at the beginning. It follows that Sarah Palin's complimentary determination to press or together with then buttons that influence a component subdivision of the entangled subsystem together with the subsequent collapse of the Earth vector or density matrix into i of the footing options (Palin could accept affected the selection of the footing but non the detail vector, together with folks inwards Boston couldn't know either) has non influenced the probability inwards Boston at all. Because \(P_B\) as good as the laid of \(P_{A,i}\) was chosen arbitrarily, nosotros may conclude that no determination made inwards Alaska influences whatsoever holding (any probability of it) that may endure measured inwards Boston. There is no activity at a distance.

To brand this give-and-take complete, i should revisit the corresponding technical department from basic quantum land theory, to explicate why operators inwards spacelike-separated regions precisely commute alongside each other. And nosotros could add together a give-and-take how \(P_{A,i}\) together with \(P_B\) are constructed as functionals of the quantum fields inwards Alaska or Boston, respectively, together with then they commute alongside each other, too. Also, to endure safe, i should add together a give-and-take explaining why the non-relativistic description of several particles may endure embedded inside a quantum land theory, together with then the commutators are apparently vanishing inwards the non-relativistic description of the subsystems, as good (there are other arguments to demonstrate this uncomplicated fact).

At whatsoever rate, at that topographic point exists a valid proof shown or sketched inwards a higher house that implies that the people's playing alongside the subsystems of entangled systems implies no activity at a distance at all – the influence is precisely zero. All of the people who say that at that topographic point is an activity at a distance or non-local influence are just wrong. Most of the time, their demonstration of the incorrect conclusion has zip whatever to practise alongside the proof inwards a higher house because it has zip to practise alongside quantum mechanics. These people aren't thinking quantum mechanically, their brains are confined somewhere inwards the 17th century.