What makes entanglement so interesting?

[dsaun777]

Isn't entanglement just the conservation of spin, momentum, and position along with Heisenberg's uncertainty principle? I don't really see the big mystery or maybe I am just not understanding a key part of what makes entanglement so interesting. Can someone pinpoint how this receives so much attention in popular science?

 

[PeterDonis]

Isn't entanglement just the conservation of spin, momentum, and position along with Heisenberg's uncertainty principle?

No. Conservation laws (btw, position isn't a conserved quantity) and the HUP apply to all states, not just entangled states.

Can someone pinpoint how this receives so much attention in popular science?

It's not just in popular science, it's in actual physics. Entangled states are the ones that show phenomena like violations of the Bell inequalities, which raise serious questions about our intuitive view of how the world works. That's why physicists spend so much time studying them.

 

[dsaun777]

No. Conservation laws (btw, position isn't a conserved quantity) and the HUP apply to all states, not just entangled states.
It's not just in popular science, it's in actual physics. Entangled states are the ones that show phenomena like violations of the Bell inequalities, which raise serious questions about our intuitive view of how the world works. That's why physicists spend so much time studying them.

I still don't see the what is so bewildering.

 

[PeterDonis]

I still don't see the what is so bewildering.

Do you understand what violation of the Bell inequalities implies? Or, rather, rules out?

There are plenty of threads in this forum discussing that.

 

[andresB]

Take a look at

https://aapt.scitation.org/doi/10.1119/1.12594

 

[phinds]

Take a look at

https://aapt.scitation.org/doi/10.1119/1.12594

Paywall

 

[Nugatory]

I still don't see the what is so bewildering.

Let’s try a classical analogy. We have a room full of couples, one man and one woman. Remarkably, in every couple exactly one of the two smokes cigarettes and exactly one of the two has blue eyes (the other always has brown eyes).

Now you and I play a game: for each couple, you meet one member and I meet the other. We each record one randomly chosen fact about the person we meet: whether they smoke or not, whether they have blue eyes or not, whether they are male or female - but only one of these three possible observations.

Then we get together and compare notes. If you met a woman and I met a smoker, we will conclude that your person was a non-smoking woman and mine was a smoking man and we don’t know anything about the color of their eyes, and similarly for all the other possible pairs of measurements. So far so good, and that’s just the macroscopic analog of entanglement - one particle is up and the other is down, one spouse is male and the other is female.

But then we look at all of our measurements, and we discover that the number of male smokers we’ve found is greater than the number of blue-eyed smokers plus the number of brown-eyed men. That is totally bewildering, but it’s how entangled particles behave when we look at more than one of their properties.

 

[vanhees71]

I still don't see the what is so bewildering.

You are right, there's nothing "bewildering" about quantum mechanics as soon as you accept the world view that Nature behaves "irreducibly random". In contradistinction to classical physics, quantum theory is an indeterministic description of the phenomena we observe in Nature.

This means you can have complete knowledge about a system, i.e., you can prepare it (in principle) in a pure state. E.g., you can determine the values of a complete set of compatible observables precisely. Then you system must be in a uniquely defined pure state described by a normalized common eigenvector of the corresponding self-adjoint operators describing these observables. This eigenvector $|\psi \rangle$ is determined up to a phase factor, but the state itself is then unique, described then by the statistical operator of the corresponding pure state, which is the projection operator $\hat{\rho}=|\psi \rangle \langle \psi|$.

This is the best in the sense of knowledge you can get about your system, i.e., the state is completely known in this case in the sense of quantum theory. Now it's clear that this does not imply that you know precisely the values of observables which are not compatible with all the observables taking determined values through this complete preparation of the system. All you can know about the outcomes of measurements of such observables are (usually) the probabilities to get a certain possible value this observable can take (an eigenvalue of the corresponding self-adjoint operator describing it in the quantum-theoretical formalism). If you then measure this observable and filter out all systems which have a certain value, the determination of the before exactly prepared values of the compatible set of operators in general gets lost, i.e., after preparation of the system to have a certain value of the other incompatible observable destroys the preparation of the before determined observables.

This is reflected by the uncertainty relations. The most famous is the one between a position-vector component and the momentum component in the same direction, which says that the standard deviations of these quantities obey the uncertainty relation $\Delta x_1 \Delta p_1 \geq \hbar/2$. First of all you can never really exactly determine position or momentum, because the operators have continuous eigenvalue spectra. Nevertheless you can, say, localize the particle very precisely, i.e., make $\Delta x_1$ very small (given by the resolution of your particle detector, which is not principally limited, i.e., you can make it as precise as you like modulo the technical problems you have to solve and how much effort you put in), but then necessarily $\Delta p_1 \geq \hbar/(2 \Delta x_1)$ is very large, i.e., the momentum component $p_1$ is necessarily very indetermined. If you know use a "velocity filter" (like crossed electric and magnetic fields) and use only particles with a very well determined velocity (and thus very well determined momentum) you necessarily loose the localization of the particle, because now due to the uncertainty relation the particle's position has a very large standard deviation.

All this is not so mind-boggling except for the fact that we cannot, according to QT, prepare particles in such a way that all observables take simultaneously precise values, and in general we only know probabilities for the outcome of measurements of observables even if we know the state completely in the sense that we have prepared the system in a pure state.

Now entangled states have however a completely surprising property, which Einstein called inseparability. An extreme example, which nowadays can easily be prepared are polarization-entangled photon pairs. You have two photons with quite well defined different momenta $\vec{k}_1$ and $\vec{k}_2$ and polarizations $h_1$ and $h_2$ ($h_1,h_2$ standing for helicities for convenience) prepared in a state
$$|\Psi \rangle =N [\hat{a}^{\dagger} (\vec{k}_1,1) \hat{a}^{\dagger}(\vec{k}_2,-1)-\hat{a}^{\dagger} (\vec{k}_1,-1) \hat{a}^{\dagger}(\vec{k}_2,1)]|\Omega \rangle,$$
where $\hat{a}^{\dagger}(\vec{k},h)$ is the creation operator for a photon with momentum $\vec{k}$ and helicity $h$ (with a certain uncertainty in momentum such that we have a normalized state rather than generalized exact momentum eigenstates, i.e., we can choose $N$ such that $\langle \Psi|\psi \rangle=1$).

Now the photons fly appart after being prepared in this state (e.g., by a process known as "parametric downversion", where a photon from a laser is interacting with a certain kind of birefringent crystal such that it gets split in a photon pair in a state of the above given type). Now wait a while and put detectors at far-distant places $A$ and $B$ (standing for the corresponding observers Alice and Bob) where Alice and Bob measure the polarization state of their photon. It is clear that the polarization of either of these single photons is not determined, because it can be either $h=1$ or $h=-1$. QT tells us (by calculating what's known as the reduced state) that indeed the polarization of each of A's or B's photon prepared in the above state is completely unknown, i.e., all Alice and Bob measure when getting a lot such prepared photons is that the photons are precisely unpolarized, i.e., the polarization state of each of the photons is the maximally uncertain unpolarized state described by the statistical operators
$$\hat{\rho}_A=\hat{\rho}_B=\frac{1}{2} \hat{1}.$$
But know Alice and Bob can do coincidence measurements, i.e., they can use clocks to make sure that they always measure the polarizations of their photons always from a the photon pairs prepared in this specific entangled state. Then you can immidiately see that if Alice measures the polarization $h_A=+1$ (which happens with 50% probability) then Bob necessarily finds $h_B=-1$ and vice versa.

This is mind-boggling for some physicists, who believe in certain interpretations of quantum theory. I'm a proponent of the ensemble interpretation (the interpretation which Einstein preferred, in saying that quantum theory only makes predictions concerning the probabilities for the outcome of measurements which can be checked only by preparing an ensemble of to be measured systems in a certain way described by the quantum state, and which was formulated most clearly by Ballentine in the earily 1970ies), and thus have no quibbles with this feature of quantum theory (today established with very high precision by experiment), because accepting this purely probabilistic interpretation of the quantum state the complete indetermination of the single-photon polarizations with 100% correlation of the outcome of measurements by Alice and Bob when measuring the same polarization state (in my example by measuring the helicities of the photons) is due to the state preparation in the very beginning.

If you however follow a certain kind of Copenhagen interpretation, including a socalled "collapse of the state" you get into trouble with relativistic causility, because within this preparation the polarization measurement of A's photon instantaneously causes (sic!) B's photon to collapse into a new state such that its polarization gets determined in the opposite helicity than that measured by Alice for her photon, but the two photons when measured by Alice and Bob can be arbitrarily distant when registered. So an instantaneous collapse then massively violates Einstein causality, because the information from A's measurement must arrive at B's photon faster than light, and this is impossible due to relativity (and photons are to be described relativistically, because if there's anything relativistic it's photons!).

However, if you give up these collapse assumption and just accept the probalistic interpretation the two photon's already carried the correlation between the outcome of measurements by A and B with them due to the preparation in this entangled state though the individual outcome of the measurement of the single-photon polarizations are completely random.

While Einstein believed that QT is incomplete and that there must be "hidden variables" which predetermine the outcome of each single-photon polarization measurement, thanks to the work by Bell we can test this assumption, at least if you assume locality, i.e., that the measurements on the photon are due to local interactions of the photons with the measurement devices (like polarizers and photon detectors at Alice's and Bob's places), because Bell could show that a deterministic (often called "realistic") local hidden-variable theory implies uncertainties for the mutual outcome of certain polarization measurements on any photon pair fulfilling an inequality of certain correlation measures for the probability distributions over the determined but onknown hidden variables, which contradict these correlation measures when calculated from standard quantum theory when the photon pairs are prepared in such a completely entangled state as discussed above (such states are thus called "Bell states"), i.e., quantum mechanics contradicts measurably the assumption that there is a local deterministic hidden-variable theory. Such "Bell tests", including some avoiding all the suggested possible "loop holes", with an amazing statistical significance confirmed the predictions of quantum mechanics, i.e., the violation of Bell's inequality.

For me that's a clear that, no matter which interpretation beyond the minimal statistical (or ensemble interpretation) you might follow, at least local deterministic hidden-variable theories are not way to make in any way quantum theory "more complete". In fact so far no such satisfactory local deterministic hidden-variable theory is known. Also I don't think that Bohmian mechanics as a non-local deterministic theory is very convincing as far as relativistic quantum field theory is concerned (though it's consistently formulated for non-relativistic quantum mechanics). That's why I think that quantum theory is by far more complete than Einstein thought.

The real problem with quantum mechanics in my opinion are not all these socalled "interpretational problems" which upset Einstein so much, but that we have no satisfactory quantum description of the gravitational interaction since the standard quantization prescriptions working so well for the other fundamental interactions in Nature (the Standard Model of elementary particles, despite all its own problems, is amazingly robust against all efforts to disprove it; just these days a new accurate determination of the fine structure constant $\alpha_{\text{em}}$ confirmed the Standard Model once more) don't work for the gravitational field (aka General Relativity).

 

[PeroK]

I still don't see the what is so bewildering.

Whether it's bewildering or not is a matter of opinion. In any case, let's take a look at what entanglement means. Specifically relating to particle spin.

In classical physics we assume that a particle has a defnite spin angular momentum, whether we measure it or not. If we measure it, we find out what spin it had before measurement; and, if we don't then it still has a definite spin, but we just don't know it.

In Quantum Mechanics this is not the case. A particle does not have a definite spin until we measure it (and, even then, we only know the spin component about one axis; the spin is stil indefinite about the other two axes).

To be clear: if we have a machine that spits out particles with random spins, then it's not the case that each particle really does have a definite spin (which we just don't now); it's that each particle does not have a definite spin component about a given axis until we measure its spin about that axis.

This fundamental difference must be understood before you can even begin to look at entanglement. And, if you understand this, you may find even that bewildering!

Now, if we consider a system of two entangled particles. We may know (from conservation laws) that the total spin is zero. In other words, if the spin of both particles is measured, they must be equal and opposite. But, as above neither particle has a definite spin before measurement.

Now, if the two entangled particles stay local to one another this may not seem too significant. Whatever determines that the measurement of spin of one particle is one thing also determines that the measurement of spin of the other particle must be the opposite.

Entangled particles, however, may be separated before measurement - and measured simultaneously (or, to be precise, the measurement events are spacelike separated). And, still, they the measurements are always opposite.

And now it may seem bewildering to imagine how nature might achieve this. The particles cannot have definite spins all along (the experiments to test Bell's theorem have ruled that out). There cannot be any communication between the particles - it appears you would issue faster-than-light communcation.

You are left, therefore, bewildered or otherwise, to accept that this is the way nature works.

 

[vanhees71]

What's deterministic in QM is the evolution of the probabilities (statistical operators) ;-).

The Bohmian interpretation is deterministic but I've not seen a convincing extension of it to relativistic quantum field theory nor have I seen an empirical demonstration of trajectories of elementary particles. In this respect note that a trace of a charged particle in a bubble or cloud chamber doesn't count, because for this case there's an explanatation for the observation of (apparent) "trajectories" within standard (minimally interpreted) quantum mechanics (as given in an early paper by Mott).

 

[vanhees71]

In classical physics we assume that a particle has a defnite spin angular momentum, whether we measure it or not. If we measure it, we find out what spin it had before measurement; and, if we don't then it still has a definite spin, but we just don't know it.

In classical mechanics we don't have spin to begin with. So this is a somewhat questionable analogy.

 

[PeroK]

In classical mechanics we don't have spin to begin with. So this is a somewhat questionable analogy.

You've obviously never seen Shane Warne's ball of the century!

 

[stevendaryl]

Isn't entanglement just the conservation of spin, momentum, and position along with Heisenberg's uncertainty principle? I don't really see the big mystery or maybe I am just not understanding a key part of what makes entanglement so interesting. Can someone pinpoint how this receives so much attention in popular science?

Entanglement is not about conservation laws. Conservation laws apply whether or not there is entanglement. Classical physics of course has conservation laws, but does not have entanglement.

Entanglement is the situation in which two subsystems such as two particles have a composite “entangled” state, but neither particle has a pure state of its own.

Classically, you can certainly produce a pair of particles such that the total spin is zero. So if you measure the spin of one particle, you automatically know that the spin of the other particle is the opposite.

This classical correlation is explained easily by assuming that (1) each particle has an unknown spin state prior to its being measured, (2) the spin state of one particle is the opposite of the spin state of the other, and (3) measuring a particle’s spin just reveals this pre-existing value.

In quantum mechanics, in the case of particles in an entangled spin state, one can prove that it is definitely NOT the case that each particle has a pre-existing spin state. Instead, measuring the spin of one particle gives a random-appearing result. So it is more difficult to explain the correlation of the measurements of the two particles.

 

[vanhees71]

I thought we talk about particles or photons. Of course there's some analogy of quantum mechanical spin with the spin of an extended classical object.

 

[vanhees71]

You've obviously never seen Shane Warne's ball of the century!

I'd say this can be well described by classical mechanics ;-)). It's not "spin" in the sense of quantum mechanics. I don't think that there's any real analogy between the "spin" of an extended classical body with the "spin" of an (elementary) particle or photon in the quantum sense. It's something related to fields, not to mechanics.

 

[stevendaryl]

What's deterministic in QM is the evolution of the probabilities (statistical operators) ;-).

The Bohmian interpretation is deterministic but I've not seen a convincing extension of it to relativistic quantum field theory nor have I seen an empirical demonstration of trajectories of elementary particles.

But as a “toy” model, you can certainly explain an EPR type experiment that violates the Bell Inequalities by using a deterministic nonlocal theory. So the entanglement doesn’t necessarily have anything to do with irreducible randomness.

I would say that the weirdness of EPR is not the randomness in itself but the perfect correlation or anti correlation in conjunction with the apparent randomness.

 

[vanhees71]

I didn't say that entanglement has to do with irreducible randomness. I said that the apparent weirdness some people still see in quantum theory though they should know better 95 years after its discovery is due to the irreducible randomness. This is, of course, totally unimportant for the physics. It's more an explanation for the psycholgy behind the claim that quantum theory was weird to begin with, though another more mundane explanation is that popular-science writers seam to think that their writings sell better when they claim it's weird. Of course it's the opposite of what science is about!

I think what's considered so "weird" of entanglement in the EPR argument is indeed the 100% correlation, but this is "weird" only together with the randomness. If everything were determined than there'd be nothing surprising about the correlation, because this is just due to momentum conservation. I never understood the claim in the EPR paper that there's a paradox in their argument, because in fact the relevant observables to explain their example are compatible observables since the relative position $\vec{r}_1-\vec{r}_2$ and the total momentum $\vec{p}_1+\vec{p}_2$ are commuting, but that's another topic.

 

[PeroK]

I still don't see the what is so bewildering.

And, of course:

Those who are not shocked when they first come across quantum theory cannot possibly have understood it.

Niels Bohr

 

[stevendaryl]

Here’s an analogy with EPR that shows the weirdness of the predictions in classical terms.

You have three players, called Alice, Bob and Charlie, playing a game. The game consists of many rounds of play.

During each round, Charlie takes six envelopes and puts into each envelope a piece of paper, on which is written “up” or “down”. He writes the number “0” on two envelopes and gives one to Bob and one to Alice. He writes “120” on two envelopes and gives one to Bob and one to Alice. He writes “240” on the final two envelopes and gives one to Bob and one to Alice.

Each round, Alice and Bob choose one of their envelopes to open and discard the others.

Charlie chooses the messages in the envelopes to create the following statistics:

* Whenever Alice and Bob open envelopes with the same label, they always get the opposite results (either Alice has the word “up” and Bob has the word “down”, or vice-versa)

* Whenever they open differently labelled envelopes, 75% of the time they get the same result, and 25% of the time they get the opposite result.

The first rule is easy to implement: Charlie just makes sure that whatever he writes on the page put in one envelope, he writes the opposite for the other envelope with the same label. This is analogous to “conservation of spin”. The result in envelope labeled 0 is the spin of one particle in the direction in the x-y plane that makes an angle of 0 degrees relative to the x axis. The result in the envelope labelled 120 is the spin aling the direction that makes a 120 degree angle relative to the x-axis. Analogously for the envelope labelled 240.

But the second rule is harder to implement. If Charlie tries to make Alice’s 0 envelope agree with Bob’s 120 envelope 75% of the time, that means Alice’s 0 envelope must disagree with her 120 envelope 75% of the time. And it must disagree with the 240 envelope 75% of the time. And her 120 and 240 envelopes must disagree 75% of the time.

Those are impossible statistics unless something weird is going on, such as (1) Maybe Charlie knows ahead of time which envelopes will be picked, or (2) what’s written on the pages in Alice’s envelopes change, depending on which envelope Bob chooses, or vice-versa.

These two possibilities correspond to (1) superdeterminism and (2) nonlocality as two loopholes to EPR.

 

[vanhees71]

Of course entanglement contradicts such classical-statistical scenarios.

Of course neither of the two possibilities is the answer of QT (in its most comprehensive form as relativistic QFT), which gives as a 3rd possibility: it's local (in the sense of the locality of interactions) but allows for inseparability described by entanglement between far-distant subsystems of a quantum system. All the weirdness goes away as soon as you accept this 3rd possibility!

 

[DrChinese]

Isn't entanglement just the conservation of spin, momentum, and position along with Heisenberg's uncertainty principle? I don't really see the big mystery or maybe I am just not understanding a key part of what makes entanglement so interesting. Can someone pinpoint how this receives so much attention in popular science?

My comments overlap with those of several others above.

1. Entanglement, as usually described, is a state of quantum properties of 2 or more particles. In that state, the particles cannot be properly described separately (in a Product state). This leads to the situation such that when they are separated in spacetime, they display characteristics referred to in the literature as "quantum nonlocality". That's a pretty big mystery! Quantum interpretations (Copenhagen, Many Worlds, Bohmian Mechanics) are designed to explain this mystery in one manner or another.

Bell tests are evidence of quantum nonlocality. Without understanding Bell, it is difficult to answer your question satisfactorily. And in fact, entanglement - famously written about by Einstein and others in 1935 - did not get much traction until after the first major Bell tests were performed in the early 1980's. Bell's paper itself was written in 1964. Here are a couple of introductory links written by yours truly to get you started, or point you to other resources:

Bell's Theorem - An Overview
Bell's Theorem with Easy Math2. Entanglement is one of the most studied areas of quantum physics. Well over 1000 theoretical and experimental papers a year are written on this, and this has been the case for the past 20 years. Here is a sample from last month:

https://arxiv.org/find/quant-ph/1/AND+all:+nov+abs:+entanglement/0/1/0/2020/0/1?per_page=100

 

[andresB]

In classical mechanics we don't have spin to begin with. So this is a somewhat questionable analogy.

That's a common misconception. We can have an intrinsic angular momentum for point particles in classical mechanics for the same reason we can have spin in QM, as a degree of freedom that is allowed by the Galilei/Poincare algebra.

 

[vanhees71]

The entanglement is the weirdness, so no, it doesn’t go away.

That's your feeling. Mine is that there's no weirdness but an empirically utmost successful description of how Nature behaves by the formalism of quantum theory. It's only weird if you insist on oldfashioned philosophical prejudices. Nature doesn't care about our philosophy!

 

[vanhees71]

That's a common misconception. We can have an intrinsic angular momentum for point particles in classical mechanics for the same reason we can have spin in QM, as a degree of freedom that is allowed by the Galilei/Poincare algebra.

Do you have a reference for that claim? I always thought that spin is a field-theoretical notion. You have, of course, spin in classical field theory. In classical electrodynamics the electromagnetic field carries also angular momentum including spin (though you have to take this with a grain of salt since massless fields with spin $\geq 1$ are special, because they must be gauge fields and there's no gauge-invariant split of total angular momentum in spin and orbit parts; only the total angular momentum and helicity are gauge-invariant observable quantities).

 

[andresB]

Do you have a reference for that claim? I always thought that spin is a field-theoretical notion. You have, of course, spin in classical field theory. In classical electrodynamics the electromagnetic field carries also angular momentum including spin (though you have to take this with a grain of salt since massless fields with spin $\geq 1$ are special, because they must be gauge fields and there's no gauge-invariant split of total angular momentum in spin and orbit parts; only the total angular momentum and helicity are gauge-invariant observable quantities).

That's a strange notion, you don't even need field theory for the quantum case, spin arises in non-relativistic QM just as fine.

For the classical case, the references are scattered around through the decades, I would have to hunt them again since I've lost track of most of them. Though, take a look at Sudarshan & Mukunda "Classical mechanics: A modern perspective", the chapter of the canonical representation of the Galilei group.

In both cases, the spin comes from the possibility of adding a term to the total angular momentum, $J=L+s$, such that it's brackets (whether commutator or Poisson bracket) with the momentum and position vanish $\left[s_{i},x_{j}\right]=\left[s_{i},p_{j}\right]=0$. Such term does not affect the Galilei/Poincare algebra, hence it's a solution to the representation problem.

Of course, in classical situations the spin formalism only is useful as an approximation for the angular momentum from the rotation of extended bodies. We now that elementary particles are quantum objects, with quantum spin so it is pointless to talk about the classical spin of the electron. But the theoretical framework is there, it's just not useful in microscopic situations.

 

[vanhees71]

That's a strange notion, you don't even need field theory for the quantum case, spin arises in non-relativistic QM just as fine.

For the classical case, the references are scattered around through the decades, I would have to hunt them again since I've lost track of most of them. Though, take a look at Sudarshan & Mukunda "Classical mechanics: A modern perspective", the chapter of the canonical representation of the Galilei group.

In both cases, the spin comes from the possibility of adding a term to the total angular momentum, $J=L+s$, such that it's brackets (whether commutator or Poisson bracket) with the momentum and position vanish $\left[s_{i},x_{j}\right]=\left[s_{i},p_{j}\right]=0$. Such term does not affect the Galilei/Poincare algebra, hence it's a solution to the representation problem.

Of course, in classical situations the spin formalism only is useful as an approximation for the angular momentum from the rotation of extended bodies. We now that elementary particles are quantum objects, with quantum spin so it is pointless to talk about the classical spin of the electron. But the theoretical framework is there, it's just not useful in microscopic situations.

Well, non-relativistic QM is a field theory at least in the formulation of wave mechanics, and that's why it's easy to describe spin. You just need to analyze the unitary representations of the physically adequate extension of the Galilei group to very naturally find the possibility of spin (i.e., an intrinsic angular momentum). I don't see how you get this in a natural way within classical mechanics. Maybe there's some analogy for extended objects (like the rigid body, where you have something pretty much analogous to spin).

I'll have a look at the cited book.

 

[DrChinese]

I have deleted my posts that were off-topic. And this thread reminded me why I left Physics Forums. The discussions are not respectful. I think I wiling leave again.

Nooo! I well know how frustrating some of these circular discussions are. Especially those with certain individuals who suck the life out of threads and completely lose sight of the bigger picture at PF. But I hope you continue reading and contributing as you have for many years.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

A number of off topic posts have been deleted (in addition to those by @stevendaryl that he already deleted himself). The OP question has been answered and the thread will remain closed.

 

[PeterDonis]

I hope you continue reading and contributing as you have for many years.

@stevendaryl I heartily concur with this hope. Please feel free to use the Report button to bring off topic posts to the attention of the moderators, in order to help keep discussions within bounds. (Also, your deleting your own posts that you felt were off topic sets an excellent example that I hope more members will try to emulate.)