There is nothing wrong with that idea. It's even what a strict standard interpretation tells you. A conscious measurement collapses the wave function spatiotemporally.

In learning about QM and amazing experiments like the delayed-choice quantum eraser experiment, I have come to think that in a way the quantum measurement precedes the history. — keystone

Measurement doesn't affect anything in the past. In the delayed-choice quantum eraser experiment, the original observed pattern remains the same regardless of the later choice of measurement.

The choice of measurement instead allows information to be decoded from the observed pattern, i.e., information revealing interference, or not.

That's the consensus amongst physicists. None of the major interpretations posit retrocausality.

However, the interference pattern can only be seen retroactively once the idler photons have been detected and the experimenter has had information about them available, with the interference pattern being seen when the experimenter looks at particular subsets of signal photons that were matched with idlers that went to particular detectors. — Delayed-choice quantum eraser: Consensus: no retrocausality - Wikipedia

Measurement doesn't affect anything in the past. — Andrew M

It does! It collapses wavef function in space and time. The past was in superposition untill we measured it. Remember Copenhagen...

In other words, if I were to simulate a quantum universe, I would start with a wave function of the universe that spans all of 'simulated time' and then as an external observer, I would make a measurement at some particular simulation time to reduce the wave function to a definite state at that instant. — keystone

That would be supernatural interference with the universe. The Wigner interpretation suggests something like that.

Such a measurement constrains the simulated reality in a way that I can deduce aspects of the history preceding that simulated moment. In this sense, the history follows the measurement, not the other way around.

Maybe try RQM instead. It doesn't involve supernatural causation, but it does involve reverse ontology such as you suggests. A measurement of an object defines its existence relative to the measurer, and the object measured is in the past light cone of the measurement, thus a sort of reverse causality where the existence of things is dependent on future measurement.

Not sure how you'd go about simulating a wave function in this way. There is no objective collapse, and even the simplest system would defy computation by classical means.

It's even what a strict standard interpretation tells you. — Landoma1

Which interpretation do you consider 'standard'?

Measurement doesn't affect anything in the past. — Andrew M

It does in some counterfactual interpretations like Bohmian mechanics. That's a pretty major interpretation.

The choice of measurement instead allows information to be decoded from the observed pattern, i.e., information revealing interference, or not. — Andrew M

I'm trying to unpack this statement. Could this be related to the Wikipedia entry where it says that "a photon in flight is interpreted as...something that has the potentiality to manifest as a particle or wave, but during its time in flight is neither." So upon flight the photon has a potential state but upon 'decoding' we deduce a history to say that it held an actual state (of a particle or a wave)? If so, the deduction of history follows the measurement, which is what I'm positing.

I don't think my current understanding assumes retrocausality since I'm not claiming that any information is being sent back in time.

That would be supernatural interference with the universe. The Wigner interpretation suggests something like that. — noAxioms

True. But if a wave function needs an actual observer to collapse it, doesn't it need an external observer to make the first observation (in my simulation example)? Until then the simulated universe is entirely a superposition of potential states.

Maybe try RQM instead. — noAxioms

From my amateur position, the idea of there being no objective collapse doesn't sit well with me. I'll keep this in mind but right now I'm still grappling with vanilla QM.

Measurement doesn't affect anything in the past. — Andrew M


It does! It collapses wave function in space and time. The past was in superposition untill we measured it. Remember Copenhagen... — Landoma1

Welcome back, quantum mysticism. "Collapse of the wave function!" carries us away from Earthly distractions into the cirque of the gods where ectoplasm interacts with aether causing spacetime curvature. Superposition is annihilated with a bolt from Zeus!

Welcome back, quantum mysticism. "Collapse of the wave function!" carries us away from Earthly distractions into the cirque of the gods where ectoplasm interacts with aether causing spacetime curvature. Superposition is annihilated with a bolt from Zeus! — jgill

It is not that QM is not mysticism that interests me; it is that they seem to be so easily conflated with each other that I find so intriguing!

Measurement doesn't affect anything in the past.
— Andrew M
It does in some counterfactual interpretations like Bohmian mechanics. That's a pretty major interpretation. — noAxioms

I don't think that's the case. From SEP (though note they do discuss a retrocausal variant there):

Foundational accounts which like Bohmian mechanics (Bohm 1952a,b) or GRW-theory (Ghirardi, Rimini, & Weber 1986) avoid postulating retrocausality do so by violating time-symmetry in some way. The GRW-theory does so by introducing explicitly time-asymmetric dynamics. In Bohmian mechanics the dynamics is time-symmetric, but the theory is applied in a time-asymmetric manner when assessing which quantum states are actually realized. — Retrocausality in Quantum Mechanics - SEP

The choice of measurement instead allows information to be decoded from the observed pattern, i.e., information revealing interference, or not.
— Andrew M

I'm trying to unpack this statement. Could this be related to the Wikipedia entry where it says that "a photon in flight is interpreted as...something that has the potentiality to manifest as a particle or wave, but during its time in flight is neither." So upon flight the photon has a potential state but upon 'decoding' we deduce a history to say that it held an actual state (of a particle or a wave)? If so, the deduction of history follows the measurement, which is what I'm positing.

I don't think my current understanding assumes retrocausality since I'm not claiming that any information is being sent back in time. — keystone

Fair enough. The important (non-spooky) point with the delayed-choice quantum eraser experiment is that the original observed pattern for the signal photon (upper red and blue paths in Figure 2) is the same, regardless of what happens later to the entangled idler photon (lower red and blue paths) with beam splitters and what not. Which is to say, one's delayed choice has no effect on that signal photon pattern, and nothing is ever erased in that signal photon pattern. Instead, if the which-way information for the idler photon is erased then you can use the later detector recordings to reveal two phase-shifted interference patterns in the signal photon pattern, per Figure 5 (i.e., R01 and R02).

So it's different to the double-slit experiment where the choice to detect the photons, or not, at the slits does affect what is later observed on the back screen.

I'm not sure anything can be deduced about the history of the photon in either experiment since one's assumptions will be interpretation-dependent.

Maybe try RQM instead.
— noAxioms

From my amateur position, the idea of there being no objective collapse doesn't sit well with me. I'll keep this in mind but right now I'm still grappling with vanilla QM. — keystone

Interestingly, that would be a different physical theory which can be potentially tested. That's given rise to the Wigner's Friend-style thought experiments over the past few years.

4 Wigner-Deutsch thought experiment

Two central questions that came up repeatedly in our discussions so far are (i) “Does Alice see a definitive measurement outcome?” and (ii) “Is Alice’s lab after the measurement indeed in a superposition state?” Deutsch [23] proposed an extension of the thought experiment described in Section 3, which turns these questions into (in principle) experimentally testable statements. — Testing quantum theory with thought experiments, p17 - Nurgalieva, Renner

The important (non-spooky) point with the delayed-choice quantum eraser experiment is that the original observed pattern for the signal photon (upper red and blue paths in Figure 2) is the same, regardless of what happens later to the entangled idler photon (lower red and blue paths) with beam splitters and what not. — Andrew M

What would you say happens to the signal pattern in the unlikely case that all idler photons are (by a stroke of luck) erased?

So it's different to the double-slit experiment where the choice to detect the photons, or not, at the slits does affect what is later observed on the back screen. — Andrew M

Is it fair to say that the difference is that double-slit experiments with and without measurement are performed sequentially while the delayed-choice quantum eraser experiment with and without measurement are performed in parallel. Because it's performed in parallel, it has one extra step at the end to separate the measured from the unmeasured data sets. With this view, it seems that the essence of the experiments are the same. Might you be discrediting the notion of true delayed-choice by attacking an insignificant aspect of the experiment (that it was conducted in parallel)?

I'm not sure anything can be deduced about the history of the photon in either experiment since one's assumptions will be interpretation-dependent. — Andrew M

If one sees an interference pattern can we deduce that there must be some element of waviness about the photon's flight through the double slits?

What would you say happens to the signal pattern in the unlikely case that all idler photons are (by a stroke of luck) erased? — keystone

It doesn't matter what happens to the idler photons, the signal pattern will remain the same. From the Wikipedia page (especially the bolded part):

The total pattern of all signal photons at D0, whose entangled idlers went to multiple different detectors, will never show interference regardless of what happens to the idler photons. One can get an idea of how this works by looking at the graphs of R01, R02, R03, and R04, and observing that the peaks of R01 line up with the troughs of R02 (i.e. a π phase shift exists between the two interference fringes). — Delayed-choice quantum eraser - Significance - Wikipedia

That is, if the idler photon which-way information is always erased, then the pattern is D0 = R01 + R02. If it is never erased, then the pattern is D0 = R03 + R04. Either way the pattern at D0 is the same and never directly shows interference. Information has to be gleaned from the idler photon detectors (and matched via the coincidence counter to the signal photon partners) to reveal R01 through R04.

Is it fair to say that the difference is that double-slit experiments with and without measurement are performed sequentially while the delayed-choice quantum eraser experiment with and without measurement are performed in parallel. — keystone

As the experiment is performed there, yes. Though the experiment could be changed to keep them separate.

Because it's performed in parallel, it has one extra step at the end to separate the measured from the unmeasured data sets. With this view, it seems that the essence of the experiments are the same. Might you be discrediting the notion of true delayed-choice by attacking an insignificant aspect of the experiment (that it was conducted in parallel)? — keystone

No, as noted above. R01 through R04 are always there. It just takes different kinds of measurements on the idler photon to reveal them.

If one sees an interference pattern can we deduce that there must be some element of waviness about the photon's flight through the double slits? — keystone

Yes. But, as the DCQE experiment shows, there is an element of waviness even when an interference pattern is not observed. It just requires the use of entanglement to reveal it.

I'm trying to unpack this statement. Could this be related to the Wikipedia entry where it says that "a photon in flight is interpreted as...something that has the potentiality to manifest as a particle or wave, but during its time in flight is neither." So upon flight the photon has a potential state but upon 'decoding' we deduce a history to say that it held an actual state (of a particle or a wave)? If so, the deduction of history follows the measurement, which is what I'm positing.

I don't think my current understanding assumes retrocausality since I'm not claiming that any information is being sent back in time. — keystone

I find that the terms "potential" and "actual" make more sense to me, as a layman, than "superposition" and "collapse". From that perspective, an unmeasured (undefined) Photon does not exist as a particle, but only a propagating "wave" of Possibility in an oceanic (holistic ; entangled) system of Potential Energy. When traveling at light-speed, It has no mass (matter) because it's not yet "manifest" as an individual "thing". Only when something slows down the quantum wave, by interference from the classical (macro) environment, does the wave begin to show specific properties, such as heat & mass.

The "interference" is like a pier post in the water (or a slit in a barrier) , it disturbs the incoming general (holistic) waveform, forcing it to take-on a specific (particular) form (i.e. disentanglement). In Beyond Weird, by Phillip Ball, he says, "we destroy the quantumness [entangled potential state] in proportion to the amount of information [energy] we import from the system [undefined state] into the environment [actual matter]". [my brackets]. When a massless photon smacks into a hunk of enformed matter, it transforms potential Energy into actual Force, causing changes in its structure.

Ball goes on to say that "the more information the environment 'absorbs' about a dust grain in superposition of position states, the more the grain becomes localized". Hence, Potential is non-local, and Actual is localized. In visual terms, the "grain" becomes less entangled in the oceanic system, and stands-out from the background as an individual object. However, we "deduce" that the particle was there all along, even before it became visible. You could say that it was there "in principle" (ideally ; theoretically) but not actually, until an observation reveals its location. This is the counter-intuitive Observer Effect, that Einstein questioned, by asking if the moon is still there even when no-one is looking at it.

You know the moon, or particle is there now, during the observation, but where was it in the past, unobserved? And what is the temporal relation between Actual "Now" and Potential "Then" ( a future state)? From a classical viewpoint, it's an apparent paradox. But in terms of abstract principles (potential & actual) it makes sense. That may be why Einstein proposed the counter-intuitive notion of Block-Time, in which all temporal states exist simultaneously in a static timeless eternity. Hence, "retro-causality" is not Actual, but merely Ideal (a mental image).

So, it's all relative. From one point-of-view, "history follows the measurement", as meaning follows a query. But from another angle, the pre-measured object was there all the time; you just didn't know it. The Greek root (metr- to measure) meant to extract information into a mind (L. mensura- to measure, from mens- mind or intention). In Quantum Theory, to extract information is equivalent to removing some matter/energy stuff from the thing observed (to abstract). But like pressing a rock into clay, the concave impression is not the thing, but it contains information about the thing. Ball calls that extracted information an "imprint" or a "replica". But that abstract knowledge is not the ding an sich. :nerd:

PS__My conclusion is that "reality" is both Potential and Actual. But we only know the actual stuff by means of our physical senses. The Potential stuff is only known by Reasoning backward from Actual Effects to Potential Causes. The relationship is similar to Plato's Ideal (mental) Form and Real (physical) things.

The theory of Forms or theory of Ideas is a philosophical theory, concept, or world-view, attributed to Plato, that the physical world is not as real or true as timeless, absolute, unchangeable ideas. ___Wikipedia

PPS__The Observer's Choice "frames" reality as a personal interpretation of what's out there.
Einstein once said: “Reality is an illusion, albeit a persistent one ”. So, what we think of as reality is, in fact, just a local-personal-relative interpretation of it. In society, and in science, we share our particular frames in order to create a general conventional view of the world.

It doesn't matter what happens to the idler photons, the signal pattern will remain the same. — Andrew M

What if all idler photons strike D2? This requires an element of pure luck as the photons pass through BS3, but it also requires an element of choice in deciding what happens after the photons pass through BSa and BSb. In this case, doesn't the element of choice have an impact on the signal pattern?

It is not that QM is not mysticism that interests me; it is that they seem to be so easily conflated with each other that I find so intriguing! — Agent Smith

Many of the pioneers of Quantum Theory -- (cat-killer) Schrodinger ; (buddha) Heisenberg ; Pauli ; Bohr ; Bohm ; Wigner ; Capra ; Seife ; etc. -- were slandered as "mystics", in part due to the mental metaphors (observation, choice, etc.) they used to explain & understand the "spooky" quantum paradoxes compared to "realistic" Classical science. Ironically, realist Einstein was proven wrong, and Quantum Queerness came to be taken for granted, as the weird way of the underworld. Perhaps, "lucid mysticism" is the "conflation" you had in mind. :smile:

Quantum Mysticism :
Does mysticism have a place in quantum mechanics today, or is the idea that the mind plays a role in creating reality best left to philosophical meditations? Harvard historian Juan Miguel Marin argues the former - not because physicists today should account for consciousness in their research, but because knowing the early history of the philosophical ideas in quantum mechanics is essential for understanding the theory on a fundamental level. . . . .
Pauli favored a hypothesis of “lucid mysticism,” a synthesis between rationality and religion.
https://phys.org/news/2009-06-quantum-mysticism-forgotten.html

Quantum Quacks :
https://en.wikipedia.org/wiki/Quantum_mysticism

Potential vs Actual :
In his earlier work, Stapp (1993) started with Heisenberg’s distinction between the potential and the actual (Heisenberg 1958), thereby taking a decisive step beyond the operational Copenhagen interpretation of quantum mechanics. While Heisenberg’s notion of the actual is related to a measured event in the sense of the Copenhagen interpretation, his notion of the potential, of a tendency, relates to the situation before measurement, which expresses the idea of a reality independent of measurement.
https://plato.stanford.edu/entries/qt-consciousness/

My conclusion is that "reality" is both Potential and Actual. But we only know the actual stuff by means of our physical senses. The Potential stuff is only known by Reasoning backward from Actual Effects to Potential Causes. — Gnomon

I'll leave it to the more informed to comment on your other thoughts, but I like this conclusion. Classically there is a desire for everything to be actualized but with QM there is more room for the potential. And if I understand the quantum zeno effect correctly, without the existence of potential, change is impossible.

What if all idler photons strike D2? This requires an element of pure luck as the photons pass through BS3, but it also requires an element of choice in deciding what happens after the photons pass through BSa and BSb. In this case, doesn't the element of choice have an impact on the signal pattern? — keystone

No, those two beam splitters just reduce the probability of all the idler photons striking D2. If the BSa and BSb beam splitters were removed, then the odds of 20 idler photons all striking D2 would be 1/2^40 = one in a million. In that case, the signal photons would have created the interference pattern R02 which we would observe and be puzzled by (assuming 20 signal photons is enough for a visible pattern to emerge - looking at Figure 4, presumably not, but that's the basic idea).

Adding back the BSa and BSb beam splitters just reduces the probability of an idler photon striking D2 from 1/2 to 1/4, with the odds of 20 idler photons all striking D2 being 1/4^20 = one in a trillion. That doesn't affect the signal pattern which would still be R02. If there were no beam splitters at all and the idler photons sped off into space, that also wouldn't affect the signal pattern.

In every case, we would simply be left with the puzzle of how that extremely unlikely signal pattern occurred. Like the particles in a box all randomly moving to one side of the box, or an egg unscrambling, it's not something we would expect to observe.

In every case, we would simply be left with the puzzle of how that extremely unlikely signal pattern occurred. Like the particles in a box all randomly moving to one side of the box, or an egg unscrambling, it's not something we would expect to observe. — Andrew M

For the double-slit experiment, are there any areas on the back screen which have exactly 0% chance of being hit by the photon when path information is known but >0% when path information is unknown. Or do all areas have >0% chance for both scenarios? I'm trying to understand if we're talking about an unlikely scenario or an impossible scenario.

By the way, I really appreciate your comments so far!!

So much confusion about Quantum theory occurs because those who are not physicists try to take mathematical descriptions (they probably do not understand) and turn them into physical objects.

The wave function is a probability amplitude form that has complex values - not something traveling through space. Superposition arises from a linear differential equation (DE) having linear combinations of solutions, not some mystical process hovering in the aether ready to appear after some sort of magical "collapse".

The concept of "measurement", which we all assume is something like an electronic ruler or scale, seems to be at the heart of understanding QM and has no easily understood definition. Wiki:

In quantum mechanics, each physical system is associated with a Hilbert space, each element of which is a wave function that represents a possible state of the physical system. The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable"

As I have pointed out before, a stripped down version of the Schrödinger equation is nothing more than a very simple DE describing, for example, continuous compounding from your local bank. Somewhere in all of this is the miasma of entanglement, which only God seems able to unravel. :roll:

Well, if what you say is true then the law of gravitation as discovered and elucidated by Newton should've been associated with alchemy because Newton was quite deeply interested in that subject.

It seems the association between QM and mysticism was merely an accident - QM heavyweights like Heisenberg, etc. were drawn to Hindu mysticism and people jumped to conclusions ( :roll: ). This QM-Mysticism link was reinforced by "coincidental similarities of language rather than genuine connections".

Nevertheless, no smoke without fire...there maybe some connection between the two that begs for greater scrutiny.

It seems the association between QM and mysticism was merely an accident - QM heavyweights like Heisenberg, etc. were drawn to Hindu mysticism and people jumped to conclusions ( :roll: ). This QM-Mysticism link was reinforced by "coincidental similarities of language rather than genuine connections". — Agent Smith

The primary difference between Classical and Quantum physics is that on the sub-sensory level (e.g sub-atomic) your physical senses can't detect objects smaller than the wavelength of the the visible spectrum. An optical microscope is useless for viewing atomic-scale objects -- it's all just an undifferentiated blur; like the surface of the ocean concealing the myriad lifeforms in the deep. Consequently, scientists were forced to view their minuscule subjects Holistically (entangled in a group) instead of in the Classical Reductive manner (chop the system into its constituent parts). Coincidentally, Eastern philosophy -- which was just-then entering the consciousness of the colonizing West -- had, long before modern technology, already developed techniques of dealing with whole systems, in which the parts are unknown, hence mysterious.

So, the pioneers of Quantum physics merely borrowed some of the Hindu & Buddhist metaphors to describe the quirky quantum behaviors of entangled particles. To paraphrase the spoon-bender in The Matrix, a particle can pass through a barrier, because "there is no wall". The "wall" and the "particle" are One. When the Buddha said there is "no self" he probably meant that your personal "self" is an integral part of a larger system (the world soul). Again, coincidentally, those spiritual Eastern concepts were found to be useful for dealing with physical Quantum concepts. There was "no genuine connection" though, because you didn't have to become a practicing Buddhist or Transcendental Meditator to appreciate the value of a holistic perspective.

However, those same concepts were adopted by New Agers, who were looking for a "genuine connection" (spiritual instead of physical). They were tired of the fragmentation of Western societies, and the reduction of Capitalism to "cash is king" as a tool for exploitation. Unfortunately, my own application of Holism in the Enformationism worldview is often dismissed, by loyal believers in reductive Scientism, as New Age woo-woo. However, rather than rejecting Western Science or adopting Eastern Religions, I merely try to have the best of both worlds : Hence, the BothAnd philosophy. There's nothing inherently mystical in Holism, unless you want to worship the Mystery of Oneness. For me, Holism & Reductionism are merely two sides of the single coin of philosophical Wisdom. :nerd:

Most interesting. — Ms. Marple

A bit disappointing that a mystical interpretation of QM is considered woo-woo; just when it was getting interesting.

What you point is very much true. Those of us not gifted in math, have no choice but to rely on physicists to try and explain this at some level of generality and simplification.

But how do we make sense of math in relation to the external world, if we (non-specialists) can't visualize the information in any way?

It seems as if we are more satisfied when we can picture things than when we are left with no choice than write equations.

It's a tough topic to talk sensibly about, you are correct.

It seems as if we are more satisfied when we can picture things than when we are left with no choice than write equations — Manuel

You and me both. The math I explore has strong visual components, but not from nature. Patterns in the complex plane mostly. Look at my icon.

For the double-slit experiment, are there any areas on the back screen which have exactly 0% chance of being hit by the photon when path information is known but >0% when path information is unknown. Or do all areas have >0% chance for both scenarios? I'm trying to understand if we're talking about an unlikely scenario or an impossible scenario. — keystone

See Figure 1-3 and Figure 1-4 of Feynman's lectures. Figure 1-4 (c) (path information known) shows that photons can strike anywhere on the back screen. Whereas Figure 1-3 (c) (path information unknown) shows several areas where photons will not strike (due to destructive interference).

By the way, I really appreciate your comments so far!! — keystone

Yours as well! It's been a good discussion.

(path information known) shows that photons can strike anywhere on the back screen. — Andrew M

Thank you. I now understand that nothing about the idler photons can be deduced by looking at the signal pattern. However, I'm still reluctant to say that the signal pattern is unaffected by the idler photons. You say:

The choice of measurement instead allows information to be decoded from the observed pattern, i.e., information revealing interference, or not. — Andrew M

When is this information encoded?

The choice of measurement instead allows information to be decoded from the observed pattern, i.e., information revealing interference, or not.
— Andrew M

When is this information encoded? — keystone

This information is encoded (or, better, implicit) at the time of entanglement, i.e, when the entangled pair of photons are created at the BBO in Figure 2. However the key information needed to decode it is not available until a measurement occurs (i.e., when an idler photon is detected).

--

Here's the mathematical basis for that answer.

The entangled signal and idler photons are initially in the following Bell state (ignoring the square root of 2 factor):

$|00\rangle + |11\rangle$

The first qubit (i.e., the first bit in each component of the state) represents the signal photon, the second qubit represents the idler photon. 0 indicates the red path, 1 indicates the blue path.

Both the signal and idler photons are in superposition and correlated in the z-basis (also called the computational or standard basis).

When the idler photon is detected at D3 or D4, that constitutes a measurement in the z-basis. If detected at D4 (the red path detector), the state collapses to:

$|00\rangle$

Since the idler photon was on the red path, that means that the signal photon was also on the red path.

Now suppose that the idler photon was instead detected at D1 or D2. In this case, the idler photon first passed through the beam splitter BSc. This is, in effect, a Hadamard operation that rotates the idler qubit from the z-basis to the x-basis (also called the plus-minus basis). $|0\rangle$ and $|1\rangle$ in terms of the x-basis are:

$|0\rangle = \frac{|+\rangle + |-\rangle}{\sqrt 2}, |1\rangle = \frac{|+\rangle - |-\rangle}{\sqrt 2}$

Substituting the above into our Bell state (again ignoring the square root of 2 factors) gives:

$\begin{aligned}|00\rangle + |11\rangle & = |0\rangle(|+\rangle + |-\rangle) + |1\rangle(|+\rangle - |-\rangle) \\& = |0+\rangle + |0-\rangle + |1+\rangle - |1-\rangle \\& = (|0\rangle + |1\rangle)|+\rangle + (|0\rangle - |1\rangle)|-\rangle \end{aligned}$

Note that the beam splitter only operates on the idler photon, but it transforms the way the signal photon is represented (i.e., as in a superposition of the red and blue paths).

When the idler photon is detected at D1 or D2, that constitutes a measurement in the x-basis. If detected at D1 (the red path + blue path detector), the state collapses to:

$(|0\rangle + |1\rangle)|+\rangle$

Thus the signal photon was in a superposition of being on the red path and on the blue path when it struck detector D0. So the signal photons that correspond to the idler photons detected at D1 will reveal an interference pattern.

Conversely, if the idler photon is detected at D2 (the red path - blue path detector), the state collapses to:

$(|0\rangle - |1\rangle)|-\rangle$

Thus the signal photon was also in superposition (with a relative phase of -1, or pi radians). It similarly reveals an interference pattern which, when added to the D1 interference pattern, results in the observed signal pattern.

--

For an imperfect visual metaphor, consider a round dining room table. In the z-basis (i.e., looking at it from the top or bottom) it would look like a circle. In the x-basis (i.e., looking at it from the side), it would look like a line. It's the same table but perceived from different vantage points.

Note that the beam splitter only operates on the idler photon, but it transforms the way the signal photon is represented (i.e., as in a superposition of the red and blue paths). — Andrew M

Thank you for the detailed response. I assume you are being fully clear and it is just me not fully comprehending your answer, but let me ask this: how can the idler photon have any impact on the signal photon representation if the signal photon hits D0 before the idler photon hits PS?

how can the idler photon have any impact on the signal photon representation if the signal photon hits D0 before the idler photon hits PS? — keystone

When the idler photon (qubit) is measured in the z-basis or the x-basis, then we can infer the path or the phase of the signal photon respectively. Making a local measurement of the idler photon has no impact on the signal photon. As you note, it has already hit D0 and contributed to a non-interference pattern independent of which basis the idler photon is measured in. But the choice of measurement does have an impact on what we learn about the signal photon and also on what we learn about the signal pattern that builds up over multiple runs of the experiment.

In terms of the dining room table metaphor, measuring in the z-basis is like perceiving the table from either the top or the bottom. Measuring in the x-basis is like perceiving the table from the side, say, from the north end or the south end. When we do the latter, we have north or south information, but not top or bottom information. But, further, there is no top/down information from that orientation. This is analogous to the Heisenberg Uncertainty Principle. You can't measure both the top/bottom and the north/south orientations of the table at the same time - that information doesn't exist.

When the idler photon (qubit) is measured in the z-basis or the x-basis, then we can infer the path or the phase of the signal photon respectively. Making a local measurement of the idler photon has no impact on the signal photon. — Andrew M

I'm finding it hard to see how these two sentences are compatible. The second sentence suggests that measuring the idler photon does not impact the signal pattern at all. The first sentence suggests that measuring the idler photon does give information on subsets of the signal pattern. If the second sentence were true, I would expect all 4 subsets to produce the same signal pattern.

IMHO, it seems like your mathematical explanation only supports your first sentence. How does it support your second sentence? Or even better for a layman like me, how does your dining room table metaphor support your second sentence?