Are *all* quantum states indeterminate if they are outside the observer's light cone?

Question

Is the state of a particle arriving from outside an observer's light cone merely unknown, or is it quantum mechanically indeterminate?

I guess the question needs a background in the form of a gedankenexperiment, so let's assume that someplace far away and long ago, someone (person B) built a laser that emits individual vertically polarized photons spaced at uniform time intervals, then passed the laser beam through a polarizing beamsplitter tilted at 45 degrees, then directed the emerging beam in our direction. Person B also sets up a detector to detect any photons deflected by the beamsplitter, thereby obtaining a set of measurements we can represent as a random string of 1's and 0's. Person B knows that the beam on its way toward us is the complement of the beam he has detected, and can be described as the complement of the string of 1's and 0's he detected. So, from Person B's perspective the photons in the beam are sent toward us in a way that corresponds deterministically to the 1's and 0's he detected.

Because it takes time for Person B's light beam to get to us, we know that the beam was emitted at a point outside our light cone. Because the beam eventually does get to us, we know that at that point of arrival our light cone intersects with the light cone of the emission event.

IF we know (perhaps because Person B sent us a text message on his laser beam) that Person B set up his experiment the way he did, we can detect the timing with which the incoming photons arrive and thereby infer the string of 1's and 0's Person B has recorded. So far, this is essentially identical to establishing a quantum key (can't correctly say that the key is "transmitted"). Person B's random string is entangled with the random string we detect at our end.

Edit 8/11/18: Now reduce the number of photons in the beam to two. That is, we can receive only (1,1),(1,0),(0,1) or (0,0), and Person B can, correspondingly, only have (0,0),(0,1),(1,0), or (1,1) at his end. In effect, Person B is sending us the value of two qbits which can each have either the value 1 or the value 0. (I know-- some more photons would need to be sent to tell us when to start our clock, etc., but I assume we can ignore that without confusing the results.) The point is that Person B can send us a quantum object whose state he knows, and he is sending it from outside our light cone.

This seems to be a significantly different situation than the standard EPR scenario, in which travelers A and B carry entangled quantum objects to widely separated points and then perform measurements, because in the standard scenario the entangled objects and the travelers have a common origin so they are always within each others' light cones.

Here is the first part of my question, which seems easy to answer:

Does the quantum object (the (1,1),(1,0),(0,1) or (0,0) signal) arrive in a mixed state? My first inclination is to say "no", because Person B knows what it's going to be. But in an EPR experiment, the order in which the measurements occur depends on the frame of reference -- so it's improper to say that one traveler's measurement causes the other traveler's measurement to have the same value. So, in some sense, in the scenario I'm proposing, the combined states of Person B's recorded signal and our detected signal are a single quantum object. That is, from our perspective Person B's recorded signal is not determined until we detect the incoming signal (billions of years later). In other words, the signal should arrive in a mixed state.

The second part of my question is: "Does every particle arriving from outside our light cone arrive in a mixed state?". I think the answer is "yes" (complicated somewhat by possible entanglements between the arriving particles), but haven't come up with convincing arguments. The closest I've come to an argument is to point out that the state of a particle arriving from outside our light cone is inherently unknowable in advance -- which sounds a lot like the definition of a mixed state.

Answer 1

To answer this, I have to be careful. We are measuring things, which may have a state described by a superposition, and then that measurement provides classical information.

I think the answer to your question is that the experiment is flawed because the 2 photon system can transmit more than 2 symbols. It can transmit (0, 0) (0, 1) (1, 0) and (1, 1). However, to cover my bases, I've got a bit more written down here to see if things can be straightened out. I don't think any actual quantum weirdness enters the story. Nothing is in a meaningful mixed state until we add some additional constraints to your problem which weren't there in your original wording.

The light cone details turn out to not be important for understanding this case. Light cones only become important when you start to get into systems which appear to transmit classical information faster than the speed of light. Nothing in your system transmits such information.

You mention a random string being entangled. I think that may be causing confusion. A string is classical information. It is not entangled. Particles can be entangled. The photons which encode the string may get entangled.

So, from a QM perspective, Person B's machine emits two photons. Each of these reach the beam splitter, and each becomes an entangled pair of photons, such that if the photon is on one path it is not on the other. Person B observes which path the photon took, and acts accordingly. The other half of each entangled pair travels to Person A. Person A also observes which path the photon took, and acts accordingly. The laws of QM indicate that these two observations should be consistent. Either both people agree that the photon took the path to Person A, or both people agree that the photon took the path to Person B. Fortunately for us, this is also consistent with or normal understanding of physics, so this doesn't create any weird behaviors which need QM to explain.

What makes this a little confusing is that you ask if a signal arrives in a mixed state. To the best of my knowledge, when people start using the word "signal," they are referring to something classical. That classical signal is not in a mixed state. We can say that the two photons are in a superposition of states until they are observed.

Now, where I think this breaks down is that you state that the photon-pair can only be in one of two states (0, 1) or (1, 0). Actually, there are two other states which can occur if the photons are independent: (0, 0) and (1, 1). This is a natural probabalistic outcome.

So how could we ensure that the pair is never in those (0, 0) or (1, 1) states? One approach is a purely classical one: we simply reject pairs like that when we see them, and both Person A and Person B continue the process until we see (0, 1) or (1, 0). Such an approach does not need any quantum mechanics. (As an aside, this is actually a procedure is a classic trick for unbiasing a random number source which is accredited to von Neumann).

A second approach is to entangle the photons before they hit the beam splitter such that they must take opposite paths. This is easy to do with electrons and spins, so a lot of experiments use them to do this. Now both parties recieve an entangled pair of photons instead of receving two independent photons. Now we will be able to observe a "mixed state." Of course we can use classical measurements to see this probalistically (like we always can with mixed states), or we can do quantum measurements (such as weak measurement) which yield fun behaviors.

Now, fun things happen if B does not act immediately. What if B doesn't observe this entangled pair right away. What if B finds a way to hold onto them until after A observes it classically, but before A can use a classical channel to provide B with the results of that observation? What does A see? The answer is that A sees something consistent with B's future decision to observe the particles or not observe them.

This situation is the delayed choice quantum eraser. It's behaviors are well understood, and really nonintuitive unless you are comfortable with QM.

Answer 2

In this question I asked if a particle arriving from outside our light cone has a totally undetermined state. I've come to the conclusion that the question was poorly posed.

The problem is that, even if the answer were "yes", it is impossible to know by measurement what the full quantum description of an arriving particle is. We can't even know if the particle arrives, much less its type or momentum or spin, without measuring it; and measurement "fixes" its state. If the particle's wave packet is totally undetermined before measurement, then the identity of the particle and its other descriptors exist only in a probabilistic sense prior to measurement.

Some of the experiments I've been able to imagine that might answer the question require a stream of identical particles to arrive from outside the light cone: if they all turn out to be electrons, or all spin-up, then we can conclude that their quantum properties are at least partially determined before they arrive. However, this kind of experiment would require prior knowledge (from outside the light cone) that the arriving particles really are identical.

Another class of experiments that might answer the question would simply look for patterns or correlations among measurements of vast numbers of non-identical particles arriving from outside the light cone – and in fact we're doing that type of experiment all the time: all optical and radio astronomy is measurement of non-identical particles arriving from outside the light cone. And, we do observe correlations among those measurements: photons are highly correlated with respect to direction of arrival, time of arrival, frequency, and even polarization. Simply put, telescopes can form images of distant stars and galaxies, which is only possible because of those correlations.

That said, the original question should be re-phrased: "Are there correlations between quantum particles arriving from outside the light cone?". The answer to that question is an obvious "yes". Those correlations reveal – or constitute- the structure of the universe that exists independently of our perspective.