Time and information

arfa brane

call me arf
Valued Senior Member
I think I might have just had a fresh insight into what information is.

A quantum state has to have a global phase, but this can never be measured, so it isn't information (?!)

So what's left? The global phase is included (it has to be) in measurement operator notation; measurement in QM is the same operation as converting the momentum into a single bit of . . . information.

So, unpick the standard double-slit experiment in terms of a set of measurements of individual particle states; the global phase vanishes for each particle that hits the screen (and was never going to be part of the measurement).

To get the probability of where each dot will appear you need an expression that accounts for the global phase which is "gauged out" by the operation of measurement; in the case of the double slit experiment, the measurement operators are what create the dots on a screen, the position of the screen is another kind of gauge factor (!).

Or Something.
Two things:
1) as long as the screen each particle will hit isn't too far away, each particle will make a dot (exchange momentum).
2) if the screen is at a distance such that an interference pattern is seen, moving the screen a short distance (some multiple or fraction of a wavelength of the field) there is still an interference pattern. That's like an invariant.

Ok, time for another coffee.
A quantum state has to have a global phase, but this can never be measured, so it isn't information (?!)
Do you think it could be possible that some information isn't accessible to measurement/observation?
In physics, information can't be created or destroyed, it has to be "conserved", much like energy is. But we already know there are any number of ways to encode information. Moreover, information can be copied so it, essentially, persists for some period of time (and is why backups of computer information are important).

But clearly, knowing something about how information is encoded (i.e. the encoding has a global phase) and being able to measure something aren't obviously the same thing. There is too much information in the universe for humans to measure.

However, there is human measurement, and there is the concept of a "quantum measurement"; this is just an interaction between field particles.

We can describe this interaction in a Hilbert space so we know that interactions have this vector space; but to measure it for humans we need to fix something, usually called a measurement basis, in the same space.

In the case of a beam of particles, the quantum interactions occur at the double slit; i.e. a quantum measurement occurs which is not recorded as information (by humans).

So it seems to be the case (although isn't clear exactly why), that our inability to record certain information (because of no-cloning and the Born rule) is why QM seems weird.
With information, we also get the concepts of reading and writing it; information can also be transmitted.
Writing information so it persists (is not erased), is the concept of storing information in a memory.

Ok, so classically, each electron in a beam writes a dot on a screen after interacting with a pair of slits (tuned to electron dimensions).
So the beam stores information in the memory which is the screen with its eventual interference pattern.

Information can be structured so it has a kind of heirarchy--the information can look different up close than it does from far away, so different that you need to encode it differently if you want to measure any of it. You need to switch contexts.

An example, the information in the message: "there is an interference pattern on the screen", depends on the individual contributions from electrons (so from 'electron-information' in a Hilbert space).
Another example: what you can read on your computer screen looks quite different to what you see with a jeweller's lens from less than an inch away. But there is still information in there (yes, really).
Information at the classical level is something that can be defined with quite arbitrary conditions. For example, Shakespeare's plays can be arbitrarily translated into different languages, even into a language that's been invented for the purpose (of rewriting).

Any information at the classical level, so even if it's what you have after reading (transmitting) some quantum information, can be arbitrarily rewritten, as long as the rewriting rules (the grammar), are useful. Information utility is what it's all about, at least, it is for us humans which is why we build computers, and other tools of calculation.

But to be information, it has to be able to be at least readable, and there must be a mechanism of transmission, ergo there must be senders and receivers.
Indeed, writing and reading imply that information in a store is transmitted from the store (via a sender) to a receiver.

In a computer, copying the contents of a register to another register, is just this kind of tranmission; information doesn't have to travel far in space for this principle to hold. Obviously, time is a factor in transmission.

The takeaway seems to be that, even when we know there is information (such as a global phase) which we can't read or transmit (so we label it "quantum information"), we know it must exist, so we can still account for it. Then the information that we can read (receive), is all the stuff not in the former category and without both, we possibly wouldn't know anything about either.

But we do, of course, thanks to our senses.
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Here is a primer on what information means in the QM context.

"...many physical states to evolve into the same state. However, this violates a core precept of both classical and quantum physics—that, in principle, the state of a system at one point in time should determine its value at any other time.[3][4] Specifically, in quantum mechanics the state of the system is encoded by its wave function. The evolution of the wave function is determined by a unitary operator, and unitarity implies that the wave function at any instant of time can be used to determine the wave function either in the past or the future."
The ah, interaction between time and information comes into focus a bit more (if you wear the right lenses), when considering Bell states and certain encodings.

So the classical picture is, you encode information, transmit it (here the physics is only relevant in terms of preserving the 'structure' of the information being sent, generally loss of signal quality is assumed and some corrective measure is needed), receive and decode it.

When you look at some of the protocols that have appeared, now that things like Bell states are well-understood, there is a bit of a hurdle with the notation. You can have a single-particle state on a Bloch sphere (this is actually the simplest it gets!), but a two-particle Bell state won't fit, it has four complex dimensions. But, there has to be a relation between the dimensionality of the quantum states, and how many bits of classical information are sent or received. The time factor also appears to be related to whether classical information exists already, at the receiver; i.e. a pair of messages exists in a memory at the receiver/decoder, the information transmitted is then about which message to read.

So, apparently, you get to transmit 2 bits of classical information with entangled (Bell) states; except the 2 bits are already transmitted. This seems to be a common error you see, a lot. I contend that the single transmitted quantum state represents a choice between two things; since you can just write these down if you haven't already, they aren't actually in the transmitted information (which is about a choice).
. . . and because this so-called superdense encoding with Bell states includes an expectation in what the receiver will receive, there's Shannon's "message entropy" to consider which says roughly that the information content of a message is proportional to the expectation of receiving it.

So, yeah . . .
I'll need to correct myself here: you can transmit two bits with superdense encoding so there's a choice of 4 messages, namely 00, 01, 10, 11. If the sender chooses randomly, each 'message' has the same expectation of being received.

The 2-message option is what you have with single qubit state transmission; 4 messages means transmitting entangled-pair quantum information.

I still maintain that the classical information exists already and the transmitted information is about which 'message' to choose.
Perhaps I can prove that it's wrong; shame if it's not even, eh?
So is the question: does quantum teleportation, however we define it, qualify as a transmission of information, in short, can a quantum state be a signal, as that word is usually understood?

It seems that, no it can't, is the answer. Physical information cannot be transmitted faster than light can; the speed of light (or say, radio waves) is the fastest any signal can propagate through space. So do we need to define a different kind of signal, one that isn't physical? How can that make sense?

Because of this concept of the persistence of entanglement across any distance, as long as the entangled particles are isolated from their environment, we 'invent' this concept of instantaneous signal transmission, but there is no physical signal. In effect, the information has already been transmitted, carried by the operation of translating one particle away from the other.

So that's seriously weird. It implies that everything we sense is ultimately information we already have. Unless say, evolution had provided an out--the idea of freedom from pre-determined consequences, free will. Ha.
Yes. See the above.

It's a bit of a toy idea. Obviously the experiments aren't what happens in nature, but particles do interact and so they become entangled; usually not for very long in a natural setting. We don't, therefore, directly observe any effects unless we do "look for" them, and to do this we need special conditions.

Humans' senses seem to be well beyond quantum entanglement; besides, photons are physical and do transmit information. Sound waves, heat and pressure are all physical signals. In general, signals produce a response in a receiver system; in physics the most general kind of system is "a system of particles". That's what humans are, for instance, in that context.

Back to: can a quantum state be a signal? Well a quantum state is a probability 'expression'. Each of two outcomes--0 or 1--has a complex probability amplitude. So can a probability be transmitted like a physical signal can? Yes, if its carried by a real particle. Hmm.
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So either we need to reconsider certain notions of space and time, in say, a signal transmission context, or it's a load of crap.

I refute it thus: with Bell-states, correlation (between measurements) does have a time-dependent factor. In a transmission context, the signal is sent before it's received--time is a component of any signal. Entanglement amounts to something that's known about a pair of particles, and about expectations, three or more are almost too complicated to think about, so we draw diagrams.

It doesn't make a lot of sense that information can be received before it got sent; quantum teleportation works if the receiver knows when to do their measurement. I wonder if there are any retrocausal quantum teleportations, where the receiver measures their state before the sender, which would then determine what they send? Or would it?

In any case, "follow the information" seems to be the rule; so in the diagrams, say a quantum algorithm drawn as a set of interconnected "gates", where is it?

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So I now need to understand what a diagram is saying.

First observation: there are inputs on the left but the lower two 'rails' depict a real measurement or projection, from a quantum to a classical space (so akin to the projection of an electron state onto a real, physical screen in a diffraction experiment).

First admission: I know already that one of the rules is that a circuit needs at least one 'fixed, real' measurement space to project states onto.

Next observation: some of the gates are quantum gates, some are classical. What's the difference? in each case there is a real physical device (a beam splitter, a pair of slits, . . .). The difference appears to be about how the gates are controlled--what is switched, and by what? type of thing.

Next admission. I know that the ket with ψ inside it is a wavefunction representing a single unknown state for a qubit. The two below this input are both in the known state 0 (in the ket basis). And so it goes. A gate labelled with an H is a Hadamard gate; this corresponds to an operation on a state vector that rotates it to an orthogonal state. The thing with a white cross on it is a 2-input gate called a CNOT.

The combination of Hadamard and CNOT operations, on one of a pair then on the pair itself resp., creates a maximally entangled state, no longer definable or knowable in terms of the state of either particle. Indeed, either of the pair will behave like unentangled particles with a random state; if it's known they're entangled it seems there is access to a kind of computational resource, but this never seems to actually ever exist, anywhere.

What does really exist is probabilities, randomness and . . . statistical measurements.

Ok, so in the above diagram, first up qubits q1 and q2 are maximally entangled. Then q0 and q1 are unentangled (?) and measured. The X and Z gates are also projecting 'something' onto both measurement axes; this is the control part of the measurement.
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Schrodinger 'toyed around' with solutions to his little problem--describing electron orbitals in a hydrogen atom. He had trouble believing his result, because of the entanglement thing, and because it meant a single electron (supposedly an indivisible, fundamental particle) could be in a superposition of two orbitals, around the same nucleus. Ergo an electron orbital has a "complex probability amplitude" of being in a superposition of states.

It's the most non-intuitive part of the deal: quantum mechanics maps a quantum state to a sphere, and to its interior. The probabilities in the quantum domain are complex, when these "become real" something weird has to happen. Nobody really knows why the squared norms of the complex "probability vectors" have to sum to 1--except that restricts them to the unit sphere; everyone knows why the sum of individual probabilities does, and why there is no sphere--you don't need one.
Let's toy around with Pauli matrices.


There are three, and each can be seen as an operation on a state vector that looks like$$ \dbinom{\alpha}{\beta} $$.

If I make $$ \sigma_1 = X $$ and $$ \sigma_3 = Z $$, then I have X + Z = $$\sqrt {2}$$H.

X and Z are real, the other one is 'pure imaginary'. But $$i$$ is a root of unity, a 4th root.

So what? what do I have there, some kind of relation? between what and what?

$$ X^2 = Z^2 = I$$, and $$ H^2 = 2I $$, where I is the identity matrix. Ok, so I'm generating a set of 2x2 matrices, using addition and multiplication of X and Z, along with certain powers of 2, namely $$ 2^{\frac 1 2}, 2^1$$. Not a lot to see, though, so far.


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