# Eigenvalues contents

1. May 21, 2015

### ephen wilb

Quantum Eigenvalues have more information contents than non-quantum eigenvalues. What are the different algorithm to extract information from them?

For example. If memory chips were made of quantum eigenvalues. How do you read and store information into them?

2. May 21, 2015

### bhobba

3. May 21, 2015

### stevendaryl

Staff Emeritus
Your use of the word "eigenvalue" seems a little odd. Just so we're on the same page, let me describe what eigenvalues mean in QM.

In classical physics, if you measure some physical quantity such as the momentum or angular momentum of a particle, you can get any real number as the result. In contrast, in quantum mechanics, observable quantities are associated with operators, and every operator has an associated set of eigenvalues. Quantum mechanics assumes that any measurement of an observable quantity can only produce an eigenvalue.

So while classically, a spinning or orbiting object can have absolutely any positive value for its angular momentum, in quantum mechanics, it can only be measured to have a discrete set of possible values: $\frac{\hbar}{2}, \hbar, \frac{3 \hbar}{2}, 2 \hbar, ...$

That's a rough idea of what "eigenvalue" means in quantum mechanics. So in light of that understanding of eigenvalues, what does it mean to say that memory chips are "made of quantum eigenvalues"?

4. May 21, 2015

### ephen wilb

In one atom, the electron has *at least* 1 billion position eigenvalues.. since time is symmetric in QM.. couldn't you put information into these position eigenvalues and store information in them and reading them? What algorithm would make it possible?

5. May 23, 2015

### ephen wilb

What I'm saying is. It's a classical bias to think that in quantum systems.. one can only read one eigenvalue at a one. But this is quantum.. and supposing quantum is informational and we won't limit it to classical access. What would it take to read and store information in each position eigenvalue of the electron in an atom? For example.. for sake of discussion.. what if you have access to all branches of the many worlds.. how would you use this to store memory in the position eigenstate of an electron in an atom?

6. May 23, 2015

### atyy

Exactly. The Copenhagen interpretation assumes that when a measurement is made, one gets a "classical" outcome, which is a single definite result.

7. May 23, 2015

### ephen wilb

So what kind of algorithm is there such that when one make an informational probe time symmetric.. one can read, store and manipulate more than one definite outcome simultaneously and use an electron in atom as memory? Jus for sake of discussions and theoretical understanding of quantum as informational system beyond the classical bias..

8. May 23, 2015

### atyy

There are none. However, you can check out the following quantum algorithms.

http://en.wikipedia.org/wiki/Grover's_algorithm
http://en.wikipedia.org/wiki/Shor's_algorithm
http://www.quantiki.org/wiki/BB84_and_Ekert91_protocols

9. May 23, 2015

### ephen wilb

I think I read in the book Schrodinger Rabbits: Many World of Quantum about computations that uses the resources of all Many Worlds. I'd like to understand it using the electron eigenpositions in the atom or what it means to use the resources of all different branches simultaneously...

10. May 23, 2015

### atyy

Yes, it is often said it that way, but I don't know if it is really correct. Here is how I would explain it. In quantum mechanics, the state evolves linearly and reversibly between measurements. When a measurement is made,we get a single definite "classical outcome, and the state evolves nonlinearly and irreversibly. In quantum computation, the final step must always be a measurement, otherwise there is no result of the computation. However, the intermediate stages can involve the linear and reversible time evolution, and these stages can at least informally be called "many worlds".

11. May 23, 2015

### bhobba

First, you need to show there is an algorithm involved. Why eigenvalues occur is right at the very foundations of an axiomatic treatment of QM - see the link in post 2 and the foundational axiom:

An observation/measurement with possible outcomes i = 1, 2, 3 ..... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.

That is why eigenvalues occur, that is why you get a single outcome. For your assertion an algorithm is used you need to rigorously prove that axiom requires an algorithm. Its pretty obvious such is not the case, but if you have such a proof, and by proof I mean a rigorous mathematical proof, not philosophical mumbo jumbo, I am all ears.

Thanks
Bill

12. May 23, 2015

### ephen wilb

I think what I'm asking is what if it were possible to probe a quantum system without classical measurements? Remember measurements are just classical bias.. we think we can only probe quantum system using measurements. What I've been thinking for two years already and I mentioned this so you can correct any wrong thinking I may have and i can stop thinking of it... I've been thinking what if the mind can probe quantum system without directly doing "measurement".. would this support Copenhagen, Bohmian, Many worlds or others.. this would be part of the solution to the QM puzzle. What physicists have written about this or works (peer reviewed) with regards to this.

13. May 23, 2015

### atyy

You can always use Copenhagen. For non-relativistic quantum mechanics, you can also use Bohmian mechanics (I ignore Bohmian treatment of relativistic quantum mechanics for simplicity, because it is not fully worked out yet). Copenhagen and Bohmian mechanics give the same predictions. I don't know if Many-Worlds really works, but if it does, it is also not supposed to give predictions that are different from Copenhagen.

And no, of course you cannot probe the quantum system without doing a measurement.

14. May 24, 2015

### ephen wilb

Do you consider each electron position eigenstate as a separate quantum state?

15. May 24, 2015

### bhobba

Of course. As is any linear combination.

Thanks
Bill

16. May 27, 2015

### ephen wilb

In Quantum Field Theory, can you say the same thing that "the state evolves linearly and reversibly between measurements".. and "When a measurement is made,we get a single definite "classical outcome, and the state evolves nonlinearly and irreversibly"?