Question RE: Susskind Entanglement Lecture 2

[meBigGuy]

In lecture 2 of the Entnglement lecture series at theoreticalminimum.com, Susskind is explaining a classical interpretation of Measureable and Observable that left me confused. I'll watch it again tonight, but thought I might get a good clarification from the excellent teachers here.

In his example he assumed 6 classical states, using a six sided die as an example. He defined the observable mapping as a 1 if the die state was 6, and 0 otherwise. This is were I became somewhat confused. He didn't address, at all, the operation that sort of converts the state to the observable. Or even, how the state is determined. I understand that his choice of mapping values is arbitrary (that is he could have chosen any mapping).

So I thought, maybe this could be likened to a die with all sides 0 except 1, which is a 1. But, in this case you don't know what state you are in. You have the answer, but not how it came to be.

It seems that you need to know the state in order to apply the mapping.

So, I'm not entirely clear how all this fits observable and measureable, and I know I'm going to need to be rock solid on the classical analogy of this for lecture 3.

I know the answer to this is simple, it is just evading me. It feels fuzzy.

Thanks

 

[Naty1]

...somewhat confused. He didn't address, at all, the operation that sort of converts the state to the observable. Or even, how the state is determined...I know the answer to this is simple,...

If only it were 'simple'!
This has been a subject of discussion and debate since QM was first formulated. [And in these forums, too.] Check out, for example, 'measurement problem' and 'measurement in quantum mechanics' anywhere, like Wikipedia... and read a few paragraphs from each...or search these forums.

From this post I suggest you remember: " linearity, superposition, and complex numbers". You will find over time that those words serve a a good reminder and introduction to quantum mechanics.

In quantum physics a quantum state is a vector in a Hilbert space, called the state vector. The state vector theoretically contains statistical information about the quantum system... if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state in a non-deterministic, but statistically predictable way...
There are also “mixed states” represented not by single vectors but rather by so called density operators...

From another discussion in these forums:

The classical von Neumann model of the measurement process:
Von Neumann also intervened decisively into the measurement problem. Summarizing earlier work, he argued that a measurement on a quantum system involves two distinct processes that may be thought of as temporally contiguous stages. In the first stage, the measured quantum system interacts with a macroscopic measuring apparatus for some physical quantity. This interaction is governed by the linear, deterministic Schrödinger equation….,Von Neumann assumes that after the first stage of the measurement process, a second non-linear, indeterministic process takes place, the “reduction (or collapse) of the wave packet”.. Since the von Neumann reduction of the wave-packet is indeterministic, there is no possibility of predicting which value will register . QM gives us additional statistical information [not a precise single measurement value].

The following quote is from Roger Penrose celebrating Stephen Hawking’s 60th birthday in 1993 at Cambridge England:

[I like it because it was given in front a room full of world famous physicists, so even those who disagree can't refute it.]

Either we do physics on a large scale, in which case we use classical level physics; the equations of Newton, Maxwell or Einstein and these equations are deterministic, time symmetric and local. Or we may do quantum theory, if we are looking at small things; then we tend to use a different framework where time evolution is described... by what is called unitary evolution...which in one of the most familiar descriptions is the evolution according to the Schrödinger equation: deterministic, time symmetric and local. These are exactly the same words I used to describe classical physics.

However this is not the entire story... In addition we require what is called the "reduction of the state vector" or "collapse" of the wave function to describe the procedure that is adopted when an effect is magnified from the quantum to the classical level...quantum state reduction is non deterministic, time-asymmetric and non local...The way we do quantum mechanics is to adopt a strange procedure which always seems to work...the superposition of alternative probabilities involving w, z, complex numbers...an essential ingredient of the Schrödinger equation. When you magnify to the classical level you take the squared modulii (of w, z) and these do give you the alternative probabilities of the two alternatives to happen...it is a completely different process from the quantum (realm) where the complex numbers w and z remain as constants "just sitting there"...in fact the key to keeping them sitting there is quantum linearity...
QUOTE]

 

[meBigGuy]

I'm just trying to apply the quantum measurement process to a classical system, kind of an analogy. But there may be fundamental problems that cause the analogy to break down. I'm trying to understand the breakdown.

First, I have a 6 state system whose values are 1,0,0,0,0,0. Is that a problem right there? Do I need a linear system where all 6 states are unique? I wonder why Susskind threw in the 1,0,0,0,0,0 mapping of the classical system.

So, can I apply a measurement matrix to such a system? Or do I need to go to 0,1,2,3,4,5 mapping or simpler yet 1,0 (to keep it simple).

 

[Naty1]

At 1 hour and about 15 minutes of this lecture Susskind describes STATES...

So it seems to me you are viewing a more advanced set of lectures...
I haven't studied those yet...saving them for this winter!

Spend 20 minutes minutes around here and see what Susskind says about states:

Susskind Quantum Physics…Lecture 1

Lecture 1 | Modern Physics: Quantum Mechanics (Stanford) - YouTube

From my notes of Susskind's explanations:

"So we use STATE instead of points to describe the [group] of possibilities...In QM States do NOT form a set as do classical states [points]... so POINTS [exact determinations] in a set are states in classical logic...
Examples of vector space over complex numbers…is a HILBERT Space...What is a STATE: a point in the set of Phase space….a probability distribution around a point is a more general form of a state... this is the maximum knowledge we can have while a probability/statistical distribution means we have less information than we might

At one hour 25+ minutes:

States of a system are a VECTOR SPACE is a new/radical/ idea doesn’t make sense at first...Vector space is a collection of vectors [not pointers, but the more abstract ‘vectors’…]
Not directional entities...In QM Ket vectors are over COMPLEX NUMBERS>…sometimes a HILBERT SPACE...Will be shown as a KET vector…..
...

First, I have a 6 state system whose values are 1,0,0,0,0,0. Is that a problem right there?

depends what you mean by 'problem'. For example, QM does not declare whether or not those state exist before measurement. Nor are these each necessarily exact values, which is a classical concept, nor are such states restricted to real numbers...

 

[Naty1]

By sheer coincidence I happened across this post of mine from another current discussion:
It relates to a possible 'problem'. Instead of the words 'particle' or 'spin' in the following, just think "state". Because all we know about a 'particle' or 'spin'are its various 'states'...measurement results.


Tom Stoer:

Particles appear in rare situations, namely when they are registered.

What it means to me: 'particles' normally exist in a superposition of wavelike states...reread my comments on 'orbitals' earlier in this post. This means trying to visualize electron spin via classical analogies has many pitfalls.

A related description from another forum expert:

Marcus:

The trouble with the particle concept is that one cannot attribute a permanent existence; It only exists at the moment it is detected. The rest of the time there is a kind of spread out thing---a cloud---a wave---a field---something that is less "particular", something that cannot be detected.

What it means to me: So we can't observe such a wave, but an atomic nucleus with orbiting electrons knows exactly the spin characteristic of every particle...and force carriers as well! That is mind boggling!

 

[meBigGuy]

"First, I have a 6 state system whose values are 1,0,0,0,0,0. Is that a problem right there?"

By that i meant that possibly the concept of 1,0,0,0,0,0, is not a valid state vector. Is it legal to have 6 states but only two values? Is this actually a two state system (1,0) with different probabilities? (1/6 and 5/6). What I don't understand is why he brought that up at all.

He discusses applying a function to the state to determine the observable, but doesn't talk much about the function. In order to apply the function, you need to know the state.

For the classical states of 1,2,3,4,5,6 with measurables of 1,0,0,0,0,0, and probabilities of 1/6 for all states, What is the measurement matrix, eigenvectors and eigenvalues. Or, is this a problematic classical example.

It is at 1:15 in lecture 2.