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bluecap
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Two particle entanglement such as bell state measurement for quantum teleportation is the example commonly used and in fact you can find everywhere. I understand it like how maximal entanglement can mean measuring one you can know the property of the other.. but it is not very useful for understanding billion-particle entanglement like what happens in macroscopic object like an apple entangling with the environment. So l want to be well versed in more than 2 particle entanglement as billion-particle entanglement is the more realistic everyday scenerio. Let me start with 2 particle entanglement and the math of it:
Entanglement is said to an extension of superposition to different systems. Here is the definition for two systems (this is the example Bhobba kept repeating over the years but he didn't mention the equations for more than 3 particle so I'll try to create the 3 particle equations after the following).
Suppose two systems can be in state |a> and |b>.
If system 1 is in state |a> and system 2 is in state |b> that is written as |a>|b>.
If system 1 is in state |b> and system 2 is in state |a> that is written as |b>|a>.
But we now apply the principle of superposition so that c1*|a>|b> + c2*|b>|a> is a possible state,
The systems are entangled – it is said neither system 1 or system 2 are in a definite state - its in a peculiar non-classical state the combined systems are in.
Let me now generalize it to many systems:
Suppose three systems can be in state |a> and |b> and |c>.
*If system 1 is in state |a> and system 2 is in state |b> and system 3 is in state |c>,that is written as |a>|b>|c>.
*If system 1 is in state |a> and system 2 is in state |c> and system 3 is in state |b>,that is written as |a>|c>|b>
*If system 1 is in state |b> and system 2 is in state |a> and system 3 is in state |c>,that is written as |b>|a>|c>
*If system 1 is in state |b> and system 2 is in state |c> and system 3 is in state |a>,that is written as |b>|c>|a>
*If system 1 is in state |c> and system 2 is in state |a> and system 3 is in state |b>, that is written as |c>|a>|b>
*If system 1 is in state |c> and system 2 is in state |b> and system 3 is in state |a>, that is written as |c>|b>|a>
But we now apply the principle of superposition so that
c1*|a>|b>|c> + c2*|a>|c>|b>+ c3*|b>|a>|c> + c4*|b>|c>|a> + c5*|c>|a>|b> + c6*|c>|b>|a>
is a possible state.
The above convention or equation is correct right? (is the equation or the terms complete). I just read about the 2 particle convention and tried generalizing it to many particle convention above. So please confirm if it is indeed correct.
I’d like to know the following:
1. When you measure only system 2. Do you form new entanglement with system 2 (like you become system 4).. or do you break the entanglement of system 2 from system 1 and 3?
How can you tell if a measurement can break the existing entanglement or form new one?
2. If your measurement produce say c3*|b>|a>|c>, what happens to the other terms? Does the equation means only one term can happen and rest discarded?
3. Measuring c3*|b>|a>|c> means system 1 has state |b>, system 2 has state |a>, system 3 has state |c> right? So if you measure system 2, that means it is in |a>.
4. Let’s assume the state means up or down.. if you measure the system 2 and come up with c3*|b>|a>|c> and |a> is up. How do you know if |b> is up or down.. how do you know if |c> is up or down?
5. Now Many worlds or Branches.. for one particle superposition (of say spin up and down.. I understand the entanglement to observers or other systems is many worlds as taught by Peterdonis). But in the case of c1*|a>|b>|c> + c2*|a>|c>|b>+ c3*|b>|a>|c> + c4*|b>|c>|a> + c5*|c>|a>|b> + c6*|c>|b>|a>, is the many worlds each term?
6. Please answer using the above example and math.. I still don’t understand the GHZ entanglement even after reading many website about it but understanding the math above would enable me to start attempting to understand it. Thanks.
7. Thanks very much for the help
Entanglement is said to an extension of superposition to different systems. Here is the definition for two systems (this is the example Bhobba kept repeating over the years but he didn't mention the equations for more than 3 particle so I'll try to create the 3 particle equations after the following).
Suppose two systems can be in state |a> and |b>.
If system 1 is in state |a> and system 2 is in state |b> that is written as |a>|b>.
If system 1 is in state |b> and system 2 is in state |a> that is written as |b>|a>.
But we now apply the principle of superposition so that c1*|a>|b> + c2*|b>|a> is a possible state,
The systems are entangled – it is said neither system 1 or system 2 are in a definite state - its in a peculiar non-classical state the combined systems are in.
Let me now generalize it to many systems:
Suppose three systems can be in state |a> and |b> and |c>.
*If system 1 is in state |a> and system 2 is in state |b> and system 3 is in state |c>,that is written as |a>|b>|c>.
*If system 1 is in state |a> and system 2 is in state |c> and system 3 is in state |b>,that is written as |a>|c>|b>
*If system 1 is in state |b> and system 2 is in state |a> and system 3 is in state |c>,that is written as |b>|a>|c>
*If system 1 is in state |b> and system 2 is in state |c> and system 3 is in state |a>,that is written as |b>|c>|a>
*If system 1 is in state |c> and system 2 is in state |a> and system 3 is in state |b>, that is written as |c>|a>|b>
*If system 1 is in state |c> and system 2 is in state |b> and system 3 is in state |a>, that is written as |c>|b>|a>
But we now apply the principle of superposition so that
c1*|a>|b>|c> + c2*|a>|c>|b>+ c3*|b>|a>|c> + c4*|b>|c>|a> + c5*|c>|a>|b> + c6*|c>|b>|a>
is a possible state.
The above convention or equation is correct right? (is the equation or the terms complete). I just read about the 2 particle convention and tried generalizing it to many particle convention above. So please confirm if it is indeed correct.
I’d like to know the following:
1. When you measure only system 2. Do you form new entanglement with system 2 (like you become system 4).. or do you break the entanglement of system 2 from system 1 and 3?
How can you tell if a measurement can break the existing entanglement or form new one?
2. If your measurement produce say c3*|b>|a>|c>, what happens to the other terms? Does the equation means only one term can happen and rest discarded?
3. Measuring c3*|b>|a>|c> means system 1 has state |b>, system 2 has state |a>, system 3 has state |c> right? So if you measure system 2, that means it is in |a>.
4. Let’s assume the state means up or down.. if you measure the system 2 and come up with c3*|b>|a>|c> and |a> is up. How do you know if |b> is up or down.. how do you know if |c> is up or down?
5. Now Many worlds or Branches.. for one particle superposition (of say spin up and down.. I understand the entanglement to observers or other systems is many worlds as taught by Peterdonis). But in the case of c1*|a>|b>|c> + c2*|a>|c>|b>+ c3*|b>|a>|c> + c4*|b>|c>|a> + c5*|c>|a>|b> + c6*|c>|b>|a>, is the many worlds each term?
6. Please answer using the above example and math.. I still don’t understand the GHZ entanglement even after reading many website about it but understanding the math above would enable me to start attempting to understand it. Thanks.
7. Thanks very much for the help