How to compute Von Neumann entropy?

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naima
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I know how to get Von Neumann entropy from a density matrix. I want to get a real number from measurements that give real numbers as outcomes. (there are complex numbers in a density matrix).
So suppose Charlie sends 1000 pairs of particles in the same state to Bob and Alice. They agree to measure the spin along two directions differing by an angle theta. (They can agree with Charlie for another set of 1000 pairs with another angle theta' if they want).
When they meet again, they compare their results and build a square table with their numbers of uu ud du and dd (the same for theta')
How can they estimate the Von Neuman entropy and know if Charlie sent them strongly entangled pairs?
 

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  • #4
naima
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Each time Bob and Alice receive one of the particle pair, they make a measurement and write on a line of a list what they measured and the value of the outcome.
At the end they can obtain ONE number from these lists. There are many ways to get such a number.
Bob can caculate one number from his list by averaging values, Alice can choose the maximun in her own, they can add these results and so on.
Look at page 3 of Atyy's link.
they compute Tr (ρ BCHSH ) where BCHSH = a · σ ⊗ ( b + b' ) · σ + a · σ ⊗ ( b − b' ) · σ (eq 4)
They do not know the density matrix ρ but the trace is the mean value so it is something they can get from their lists.
Now the author give us in B three "popular entanglement measures": negativity, concurrence and REF.
they are defined with eigenvalues of operators depending on the density matrix..
I could say take the eigenvalues of pi of ρ, compute pi log pi add them and you have the Von Neumann entropy! but i do not know the eigenvalues.
How can he get these three measures from the lists?
 
  • #6
naima
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The answer was in Oregon

Alice does not need Bob's help to measure the Von Neumann entropy.
As it is said in the paper, she only has to measure the mean value of the spin along the 3 axes x, y, and z.
She can compute the reduced density matrix, find its eigenvalues and compute the entropy.
The author highlights the fact that a density matrix is measurable (a state vector is not measurable)

Prepare the computer simulation by double clicking on the .jar file called “spins.
Read the accompanying “Notes for SPINS program.
Where can we get this fike?
 
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naima
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Thank you. Oregon is far from France.
 

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