PROOF: Quantum Fidelity with pure states

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  • Thread starter Alex Dingo
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Hi, I'm currently working on showing the relation of quantum fidelity:

The quantum “fidelity” between two pure states ρ1 = |ψ1⟩⟨ψ1| and ρ2 = |ψ2⟩⟨ψ2| is given by |⟨ψ1|ψ2⟩|^2.
Show that this quantity may be written as Tr(ρ1ρ2).

I've been following the wikipedia page on fidelity but can't understand it.

If I assume the statement is true and start from the Tr(ρ1ρ2) = Tr(|ψ1⟩⟨ψ1|ψ2⟩⟨ψ2|) I'm trying to find how I can reduce this, perhaps with summations to arrive at |⟨ψ1|ψ2⟩|^2 - can anyone suggest which direction to go in?
 

Answers and Replies

  • #2
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Try expressing both states in the same orthonormal basis.

Thanks
Bill
 
  • #3
Try expressing both states in the same orthonormal basis.

Thanks
Bill
Hi Bill,

If I express Ψ1=∑√(p(x))|x> and Ψ2=∑√(q(x))|x> such that they are both expressed in the basis of x and compute |⟨ψ1|ψ2⟩|^2 I find Is equal to ∑[√p(x)√q(x)]^2 <x|x> such that I am left with Σp(x)q(x) which is a summation of their probabilities over x. Can I then relate the relation that the sum of p(x) and p(x) is 1 to make the link between the final result of Tr(p1p2) or have I made a mistake?
Thanks,
Alex.
 
  • #4
DrClaude
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If I express Ψ1=∑√(p(x))|x> and Ψ2=∑√(q(x))|x> such that they are both expressed in the basis of x and compute |⟨ψ1|ψ2⟩|^2 I find Is equal to ∑[√p(x)√q(x)]^2 <x|x> such that I am left with Σp(x)q(x) which is a summation of their probabilities over x.
First, get rid of those square roots, which don't make any sense, since the expansion coefficients are complex numbers:
$$
\psi_1 = \sum_n p_n | n \rangle
$$
with ##p_n \in \mathbb{C}##.

After you have done that and calculated ##| \langle \psi_1 | \psi_2 \rangle |^2##, do it also for the trace of ##\rho_1 \rho_2## over the same basis.
 
  • #5
First, get rid of those square roots, which don't make any sense, since the expansion coefficients are complex numbers:
$$
\psi_1 = \sum_n p_n | n \rangle
$$
with ##p_n \in \mathbb{C}##.

After you have done that and calculated ##| \langle \psi_1 | \psi_2 \rangle |^2##, do it also for the trace of ##\rho_1 \rho_2## over the same basis.
Hi DrCloud,

So the result for calculating ##| \langle \psi_1 | \psi_2 \rangle |^2## I found it equat to ##\Sigma p^2_n q^2_n ##. As for the calculation with the trace. I find
## Tr(\rho_1 \rho_2) = Tr(\langle n | \Sigma p^2_n |n\rangle \langle n | \Sigma q^2_n |n\rangle ##

From this point I can take the summations outside the trace and which leave ## \Sigma p^2_n q^2_n Tr( \langle n|n \rangle \langle n|n \rangle) ##
Although this final result with a number inside the trace i'm not sure I've done correctly.

Thanks,
Alex.
 
  • #6
DrClaude
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7,061
3,213
From this point I can take the summations outside the trace and which leave ## \Sigma p^2_n q^2_n Tr( \langle n|n \rangle \langle n|n \rangle) ##
Although this final result with a number inside the trace i'm not sure I've done correctly.
You seem to be holding on too long to the trace operation. Consider
$$
\mathrm{Tr}(\hat{A}) = \sum_n \langle n | \hat{A} | n \rangle
$$
 
  • #7
You seem to be holding on too long to the trace operation. Consider
$$
\mathrm{Tr}(\hat{A}) = \sum_n \langle n | \hat{A} | n \rangle
$$
Hi,
Okay this makes more sense, I forgot about this relationship.

I have computed this with ##\hat{A} = p1p2##
I find that ## \mathrm{Tr}(p1p2)=\Sigma\langle n | p_n|n\rangle \langle n | p_n q_n |n \rangle \langle n| q_n | n \rangle ##
which reduces to ## \Sigma p_n^2 q_n^2 ## as desired ?
 
  • #8
DrClaude
Mentor
7,061
3,213
I find that ## \mathrm{Tr}(p1p2)=\Sigma\langle n | p_n|n\rangle \langle n | p_n q_n |n \rangle \langle n| q_n | n \rangle ##
I'm not sure how you got there (that is, I can't vouch that you did everything right), but the final result is correct.
 

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