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NeophyteinPhysics

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- Homework Statement
- **2.1 Fidelity of measurement**

**a**) For two states ##|ψ_1\rangle## and ##|ψ_2\rangle## in an N-dimensional Hilbert space,

define the relative angle ##θ## between the states by

##|\langleψ_2|ψ_1\rangle| ≡ \cos θ##, (2.164)

where ##0 ≤ θ ≤ π/2##. Suppose that the two states are selected

at random. Find the probability distribution ##p(θ)dθ## for the

relative angle.

**Hint**: We can choose a basis such that

##|ψ_1\rangle = (1,\vec{0})##

##|ψ_2\rangle = (e^{iϕ}\cos θ, ψ^⊥_2)##

“Selected at random” means that the probability distribution

for the normalized vector ##|ψ_2\rangle## is uniform on the (real) (2N −1)-

sphere (this is the unique distribution that is invariant under

arbitary unitary transformations). Note that, for fixed ##θ##, ##e

^{iϕ}##

parametrizes a circle of radius ##\cos θ##, and ##|ψ^⊥_2\rangle## is a vector that

lies on a 2N − 3 sphere of radius ##\sin θ##.

**b**) A density operator ρ is said to approximate a pure state ##|ψ\rangle## with

fidelity

F = ##\langle ψ|ρ|ψ\rangle## . (2.167)

Imagine that a state ##|ψ_1\rangle## in an N-dimensional Hilbert space is selected at random, and we guess at random that the state is ##|ψ_2\rangle##. On the average, what will be the fidelity of our guess?

**c**) When we measure, we collect information and cause a disturbance – an unknown state is replaced by a different state that

is known. Suppose that the state ##|ψ\rangle## is selected at random,

and then an orthogonal measurement is performed, projecting

onto an orthonormal basis {##|E_a\rangle##}. After the measurement, the

state (averaged over all possible outcomes) is described by the

density matrix

##ρ = \sum_a E_a|ψ\rangle\langle ψ|E_a## , (2.168)

where ##Ea = |E_a\rangle\langle E_a|##; this ρ approximates ##|ψ\rangle## with fidelity

##F = \sum_a (\langle ψ|E_a|ψ\rangle)^2 ## (2.169)

Evaluate F, averaged over the choice of ##|ψ\rangle##. Hint: Use Bayes’s

rule and the result from (a) to find the probability distribution

for the angle ##θ## between the state ##|ψ\rangle## and the projected state

##E_a|ψ\rangle /||E_a|ψ\rangle||##. Then evaluate ##<\cos^2 θ>## in this distribution.

Remark: The improvement in F in the answer to (c) compared to

the answer to (b) is a crude measure of how much we learned by

making the measurement.

- Relevant Equations
- ## ρ = \sum_a E_a|ψ\rangle\langle ψ|E_a ## , (2.168)

## F = \sum_a (\langle ψ|E_a|ψ\rangle)^2 ## (2.169)

F = ##\langle ψ|ρ|ψ\rangle## . (2.167)

The above question is adopted from the exercise of Preskill's quantum information lecture note

My attempt:

(a) From the condition, ## p(\theta)\propto \sin^{(2N-4)}\theta \cos\theta ##. Normalizing the probability distribution would give the answer. This is because the weight of the phase of the first coordinate of ##\phi_2## is proportional to ##\cos{\theta}## and that of the angle of the second coordinate is to ## \sin^{(2N-4)} ##, since it lies on the surface of a 2N-3 sphere.

(b) From the calculation, ##\frac{2}{2N-1}##---------------- (this answer seems wrong, since it does not reproduce 1/2, a desired value for N=2 as far as I know.)

(c) I tried to concatenate the procedure of labeling ##|E_a\rangle## as the z axis of (2k-1) dimensional sphere, such that for each ##E_{a,i}##, ##p(\theta_{a,i})\propto \sin^{(2i-4)}\theta_{a,i} \cos\theta_{a,i}##. Then

## \begin{align*}

F=&\sum_k\int cos^4\theta (2k-3)\left(\frac{1}{2k-3}-\frac{1}{2k-1}+\frac{1}{2k+1}\right)\\

=&\sum_k \left(1-\frac{1}{4k^2-1}\right)

\end{align*}##

This result is obviously larger than 1, so this is also wrong.

Since I find it hard to find any attempt on this exercise, I humbly request the possible answer to this question. In addition, if possible, please let me know errors in my attempt.

Thank you very much.

My attempt:

(a) From the condition, ## p(\theta)\propto \sin^{(2N-4)}\theta \cos\theta ##. Normalizing the probability distribution would give the answer. This is because the weight of the phase of the first coordinate of ##\phi_2## is proportional to ##\cos{\theta}## and that of the angle of the second coordinate is to ## \sin^{(2N-4)} ##, since it lies on the surface of a 2N-3 sphere.

(b) From the calculation, ##\frac{2}{2N-1}##---------------- (this answer seems wrong, since it does not reproduce 1/2, a desired value for N=2 as far as I know.)

(c) I tried to concatenate the procedure of labeling ##|E_a\rangle## as the z axis of (2k-1) dimensional sphere, such that for each ##E_{a,i}##, ##p(\theta_{a,i})\propto \sin^{(2i-4)}\theta_{a,i} \cos\theta_{a,i}##. Then

## \begin{align*}

F=&\sum_k\int cos^4\theta (2k-3)\left(\frac{1}{2k-3}-\frac{1}{2k-1}+\frac{1}{2k+1}\right)\\

=&\sum_k \left(1-\frac{1}{4k^2-1}\right)

\end{align*}##

This result is obviously larger than 1, so this is also wrong.

Since I find it hard to find any attempt on this exercise, I humbly request the possible answer to this question. In addition, if possible, please let me know errors in my attempt.

Thank you very much.