Quantum Information HT24, Basic measurements
Flashcards
What does it mean for two orthonormal bases $\\{ \vert \psi _ n \rangle : n = 0, \cdots, d-1\\}$ and $\\{ \vert \psi _ n' \rangle : n = 0, \cdots, d-1\\}$ to be equal up to global phase?
For each $n$, $\exists \gamma _ n \in [0, 2\pi]$ such that
\[ \vert \psi' _ n \rangle = e^{i\gamma _ n} \vert \psi _ n \rangle\]What are the possible types of basic measurements on a $d$-dimensional quantum system?
Orthonormal bases in $\mathbb C^d$, up to global phases.
Suppose:
- We have a quantum system in a state $ \vert \psi \rangle$
- We perform the basic measurement corresponding to the orthonormal basis $\\{ \vert \psi _ k \rangle : k = 0, \cdots, d-1\\}$.
What is the probability we obtain the outcome $ \vert \psi _ k\rangle$, and what is the state of the system immediately after the measurement?
If we obtain the outcome $\psi _ l$, the state of the system immediately after the measurement is $ \vert \psi _ n\rangle$.
Can you state the Born rule?
Suppose:
- We have a quantum system in a state $ \vert \psi \rangle$
- We perform the basic measurement corresponding to the orthonormal basis $\\{ \vert \psi _ k \rangle : k = 0, \cdots, d-1\\}$.
Then the probability we obtain the outcome $ \vert \psi _ k\rangle$ is
\[p _ k = \vert \langle \psi _ k \vert \psi\rangle \vert ^2\]Suppose $\\{ \vert \psi _ n \rangle : n = 0, \cdots, d-1\\}$ and $\\{ \vert \psi _ n' \rangle : n = 0, \cdots, d-1\\}$ are orthonormal bases equal up to global phase. What can we say about the probabilities of obtaining each outcome $\psi _ k$ and $\psi' _ k$, and the post-measurement state?
The probabilities are the same
\[ \vert \langle \psi _ n \vert \psi\rangle \vert ^2 = \vert \langle \psi' _ n \vert \psi \rangle \vert ^2\]and the post-measurement state will always be the same.
What is the computational basis for $\mathbb C^d$?
The orthonormal basis given by
\[\\{ \vert k \rangle : k = 0, \cdots, d - 1\\}\]What is the Fourier basis for $\mathbb C^d$?
The orthonormal basis given by
\[ \vert e _ k \rangle := \frac{1}{\sqrt d} \sum^{d-1} _ {m=0} \exp\left[ \frac{2\pi i m k}{d} \right] \vert m\rangle\]Can you define the vectors $ \vert +\rangle$ and $ \vert -\rangle$, and how do these relate to the Fourier basis?
These are the Fourier basis vectors in dimension $2$.
Suppose we have the ONB $\\{ \vert \phi _ 0\rangle, \vert \phi _ 1\rangle, \vert \phi _ 2\rangle\\}$ where
\[\begin{aligned}
\vert \phi _ 0 \rangle &:= \frac{1}{\sqrt 2} ( \vert 0\rangle + i \vert 2\rangle) \\\\
\vert \phi _ 1 \rangle &:= \vert 1\rangle \\\\
\vert \phi _ 2 \rangle &:= \frac{1}{\sqrt 2} ( \vert 0\rangle - i \vert 2\rangle)
\end{aligned}\]
What state gives rise to the following probability distribution
\[\begin{aligned}
p(0) &= 1 / 5 \\\\
p(1) &= 1 /3 \\\\
p(2) &= 7/15
\end{aligned}\]
for each of the outcomes?
Use the ONB and give coefficients that are square roots of the corresponding amplitudes, so
\[ \vert \psi\rangle = \sqrt{1/5} \vert \phi _ 0\rangle + \sqrt{1/3} \vert \phi _ 1\rangle + \sqrt{7/15} \vert \phi _ 2\rangle\]