# Notes - 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

\[|\psi'_n \rangle = e^{i\gamma_n} |\psi_n \rangle\]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

\[p_k = |\langle \psi_k | \psi\rangle|^2\]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

\[|\langle \psi_n | \psi\rangle|^2 = |\langle \psi'_n | \psi \rangle|^2\]and the post-measurement state will always be the same.

What is the computational basis for $\mathbb C^d$?

The orthonormal basis given by

\[\\{|k \rangle : k = 0, \cdots, d - 1\\}\]What is the Fourier basis for $\mathbb C^d$?

The orthonormal basis given by

\[|e_k \rangle := \frac{1}{\sqrt d} \sum^{d-1}_{m=0} \exp\left[ \frac{2\pi i m k}{d} \right] |m\rangle\]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

\[\begin{aligned}
|\phi_0 \rangle &:= \frac{1}{\sqrt 2} (|0\rangle + i|2\rangle) \\\\
|\phi_1 \rangle &:= |1\rangle \\\\
|\phi_2 \rangle &:= \frac{1}{\sqrt 2} (|0\rangle - i|2\rangle)
\end{aligned}\]
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

\[|\psi\rangle = \sqrt{1/5} |\phi_0\rangle + \sqrt{1/3} |\phi_1\rangle + \sqrt{7/15} |\phi_2\rangle\]