CHSH-Ineq

Research project on the CHSH inequality

Member Names: Seaena Kim, Adam Jurkowski, Mark Agib

Github content

After cloning the repo run pip install requirments.txt so we can all have the same libraries installed

All of the code has an associated README.md for documentation purposes and code/nisq-entaglment/code.ipynb has some more markdown for explanations of specfic descions.

Research

(Working) Research Question: How can the CHSH parameter be used to measure the quality of entanglement on distant qubits based on noisy simulations of quantum computers?

Some prelim results. Noisy simulation with noise model from ibm_brisbane with diffrent phases for entanglemnt 6 qubits away from each other:

Had to regenerate data becuase I realized that the qubits it automaically chose were not next to each other. Do this by using layout_method="trivial" in the transpiler.

Image of which qubits are being used on the physical device and their layout on ibm_brisbane's heavy hex topology

Notes to the reader

Some papers denote the CHSH inequality a little diffrently. It is denoted as

$$\left|\text{Tr}\left(\rho \mathcal{B}_{CHSH}\right) \right| \leq 2$$

(Note: This stems from the fact that the expectation value of a observable is the trace of the density matrix and the obervable.)They describe the CHSH parameter or operator as

$$\mathcal{B}_{CHSH} = \vec{a} \cdot \vec{\sigma} \otimes \left(\vec{b} +\vec{b}' \right) \cdot \vec{\sigma} + \vec{a}' \cdot \vec{\sigma} \otimes \left(\vec{b} - \vec{b}'\right) \cdot \vec{\sigma}$$

where $\vec{a}$, $\vec{a}'$ and $\vec{b}$, $\vec{b}'$ are unit vecotrs in $\mathfrak{R}^3$ and describe the measurments on Alice and Bob's sides, respectively and $\sigma$ are vectors of the pauli matrices. (Note: Taking the dot product of alice's unit vecotr and the pauli matrix is an oberservable and it corresponds to measuring on the pauli matrice's axis.) On a futher side note (skip ahead if you understand) the correlation functions described in the classic CHSH expression $E(a, b)$ can be written mathematically as

$E(\vec{a},\vec{b}) = \text{Tr} \left[\rho \left(\vec{a} \cdot \vec{\sigma} \otimes \vec{b} \cdot \vec{\sigma} \right) \right]$

and so when we substitute in for those terms we get

$$ \text{Tr} \left[ \rho \left(\vec{a} \cdot \vec{\sigma} \otimes \vec{b}\cdot \vec{\sigma} + \vec{a} \cdot \vec{\sigma} \otimes \vec{b}' \cdot\vec{\sigma} + \vec{a}' \cdot \vec{\sigma} \otimes \vec{b} \cdot \vec{\sigma} - \vec{a}' \cdot \vec{\sigma} \otimes \vec{b}' \cdot \vec{\sigma} \right) \right] $$

(pretty long lol). But once we group (try this yourself) we get

$$\vec{a} \cdot \vec{\sigma} \otimes \left(\vec{b} +\vec{b}' \right) \cdot \vec{\sigma} + \vec{a}' \cdot \vec{\sigma} \otimes \left(\vec{b} - \vec{b}'\right) \cdot \vec{\sigma}$$

which happens to be excatly what the CHSH operator was (check for yourself). That's becuase these are equivalent ways to write the same thing.

And the density matrix $\rho$ can be expressed as

$$\rho = \frac{1}{4}\left(I\otimes I + \vec{r} \cdot \vec{\sigma} \otimes I + I \otimes \vec{s} \cdot \vec{\sigma} + \sum_{n, m = 1}^{3}{T_{n,m}\sigma_n \otimes \sigma_m}\right) $$

where $T_{n,m} = \text{Tr} \left[\rho \left(\sigma_i \otimes \sigma_j \right)\right]$ is the corrleation matrix and $\vec{r}$ and $\vec{s}$ are unit vectors in $\mathfrak{R}^3$. Now you might be wondering what does that all mean. Well its based on a represenation of the bloch sphere for two qubits. The $I \otimes I$ is the identiy operator applied on the first and second qubits. Its then added to the the dot product of the pauli matrices on the unit vector tensored with the identiy operator like $(r_1 \sigma_1 +r_2 \sigma_2 + r_3 \sigma_3) \otimes I$ that. You than do the same for the second qubit (on $\vec{s}$).

Finally in the summation $\sigma_n \otimes \sigma_m$ is all the possible combinations of pauli operators on $1$ and $2$ (hence the sum to $3$) and the correlation matrix descibes how strongly the qubits are coorleated along the axes. Since $ \text{Tr}(\rho) = 1$ the factor $\frac{1}{4}$ keeps the trace $1$.

Sources: arXiv:1306.6504v2 and arXiv:2501.03373

TODO: write about other measures of entanglment (negativity, concurrence, and REE) and each paper.