BellInequalityMaxQubits

BellInequalityMaxQubits is a function that computes an upper bound for the maximum possible value of a given Bell inequality in a quantum mechanical setting where the two parties each have access to qubits (i.e., 2-dimensional quantum systems). This bound is computed using the method presented in.

Syntax

 * BMAX = BellInequalityMaxQubits(JOINT_COE,A_COE,B_COE,A_VAL,B_VAL)
 * [BMAX,RHO] = BellInequalityMaxQubits(JOINT_COE,A_COE,B_COE,A_VAL,B_VAL)

Input arguments

 * JOINT_COE: A matrix whose $(i,j)$-entry gives the coefficient of $\langle A_i B_j \rangle$ in the Bell inequality.
 * A_COE: A vector whose $i$-th entry gives the coefficient of $\langle A_i \rangle$ in the Bell inequality.
 * B_COE: A vector whose $i$-th entry gives the coefficient of $\langle B_i \rangle$ in the Bell inequality.
 * A_VAL: A vector whose $i$-th entry gives the value of the $i$-th measurement result on Alice's side.
 * B_VAL: A vector whose $i$-th entry gives the value of the $i$-th measurement result on Bob's side.

Output arguments

 * BMAX</tt>: An upper bound on the qubit value of the Bell inequality.
 * RHO</tt>: A many-qubit quantum state that acts as a witness that verifies the bound provided by BMAX</tt>. This is the positive-partial-transpose (PPT) state described by.

The I3322 inequality
The I3322 inequality is a Bell inequality that says that if $\{A_1,A_2,A_3\}$ and $\{B_1,B_2,B_3\}$ are $\{0,1\}$-valued measurement settings, then in classical physics the following inequality holds:
 * $$\langle A_1 B_1 \rangle + \langle A_1 B_2 \rangle - \langle A_1 B_3 \rangle + \langle A_2 B_1 \rangle + \langle A_2 B_2 \rangle + \langle A_2 B_3 \rangle - \langle A_3 B_1 \rangle + \langle A_3 B_2 \rangle - \langle A_2 \rangle - \langle B_1 \rangle - 2\langle B_2 \rangle \leq 0.$$

It is straightforward to check that a 2-qubit maximally-entangled Bell state allows for a value of $1/4$ in this Bell inequality. The following code verifies that it is not possible to get a value of larger than $1/4$ using $2$-dimensional systems:

It is worth taking a look at the $I_{3322}$ example at the BellInequalityMax</tt> page to compare the computations provided there.