# BellInequalityMaxQubits

BellInequalityMaxQubits | |

Approximates the optimal value of a Bell inequality in qubit (i.e., 2-dimensional quantum) settings | |

Other toolboxes required | CVX |
---|---|

Related functions | BellInequalityMax NonlocalGameValue XORGameValue |

Function category | Nonlocality and Bell inequalities |

Usable within CVX? | no |

` 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

^{[1]}.

## 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)`

## Argument descriptions

### 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`: An upper bound on the qubit value of the Bell inequality.`RHO`: A many-qubit quantum state that acts as a witness that verifies the bound provided by`BMAX`. This is the positive-partial-transpose (PPT) state described by^{[1]}.

## Examples

### The I_{3322} inequality

The I_{3322} inequality^{[2]}^{[3]} 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:

```
>> BellInequalityMaxQubits([1 1 -1;1 1 1;-1 1 0], [0 -1 0], [-1 -2 0], [0 1], [0 1])
ans =
0.2500
```

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

## Source code

Click here to view this function's source code on github.

## References

- ↑
^{1.0}^{1.1}M. Navascués, G. de la Torre, and T. Vértesi. Characterization of quantum correlations with local dimension constraints and its device-independent applications.*Phys. Rev. X*, 4:011011, 2014. E-print: arXiv:1308.3410 [quant-ph] - ↑ M. Froissart. Constructive generalization of Bell's inequalities.
*Nuov. Cim. B*, 64:241, 1981 - ↑ D. Collins and N. Gisin. A relevant two qubit Bell inequality inequivalent to the CHSH inequality. J. Phys. A: Math. Gen., 37(5):1175, 2004. E-print: arXiv:quant-ph/0306129