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 I3322 inequality
The I3322 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:
- <math>\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.</math>
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 =
0.2500It 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