Difference between revisions of "BellInequalityMaxQubits"

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|desc=Approximates the optimal value of a Bell inequality in qubit (i.e., 2-dimensional quantum) settings
 
|desc=Approximates the optimal value of a Bell inequality in qubit (i.e., 2-dimensional quantum) settings
 
|rel=[[BellInequalityMax]]<br />[[NonlocalGameValue]]<br />[[XORGameValue]]
 
|rel=[[BellInequalityMax]]<br />[[NonlocalGameValue]]<br />[[XORGameValue]]
|req=[http://cvxr.com/cvx/ cvx]
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|req=[http://cvxr.com/cvx/ CVX]
 
|cat=[[List of functions#Nonlocality_and_Bell_inequalities|Nonlocality and Bell inequalities]]
 
|cat=[[List of functions#Nonlocality_and_Bell_inequalities|Nonlocality and Bell inequalities]]
 
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Latest revision as of 16:38, 13 June 2018

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: \[\langle A_1 B_1 \rangle + \langle A_1 B_2 \rangle - \langle A_1 B_3 \rangle + \langle A_2 B_1 \rangle + \langle A_2 B_2 \rangle + \langle A_2 B_3 \rangle - \langle A_3 B_1 \rangle + \langle A_3 B_2 \rangle - \langle A_2 \rangle - \langle B_1 \rangle - 2\langle B_2 \rangle \leq 0.\] 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 on "expand" to the right to view the MATLAB source code for this function.

References

  1. 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]
  2. M. Froissart. Constructive generalization of Bell's inequalities. Nuov. Cim. B, 64:241, 1981
  3. 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