Difference between revisions of "BellInequalityMax"
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===The I<sub>3322</sub> inequality=== | ===The I<sub>3322</sub> inequality=== | ||
| − | The I<sub>3322</sub> inequality<ref>M. Froissart. Constructive generalization of Bell's inequalities. <em>Nuov. Cim. B</em>, 64:241, 1981</ref> 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: | + | The I<sub>3322</sub> inequality<ref>M. Froissart. Constructive generalization of Bell's inequalities. <em>Nuov. Cim. B</em>, 64:241, 1981</ref><ref>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: [http://arxiv.org/abs/quant-ph/0306129 arXiv:quant-ph/0306129]</ref> 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> | : <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> | ||
Similarly, it is known that a value of (strictly) larger than $1/4$ is possible in quantum mechanics<ref>K.F. Pál and T. Vértesi. Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems. <em>Phys. Rev. A</em>, 82:022116, 2010. E-print: [http://arxiv.org/abs/1006.3032 arXiv:1006.3032] [quant-ph]</ref>, and the best possible bound is $1$ in no-signalling theories. The following bound computes the best possible classical and no-signalling bounds exactly, and computes two different upper bounds for the quantum mechanical setting: | Similarly, it is known that a value of (strictly) larger than $1/4$ is possible in quantum mechanics<ref>K.F. Pál and T. Vértesi. Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems. <em>Phys. Rev. A</em>, 82:022116, 2010. E-print: [http://arxiv.org/abs/1006.3032 arXiv:1006.3032] [quant-ph]</ref>, and the best possible bound is $1$ in no-signalling theories. The following bound computes the best possible classical and no-signalling bounds exactly, and computes two different upper bounds for the quantum mechanical setting: | ||
Revision as of 14:23, 19 December 2014
| BellInequalityMax | |
| Computes the maximum value of a Bell inequality | |
| Other toolboxes required | cvx |
|---|---|
| Related functions | NPAHierarchy |
| Function category | Nonlocality and Bell inequalities |
| Usable within CVX? | no |
BellInequalityMax is a function that computes the maximum possible value of a given Bell inequality under either classical mechanics, quantum mechanics, or general no-signalling theories. In the classical and non-signalling cases, an exact value is computed, whereas the value computed in the quantum case is only an upper bound (found using the NPA hierarchy).
Syntax
- BMAX = BellInequalityMax(JOINT_COR,A_COR,B_COR,A_VAL,B_VAL)
- BMAX = BellInequalityMax(JOINT_COR,A_COR,B_COR,A_VAL,B_VAL,MTYPE)
- BMAX = BellInequalityMax(JOINT_COR,A_COR,B_COR,A_VAL,B_VAL,MTYPE,K)
Argument descriptions
- JOINT_COR: A matrix whose $(i,j)$-entry gives the coefficient of $\langle A_i B_j \rangle$ in the Bell inequality.
- A_COR: A vector whose $i$-th entry gives the coefficient of $\langle A_i \rangle$ in the Bell inequality.
- B_COR: 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.
- MTYPE (optional, default 'classical'): A string indicating which type of theory should be used when computing the maximum value of the Bell inequality. Must be one of 'classical', 'quantum', or 'nosignal'. If MTYPE = 'quantum' then only an upper bound on the Bell inequality is computed, not necessarily is best upper bound (see the argument K below).
- K (optional, default 1): If MTYPE = 'quantum' then this is a non-negative integer indicating what level of the NPA hierarchy should be used when bounding the Bell inequality. Higher values of K give better bounds, but require more memory and time. If MTYPE is anything other than 'quantum' then K has no effect.
Examples
The CHSH inequality
One formulation of the CHSH inequality[1] says that if $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are $\{-1,+1\}$-valued measurement settings, then in classical physics the following inequality holds: \[\langle A_1 B_1 \rangle + \langle A_1 B_2 \rangle + \langle A_2 B_1 \rangle - \langle A_2 B_2 \rangle \leq 2.\] Similarly, the best bound on this quantity is $2\sqrt{2}$ in a quantum mechanical setting (this is Tsirelson's bound)[2], and it is $4$ in no-signalling theories. All three of these bounds can be found as follows:
>> BellInequalityMax([1 1;1 -1], [0 0], [0 0], [-1 1], [-1 1], 'classical')
ans =
2
>> BellInequalityMax([1 1;1 -1], [0 0], [0 0], [-1 1], [-1 1], 'quantum')
ans =
2.8284
>> BellInequalityMax([1 1;1 -1], [0 0], [0 0], [-1 1], [-1 1], 'nosignal')
ans =
4.0000The CHSH inequality again
Another (equivalent) formulation of the CHSH inequality says that if $\{A_1,A_2\}$ and $\{B_1,B_2\}$ are $\{0,1\}$-valued measurement settings, then in classical physics the following inequality holds: \[\langle A_1 B_1 \rangle + \langle A_1 B_2 \rangle + \langle A_2 B_1 \rangle - \langle A_2 B_2 \rangle - \langle A_1 \rangle - \langle B_1 \rangle \leq 0.\] Similarly, the best bound on this quantity is $1/\sqrt{2} - 1/2$ in a quantum mechanical setting, and it is $1/2$ in no-signalling settings. These three bounds can be verified as follows:
>> BellInequalityMax([1 1;1 -1], [-1 0], [-1 0], [0 1], [0 1], 'classical')
ans =
0
>> BellInequalityMax([1 1;1 -1], [-1 0], [-1 0], [0 1], [0 1], 'quantum')
ans =
0.2071
>> BellInequalityMax([1 1;1 -1], [-1 0], [-1 0], [0 1], [0 1], 'nosignal')
ans =
0.5000The I3322 inequality
The I3322 inequality[3][4] 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: \[\langle A_1 B_1 \rangle + \langle A_1 B_2 \rangle - \langle A_1 B_3 \rangle + \langle A_2 B_1 \rangle + \langle A_2 B_2 \rangle + \langle A_2 B_3 \rangle - \langle A_3 B_1 \rangle + \langle A_3 B_2 \rangle - \langle A_2 \rangle - \langle B_1 \rangle - 2\langle B_2 \rangle \leq 0.\] Similarly, it is known that a value of (strictly) larger than $1/4$ is possible in quantum mechanics[5], and the best possible bound is $1$ in no-signalling theories. The following bound computes the best possible classical and no-signalling bounds exactly, and computes two different upper bounds for the quantum mechanical setting:
>> BellInequalityMax([1 1 -1;1 1 1;-1 1 0], [0 -1 0], [-1 -2 0], [0 1], [0 1], 'classical')
ans =
0
>> BellInequalityMax([1 1 -1;1 1 1;-1 1 0], [0 -1 0], [-1 -2 0], [0 1], [0 1], 'quantum', 1)
ans =
0.3660
>> BellInequalityMax([1 1 -1;1 1 1;-1 1 0], [0 -1 0], [-1 -2 0], [0 1], [0 1], 'quantum', 2)
ans =
0.2518 % this bound is better than the previous bound because we increased K
>> BellInequalityMax([1 1 -1;1 1 1;-1 1 0], [0 -1 0], [-1 -2 0], [0 1], [0 1], 'nosignal')
ans =
1.0000We note that the exact value of the best upper bound in the quantum setting is an open problem, but is approximately $0.250875$.
Source code
Click here to view this function's source code on github.
References
- ↑ J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23(15):880–884, 1969.
- ↑ B.S. Tsirelson. Quantum generalizations of Bell's inequality. Lett. Math. Phys., 4:93, 1980. E-print
- ↑ 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
- ↑ K.F. Pál and T. Vértesi. Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems. Phys. Rev. A, 82:022116, 2010. E-print: arXiv:1006.3032 [quant-ph]