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# 與Bell's inequality的約會

Consider arbitrary conditional probability distribution $\{\bbP[a_1,\dots,a_k\vert x_1,\dots,x_k]\}$, we can categorize them into the following sets:

• Local set: There exists hidden variable $\lambda$ such that for any $a_i,x_i$, $\bbP[a_1,\dots,a_k\vert x_1,\dots,x_k]=\sum_{\lambda}\bbP[\lambda]\cdot\prod_i\bbP[a_i\vert x_i,\lambda]$
• Quantum set: There exists a quantum state $\vert \sigma\rangle$ and a measurement $M$ such that the conditional distribution is the distribution after using $M$ to measure $\vert \sigma\rangle$
• No-signaling set: The marginal probabilities are all the same. That is, for any $i$ and $x_1,\dots,x_k,x_1’,\dots,x_{i-1}’,x_{i+1}’,\dots,x_k’$, $$\sum_{a_j,j\neq i}\bbP[a_1,\dots,a_k\vert x_1,\dots,x_i,\dots,x_k]=\sum_{a_j,j\neq i}\bbP[a_1,\dots,a_k\vert x_1’,\dots,x_i’,\dots,x_k’].$$
• Signaling set: Others.

Bell’s inequality provides a condition to identify local set. Concretely, its a half-space in the probability simplex that contains the local set.

The goal no-locality is to find some characterization of the quantum set.