Here is a good video about Bell's Test that rule out the world is "local and has hidden property".
Game: a co-op game
RED
, YELLOW
, and based on the result, pick a response a
ORANGE
, GREEN
, and based on the result, pick a response b
RED
and GREEN
, they win iff a \neq bRED
and GREEN
, they win iff a = bNotice
Alice(R/Y)
,Bob(O/G)
can be viewed as a deterministic function with fixed global variables. Classically, the best chance of winning is 75\%.
Strategy:
If Alice see RED
, rotate 0^\circ
If Alice see YELLOW
, rotate 45^\circ
If Bob see ORANGE
, rotate 22.5^\circ (equivalent to Alice rotate -22.5^\circ)
If Bob see GREEN
, rotate 67.5^\circ (equivalent to Alice rotate -67.5^\circ)
Alice set a
equal to the measurement of qubit A
Bob set b
equal to the measurement of qubit B
Observe when we don't get RED
and GREEN
, the overall rotation is like Alice rotate \pm 22.5^\circ. When we get RED
and GREEN
, the overall rotation is like Bob rotate 67.5^\circ.
Therefore, when we get RED
and GREEN
, the outcome of qubit A and B are less correlated. We win with overall probability 85\%. (which is proven to be optimal by Boris Tsirelson)
Let U be a unitary operation on \mathbb{R}^N, then there exists
orthogonal 2D subspace P_1, P_2, ... such that U will rotate all vectors in the subspace by \theta_1, \theta_2, ...
orthogonal 1D subspace (that is the rotational axis) such that U will do nothing to all vectors in the subspace
remaining 1D subspace such that U will negate (180^\circ rotation) all vectors in the subspace
It is fine to miss some categories
Let U be a unitary operation on |v\rangle \in \mathbb{R}^N, then
U is a rotation \iff (\exists k \neq 0)(U^k |v\rangle = |v\rangle)
U rotate |v\rangle in 2D subspace \iff \text{avg}\{UU|v\rangle, |v\rangle\} = U|v\rangle
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