Tuesday, November 12th 2019

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Monday, November 11th 2019

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Friday, November 8th 2019

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1Vanessa Kosoy3d It seems useful to consider agents that reason in terms of an unobservable
ontology, and may have uncertainty over what this ontology is. In particular, in
Dialogic RL
[https://www.alignmentforum.org/posts/dPmmuaz9szk26BkmD/vanessa-kosoy-s-shortform#Wi65Ahs9abL63gPSe]
, the user's preferences are probably defined w.r.t. an ontology that is
unobservable by the AI (and probably unobservable by the user too) which the AI
has to learn (and the user is probably uncertain about emself). However,
onotlogies are more naturally thought of as objects in a category than as
elements in a set. The formalization of an "ontology" should probably be a POMDP
or a suitable Bayesian network. A POMDP involves an arbitrary set of states, so
it's not an element in a set, and the class of POMDPs can be naturally made into
a category. Therefore, there is need for defining the notion of a probability
measure over a category. Of course we can avoid this by enumerating the states,
considering the set of all possible POMDPs w.r.t. this enumeration and then
requiring the probability measure to be invariant w.r.t. state relabeling.
However, the category theoretic point of view seems more natural, so it might be
worth fleshing out.
Ordinary probably measures are defined on measurable spaces. So, first we need
to define the analogue of "measurable structure" (σ-algebra) for categories. Fix
a category C. Denote Meas the category of measurable spaces. A measurable
structure on C is then specified by providing a Grothendick fibration
[https://ncatlab.org/nlab/show/Grothendieck+fibration] B:MFC→Meas and an
equivalence E:B−1(pt)→C. Here, B−1(pt) stands for the essential fiber
[https://ncatlab.org/nlab/show/essential+fiber] of B over the one point space
pt∈Meas. The intended interpretation of MFC is, the category of families of
objects in C indexed by measurable spaces. The functor B is supposed to extract
the base (index space) of the family. We impose the following conditions on MFC
and B:
Given A∈

Thursday, November 7th 2019

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Monday, November 4th 2019

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1[comment deleted]7d (13/?)
Global phase.
Consider a quantum system that's even smaller than a qubit. Something with only
one possible measurement outcome, instead of two. Let's call it "quantum nit",
or "qunit", by analogy with "unit" in typed programming, which means a type with
one element.
A qunit has only one possible mixed state, with the density matrix ((1)). But
its pure states are more complicated: a one-qunit pure state is a number whose
norm is 1, so either 1 or -1 (or more if we allow complex numbers). The same is
true for an N-qunit pure state - it's also a single number whose norm is 1! The
reason is that the tensor product of N 1-dimensional spaces is 1-dimensional,
not N-dimensional. So the joint state of all qunits in the world is a single
number, which is called "global phase".
Now imagine we have a system with a bunch of qubits, and take its tensor product
with a qunit. We can move a scalar factor from one side of a tensor product to
the other, and that shouldn't change anything. So global phase can't affect any
measurement, it's just an extra degree of freedom in the theory.

18d (12/?)
The uncertainty principle, or something like it.
When you measure a qubit cos(φ)|0> + sin(φ)|1>, the result has variance
(1-cos(4φ))/4. (I'm skipping over the trig calculations here and below.) If you
have a choice between either measuring the qubit, or rotating it by an angle ψ
and then measuring it, then the sum of variances of the two operations is at
least (1-|cos(2ψ)|)/2. In particular, if ψ=π/4, the sum of variances is at least
1/2.
So no matter what state a qubit is in, and how precisely you know its state,
there are two possible measurements such that the sum of variances of their
results is at least 1/2. The reason is that the bases corresponding to these
measurements are at an angle to each other, not aligned. Apparently in the real
world, an object's "position" and "momentum" are two such measurements, so
there's a limit on their joint precision.
You can also carry out one measurement and then the other, but that doesn't help
- after the first measurement you have variance in the second. Moreover, after
the second you have fresh variance in the first. This lets you get an infinite
stream of fair coinflips from a single qubit: start with |0>, rotate by π/4,
measure, rotate by π/4, measure...

Sunday, November 3rd 2019

Sun, Nov 3rd 2019