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November 2019

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2Vladimir Slepnev8mo(11/?) Superdense coding. Alice is told two classical bits of information and sends a qubit to Bob, who can then recover the two classical bits. Again, it relies on Alice and Bob sharing a prepared pair beforehand. It's the opposite of quantum teleportation, where Alice sends two classical bits and Bob can recover a qubit. First let's talk about bases. This is the usual basis: |00>, |01>, |10>, |11>. This is the Bell basis: (|00> + |11>)/√2, (|00> - |11>)/√2, (|10> + |01>)/√2, (|10> - |01>)/√2. Check for yourself that each two vectors are orthogonal to each other. To "measure a state in a different basis" means to apply a rotation from one basis into another, then measure. For example, if you have a state and you know that it's one of the Bell basis states, you can figure out which one, by rotating into the usual basis and measuring. One cool thing about the Bell basis is that you can change any basis vector into any other basis vector by operations on the first qubit only. For example, rotating the first qubit by 90 degrees turns (|00> + |11>)/√2 into (|10> - |01>)√2. Flipping the first qubit gives (|10> + |01>)√2, and so on. Now superdense coding is easy. Alice and Bob start by sharing two halves of a Bell state. Then depending on which two classical bits Alice needs to send, she either leaves the state alone or rotates into one of the other three, by operating only on her qubit. Then she sends her qubit to Bob, who now has both qubits and can rotate them into the usual basis and measure them, recovering the two classical bits.
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1Vladimir Slepnev8mo(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...
1Vanessa Kosoy8moIt 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 [] , 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 [] B:MFC→Meas and an equivalence E:B−1(pt)→C. Here, B−1(pt) stands for the 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