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6

Yes, it is possible to conceive theories with "stronger correlations" than those given by quantum mechanics. One way to make this statement precise is to consider some kind of "measurement apparatus" (you can think of it as simply a black a box with some buttons that you can push and different LEDs that correspond to different possible outputs), and analyse ...


6

Basically, it means that the correlations could be used to send a message. Or simply that Bob’s measurement outcomes can reveal some details of Alice’s actions. This is impossible when Alice and Bob each hold one qubit of a Bell pair. Despite the entanglement present, as well as contextuality, signaling in this case would result faster than light ...


5

I think that I can explain the definition through the following simple example: Suppose that you perform two experiments in the same house in two separate rooms. In the first you measure the observable $A$ and instantaneously in the second you measure the observable $B$. The measurements are afflicted with noise, so you do not get a definite answer every ...


5

Statistical independence of random variables means that their joint probability is the product of their marginal probabilities: $$P(A∩B) = P(A) P(B)$$ The following Venn diagram, in which the event $A$ occupies the left half of the big square, while the event $B$ occupies the lower half, exemplifies the above relation. ($A^c$ and $B^c$ denote the complements)...


4

I think you're doing things a little bit backwards. You probably shouldn't be calculating $P(a|x)$ or $P(b|y)$ in advance, because you're simply trying to ask: Given a set of $\{P(ab|xy)\}$, do there exist assignments to $P(a|x)$ and $P(b|y)$ that satisfy $P(ab|xy)=P(a|x)P(b|y)$ for all $a,b,x,y$? So, how do you evaluate the probability of getting ...


4

EPR pairs are a particular case of entangled pairs of qubits. From Wikipedia: "Quantum entanglement is a physical phenomenon which occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot be described independently of the state of the other." More to the point regarding your question, ...


3

It perhaps helps to express $P(ab|xy)$ in words: the probability that Alice gets answer A and Bob gets answer B given that choices x and y were made Now independence in classical probability holds if and only if $$ P(e_1\text{ and }e_2)=P(e_1)P(e_2) $$ where $e_1$ and $e_2$ are events, and practically, you can see what it means through Bayes' theorem $$ ...


3

The equation $P(ab|xy) = P(a|x)P(b|y)$ would imply that any dependence that the output $ab$ has on the inputs $xy$ (expressed by the lhs) is solely due to $a$ depending on $x$ alone, and $b$ depending on $y$ alone. This is expressed by the rhs by treating the value of $a$ and its dependence on $x$ as an independent event from the value of $b$ and its ...


3

It is the tensor product of the Pauli X with itself. Preskill is specifying the kind of correlation by giving you a matrix with a +1 eigenspace corresponding to the desired set of states. States where the simultaneous application of an $X$ gate to each qubit has no effect, including phase kickback when conditioned on an ancilla qubit. This notation is very ...


2

It's fundamentally similar to/the same as Baker–Campbell–Hausdorff (BCH). Generally, in quantum physics, this is most often used (or at least taught) with commuting Hamiltonians: $$e^{-i\left(H_1+H_2\right)t} = \sum_{n=1}^\infty\frac{\left(-it\right)^n}{n!}\left(H_1+H_2\right)^n = e^{-iH_1t}e^{-iH_2t}e^{\frac{1}{2}\left[H_1, H_2\right]t^2}\cdots$$ where the ...


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... and the question changes again. Re Update 4: I'm not sure what you intended the variable $a$ to be in the question (I think part of the problem is that you've got muddled between the chosen questions and the given answers), but I have used it very carefully, to always be the $\pm1$ answer given. In a quantum setting, $a$ is the answer, a value $\pm 1$ ...


2

I do understand that the sum of these three probabilities is greater than one because there are some constraints already involved; like if we uncover all three coins at least two have to be the same. So naturally, there's some redundancy leading to a sum of probabilities that is greater than one! I would say the explanation is simpler than that. ...


2

If you were to try and imagine the simplest form of correlation, you might think of two bits that were randomly either both $0$, or both $1$. Bell states are just this, but quantum. We have bits instead of qubits, and the randomness is due to superposition. Since they are most conceptually simple form of entanglement, and the easiest to describe using ...


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