PBR Paper, Calculation of Outcome Probabilities

[msumm21]

I'm trying to understand the PBR proof as explained in their paper on arXiv (http://arxiv.org/abs/1111.3328). I'm having trouble following the calculation of the probability of measurement outcomes on page 6 (A7). Specifically, going from the first line to the second line of A7, they seem to equate $\langle x_1...x_n|H^{\otimes n}$ with $(\Sigma_z (-1)^{x\cdot z}\langle z |)/\sqrt{2^n}$. I don't understand that equality, probably because I don't understand what the $z$'s are or where they came from. Doesn't seem to be defined in the paper -- later in the paper it says $z$ is the sum over $i$ of $z_i$, but the $z_i$'s don't appear to be defined.

If someone could clarify it would be much appreciated.

 

[msumm21]

I think I might see what it is now. It looks like they are treating tensor states like $|\psi_{x_1}\rangle \otimes \ldots\otimes |\psi_{x_n}\rangle$ as vectors in $\mathbb{R}^n$ with coordinates $(x_1,...,x_n)$ (which works out here despite the fact that the tensor space is really $2^n$ dimensional because all the states of the component systems are either (complex) multiples of |0> or |1>, no combinations thereof). Then the z's are the typical orthonormal basis elements of this n-dimensional space, and similarly the quantum state is "encoded" as an element of this basis denoted x, and the usual n-dimensional dot product is used. The math seems to work out if you assume this.

 

[Mathematech]

I'm catching up on a huge backlog of papers I've been meaning to read for ages and only reached the PBR paper yesterday. I'm stuck on Appendix A, the stuff labelled A7. Maybe I don't understand how the inner product on the tensor product space works, but I can't see how they got from line 4 to line 5. According to my understanding it looks like a plus and some parentheses have gone missing.

 

[Mathematech]

Scratch that, I googled a bit and now it makes sense :) It also helped when I bothered to read a few lines lower that they are using |z| for what most people would have written as |z|^2.