# Beam splitter action on an entangled state : how do photons recombine?

Context:

Hello, I have to present this article in class next week. I encounter some difficulty with the supplementary material at the end of section I.

There, the authors say they have a state of the form : $$\label{eq:1} \alpha |g⟩ [|10⟩ + |01⟩] + \alpha |e⟩ [|10⟩ − |01⟩] + β|g⟩ [|10⟩ − |01⟩] + β|e⟩ [|10⟩ + |01⟩]$$ Then, by passing this state through a 50:50 Beam Splitter, they obtain : $$α|g⟩|10⟩ + α|e⟩|01⟩ + β|g⟩|01⟩ + β|e⟩|10⟩$$

What I tried:

However, I have difficulty seeing how to pass from the first state to the final one. In effect, I use the BS action : $$a^\dagger \rightarrow \frac{1}{\sqrt{2}}(c^\dagger + id^\dagger)$$ $$b^\dagger \rightarrow \frac{1}{\sqrt{2}}(ic^\dagger + d^\dagger)$$

Let's say I take only the first term of the first expression I wrote above, I can rewrite it as follows : $$\alpha |g⟩ [|10⟩ + |01⟩] \equiv \alpha |g⟩ [(a^\dagger +b^\dagger)|0⟩_a|0⟩_b|0⟩_c|0⟩_d]$$ which transforms under the action of the BS as : $$\alpha |g⟩ [\frac{1}{\sqrt{2}}(c^\dagger + id^\dagger + ic^\dagger + d^\dagger)|0⟩_a|0⟩_b|0⟩_c|0⟩_d]$$ This is quite different from what is expected from the final expression of the authors of the article, I should rather obtain something like : $$\alpha |g⟩ [c^\dagger|0⟩_a|0⟩_b|0⟩_c|0⟩_d]$$ but I don't see how to get rid of the imaginaries in my former expression. What am I doing wrong ?

• '50/50 beamsplitter' is an ambiguous description. It could mean $\sqrt{1/2} \begin{bmatrix} 1&i\\i&1\end{bmatrix}$, the operation you described, or it could mean a Hadamard operation, $\sqrt{1/2} \begin{bmatrix} 1&1\\1&-1\end{bmatrix}$, or really it could mean any other 2x2 unitary matrix where the entries all have equal magnitude. If the paper doesn't specify which, you'll have to figure it out from how they say things are acting. Commented Jan 28, 2017 at 18:28
• Thank you very much @CraigGidney, using the Hadamard operation indeed seems to be working. I just have a weird $2/\sqrt{2}$ constant in front but the general form seems to be respected. Do you have an idea how this constant could cancel ?
– Mary
Commented Jan 28, 2017 at 19:36
• Gaining factors of $\sqrt 2$ is the equivalent of a sign-error when beam splitters are involved. Just recheck the math. In particular, make sure that all your operations are properly unitary. Commented Jan 28, 2017 at 21:18
• I found that actually, I should use the $H_2$ Hadamard matrix (it's a 4x4) due to the fact that I have a tensor product of two photons states. The $H_2$ matrix to be unitary, has a prefactor that is not $1/\sqrt{2}$ but rather $1/2$. Several links helped me : physics.stackexchange.com/questions/191344/… and physics.stackexchange.com/questions/55696/… and en.wikipedia.org/wiki/Hadamard_transform#Definition
– Mary
Commented Jan 28, 2017 at 23:28
• @Mary Ideally, you should write an answer to your post, explaining how to do it, and accept it, so it can serve as a future reference. Commented Jan 29, 2017 at 14:37