Reconstructing an entangled state

In a nonlinear crystal such as BBO, a pump photon at 405 nm now and then splits into two photons at 810 nm. This is spontaneous parametric down-conversion. Stack two thin crystals with their axes at right angles and the pair comes out in a superposition: both photons horizontal, or both vertical.

|ψ⟩ = cos θ |HH⟩ + e^iφ sin θ |VV⟩

Here you set that state yourself. θ balances the two terms, φ is the phase between them, and the visibility V mixes in white noise, the way imperfect alignment and timing do in a real source. The full state is then

ρ = V |ψ⟩⟨ψ| + (1 − V) 𝟙/4

We call one photon Alice's and the other Bob's.

One detector click tells you very little. To learn a polarisation state you project each photon onto one of six states: horizontal, vertical, diagonal, anti-diagonal, and right- and left-circular. They sit in opposite pairs on the Poincaré sphere, one pair per axis.

In the lab each arm has a quarter-wave plate, a half-wave plate and a polarising beam splitter that passes H. Turning the two plates picks which state reaches the detector. The angles below come from the same model the simulation uses.

Set Alice's analyser to one state and Bob's to another, then count how often both detectors click within a few nanoseconds of each other. That count is proportional to the probability ⟨ab|ρ|ab⟩ of finding the pair in |a⟩|b⟩.

Photon counting is random. A setting that should give 400 counts gives 400 ± 20 or so, because the noise grows as √N. On top of that, unrelated photons sometimes arrive together by chance. These accidentals add a flat background, which is usually subtracted.

Click a cell to measure that pair of settings. Each cell is one experiment: rows are Alice's setting, columns are Bob's.

Any two-photon polarisation state can be written with 16 real numbers, the two-photon Stokes parameters S_ij:

ρ = ¼ Σ_i,j S_ij σ_i ⊗ σ_j

Each coincidence count is a fixed linear combination of these 16 numbers. If your settings give 16 independent equations, you can solve for all of them with ordinary linear algebra. With more than 16 the page takes the least-squares fit.

There's a catch. Noise can push the answer outside the set of physical states. A real density matrix never has a negative eigenvalue, but linear inversion can produce one, especially when counts are low.

Instead of solving the equations exactly, ask a different question: of all the physical states, which one makes the counts you actually saw most likely? The answer is always a valid state, with no negative eigenvalues, and it uses the data more sensibly when counts are low.

This page finds it with the iterative R·ρ·R method. It starts from the completely mixed state and repeatedly nudges ρ toward the data, keeping it positive at every step.

Three numbers sum up the reconstruction. Fidelity says how close it is to the true state, from 0 to 1. Purity, Tr ρ², is 1 for a pure state and ¼ for complete noise. Concurrence measures entanglement: 0 for a state that could be made without it, 1 for a Bell state.