I am studying the Quantum Teleportation Tutorial [Basic tutorial: quantum teleportation — Strawberry Fields] and I don’t comprehend how those results (probabilities of Fock states) show that the state α=1+0.5i was indeed teleported. What am I missing?

Hi @Moniquewg,

Notice how inputs states to modes 1 and 2 are squeezed states: the |0\rangle_p and |0\rangle_q. The input state, the one being teleported from mode 0 to mode 2, is a coherent state. In the experimental setup you cannot access the state but the measurement outcomes and then run statistics on them. The frequency on which a certain photon number appears is encoded in the Fock amplitudes — in the tutorial, the last plot shows that Fock amplitudes from the output state on mode 2 agree with those of a coherent state with \alpha = 1+0.5i, proving that the state has been effectively teleported.

Hi @Moniquewg,

Just to complement Sebastian’s answer, in the StrawberryFields documentation you can find the expression for finding the decomposition of a coherent state in the Fock basis.

Each coefficient in the decomposition will determine the probability of measuring each of the Fock states. The probability, which is the height of each bar in the histogram, is the squared norm of each of the coefficients. You can calculate the decomposition and probabilities for the coherent state `1+0.5i`

as follows:

```
import strawberryfields as sf
import math
a = 1+0.5j # Coherent state
coef1 = math.exp(-abs(a)**2/2) # Exponential part in the coefficients
# List of coefficients
coef = [coef1*a**n/math.sqrt(math.factorial(n)) for n in range(7)]
# Probabilities
probs = [abs(c)**2 for c in coef]
print(probs)
```

So if you print the histogram and compare it with the one obtained in the quantum teleportation tutorial you will see that they’re very similar.

```
from matplotlib import pyplot as plt
plt.bar(range(7), probs[:7])
plt.xlabel('Fock state')
plt.ylabel('Marginal probability')
plt.title('Mode 2')
plt.show()
```

Thank you very much to @Sebastian_Duque_Mesa for the explanation!

And thank you @Moniquewg for the great question.