# Other¶

Note

In Strawberry Fields we use the convention $$\hbar=2$$ by default, but other conventions can also be chosen by setting the global variable sf.hbar at the beginning of a session. In this document we keep $$\hbar$$ explicit.

## Loss channel¶

Loss is implemented by a CPTP map whose Kraus representation is

$\mathcal{N}(T)\left\{\ \cdot \ \right\} = \sum_{n=0}^{\infty} E_n(T) \ \cdot \ E_n(T)^\dagger , \quad E_n(T) = \left(\frac{1-T}{T} \right)^{n/2} \frac{\a^n}{\sqrt{n!}} \left(\sqrt{T}\right)^{\ad \a}$

Definition

Loss is implemented by coupling mode $$\a$$ to another bosonic mode $$\hat{b}$$ prepared in the vacuum state, by using the following transformation

$\a \to \sqrt{T} \a+\sqrt{1-T} \hat{b}$

and then tracing it out. Here, $$T$$ is the energy transmissivity. For $$T = 0$$ the state is mapped to the vacuum state, and for $$T=1$$ one has the identity map.

Tip

Implemented in Strawberry Fields as a quantum channel by strawberryfields.ops.LossChannel

One useful identity is

$\mathcal{N}(T)\left\{\ket{n}\bra{m} \right\}=\sum_{l=0}^{\min(n,m)} \left(\frac{1-T}{T}\right)^l \frac{T^{(n+m)/2}}{l!} \sqrt{\frac{n! m!}{(n-l)!(m-l)!}} \ket{n-l}\bra{m-l}$

In particular $$\mathcal{N}(T)\left\{\ket{0}\bra{0} \right\} = \pr{0}$$.

## Thermal loss channel¶

Definition

Thermal loss is implemented by coupling mode $$\a$$ to another bosonic mode $$\hat{b}$$ prepared in the thermal state $$\ket{\bar{n}}$$, by using the following transformation

$\a \to \sqrt{T} \a+\sqrt{1-T} \hat{b}$

and then tracing it out. Here, $$T$$ is the energy transmissivity. For $$T = 0$$ the state is mapped to the thermal state $$\ket{\bar{n}}$$ with mean photon number $$\bar{n}$$, and for $$T=1$$ one has the identity map.

Tip

Implemented in Strawberry Fields as a quantum channel by strawberryfields.ops.ThermalLossChannel

Note that if $$\bar{n}=0$$, the thermal loss channel is equivalent to the loss channel.

## Commutation relations¶

A collection of commutation relations between the gates.

$B^\dagger(\theta,\phi) D(z) B(\theta,\phi) = D(z \cos \theta) \otimes D(z e^{-i\phi} \sin \theta)$