Other

Note

In Strawberry Fields we use the convention \(\hbar=2\) by default, but other conventions can also be chosen on engine initialization. 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.

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