.. raw:: html Other ====== .. note:: In Strawberry Fields we use the convention :math:\hbar=2 by default, but other conventions can also be chosen on engine :func:initialization . In this document we keep :math:\hbar explicit. .. _loss: Loss channel --------------------------------------------- Loss is implemented by a CPTP map whose Kraus representation is .. math:: \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} .. admonition:: Definition :class: defn Loss is implemented by coupling mode :math:\a to another bosonic mode :math:\hat{b} prepared in the vacuum state, by using the following transformation .. math:: \a \to \sqrt{T} \a+\sqrt{1-T} \hat{b} and then tracing it out. Here, :math:T is the *energy* transmissivity. For :math:T = 0 the state is mapped to the vacuum state, and for :math:T=1 one has the identity map. .. tip:: *Implemented in Strawberry Fields as a quantum channel by* :class:strawberryfields.ops.LossChannel One useful identity is .. math:: \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 :math:\mathcal{N}(T)\left\{\ket{0}\bra{0} \right\} = \pr{0}. .. _thermal_loss: Thermal loss channel --------------------------------------------- .. admonition:: Definition :class: defn Thermal loss is implemented by coupling mode :math:\a to another bosonic mode :math:\hat{b} prepared in the thermal state :math:\ket{\bar{n}}, by using the following transformation .. math:: \a \to \sqrt{T} \a+\sqrt{1-T} \hat{b} and then tracing it out. Here, :math:T is the *energy* transmissivity. For :math:T = 0 the state is mapped to the thermal state :math:\ket{\bar{n}} with mean photon number :math:\bar{n}, and for :math:T=1 one has the identity map. .. tip:: *Implemented in Strawberry Fields as a quantum channel by* :class:strawberryfields.ops.ThermalLossChannel Note that if :math:\bar{n}=0, the thermal loss channel is equivalent to the :ref:loss channel . Commutation relations --------------------------------------------- A collection of commutation relations between the gates. .. math:: B^\dagger(\theta,\phi) D(z) B(\theta,\phi) = D(z \cos \theta) \otimes D(z e^{-i\phi} \sin \theta)