In this section, we develop the necessary formalism to discuss CV entanglement swapping. A CV system is a canonical infinite dimensional quantum system comprised of

N bosonic modes with an associated Hilbert space

$\mathscr{H}={\otimes}_{k=1}^{N}{\mathscr{H}}_{k}$. Each of these modes

${\mathscr{H}}_{k}$ has an associated annihilation and creation operator

â,

â^{†} respectively. These operators obey the commutation relations

$[{\widehat{a}}_{i},{\widehat{a}}_{j}^{\u2020}]={\delta}_{ij}$ and

$[{\widehat{a}}_{i},{\widehat{a}}_{j}]=[{\widehat{a}}_{i}^{\u2020},{\widehat{a}}_{j}^{\u2020}]=0$. The space

${\mathscr{H}}_{k}$ is spanned by the Fock basis

${\left\{|{n}_{k}\rangle \right\}}_{n=0}^{\infty}$ of eigenstates of the number operator

${\widehat{n}}_{k}={\widehat{a}}_{k}^{\u2020}{\widehat{a}}_{k}$. These eigenstates have the property that

$\widehat{n}|n\rangle =n|n\rangle $,

$\widehat{a}|n\rangle =\sqrt{n|}n-1\rangle $,

${\widehat{a}}^{\u2020}|n\rangle =\sqrt{n+1}|n+1\rangle $, as well as the fact that the vacuum state |0⟩ is annihilated by

â|0⟩ = 0. In the absence of any interactions, these modes evolve according to the Hamiltonian

$H={\displaystyle {\sum}_{k=1}^{N}({\widehat{a}}_{k}^{\u2020}{\widehat{a}}_{k}+1/2)}$. We can define two quadrature operators

${\widehat{q}}_{k}={\widehat{a}}_{k}+{\widehat{a}}_{k}^{\u2020}$ and

${\widehat{p}}_{k}=i({\widehat{a}}_{k}^{\u2020}-{\widehat{a}}_{k})$ which act in a similar fashion to the position and momentum operators in the quantum harmonic oscillator. These commutation relations can be compactly written by defining

so that we have

$[{\widehat{x}}_{i},{\widehat{x}}_{j}]=2i{\mathrm{\Omega}}_{ij}$ where

is the symplectic form.

Of particular interest in CV systems are Gaussian states and operations [

12]. A Gaussian state, with density matrix

$\widehat{\rho}$, is completely characterized by its first two moments; the displacement vector

and the covariance matrix

where

$\Delta {\widehat{x}}_{i}={\widehat{x}}_{i}-\langle {\widehat{x}}_{i}\rangle $ and {.

,.} is the anticommutator. In this form, the uncertainty relation can be expressed as

V +

iΩ ≥ 0. We can fully represent an arbitrary quantum state

$\widehat{\rho}$ by its Wigner function, defined as

where

$\chi (\mathbf{\xi})=\mathrm{Tr}[\widehat{\rho}D(\mathbf{\xi})]$ is the characteristic function and

$D(\mathbf{\xi})=\mathrm{exp}(i{\widehat{\mathrm{x}}}^{\mathrm{T}}\mathrm{\Omega}\mathbf{\xi})$ is the Weyl operator. For Gaussian states the Wigner function can be written as

and thus we have complete knowledge of the state from only the first two moments. It is convenient for many purposes to deal only with the displacement vector and covariance matrix directly with the knowledge that we can always express the state in another form if we so choose.

Gaussian operations are those that take Gaussian states to other Gaussian states. They correspond to Hamiltonians which are linear or quadratic in the quadrature operators. To every unitary transformation U_{S,d} generated from a quadratic Hamiltonian there exists a symplectic operator S and vector d which generate the mapping
$\widehat{\mathrm{x}}\to S\widehat{\mathrm{x}}+\mathrm{d}$. This transforms the moments of a Gaussian state as
$\overline{\mathrm{x}}\to S\overline{\mathrm{x}}+\mathrm{d}$ and V → SV S^{T}.

The Einstein-Podolsky-Rosen (EPR) state

with

λ = tanh

r ∈ [0, 1], where

r is the squeezing parameter, is one important example of a Gaussian state. This two-mode state is entangled, and often used as a resource state in CV protocols. The EPR state has zero mean and covariance matrix given by

where

ν = cosh 2

r,

$\mathrm{I}$ is the 2

×2 identity matrix and

Z is the Pauli matrix

Z = diag(1,

−1).

Quantum teleportation can be conveniently described in the Wigner function formalism, where one makes a Bell measurement between the input state and half of the entangled pair [

5]. Suppose we have an input state described by a Wigner function

${W}_{in}(\widehat{x},\widehat{p})$ and an EPR pair described by

${W}_{EPR}({\widehat{x}}_{1},{\widehat{p}}_{1},{\widehat{x}}_{2},{\widehat{p}}_{2})$. To teleport the state Alice mixes the input state and the first mode of the EPR pair on a 50:50 beam splitter which performs the transformations:

$\widehat{x}\to \widehat{X}=(\widehat{x}-{\widehat{x}}_{1})/\sqrt{2}$,

$\widehat{p}\to \widehat{k}=(\widehat{p}-{\widehat{p}}_{1})/\sqrt{2}$,

${\widehat{x}}_{1}\to q=(\widehat{x}+{\widehat{x}}_{1})/\sqrt{2}$, and

${\widehat{p}}_{1}\to \widehat{P}=\widehat{P}=(\widehat{p}+{\widehat{p}}_{1})/\sqrt{2}$. Alice then measures the pair

$(\widehat{X},\widehat{P})$ and sends the result

$(\overline{X},\overline{P})$ to Bob who displace his mode as

${\widehat{x}}_{2}\to {\widehat{x}}_{out}={\widehat{x}}_{2}+\sqrt{2}g\overline{X}$, for some possible gain

g > 0, and similarly for

${\widehat{p}}_{out}$. The final state is obtained by integrating over all measurement outcomes as

where

${W}_{t}(\zeta ,{\zeta}_{A},{\zeta}_{B})={W}_{in}[(\zeta +{\zeta}_{A})/\sqrt{2}]{W}_{EPR}[({\zeta}_{A}-\zeta )/\sqrt{2},{\zeta}_{B}]$ is the Wigner function for the total state with

${\zeta}_{i}=({\widehat{x}}_{i},{\widehat{p}}_{i})$.