\section{Appendix}

\noindent The experimental apparatus was comprised of five sections:\\
\indent 1. Entangled state generation,\\
\indent 2. Tomographic setup for photon B,\\
\indent 3. Delay line for photon A,\\
\indent 4. Generation and state preparation for photon 3, and\\
\indent 5. Bell state measurement.\\
Results and analyses are presented for:\\
\indent 6. Experimental results for single qubit tomography, and\\
\indent 7. Experimental results for process tomography.
 
\subsection{1. Entangled state generation}
Pulses from a mode-locked Ti-sapphire laser with a duration of 140\,fs, a repetition rate of 76\,MHz, and a central wavelength of 780\,nm first passed through a frequency doubler.
The resultant ultraviolet pulses with a power of 200\,mW then (at t=0\,ns) pumped a pair of sandwiched beam-like-type cut BBO crystals, each of thickness 1\,mm,
to produce a stream of photon pairs, entangled in their polarisation.
Temporal and spatial compensation (TSC) was made with LiNbO$_3$ and YVO$_4$ crystals \cite{Zhan15experimental}.
The photonic qubits were then distributed to Alice and Bob via a pair of 3\,m single mode fibres (SMF).
Half waveplates (HWP) and tilted quarter waveplates (QWP) at both ports of the fibres were used to maintain the polarisation reference frame.

With the photonic qubits on Alice's side having passed through the 31\,m delay line and a 2\,m SMF,
both qubits were measured in the Pauli basis (details below),
and their density matrix reconstructed by the maximum likelihood estimation algorithm \cite{Pari04quantum}.
The quantum state that generates the experimental data with highest probability is
\begin{equation}
\rho_\text{MLE}=\left(
         \begin{array}{cccc}
           0.486 & 0.026 + 0.007i & -0.031 - 0.009i & 0.446 + 0.112i \\
           0.026 - 0.007i & 0.018 & -0.001 + 0.015i & 0.035 + 0.014i \\
            -0.031 + 0.009i & -0.001 - 0.015i & 0.021 & -0.020 - 0.017i \\
           0.446 - 0.112i & 0.035 - 0.014i & -0.020 + 0.017i & 0.475 \\
         \end{array}
       \right),
\end{equation}
which has a fidelity with the ideal state of $0.927\pm0.001$, where the error is the standard deviation from 100 numerical simulations with Poissonian statistics.

\subsection{2. Tomographic setup for photon B}
From each pair, the photon on Bob's side, photon B, was immediately injected into its tomographic set up and measured at t=17\,ns,
due to the overall optical path length from photon generation, including the approx. 0.6\,m in this section.
The module contained a HWP, a QWP, a polarisation beam splitter (PBS), and a single photon detector (SPD), to realise arbitrary projective measurements on qubits encoded in the polarisation of photons.
This photon was finally passed through an interference filter (FWHM of 8\,nm ) before detection.
Waveplate settings were chosen to implement measurements in the three Pauli bases,
and results were stored with respect to these settings.

\subsection{3. Delay line for photon A}
The photon on Alice's side, photon A, was directed to a delay line, comprised of eleven mirrors which reflected the photon through 31\,m of free-space before coupling to 2\,m of SMF.
A telescope was implemented with three concave lenses, resulting in an overall collection efficiency of approx. 50\% including losses from reflection and fibre insertion.

\subsection{4. Generation and sate preparation for photon 3}
After 8 consecutive pump pulses (105 ns), a second beam-like BBO crystal was pumped to generate pairs of (unentangled) photons,
with one photon from each pair directly detected as a trigger, after passing through an interference filter (FWHM of 8\,nm).
The other photon from this pair, photon 3, passed through a 5\,m SMF, after which its state was prepared by a combination of HWP and QWP.

\subsection{5. Bell state measurement}
Photon 3 and photon A impinged simultaneously on the first PBS.
Two HWPs, one at each PBS exit port, had optics axes set at 22.5$^\circ$, while a tilted QWP ensured the correct phase was implemented.
This configuration \cite{MegiPRL13}, deterministically resolves two of the four Bell states, such that even ($+ + , - -$) or odd ($+ - , - +$) measurement statistics are observed for states 
$\ketbra{\Phi^+}{\Phi^+}$
and
$\ketbra{\Phi^-}{\Phi^-}$
respectively.
The SMF and interference filters (FWHM of 2 nm) prior to detection reduced spatial and temporal difference between the two photons to produce a Hong-Ou-Mandel \cite{HongOM87} interference visibility of 0.89$\pm$0.01 at a pumping power of 200 mW.

\subsection{6. Experimental results for single qubit tomography}
Tomographic data for photon B
were collected over 150\,s at a pumping power of 200 mw
for each of the six states prepared on photon 3 and for each Pauli basis.
Data were grouped into two sets based on the result of the associated Bell state measurement,
$\ketbra{\Phi^\pm}{\Phi^\pm}$,
and respective single qubit density matrices, $\rho_{\Phi^{+}}$ and $\rho_{\Phi^{-}}$, were calculated with maximum likelihood estimation.
Table~\ref{table:states} shows these, while Table~\ref{table:statesC} shows density matrices from the combined data sets as described in the main text. Errors on fidelities are the standard deviations from 100 simulated results with Poissonian statistics. The average fidelity over all six states and two Bell state measurements is $0.90\pm0.01$.
\begin{table}[h]
  \caption{\label{table:states}
  Reconstructed single qubit density matrices $\rho_{\Phi^{+}}$ and $\rho_{\Phi^{-}}$ for all six states $\ket{\phi_{3}}$ prepared on photon 3,
  alongside their corresponding fidelities with errors.}
  \begin{tabular}{c c c c c }
    \hline
    $\ket{\phi}$ & $\rho_{\Phi^+}$ & $F(\rho_{\Phi^+},\ket{\phi})$ & $\rho_{\Phi^-}$ & $F(\sigma_{3}\rho_{\Phi^-}\sigma_{3},\ket{\phi})$\\
    \hline
    $\ket{H}$ &        $\left(
                       \begin{array}{cc}
                         0.94 & -0.02 + 0.06i \\
                         -0.02 - 0.06i & 0.06 \\
                       \end{array}
                     \right)      $          & 0.94$\pm0.03$
                     &        $\left(
                       \begin{array}{cc}
                         0.96 & 0.11 - 0.02i \\
                         0.11 + 0.02i & 0.04 \\
                       \end{array}
                     \right)      $          & 0.96$\pm0.03$\\
                     
    $\ket{V}$ &        $\left(
                       \begin{array}{cc}
                         0.06& -0.13 - 0.09i \\
                         -0.13 + 0.09i & 0.94 \\
                       \end{array}
                     \right)      $          & 0.94$\pm0.02$
                     &        $\left(
                       \begin{array}{cc}
                         0.08& -0.11-0.00i \\
                         -0.11+0.00i & 0.92 \\
                       \end{array}
                     \right)      $          & 0.92$\pm0.02$\\
    $\ket{D}$ &        $\left(
                       \begin{array}{cc}
                         0.40 & 0.38 + 0.01i \\
                         0.38 - 0.01i & 0.60 \\
                       \end{array}
                     \right)      $          & 0.88$\pm0.03$
                     &        $\left(
                       \begin{array}{cc}
                         0.40 & -0.39 - 0.11i \\
                         -0.39 + 0.11i & 0.60 \\
                       \end{array}
                     \right)      $          & 0.89$\pm0.03$\\
    $\ket{A}$ &        $\left(
                       \begin{array}{cc}
                         0.53 & -0.37 - 0.02i \\
                         -0.37 + 0.02i & 0.47 \\
                       \end{array}
                     \right)      $          & 0.87$\pm0.03$
                     &        $\left(
                       \begin{array}{cc}
                         0.52 & 0.38 + 0.12i \\
                         0.38 - 0.12i & 0.48 \\
                       \end{array}
                     \right)      $          & 0.88$\pm0.03$\\
    $\ket{R}$ &        $\left(
                       \begin{array}{cc}
                          0.46 & -0.01 - 0.40i \\
                         -0.01 + 0.40i & 0.54 \\
                       \end{array}
                     \right)      $          & 0.90$\pm0.03$
                     &        $\left(
                       \begin{array}{cc}
                          0.50 & -0.14 + 0.33i \\
                         -0.14 - 0.33i & 0.50 \\
                       \end{array}
                     \right)      $          & 0.83$\pm0.02$\\
    $\ket{L}$ &        $\left(
                       \begin{array}{cc}
                         0.40 & -0.12 + 0.38i \\
                         -0.12 - 0.38i & 0.60 \\
                       \end{array}
                     \right)      $          & 0.88$\pm0.04$
                     &        $\left(
                       \begin{array}{cc}
                         0.45 & 0.10 - 0.37i \\
                         0.10 + 0.37i & 0.55 \\
                       \end{array}
                     \right)      $          & 0.87$\pm0.03$\\
\hline
  \end{tabular}
  \end{table}
    \begin{table}[h]
 \centering{
  \caption{\label{table:statesC}
  Reconstructed single qubit density matrices $\rho_3$ for all six states $\ket{\phi_{3}}$ prepared on photon 3,
  alongside their corresponding fidelities with errors.}
  \begin{tabular}{c c c }
    \hline
    $\ket{\phi_3}$ & $\rho_3$ & $F(\rho_3,\ket{\phi_3})$\\
    \hline
    $\ket{H}$ &        $\left(
                       \begin{array}{cc}
                         0.95 & -0.06 + 0.04i \\
                         -0.06 - 0.04i & 0.05 \\
                       \end{array}
                     \right)      $          & 0.95$\pm0.02$\\
    $\ket{V}$ &        $\left(
                       \begin{array}{cc}
                         0.07& -0.01 - 0.05i\\
                         -0.01+ 0.05i & 0.93 \\
                       \end{array}
                     \right)      $          & 0.93$\pm0.02$\\
    $\ket{D}$ &        $\left(
                       \begin{array}{cc}
                         0.40 & 0.39 + 0.06i \\
                         -0.39 - 0.06i & 0.60 \\
                       \end{array}
                     \right)      $          & 0.89$\pm0.02$\\
    $\ket{A}$ &        $\left(
                       \begin{array}{cc}
                         0.52 & -0.37 - 0.07i \\
                         -0.37 + 0.07i & 0.48 \\
                       \end{array}
                     \right)      $          & 0.87$\pm0.02$\\
    $\ket{R}$ &        $\left(
                       \begin{array}{cc}
                          0.48 & 0.06 - 0.37i \\
                         0.06 + 0.37i & 0.52 \\
                       \end{array}
                     \right)      $          & 0.87$\pm0.02$\\
    $\ket{L}$ &        $\left(
                       \begin{array}{cc}
                         0.43 & -0.11 + 0.38i \\
                         -0.11 - 0.38i & 0.57 \\
                       \end{array}
                     \right)      $          & 0.88$\pm0.02$\\
    \hline
  \end{tabular}}
  \end{table}

\subsection{7. Experimental results for process tomography}
Following \cite{Jeif03quantum}, $S$ is a positive semidefinite operator on a Hilbert space $\mathcal{H}\otimes\mathcal{K}$
that maps the quantum state $\rho_\text{in}$ on a Hilbert space $\mathcal{H}$
to the output state $\rho_\text{out}$ on a Hilbert space $\mathcal{K}$,
\begin{equation}\label{EQ: process tomo}
  \rho_\text{out}=\operatorname{Tr}_H[S\rho_\text{in}^T\otimes1_\mathcal{K}],
\end{equation}
where $1_\mathcal{K}$ is an identity operator on $\mathcal{K}$, and $T$ denotes the matrix transposition.
$S_\text{MLE}$ is found from an iterative method to generate the experimental data with maximum probability.
As described in the main text, two different processes $S^\pm$ are respectively associated with
the two Bell measurement results $\ketbra{\Phi^\pm}{\Phi^\pm}$.
The ideal and maximum likelihood estimators of $S^\pm$ are obtained via the algorithm in \cite{Jeif03quantum}
using the experimental data that produced the density matrices in Table \ref{table:states}.

The $S^\pm$ matrices are simply transformed into $\chi^\pm$
from the standard definition \cite{Obri04quantum}
\begin{equation}\label{Eq:process tomo other}
  \rho_\text{out}=\sum_{m,n=0}^{d^2-1}\chi_{mn}\sigma_m\rho_\text{in}\sigma_n
\end{equation}
with Identity and Pauli operators $\sigma_{m}$,
such that
\begin{equation}\label{Eq:S chi connection2}
  \chi=U_2^\dagger U_1^\dagger \, S \,\, U_1U_2
\end{equation}
with
\begin{equation}
  U_1=\left(
          \begin{array}{cccc}
            1 & 0 & 0 & 0 \\
            0 & 0 & 1 & 0 \\
            0 & 1 & 0 & 0 \\
            0 & 0 & 0 & 1 \\
          \end{array}
        \right),
        \;\;\;\;\;\;\;\;
 U_2=\frac{1}{2}\left(
    \begin{array}{cccc}
      1 & 0 & 0 & 1 \\
      0 & 1 & -i & 0 \\
      0 & 1 & i & 0 \\
      1 & 0 & 0 & -1 \\
    \end{array}
  \right)
\end{equation}
to give
\begin{equation}\label{Eq:chi+ MLE}
  \begin{aligned}
      \chi^+_\text{ideal}=\left(
          \begin{array}{cccc}
            1& 0 & 0 & 0 \\
            0 & 0 & 0 & 0 \\
            0 & 0& 0 & 0 \\
            0 & 0 & 0 & 0 \\
          \end{array}
        \right),\quad
    \chi^+_\text{MLE}
    =\left(
          \begin{array}{cccc}
            0.84 & -0.01+0.06i & 0.00+0.06i & -0.01-0.03i \\
            -0.01-0.06i & 0.03 & 0.02+0.01i & -0.01+0.00i \\
            0.00-0.06i & 0.02-0.01i & 0.04 & -0.02+0.01i \\
            -0.01+0.03i & -0.01-0.00i & -0.02-0.01i & 0.09 \\
          \end{array}
        \right),
  \end{aligned}
\end{equation}
and
\begin{equation}\label{Eq:chi-MLE}
  \begin{aligned}
      \chi^-_\text{ideal}=\left(
          \begin{array}{cccc}
            0& 0 & 0 & 0 \\
            0 & 0 & 0 & 0 \\
            0 & 0& 0 & 0 \\
            0 & 0 & 0 & 1 \\
          \end{array}
        \right),\quad
    \chi^-_\text{MLE}
    =\left(
          \begin{array}{cccc}
            0.10 & -0.00+0.01i & 0.01+0.07i & 0.00+0.12i \\
            -0.00-0.01i & 0.01 & 0.00-0.00i & 0.03+0.01i \\
            0.01-0.07i & 0.00+0.00i & 0.05 & 0.02+0.00i \\
            0.00-0.12i & 0.03-0.01i & 0.02-0.00i & 0.83 \\
          \end{array}
        \right).
  \end{aligned}
\end{equation}

The process fidelities are immediately available in this representation as
$0.84 \pm 0.02$ and $0.83 \pm 0.02$ for the Identity and $\sigma_{3}$ respectively
where the errors on the fidelities are the standard deviations from 100 simulated results with Poissonian statistics.

\end{document}
