The goal of quantum teleportation in this setup is to transfer the most general polarization state|χic = α|Hic+β|V ic with arbitrary amplitudes α, β of the photon in mode c onto the photon in mode a. This requires a maximally entangled Bell state in modes a and b, as well as the complete Bell state projection measurement between the photons in mode b
and c. The Bell state is obtained by proper alignment of the photon pair originating from the forward down conversion which is adjusted to yield state|φ+i. The photon which will carry the state |χic is provided by the backward emission of the down conversion which is operated as a heralded single photon source [165–167] with the photon in modedinitializing the trigger. The polarization state|χicis prepared by a polarizer in front of the fiber coupler in modecand proper alignment of the fibre’s polarization controller.
Teleported states
Four different input states|Hi,|V i,|+i,|Ri are chosen to be teleported. For each of them a single qubit tomography of the corresponding output state is carried out in modea. This yields the density matricesk, (k∈ {H, V,+, R}), of the experimentally teleported states and allows the calculation of the fidelities,Fk =hk|m1,m2k
†
m1,m2|kito the input states. The
(a)<e(H) (b) =m(H) (c) <e(V) (d) =m(V)
(e) <e(+) (f ) =m(+) (g) <e(R) (h)=m(R) Figure 3.3: Experimentally determined density matrices of various teleported states. For each input state the data at the output was averaged over the four possible detections in the cphasegate. Prior to the averaging, the proper unitary transformation was applied to the data. (a) Real part and (b) imaginary part of the output density matrix for teleportation of |Hi. In this instance, lack of quality in the output state is primarily caused by imperfect input states. (c) Real part and (d) imaginary part of the output density matrix for teleportation of |Vi. This state is expected to be teleported worst as its quality is impaired at most for incoherentcphaseoperation. This is observable in the comparably high|HihH| noise. (e) Real part and (f) imaginary part of the output density matrix for teleportation of|+iand (g) real part and (h) imaginary part of the output density matrix for teleportation of|Ri. These two states are supposed to be teleported approximately with the same quality. This is not exactly the case in the experiment due to additional imperfections besides lack of coherence in the gate and lack of quality of the input states.
to the data during the evaluation process. After averaging for each state over the different projection outcomes, m1, m2, this results in FH = 0.93±0.02, FV = 0.75±0.05, F+ = 0.79±0.02, FR= 0.84±0.03. The real- and imaginary parts ofk are graphically displayed in Fig. 3.3(a) – Fig. 3.3(h). As can be seen, the quality of the output states differs for the various input states. This can be understood by considering the influence of imperfect gate operation. In Sec. 2.1 it was concluded that for the experimental gate the main source of errors is a lack of coherence. Taking that into account, it is obvious that the teleportation works best for the state|Hi, as in this instance no interference is required. In contrast, from this point of view the output state for the input|V i is expected to be the worst. The states
|+i and |Ri should be teleported approximately at the same quality on average. However, for the state|+ithe fidelity of the output state depends on the result of the projection in the cphasegate. It is worse for m1 = 0 and better for m1 = 1. A calculation using Eqn. (2.8) withQ0 = 0.90 is shown in Tab. 3.1(a). For comparison the measured results are summarized in Tab. 3.1(b). As can be seen, the measured fidelities exhibit roughly the expected behavior. Still, they are significantly worse than calculated. This is due to the fact that, naturally in
3.1 Teleportation and entanglement swapping 73
the experiment other effects occur in addition to the lack of coherence. Tab. 3.1(a) shows in square brackets the results of the output state fidelities for a calculation which uses theqpt matrixχfit and allows for imperfect input states a,b⊗cof the form
a,b = f1|φ+ihφ+|+1−4f1 1⊗2 (3.5a)
c = f2|kihk|+ 1−f2
2 1, (3.5b)
withf1 = 0.93 andf2 = 0.95. Comparing these values again with the experimental ones points up that the loss in quality for |Hi is indeed not impaired by lack of coherence but mainly determined by impurity of the input states. For|Riobviously both effects are relevant. The model is not capable of explaining why the measured fidelities for every state, not only for
|+i, fluctuate for the different projections in the cphase gate. This must be due to noise effects which are not accounted for in χfit. These effects seem to be also responsible for the worse than expected fidelity of|Vi and |+i.
Despite all imperfections, it is important to note that the average fidelities are all well above the optimal classical limit2 of 23.
Quantum process tomography
The four input states,|Hi,|V i,|+i,|Ri, represent a tomographic set out of which a tele- portation process tomography can be evaluated. Similarly as described for thecphasegate (see Sec. 2.1.3), such a tomography yields a matrixMij which characterizes the performance of the teleportation process according to
b
(|kihk|) =k = X
ij
Mij i|kihk| †j. (3.6)
For the linear decomposition of the superoperator, , the basis of unitary transformationsb i is here chosen to be
1 =1, 2 =x, 3 =−iy, 4 =z. (3.7) The experimentally measured matrix Mexp is shown in Fig. 3.4. In this representation an ideal teleportation (Mth) corresponds to the identity operation. Thus the height of the ( 1, 1)-entry of Mexp directly gives the process fidelity,
Fp = tr(MthMexp), (3.8)
which is the overlap between the experimentally obtained and the theoretically expected matrix. It measures the quality of the implemented teleportation process and reaches in the experiment a value ofFp = 0.75. The limiting factor of the process fidelity is the fidelity of the state which is teleported worst. Following the discussions of the previous section this is the state|V i for which the output state fidelity reaches an average value comparable toFp.
The inherent quantum features of the teleportation process are best seen by performing entanglement swapping. In the experiment described before, the teleportation of a polarized
2The best fidelity which can be achieved by a local measurement of the state and a remote preparation
after the exchange of classical information is 2
3 for a single qubit averaged over all possible input states. For a
(a)Calculated fidelities of the teleported output states
state Fidelity for different projection results: m1, m2 Average
0,0 0,1 1,0 1,1 fidelity
|Hi 1.00 [ 0.93 ] 1.00 [ 0.93 ] 1.00 [ 0.93 ] 1.00 [ 0.93 ] 1.00 [ 0.93 ] |V i 0.83 [ 0.81 ] 0.83 [ 0.81 ] 0.83 [ 0.81 ] 0.83 [ 0.81 ] 0.83 [ 0.81 ] |+i 0.83 [ 0.77 ] 1.00 [ 0.94 ] 0.83 [ 0.77 ] 1.00 [ 0.94 ] 0.915 [ 0.855 ] |Ri 0.91 [ 0.85 ] 0.91 [ 0.85 ] 0.91 [ 0.85 ] 0.91 [ 0.85 ] 0.91 [ 0.85 ]
(b) Measured fidelities of the teleported output states
state Fidelity for different projection results: m1, m2 Average
0,0 0,1 1,0 1,1 fidelity
|Hi 0.92±0.05 0.94±0.04 0.95±0.04 0.93±0.05 0.93±0.02 |Vi 0.77±0.06 0.80±0.06 0.71±0.07 0.71±0.06 0.75±0.05 |+i 0.74±0.04 0.96±0.02 0.66±0.05 0.86±0.04 0.79±0.02 |Ri 0.81±0.05 0.85±0.05 0.88±0.04 0.81±0.05 0.84±0.03
Table 3.1: Table of the teleportation output states’ fidelity for the different input states depending on the result of the Bell state projection measurement in the cphasegate. (a) Calculated fidelities of the output states using thecphasemodel of Eqn. (2.8) and assuming a quality parameterQ0 = 0.90. Values in square brackets are for a similar calculation using
thecphase qptmatrixχfit and allowing additionally for imperfect input states. The state |Hiis expected to be teleported best as it is not influenced byQ0. In contrast the state|Vi
is maximally affected by Q0 and therefore teleported worst. Except for the state|+i, the
output fidelities are not dependent on the results of the Bell state projection. (b) Measured fidelities of the teleported output states. In the experiment the fidelities depend on the results of the Bell projection measurement. This effect cannot be explained by lack of coherence in thecphasegate or admixture of white noise in the input states.
3.1 Teleportation and entanglement swapping 75
(a)<e(Mexp) (b)=m(Mexp)
Figure 3.4: (a) Real part and (b) imaginary part of the experimentally reconstructed quantum process tomography matrix for the teleportation process. An ideal teleportation corresponds to the identity. Thus, the height of the ( 1, 1) entry directly gives the process fidelity Fp which is a measure for the performance of the protocol. It is bounded by the fidelity of the state which is teleported worst and reaches hereFp= 0.75.
photon does not succeed always, e.g., due to experimental restrictions like limited detection efficiencies etc. Hence it could be argued that the observed teleportation fidelities are a result of statistical averaging over many measurements. Such arguments can be directly refuted for entanglement swapping. Here, the teleported photon is part of an entangled pair, in that sense it is not polarized. Therefore, the outcome of a measurement on this photon considered apart is completely random. Thus, if the observed teleportation results for individual one-photon output states were attributed to statistical averaging, the analogue experimental procedure would unavoidably lead to a random result for the correlation measurements on two-photon output states. In the following, however, it will be proven that indeed quantum correlations can be observed. This confirms the entanglement contained in the swapped photon pair and proves that teleportation succeeds for every single instance.