We investigate the benefits of adaptive Bayesian quantum tomography in laboratory experiments on a single photon light source. In a real world setup there are additional sources of noise. After presenting the experimental setup we describe how to model this noise and show our findings.
Figure 4.4: Experimental setup. An attenuated laser is used as a source, the po- larization state is prepared by a custom waveplate, and analyzed by a sequence of a quarter- and half-wave plates, followed by a polarizing beam-splitter and two single- photon counters. Waveplates are rotated by electronically controlled step-motor drivers to allow for adaptivity.
4.6.1 Setup and Apparatus
We perform polarization tomography on single photons of light emitted from an atten- uated laser. A measurement is made by passing the light through a filter called the polarization beam splitter (PBS). Depending on the state, the photon will pass though the PBS or be reflected. Detectors, called single-photon counting modules (SPCMs), count the photons that follow each path. Recall that in single qubit tomography, a pro- jective measurement is characterized by two degrees of freedom, the polar and azimuth angles in the Bloch sphere. The different measurements are achieved by rotating the photon twice, using a quarter-wave plate (QWP) and a half-wave plate (HWP). Their orientation is set using motors, and during adaptive tomography, the wave plates are rotated to achieve the optimal measurements.
Figure4.4depicts the setup. In detail, we use a CW 850 nm vertical-cavity surface- emitting laser (VCSEL) diode laser coupled to a single-mode fibre as the light source. The radiation is attenuated to the single-photon level by a set of neutral density filters (F) and additionally spatially filtered with small iris apertures. The input polarization state is defined by a Glan-Taylor prism GP with high extinction ratio (more than 6000:1), the prism transmits horizontally polarized light, which may be transformed to an arbitrary state with a proper choice of a quartz wave plate (WP).
The measurement scheme consists of an effective zero-order QWP and a HWP. The plates are rotated by step-motor-driven stages, with minimal angular step of 0.1◦. The
zero position is controlled by a Hall sensor providing uncertainty of 0.2◦. We clean up the polarization states in the output channels of the PBS cube with two additional Glan-Taylor prisms to ensure high extinction ratio. Effectively this is equivalent to introducing some losses in the ideal PBS cube without altering the output polarization states. In each channel photons are coupled to multi-mode fibres (MMF) and detected by single photon counting modules D1 and D2 (Perkin-Elmer). Electronic pulses from SPCM’s are sent to a counter built in-house which may operate in two ways - count for a fixed period of time or until a specified number of counts is reached.
4.6.2 Modelling Experimental Imperfections
In practice quantum tomography is subject to experimental noise. This noise is not modelled in the likelihood function given by Born’s rule (4.2). In our experiment we identified two major additional sources of noise: dark counts with detector-specific rates and attenuation in both channels due to detector inefficiency and losses at the optical elements.
Dark and Background Counts
Dark and background counts are false positive observations that are detected even when there is no photon present. A popular approach to account for dark counts is to model the observed state as a linear mixture of the true state and the maximally mixed state [Lvovsky et al.,2001]. With this approach one can describe certain simple sources of noise, such as dark counts being generated at each detector with equal rates. We take a more flexible approach and model the noise directly in the likelihood function.
We assume that photons produced at the laser source and dark counts are all generated independently. In particular, we assume that the production of photons by the laser source, and generation of dark counts by the detectors can be modelled using independent homogeneous Poisson processes with rate parameters λs for the
source and λγd for each detector γ. We assume that the rates of the Poisson processes remain constant over time. This homogeneity assumption is likely to be violated due to parameter drift in the apparatus, but by re-calibrating the system periodically we ensure that the drift is small. Audenaert & Scheel[2009] consider more general noise scenarios, but the resulting computations are more complex and may require numerical methods. The rates Λ = {λs, {λγd}Γγ=1} are estimated from prior experimentation.
The new likelihood function follows directly from these assumptions and Born’s rule. The total rate of photons (including dark counts) entering the system follows a
Poisson process with rate λs+PΓγ=1λ γ
d. Therefore the probabilities that a detection
is from the source or a dark count are given by
P (source|Λ) = λs λs+PΓγ=1λγd , P (dark|Λ) = PΓ γ=1λ γ d λs+PΓγ=1λγd .
The likelihood follows from Born’s rule (4.1),
P (γ|ρ, α, Λ) = P (γ|source, ρ, α)P (source|Λ) + P (γ|dark)P (dark|Λ) = tr[Mαγρ] λs λs+PΓγ=1λγd + λ γ d PΓ γ=1λ γ d PΓ γ=1λ γ d λs+PΓγ=1λγd = tr[Mαγρ]λs+ λ γ d λs+Pγλ γ d . (4.8)
When there are no dark counts, λγd = 0, ∀γ, then Equation (4.8) reduces to Born’s rule (4.1).
Channel Inefficiency
As well as dark counts, the detectors can produce false negatives. Photons may also be reflected at the optical elements, such as the wave-plates and PBS. Furthermore, the GT prisms may have different attenuation factors. To model these channel-specific losses, each detector is assigned an efficiency ηγ ∈ [0, 1]. the probability of a photon
being ‘lost’ in the channel ending in detector γ is given by 1 − ηγ. Therefore, the
probability of observing a measurement at detector γ is proportional to tr[Mαγρ]ηγ.
The likelihood is straightforward,
P (γ|ρ, α, η1, . . . , ηγ) =
tr[Mαγρ]ηγ
P
γtr[Mαγρ]ηγ
. (4.9)
In Equations (4.8) and (4.9), both the numerator and denominator contain only linear terms in the additional parameters (Λ, {ηγ}). Therefore, one only requires esti-
mates of the ratio of the dark count rates to the source rate λγd/λs, and, for single-qubit
tomography, the ratio of the efficiencies of the two channels η1/η2. For this reason we
10 2 10 3 10 4 10 -3 10 -2 Adaptive Random MUB a ve r a g e i n f i d e l i t y, 1 - F number of measurements, N
Figure 4.5: Experimental results: mean infidelity 1 − Ep(ρ|D)F (ρ, ¯ρ) with true state ¯ρ
for random measurements – red (middle) line, adaptive measurements – black (lower) line, and measurements in MUBs – blue (upper) line. We average over 10 experimental runs, shaded areas show the standard deviation. Dashed straight lines indicate the power law fits.
Block Sampling
The time taken to rotate the WPs into position is longer than rate of generation of the states or the time required to run SIS or BALD. Therefore, we adjust the apparatus after blocks of measurements that increase in size with amount of data collected as dN/100e. In simulation we found no statistical difference between this strategy and adjusting after every measurement.
4.6.3 Results
In a real world application of tomography the true state is unknown, so we estimate the prepared state by averaging over many runs of adaptive protocol. Figure4.5gives the mean infidelity to the (estimated) true state. Power law fits give a = −0.64 ± 0.02 and a = −0.60 ± 0.05 for random and MUB protocols, respectively, while adaptive strategy yields a = −0.92 ± 0.03.
Within the errors bands, the scaling laws obtained in the experiments agree with the simulations in Section4.5.1. This demonstrates that we were able to realize in practice the advantages of using BALD for adaptive Bayesian tomography. Our model does not
take into account systematic errors, such as imprecise waveplate rotations. However, for the infidelities values that we reached, 10−4− 10−3, we did not observe deviations
from the expected behaviour and could not identify the influence of systematic errors.