Distinguishing between statistical and systematic errors in quantum process tomography

It is generally assumed that every process in quantum physics can be described mathematically by a completely positive map. However, experimentally reconstructed processes are not necessarily completely positive due to statistical or systematic errors. In this paper, we introduce a test for discriminating statistical from systematic errors which is necessary to interpret experimentally reconstructed, non-completely positive maps.Wedemonstrate the significance of the test using several examples given by experiments and simulations. In particular, we demonstrate experimentally how an initial correlation between the system to be measured and its environment leads to an experimentally reconstructed map with negative eigenvalues. These experiments are carried out using atomic 171Yb+ ions confined in a linear Paul trap, addressed and coherently manipulated by radio frequency radiation.

It is generally assumed that every process in quantum physics can be described mathematically by a completely positive map. However, experimentally reconstructed processes are not necessarily completely positive due to statistical or systematic errors. In this paper, we introduce a test for discriminating statistical from systematic errors which is necessary to interpret experimentally reconstructed, non-completely positive maps. We demonstrate the significance of the test using several examples given by experiments and simulations. In particular, we demonstrate experimentally how an initial correlation between the system to be measured and its environment leads to an experimentally reconstructed map with negative eigenvalues. These experiments are carried out using atomic 171 Yb + ions confined in a linear Paul trap, addressed and coherently manipulated by radio frequency radiation.

I. INTRODUCTION
Quantum process tomography [1] is an important tool to experimentally verify quantum gates [2,3] and to investigate complex quantum systems [4]. Quantum process tomography is the most detailed characterization of gates, but it is very resource-intensive [5]. The effort can be reduced for matrices with low rank by using methods from compressed sensing [6,7]. Another possibility is a process certification with the help of the Monte Carlo method [8][9][10]. Here, the average output fidelity compares the experimentally realized process with the target unitary. For tomography of a quantum gate, the gate is applied to N states which are eigenstates of a random combination of local Pauli operators. For each state, the fidelity between the ideal output and the experimentally realized output is estimated. With the help of these state fidelities, the average output fidelity between the experimentally realized gate and the ideal gate can by estimated with an uncertainty which decreases as 1/N .
Another way to approximate the quantum process fidelity, suggested by H. Hofmann [11], uses two sets of mutually unbiased bases. Applying an ideal unitary quantum gate on each basis leads to an orthogonal output basis, which makes the measurement of the fidelity between the ideal output and the experimentally realized output easy. For each of the two bases, the average state fidelity is calculated, which are upper bounds of the process fidelity. A lower bound is given by the sum of both minus one. This method was used to characterize a CNOT gate realized with a four-photon six-qubit cluster state [12]. In a similar fashion, other properties of channels can also be characterized [13].
All these methods are based on the assumption that the time evolution of the system state ρ S can be described by a completely positive (CP) map E. In this case, E can be represented by Kraus operators, or by the χ-matrix which allows for the expansion of using a complete basis of operators {M m } [1]. Every completely positive map can be interpreted as the time evolution of an open quantum system where we assume unitary evolution of the enlarged initial system ρ S ⊗ ρ E . Here, ρ E describing the initial state of the environment which is assumed to be independent of the system state (see Fig. 1). However in general, when considering open quantum systems, complete positivity cannot be guaranteed [14][15][16][17]. Nevertheless, one goal of experimental quantum information is to isolate the experimental systems in such a way, that it can be considered as a closed quantum system. Therefore, the assumption that quantum channels can be represented by completely positive maps is well arXiv:1808.10336v1 [quant-ph] 30 Aug 2018 justified but has to be checked for a given experimental process.
There exist several reasons for the appearance of nonpositive maps in quantum process tomography: (i) statistical errors due to the limited number of measurements, or systematic errors such as (ii) misaligned measurements or (iii) initial correlation between the system and the environment. Such initial correlation can arise if the preparation of the system also influences the environment as indicated in Fig. 1 by red dotted lines. In the first two cases, the resulting map can be non-positive meaning that the reconstructed state ρ S may have negative eigenvalues. Or, E is positive but not complete-positive meaning that ρ S itself is positive, but the time evolution of a larger (composite) system with E acting only on one part of it leads to negative eigenvalues of the state of the total system. These types of errors arise also in quantum state tomography. On the other hand, in the case of (iii), initial correlation between the system and the environment, the resulting map will be positive but not completely positive. This effect does not arise in state tomography and is therefore a new and unique feature of process tomography.
If a non-completely positive map appears in quantum process tomography, it is important to decide whether the negativity is the result of statistical or systematic errors [18]. In the first case, one may ignore the negativity, or record more data to reduce it. However, in the second case we have to find the error and modify our experiment to improve it. Indications for systematic errors can be found by just analyzing the collected data without changing the experiment as we will demonstrate in this paper. The method introduced here is an important tool for quantum process tomography, since it gives meaningful hints about possible systematic errors at a very low cost in terms of experimental resources.
In what follows, we first discuss in Sec. II the meaning of negative eigenvalues in the case of initial correlation between the system and its environment. Then, we explain in Sec. III a plausibility check testing the probability, that the non-positivity of a reconstructed quantum process is due to statistical effects. Consecutively, we test the performance of the introduced consistency test in Sec. IV. In Sec. IV A, we first concentrate on simulations. Then, we introduce and carry out an experiment where we intentionally engineer an initial correlation between the system and the environment, each given by a single trapped ion , and apply our test to the experimentally reconstructed process (Sec. IV B). Finally, we finish this article with conclusions in Sec. V.

II. NON-COMPLETELY POSITIVE MAPS
Every linear map E : H d → H d is completely characterized by the so called Choi-matrix with |Φ + = j |jj AB . The evolved state ρ S is then given by the projection of system B onto the initial state ρ * S that is ρ S = Tr B [(1 A ⊗ ρ * S )ρ E ]. Furthermore, the linear map E is completely positive, if ρ E is positive semidefinite [19]. In this case, process tomography of a linear map E acting on a d dimensional system A can be performed with the following steps: 1. Provide a d-dimensional ancilla system B.

Prepare the state |Φ
3. Prepare the state ρ E by applying (E A ⊗ 1 B ) on the state |Φ + A,B .
4. Perform a state tomography on the resulting state ρ E by projecting the state onto the basis M k ⊗ M j .
The quantum process tomography scheme described above exhibits the experimental problem, that a physical system twice as big as the system on which the map acts needs to be available and controllable. This is often not the case. Nevertheless, the scheme for process tomography described above can also be applied without an additional ancilla system, as is outlined in the next paragraph.
The expectation value of the observableÂ ⊗B of the state |Φ + AB Φ + | is equivalent to with b j being the eigenvalues ofB and |b * j being the complex conjugate of the corresponding eigenvectors (see e.g. [10]). As a consequence, instead of performing the above described quantum process tomography scheme with ancilla system, (i) we prepare the system in different basis states M * j corresponding to getting the measurement outcome M j for the measurement on system B, (ii) apply the map E on the system and finally (iii) perform the measurement M k on the system. An important difference between the two schemes is that, in the first case, the map on system A is first applied before we define the initial state of the system by the projection of system B, and in the second case, the projection of system B is equivalent to the preparation process, which is performed before the application of the map. If the initial state of the environment is independent of the system, then the preparation/projection of system B and the map E do commute. If the preparation of the system induces correlations with the environment, then the map E may depend on the preparation process, and therefore they do not commute. In this case, the description of the time evolution of the system by E is incomplete since it does not involve the preparation process.
This incompleteness of E as description of the time evolution of ρ S can be illustrated with the following example: let's assume that the preparation of the system represented by a single qubit prepares the environment, also given by a single qubit, in exactly the same state, that is ρ E = ρ S . This is a pure classical correlation, it does not involve any quantum correlations. The time evolution of the joined system is given by withẑ andx denoting the Pauli matrices. A complete basis {M j } of the system is given by the eigenstates |0 , |1 ofẑ and |+ and |i denoting the eigenstates ofx andŷ with eigenvalue +1. As a consequence, the time evolution of these states given by completely describes the map E. However, no oracle can perform such a time evolution from knowing ρ S alone without additional information, since it is impossible to distinguish the input state with only a single copy. This time evolution is only possible with additional information given here by the copy of the state provided by the environment. In this way, the time evolution described here is not linear anymore. As a consequence, the attempt to describe this time evolution with the help of a liner map, given by the resulting Choi-matrix leads to a non-physical result given by the negative eigenvalue λ E = − √ 3/2. This does not mean that the states ρ S of the system after the time evolution are non-positive (see Eq. (5)). The map describing this time evolution is still positive. However, it is not completely positive. This means, if our system S is coupled to a second system R, then the time evolution according to E S may lead to a non-positive state ρ RS = (1 R ⊗ E S )(ρ RS ). This is due to the incomplete description given by E of our system. Without the exact definition of the preparation process of ρ RS and its effects on the environment, we cannot predict the time evolution of the composite system.
Let us assume, for example, that the environmental qubit is only affected by single qubit rotations acting on the system qubit, but not by the interaction between the systems S and R. Then, a state such as (|00 RS + |11 RS )/ √ 2 can be prepared with a CNOT-gate with S being either the control-or the target qubit leading to two different initial states As a consequence, the description of the time evolution of the system by E is incomplete and may lead to nonphysical predictions expressed by the non-complete positivity of the map E.

III. CONSISTENCY TEST
As discussed in the introduction, there exist different reasons for the appearance of negative eigenvalues in experimentally reconstructed processes. In what follows, we describe a method that tests if the observed negativity might be the result of statistical effects, or if the assumed model underlying the reconstruction process should be revisited. Our method for detecting systematic errors in quantum process tomography is based on a witness test, similar to an entanglement witness, and based on certification of experimental errors in state tomography [18,20,21]. That is, we construct an observable Z w = |λ λ|, called witness, which is positive semidefinite for the assumed model. Therefore, the appearance of negative mean values with sufficient significance indicates an inconsistency with the assumed model. The significance can be tested with the help of the Hoeffding inequality [22]. Now, we first analyse quantum process tomography with the help of an ancilla system, before we describe our consistency test for quantum process tomography without ancilla system.
The expectation value Z w = λ|ρ E |λ given by the projection of ρ E on an arbitrary state λ must be pos- The basis {M k } of the operator space of H A is chosen by projectors of different measurement settings and their measurement outcomes. Measurement outcomes of the same measurement setting are not independent of each other. Therefore, we relabel the basis {M k } by M s k where s denotes the different settings and k the outcomes (see e.g. App. A). As a consequence, M s k ⊗ M r j represents a complete basis of H A ⊗ H B with r, j labeling the settings and outcomes of system B. Furthermore, we assume that each measurement setting (r, s) is used N RS times. As a result, the witness Z w can be expand by With the help of these expansion, we are able to determine the expectation values where f r,s j,k denotes the observed frequencies to get the result (j, k) for the measurement setting (r, s). If these frequencies are the result of a quantum model, then the probability P to get a negative expectation value which follows from the Hoeffding inequality [22]. Here w r,s max and w r,s min denote the maximal and minimal expansion coefficients for the measurement setting (r, s). If this probability is very low and lies below a predefined threshold α, then the consistency test fails. In this case, the assumed model is very unlikely and the experiment should be revisited. In summary, the consistency test consists of three steps: • Choose a witness Z w = |λ λ| (see Sec. IV).
• Expand Z w into the basis M r j ⊗ M s k . • If Z w > 0 then determine the probability P and compared it to the predefined threshold.
In the case of process tomography without ancilla system, the expectation value of the witness Z w is given in a similar way by Here, the system was prepared in the state M r * j , evolved in time, and measured in the basis M s k . The witness Z w determined by the coefficients w r,s j,k stays the same. Only the way in which the frequencies f are evaluated is different. Therefore, the consistency test for process tomography stays always the same no matter how we perform the process tomography.
The witness Z w depends on the map E. However, it is important not to use the same data to determine the witness Z w and to perform the witness test. If we scan a large set of data for any correlation, we will always find a correlation with high significance due to statistical fluctuations, see e.g. [23]. Therefore, the witness Z w should be determined by a different set of data or by testing theoretically assumed errors as we will demonstrate in the next section.

IV. EXAMPLES
To investigate the potential of the scheme described above for discriminating between statistical and systematic errors, we simulate and experimentally perform process tomography of several single-qubit quantum channels.
For each simulation/ experiment, we first prepare N RS of each of the states M r * j followed by a measurement described by M s k . Then, we reconstruct the state ρ E (see Appendix A). To determine the best witness Z w we use different methods. For the simulations, we divide the data set into two parts. The first part is used to determine Z w , with the second part we perform the consistency test. Another option is to guess the underlying error. In this case, a Choi-matrix ρ theo E including the assumed error can be theoretically calculated. With the help of ρ theo E the witness Z w can then be predicted. We have used this procedure to test our experimentally generated data.

A. Simulation
If the state ρ (1) E possesses negative eigenvalues, then the best witness Z w = |λ λ| is given by the eigenstate |λ min of ρ (1) E with the most negative eigenvalue λ. We determine the coefficient w r,s j,k by representing Z w as a sum over all M r j ⊗ M s k (see Eq. (9)), and evaluate C = r,s (w r,s max − w r,s min ) 2 . Afterwards, a second round with N RS preparations and measurements of each setting is performed, which leads to ρ In the following, we simulate the process tomography of three different processes: (i) a perfect process tomography with only statistical errors, (ii) a process tomography with wrong preparation and measurement directions (iii) a process tomography with initial correlation between the system and the environment.
For the first two cases, the time evolution is given by In the second case, we assume an experiment with trapped ions. Here, the preparations and measurements in x-and y-directions are performed by applying additional π/2 rotations around the y− or x− axis to the ions, followed by a measurement in the z-direction. If the RF field , used for the π/2 rotations, is detuned by δ from the qubit resonance, then the measurement directions are not perfect anymore. The detuning will lead to a different rotation angle θ = θ · √ Ω 2 + δ 2 /Ω, with Ω being the Rabi frequency, and the rotation axis n will be tilted towards the z-axis with n e z = δ/ √ Ω 2 + δ 2 . In the third case, we assume that a second ion is sitting in the trap playing the role of the environment. We simulate a strong initial correlation between system and environment by preparing the environment in the same state as the system. The time evolution is given by Eq. (4). We simulated the above described cases with the help of MATLAB. In Fig. 2-5 we summarize the percentage of process tomographies with Prob[w·f < −t] < 0.01 as well as the average negative expectation value t = Tr [Z w ρ E ] for 10 4 simulated tomographies. Each tomography corresponds to the reconstruction of the process matrix ρ E from 12 different measurement settings, each repeated N RS times.
If only (i) statistical errors are present, the percentage of discarded tomographies stays the same independent of the number of measurements per setting as shown in Fig. 2. The number of discarded events depends only on the threshold Prob[w · f < −t] < α determing when the measurement is discarded. It does not depend on the average negative mean value |t| which decreases with N RS .
However, if also systematic errors are present, such as in case (ii) and (iii), we detect them more likely the more measurements we perform. For example about N RS = 250 measurements per setting are necessary to detect a detuning of δ/Ω=0.25 reliably as demonstrated in Fig. 3. For this case, the average negative value |t| increases for small N RS until it reaches its true value |t| = |λ min | ≈ 0.56. This effect results from statistical fluctuations of the direction of the eigenstate |λ min for small N RS . In general, the mean negative value |t| and the number of measurements per setting N RS necessary to reliably detect a systematic error depend on the magnitude of the systematic error. The larger the systematic error, e.g. the detuning δ in case (ii), the fewer measurements we need to detect it as displayed in Fig. 4.
The behavior of our consistency test in the presence of (iii) initial correlations between system and environment, as shown in Fig. 5, is similar to the behavior for case (ii) since both errors are systematic errors. However, for the example in case (iii) we get higher discarding rates than in case (ii) since the minimal eigenvalue λ min ≈ −0.87 for our example in case (iii) is smaller than the one for case (ii).

B. Experimental results
In the following we describe the experimental implementation of a process tomography with initial correlation between the system and its environment. Both the system and the environment are represented by a single qubit, each realized by a single trapped 171 Yb + ion. They form a Coulomb crystal exposed to a static magnetic field gradient of 19 T/m in a linear Paul trap with an axial trap frequency of 2π × 120 kHz and a radial trap frequency of 2π × 590 kHz. The state |0 is represented by the energy level | 2 S 1/2 , F = 0 and |1 by | 2 S 1/2 , F = 1, m F = +1 [24][25][26].
The vibrational excitation is reduced by Doppler cooling followed by RF sideband cooling and is characterized by the mean vibrational quantum number of the center-of-mass mode n < 15 [27]. Then, the qubits are initialized in the state |0 by optical pumping. Singlequbit rotations with the help of RF pulses near 12.6 GHz corresponding to the respective qubit transitions are performed to prepare the system qubit and the environmental qubit in the same desired initial states given by |0 , |1 , |+ , and |i . The time evolution (see Eq. (4)) is realized with the help of MAgnetic Gradient Induced Coupling (MAGIC) [24,26]. The evolution time takes 5.8 ms matching the J-coupling between 2 ions. The qubit dephasing is protected by dynamical decoupling (DD) pulses applied to both qubits using the Universally Robust (UR) DD sequence [28]. These DD-pulses are applied during the evolution time (for more epxeri-mental details see Appendix B). Finally, the measurement on the system qubit is performed in different bases (σ x , σ y , σ z ) with the help of single qubit rotations and by detecting state selectively scattered resonance fluorescence using an electron multiplying charge coupled device (EMCCD). Detailed information about the experimental setup is available elsewhere [24][25][26].
Each preparation and measurement setting was repeated N RS = 394. The resulting experimental reconstructed Choi-matrix is given by with statistical error of ∆ρ j,k = ±0.025 and a minimal eigenvalue of λ exp = −0.70. We used the eigenstate |λ theo corresponding to the eigenvalue λ theo = − √ 3/2 ≈ −0.87 of theoretically predicted Choi-matrix Eq. (6) to determine the witness Z w = |λ theo λ theo |. The resulting expectation value Tr [Z w ρ (exp) E ] = −0.67 is with a probability P < 4 · 10 −20 the result of purely statistic effects. As a consequence, our consistency test revealed the error of the experimentally realized process with the help of the theoretically predicted witness Z w . On the other hand, the theoretically predicted witness for just simple detuning of δ = 0.25Ω, 0.5Ω or Ω does not reveal any inconsistencies.
In general, our consistency test only makes a statement about whether the assumed model is consistent with the observed data, and whether the negativity we observe is severe or not. In this sense, it can only falsify a model, but never verify it. The test itself, especially if the witness is reconstructed via a first set of data, makes no statement about the systematic error itself. To receive information about the sort of error, we have to study the influence of different possible error sources on the data. Here, it is also helpful to not only have a look on the Choi-matrix itself, but also on the reconstruction of the time evolution of test states ρ j which can be received from the same data.
In App. C we summarize the reconstructed states ρ j for the experimental data as well as for other assumed errors such as detuned RF pulses applied to the qubits. Detuned pulses lead to wrong preparation of the inital test states ρ j , a different time evolution, and wrong measurement directions. Wrong preparation and measurement directions can lead to negative eigenvalues of the Choi-matrix as well as negative eigenvalues of the reconstructed states. On the other hand, initial correlation between the system and the environment leads only to negative eigenvalues of the Choi-matrix.
Other errors, such as detection errors, dephasing, and spontaneous decay change the time evolution but will lead to a proper Choi-matrix with positive eigenvalues. However, they can explain the difference between the theoretically predicted Choi-matrix for our specially designed correlation and the observed experimental data. The purity of the reconstructed state ρ j with j = 1, 2, 4 is very high (see App. C). Therefore, we assume that dephasing and decay do not play an important role in our experiment.
A detection error ε will shift extreme expectation values such as ẑ = ±1 towards the average ẑ = 0. If the detection errors for the two eigenstates are different, the average ẑ = 0 will be additionally shifted towards the measurement value with smaller error. These are the so-called dark states |0 , |− and | − i in our case which lead to reduced mean values. However, this behavior can only be observed in some of our measurements whereas the mean values are shifted sometimes also in the other directions (see App. C). This could be the result of stray light from the fluorescence laser which leads to population trapping in the states | 2 S 1/2 , F = 1, m F = 0 and | 2 S 1/2 , F = 1, m F = −1 . This leads together with DD to increased mean values.
The overall effect of these three possible errors (detuned pulses, asymmetric detection error and stray light) on the time evolution of the test states can be seen in App. C and fits very well the experimental data.
Another error source in process tomography are drifts. These errors can be treated in the same way as in state tomography, e.g. they can be decreased by randomly switching between different measurements setups. Another method is to describe the observed data with the help of additional parameters and use the Akaike Information Criterion to judge if this model leads to a better description of the system [29,30].

V. CONCLUSION
In this paper, we discuss and demonstrate, via experiments and simulations, how non-completely positive maps can appear in quantum process tomography. Furthermore, we introduced a simple tool for data analysis to discriminate between statistical and systematic errors. Using this tool, initial correlations between the system and its environment are experimentally detected with less than 400 repetitions for each measurement setup. Furthermore, the witness constructed specifically to identify systematic errors in the preparation and measurement process (caused by detuned RF pulses) did not detect errors caused by initial correlations. This suggests that the witness test can not only discriminate between systematic and statistical errors but also between preparation/measurement errors and correlations. However, further studies on the different influence of these different errors on the Choi matrix need to be carried out to confirm this conjecture. The consistency tests introduced here for data collected in the course of quantum process tomography can be carried out with small additional effort compared to collecting the experimental data and bring significant insights.
In general, our consistency test can not only be applied to full process tomographies, but also on incomplete measurements. In this case, our test will be sensitive solely to some systematic errors. In this way, possible test states and measurements might be determined with the help of this consistency test via a first full process tomography to identify possible problems of an experiment such as drifting laser-or radio frequencies. Later on, only the determined test state needs to prepared and appropriate measurements need to be performed to observe the thus identified problem and to appropriately counteract while carrying out experiments.
with |+ = (|0 + |1 )/ √ 2 and |i = (|0 + i|1 )/ √ 2. The reconstruction of the state ρ E is given by with the probabilities p j,k = Tr [M r j ⊗ M s k ρ E ] and the Dual-basis The experimental sequence is shown in Tab. I. The system qubit and the environmental qubit are initialized in the state |00 . Then, the states |0 , |+ , |i , and |1 are prepared in step 0 by single qubit rotations given by Step 1 and the conditional evolution perform a controlled-phase gate, where the environmental qubit is the control qubit and the system qubit is the target qubit.
Step 2 to 22 describe the conditional evolution together with the pulses for dynamical decoupling (DD).
Here, J j,k describes the coupling between ion j and ion k. For our experiment, we used N p = 100 pulses for DD, which means we repeated step 2 to 22 for 10 times. We used a total conditional evolution time τ = π/(2J j,k ) = 5.8 ms.
Step 23 describes the rotation of the system qubit necessary to perform spin measurements in x, y or z direction.
Reconstruction of the time evolution from the experimental data:   (1), (2), and (12) indicate that an operation is applied to the system, to the environment, or to both qubits, respectively. I represents the identity operator. Each single qubit rotation or RF pulse is specified by a pulse area and phase given within parentheses.
To get a similar negativity of ρ E solely by detuning without initial correlations (as an example we set ρ E = |0 0|) we assume a detuning of δ = 0.4Ω leading to λ − = −0.85.The detuning influences the preparation, time evolution and the measurement directions leading to: As can be seen, the behavior for solely detuning is quite different from that resulting from initial correlation. The main difference is that now all states stay nearly pure during the time evolution and that the reconstructed states themselves may have negative eigenvalues. Furthermore, the negativity is not detected by the witness used for initial correlations. The difference between the theoretically predicted Choi-matrix and the experimentally reconstructed Choimatrix can be the result of different errors such as (i) asymmetric detection error for the bright and the dark state, (ii) stray light shelving population from the |S 1,2 , F = 1, m f = +1 to the states |S 1,2 , F = 1, m f = 0 or |S 1,2 , F = 1, m f = −1 , or (iii) small detuning.
Detection errors shift the extremal expectation values σ = ±1 towards the average σ = 0. Asymmetric errors also shift the zero-point of the expectation value towards the direction of smaller error. Typical errors in our experiments are ε B = 0.06 for the bright state (corresponding to |1 , |+ , |i ) and ε D = 0.03 for the dark state. The matrix entries are directly proportional to the expectation values ρ 11 ∼ σ z , Re (ρ 0,1 ) ∼ σ x and Im (ρ 0,1 ) ∼ −σ y . This leads to the theoretically predicted reconstructed states Stray light would lead to increased expectation values. The increase depends on the population in state |1 averaged over time. Here, we consider mainly the free evolution time, because the time for single qubit rotations (order of 10 µs) is very small in comparison. The states ρ j with 1 ≤ j ≤ 3 are always in the x-y-plane during the conditional evolution time and thereforep(|1 ) = 0.5. The state ρ 4 spends, due to the dynamical decoupling pulses, half of the time in |0 and half of the time in |1 . Therefore, we find for this state alsop(|1 ) = 0.5. As a consequence, the effect of stray light is the same for all 4 input states. An assumed population transfer of 5% would lead to the following time evolutions: A detuning of δ = 0.1Ω (influencing the preparation, time evolution, the dynamical decoupling and the measurement directions) together with the initial correlation of the system and its environment would lead to: A comparison with the experimentally reconstructed states Eq. (C13)-(C16) show remarkable similarities with different assumed errors for different states and measurements. However, this is not surprising since the experimental parameter such as the detuning or the intensity of the laser light used for state selective detection may fluctuate during a sequence of measurements. Therefore, not all errors are always present.