4.3 From clicks to density matrices
4.3.3 Evaluation of relevant values
With the density operator obtained from quantum state tomography, any information we seek to know about a quantum state can be determined. Yet, it requires, in particular for multipartite states, a big experimental effort to measure all corresponding correlations. Thus, when we are interested only in certain informations, we might be able to do with only a restricted set of measurements.
Furthermore, even if the complete set of correlations is necessary, it is advantageous to determine values directly from the measurement data because the error calculation is simplified. This is particularily important, as the measured data reproduces a physical state only within the statistical errors.
In the following, the derivation of characteristic values of a quantum state will be presented. We discuss the evaluation of observables, then of entanglement measures.
Observables
The first experiments on entangled quantum states (at least with the explicit goal to study entanglement) were Bell tests. Here, one determines a set of correlations and calculates whether the experimental results were in agreement with the predictions from classical (better: local hidden variable) theories, or whether they exceeded the predicted values in accordance with quantum mechanics. Complete knowledge about the state is not necessary.
In a similar way, the information whether a state is, for example, separable or symmetric does not require complete knowledge of a quantum state. Yet, usually we cannot measure the expectation values of the corresponding observables ˆO (witness or swap operators, respecively) directly, as they are in general non-local. In our experiments, we can only measure local observables, i.e. correlations of the type KB~ = hOˆB~i. One can, however, express ˆO as linear combination of local observables. The expectation value of ˆO can then be calculated from the different correlations due to the linearity of the trace. As this decomposition is not unique, we seek to find a minimal local decomposition, i.e. a decomposition into local observables such that as few measurement settings as possible are needed to evaluate ˆO.
Most of the entanglement witnesses we discuss are already designed such that they consist out of view measurement settings. Also the linear decomposition of the swap operator is given in equation 2.32. It may be a little unexpected that also the fidelity of an observed state relative to the desired one can be determined with reduced measurement settings. It is given by the expectation value of the projector on the expected pure state
|ψi:
Fρψ =T r[|ψihψ|ρ] =hψ|ρ|ψi (4.24) Note, here, the close relation to the generic entanglement witness:
T r[Wψρ] =T r[(α11− |ψihψ|)ρ] =α−Fρψ (4.25) Minimal local decompositions of generic entanglement witnesses have been studied in [166].
In particular the fidelity to graph states is, due to their special structure, easy to determine from 2N correlations forN qubits. This will be demonstrated on the example
of the four-qubit cluster state. The stabilizing group consists of 16 elements (which are summarized in table 6.1). All other standard bases have zero correlations. Using the decomposition of the density operator in terms of the standard bases we get:
FC4 ρ =h C4|ρ| C4i = 3 X i,j,k,l=0 Kijklh C4|σˆi⊗σˆj⊗σˆk⊗σˆl| C4i = 16 X m=1 Kmh C4|Sˆm| C4i (4.26)
whereSm are the stabilizing operators and Km are the corresponding measured correla- tions. Then, if the operators’ signs are chosen such thath C4|Sˆm| C4i= 1, the fidelity is simply the average over these 16 correlations. In dependence on their explicit form the number of measurement settings might be even smaller – in our example 9 settings are necessary. Compared to a state tomography with 81 settings this is a drastic reduction of the effort. For graph states the problem is particularly simplified, but also for other quantum states a strong reduction of settings is possible. For example, all states of the first SLOCC-family of four-qubit entanglement (section 3.1), including the state D4(2), have at maximum 40 non-zero correlations, where at maximum 21 measurement settings are necessary to evaluate the fidelity.
For the estimation of the errors we chose the straightforward approach: As the different correlations partly stem from the same measurement setting, the expectation value of the measured observable is expressed as linear combination of the different relative frequencies instead of correlations. The errors on these are due to Poissonian counting statistics plus errors on the detection efficiencies of the APDs. The total error is then calculated by gaussian error propagation.
Entanglement measures
Entanglement measures like the concurrence, the entanglement of formation, the neg- ativity and the geometric measure of entanglement, which were introduced in the first chapter, are not linearly dependent on the density matrix. Therefore, we need (to our best knowledge) the complete density matrix of the state for their evaluation, in contrast to the evaluation of the entanglement witnesses, symmetry etc. introduced before. For a lower bound on the geometric measure of entanglement we use a smaller set of data, as demonstrated in the first chapter.
The concurrence and negativity of two-qubit states, however, will be calculated explic- itly according to equation 2.45 and equation 2.49 (pages 18 and 19). With the density matrix obtained with the method demonstrated in section 4.3.2 these values can easily be obtained. As mentioned before, these matrices are not physical. Yet, we expect the correct physical matrix to be within the error bounds of the data. The estimation of the error on the entanglement measures is, however, not as straightforward as in the last section and will be outlined shortly in the following.
The error on the concurrence can be estimated via perturbation theory of non-hermitian matrices as presented in James et al. [163]. The decomposition of the density matrix they used can easily be substituted with the decomposition in equation 4.23.
4.3 From clicks to density matrices
The negativity can be calculated by using the measurement data directly via a slight modification of the presented tomography. To do so, the partial transpose with regard to one subsystem (here the second) is directly applied on our decomposition (equation 4.22):
ρP Texp = X3 k,l=1 +1 X i,j=0 ci,jk,l ³ γki ⊗γlj ´ P T = 3 X k,l=1 +1 X i,j=0 ci,jk,l ³ γki ⊗(γlj)T ´ . (4.27)
Thus, simply transposing theγ-matrices of one qubit allows the calculation of the partially transposed density matrices, where the coefficients given by the measurement data are left unchanged. This can, naturally, be also applied to multiqubit density matrices, where several qubits can be transposed this way in equation 4.23. To obtain the negativity, the eigenvalues of this density matrix are calculated. The reduction of the problem to the calculation of eigenvalues reduces the error calculation to a known problem. In [163] the calculation of the errors on the eigenvalues of a density matrix derived from tomography via perturbation theory is presented.
In this section, the determination of all values that are calculated in the experimental analyses in chapters 5, 6 and 7 were completed. The calculation of all values is reduced directly to measurement data. To do so even for entanglement measures, which are no observables, we rely on a new kind of tomographic decomposition that is suited to our strategy of state analysis (equation 4.23). In the following, the tomography of a quantum process will be demonstrated.