Figure 18 – uploaded by Lea Lautenbacher

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Figure 18 — Plot of the smallest eigenvalues A of the Choi matrix of each one of the intermediate depolarizing maps Ay, and Ag?y"°”.

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Figure 1 — State of a qubit in the Bloch sphere, parametrized in terms of spherical coordinates. 2.3. BLOCH SPHI Led es

Figure 2 — Schematic representation of an open quantum system.

It is easy to identify the Kraus operators

igure 5 — Schematic representation of the total evolution including the Petz recovery map. A initial state po is taken to a evolved state p, = A(po). The Petz is then applied to the evolved state A Petz (pt):

(0 symmetry properties of the mentioned quantum channels, the azhimutal angle was ixed, without loss of generality, to ¢ = 0. Following the r axis in Fig. 8 a) the value of (po, \petz(pz)) begins in its maximum and decreases as long as pp reaches pure states r = 1) for all the three quantum channels. In Fig. 8 b) we observe another view of the same image where it is possible to note that the initial state is better recovered for the lephasing channel in comparison to the other evolutions, since the fidelity maintains itself nore stable at its maximum. As already mentioned, the dephasing channel presents higher values for the fidelity compared to depolarizing and amplitud ed amping, which have values so close to each other, in a manner that the channels cannot be distinguished in Fig. 8, since the curves are overlapped. Now the question is what dit ffers the dephasing from the other evolutions? Based on what was developed in this work, the answer is quite simple to understand. The image set of this quantum channel, at t Ne imit of non-bijectivity, corresponds to a line that contains an infinite number of states that can be the best state on average. From another perspective, after a totally dephasing evolution we lose less information than for a totally depolarizing one, for example, where all the initial states converge to the maximally mixed state. The key point is that, the harder it is to retrieve information from a channel, the more non-invertible it can be classified.

Figure 9 — The final state is simply the state after the evolution pr = A(po) = ZA(po0)

Figure 10 — Probability distributions characterizing a non-Markovian dephasing evolution. rewritten in terms of a new probability distribution defined as

Figure 11 — Plot of the smallest eigenvalues of each one of the intermediate dephasing maps Ag; and approx Nott .

Figure 12 — Distinguishability between two states Dz,-(1, 92) over time for dephasing. With p; = |+)(+| and p2 = |—)(—| for both cases.

Figure 13 — Distance between the Choi matrices over time for both cases for a dephasing channel. Another issue to be addressed here is the similarity between the original NM dynamics and the new proposed dynamics. To do so, we use the formalism presented in Section 6.4. We observed that ||A5?9°°" — Astollo = ||J(Asvo°”) — J(Azeo)|[1- Then, for simplicity, to avoid the maximization required to compute the diamond norm, we opted to investigate the trace distance between the Choi matrices shown in Fig. 13 for both cases 1 and 2. For this calculation we still used the Fano algorithm; however, we applied the map only to one of the Bell states.

“igure 14 — Plot of the smallest eigenvalues \ of each one of the intermediate dephasing maps A», and approx Mont .

Figure 15 — Plot of the smallest eigenvalues of the Choi matrix of the intermediary maps A4,, and approx Nant 2

Figure 16 — Choi distances for both cases with final time fixed. In a) tp = 27 and b) tp = 4 For Case 2 the distance gets smaller when we fix the final time ty, and even smallet or t >> 27. This can be easily understood if we look at the probabilities in Fig. 10. Note hat pe? "(t) slightly oscillates and then starts to stabilize at its maximum value equal tc . What happens to the map in this time frame? Basically it is a maximum dephasing hat comes back and forth due to the non-Markovianity effects, represented here by these scillations. When we fix the time, it is as if we limited the dynamics to stabilize more uickly, decreasing these oscillations. The point is that now, as already mentioned, the igenvalues of the original dynamics are less positive, which makes it difficult for a CF 1ap to perfectly simulate a NCP one. For this reason, although the distance has now ecreased and does not fluctuate, it will never reaches zero, unlike what we get with a on-fixed time. For even bigger times, we have reached the regime where the dynamics 4n,t 18 even more stable as can be verified in Fig. 10. For this reason, the distance is ever

Figure 17 — Probability distributions characterizing a non-Markovian depolarizing evolution. From this point on, we maintain the classification used in the last section. We use Case 1 and Case 2 for pt?" (t) and pi?" (t), respectively. 7.3.1 Final time free in 2t

initial states p,; and p2 as in the previous section.

Figure 20 — Distance between the Choi matrices over time for both cases for a depolarizing channel. Regarding the distance between the dynamics shown in Fig. 20, it is possible to identify right away that the distance ||J(A3/o°") — J(Azzo)||1 is larger than for the dephasing channel. For Case 1, we now have the maximum value of the distance equal to 1.5. This agrees with the results presented in Chapter 5, since the Petz recovery map is in fact able to better recover states from the dephasing evolution, something justified here by the degree of non-invertibility of the channels. For Case 2, the values are also larger than for the previous quantum channel, what confirms that the maps are now a little bit more different. Basically the Petz is a little bit less effective for this channel. Apart from the numerical values, the shapes of the curves for both cases are equal compared to Fig. 13. Of course, something that is expected since we maintained the behavior of the probability distribution functions.

Figure 21 — Probability distributions characterizing a non-Markovian amplitude damping evolution. reaches its maximal value is more accentuated and less like a plateau.

$Figure 22 — Plot of the eigenvalues \ of the Choi matrix of the intermediary amplitude damping map Ages and AsPt'°*. We plotted two eigenvalues of J(AS?4"°"). For Case 1, we observe the negative eigenvalue with minimal point when t = 7. This point corresponds to the limit of non-bijectivity, when pt? = 1, as can be seen in Fig. 21. There are also time intervals, for t € [0,2.1] and again for t € [4.2,6] where Ao: is positive. When we look for Case 2, we confirm one more time what was mentioned before about the narrowing of the valleys on the negative semi-axis. This is explained due to the behavior of per "(t) which is stabilizing for t > 27. Until now, there is no brand new information about the dynamics.$

Figure 22 — Plot of the eigenvalues \ of the Choi matrix of the intermediary amplitude damping map Ages and AsPt'°*. We plotted two eigenvalues of J(AS?4"°"). For Case 1, we observe the negative eigenvalue with minimal point when t = 7. This point corresponds to the limit of non-bijectivity, when pt? = 1, as can be seen in Fig. 21. There are also time intervals, for t € [0,2.1] and again for t € [4.2,6] where Ao: is positive. When we look for Case 2, we confirm one more time what was mentioned before about the narrowing of the valleys on the negative semi-axis. This is explained due to the behavior of per "(t) which is stabilizing for t > 27. Until now, there is no brand new information about the dynamics.

Figure 18 – uploaded by Lea Lautenbacher

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