## Robustness of Fixed Points of Quantum Processes

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## Abstract

The thesis combines two independent lines of research, both of which lie in the general area of the theory of robustness of fixed points (or invariant states) of quantum processes.

In the first part of the thesis, we address the following question: Given a quantum channel and a quantum state which is almost a fixed point of the channel, can we find a new channel and a new state, which are respectively close to the original ones, such that they satisfy an exact fixed point equation? This question can be asked under many interesting constraints in which the original channel and state are assumed to have certain structures which the new channel and state are supposed to satisfy as well.

We answer this question in the affirmative under fairly general assumptions on afore-mentioned structures through a compactness argument. We then concentrate on specific structures of states and channels and establish explicit bounds on the approximation errors between the original- and new states and channels respectively. We find a particularly desirable form of these approximation errors for a variety of interesting examples. These include the structure of general quantum states and general quantum channels, unitary channels, mixed unitary channels and unital channels, as well as the structure of classical states and classical channels. On the other hand, for the setup of bipartite quantum systems for which the considered channels are demanded to act locally, we are able to lower bound the possible approximation errors. Here, we show that these approximation errors necessarily scale in terms of the dimension of the quantum system in an undesirable manner.

We apply our results to the robustness question of quantum Markov chains (QMC) and establish the following: For a tripartite quantum state we show the existence of a dimension-dependent upper bound on the distance to the set of QMCs, which decays as the conditional mutual information of the state vanishes.

In the second part of the thesis we prove the so-called quantum Zeno- and strong damping limits for infinite-dimensional open quantum systems. In the former case, which we refer to as the quantum Zeno regime, the dynamics of the open quantum system is governed by a quantum dynamical semigroup, which is repeatedly and frequently interrupted by the action of a quantum operation. The quantum operation is considered to be mixing, in the sense that if applied multiple times it converges to its fixed point space. We then analyse the effective dynamics of the overall process in the limit of the application frequency of the quantum operation going to infinity. The strong damping regime can be considered as a continuous variant of the quantum Zeno regime. Here, the discrete and frequent action of the quantum operation is replaced by an additional term in the generator of the dynamical semigroup, whose individual dynamics is mixing, in the sense that it converges to its fixed point space in the infinite time limit. We analyse the overall dynamics in the limit of infinite interaction strength.

All previous proofs of quantum Zeno limits in the literature relied on an assumption given by a certain spectral condition. We give a full characterisation of quantum operations which are mixing in the uniform topology under this assumption. Then, using a novel perturbation technique, we are able to go beyond this assumption and prove quantum Zeno- and strong damping limits in an unified way, if the mixing happens in the strong sense, i.e. pointwise for a given state. Here, we see that the effective processes converge to the fixed point spaces, on which they are governed by an effective quantum Zeno dynamics. The result is quantitative and gives a bound on the speed of convergence of the quantum Zeno- and strong damping limits, given a bound on the speed of convergence of the mixing process.