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Register a new userCommunication Cost for Non-Markovianity of Tripartite Quantum States: A Resource Theoretic Approach
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Authors
Eyuri Wakakuwa
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To quantify non-Markovianity of tripartite quantum states from an operational viewpoint, we introduce a class $Omega^*$ of operations performed by three distant parties. A tripartite quantum state is a free state under $Omega^*$ if and only if it is a quantum Markov chain. We introduce a function of tripartite quantum states that we call the non-Markovianity of formation, and prove that it is a faithful measure of non-Markovianity, which is continuous and monotonically nonincreasing under a subclass $Omega$ of $Omega^*$. We consider a task in which the three parties generate a non-Markov state from scratch by operations in $Omega$, assisted with quantum communication from the third party to the others, which does not belong to $Omega$. We prove that the minimum cost of quantum communication required therein is asymptotically equal to the regularized non-Markovianity of formation. Based on this result, we provide a direct operational meaning to a measure of bipartite entanglement called the c-squashed entanglement.
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We establish a convex resource theory of non-Markovianity under the constraint of small time intervals within the temporal evolution. We construct the free operations, free states and a generalized bona-fide measure of non-Markovianity. The framework satisfies the basic properties of a consistent resource theory. The proposed resource quantifier is lower bounded by the optimization free Rivas-Huelga-Plenio (RHP) measure of nonMarkovianity. We further define the robustness of non-Markovianity and show that it can directly be expressed as a function of the RHP measure of non-Markovianity. This enables a physical interpretation of the RHP measure.
We investigate the conditions under which an uncontrollable background processes may be harnessed by an agent to perform a task that would otherwise be impossible within their operational framework. This situation can be understood from the perspective of resource theory: rather than harnessing useful quantum states to perform tasks, we propose a resource theory of quantum processes across multiple points in time. Uncontrollable background processes fulfil the role of resources, and a new set of objects called superprocesses, corresponding to operationally implementable control of the system undergoing the process, constitute the transformations between them. After formally introducing a framework for deriving resource theories of multi-time processes, we present a hierarchy of examples induced by restricting quantum or classical communication within the superprocess - corresponding to a client-server scenario. The resulting nine resource theories have different notions of quantum or classical memory as the determinant of their utility. Furthermore, one of these theories has a strict correspondence between non-useful processes and those that are Markovian and, therefore, could be said to be a true quantum resource theory of non-Markovianity.
The Gottesman-Kitaev-Preskill (GKP) quantum error-correcting code has emerged as a key technique in achieving fault-tolerant quantum computation using photonic systems. Whereas [Baragiola et al., Phys. Rev. Lett. 123, 200502 (2019)] showed that experimentally tractable Gaussian operations combined with preparing a GKP codeword $lvert 0rangle$ suffice to implement universal quantum computation, this implementation scheme involves a distillation of a logical magic state $lvert Hrangle$ of the GKP code, which inevitably imposes a trade-off between implementation cost and fidelity. In contrast, we propose a scheme of preparing $lvert Hrangle$ directly and combining Gaussian operations only with $lvert Hrangle$ to achieve the universality without this magic state distillation. In addition, we develop an analytical method to obtain bounds of fundamental limit on transformation between $lvert Hrangle$ and $lvert 0rangle$, finding an application of quantum resource theories to cost analysis of quantum computation with the GKP code. Our results lead to an essential reduction of required non-Gaussian resources for photonic fault-tolerant quantum computation compared to the previous scheme.
We study a quantum walker on a one-dimensional lattice with a single defect site characterized by a phase. The spread and localization of discrete-time quantum walks starting at the impurity site are affected by the appearance of bound states and their reflection symmetry. We quantify the localization in terms of an effective localization length averaged over all eigenstates and an effective participation ratio after time evolution averaged over all initial states. We observe that the reduced coin system dynamics undergoes oscillations in the long-time limit, the frequencies of which are related to the unitary sublattice operator and the bound state quasi-energy differences. The oscillations give rise to non-Markovian evolution, which we quantify using the trace distance and entanglement based measures of non-Markovianity. Indeed, we reveal that the degree of the non-Markovian behavior is closely related to the emergence of bound states due to the phase impurity. We also show that the considered measures give qualitatively different results depending on the number and symmetries of supported bound states. Finally, comparing localization and non-Markovianity measures, we demonstrate that the degree of non-Markovianity becomes maximum when the walker is most localized in position space.
One of the most important topics in the study of the dynamics of open quantum system is information exchange between system and environment. Based on the features of a back-flow information from an environment to a system, an approach is provided to detect non-Markovianity for unital dynamical maps. The method takes advantage of non-contractive property of the von Neumann entropy under completely positive and trace preserving unital maps. Accordingly, for the dynamics of a single qubit as an open quantum system, the sign of the time-derivative of the density matrix eigenvalues of the system determines the non-Markovianity of unital quantum dynamical maps. The main characteristics of the measure is to make the corresponding calculations and optimization procedure simpler.
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