Consider an open quantum system with (discrete-time) Markovian dynamics. Our task is to store information in the system in such a way that it can be retrieved perfectly, even after the system is left to evolve for an arbitrarily long time. We show that this is impossible for classical (resp. quantum) information precisely when the dynamics is mixing (resp. asymptotically entanglement breaking). Furthermore, we provide tight universal upper bounds on the minimum time after which any such dynamics 'scrambles' the encoded information beyond the point of perfect retrieval. On the other hand, for dynamics that are not of this kind, we show that information must be encoded inside the peripheral space associated with the dynamics in order for it to be perfectly recoverable at any time in the future. This allows us to derive explicit formulas for the maximum amount of information that can be protected from noise in terms of the structure of the peripheral space of the dynamics.
Consider an open quantum system with (discrete-time) Markovian dynamics. Our task is to store information in the system in such a way that it can be retrieved perfectly, even after the system is left to evolve for an arbitrarily long time. We show that this is impossible for classical (resp. quantum) information precisely when the dynamics is mixing (resp. asymptotically entanglement breaking). Furthermore, we provide tight universal upper bounds on the minimum time after which any such dynamics 'scrambles' the encoded information beyond the point of perfect retrieval. On the other hand, for dynamics that are not of this kind, we show that information must be encoded inside the peripheral space associated with the dynamics in order for it to be perfectly recoverable at any time in the future. This allows us to derive explicit formulas for the maximum amount of information that can be protected from noise in terms of the structure of the peripheral space of the dynamics.
We consider an open quantum system undergoing Markovian dynamics, the latter being modelled by a discrete-time quantum Markov semigroup (Φn)n∈N, resulting from the action of sequential uses of a quantum channel Φ, with n∈N being the discrete time parameter. We find upper and lower bounds on the one-shot ε-error information transmission capacities of Φn for a finite time n∈N and ε∈[0,1) in terms of the structure of the peripheral space of the channel Φ. We consider transmission of (i) classical information (both in the unassisted and entanglement-assisted settings); (ii) quantum information and (iii) private classical information.
We study the information transmission capacities of quantum Markov semigroups acting on finite-dimensional quantum systems. We show that, in the limit of infinite time, the capacities can be efficiently computed in terms of the structure of the peripheral space of the semigroup, are strongly additive, and satisfy the strong converse property. We also establish convergence bounds to show that the infinite-time capacities are reached after time scaling quadratically with the system dimension. From a data storage perspective, our analysis provides tight bounds on the number of bits or qubits that can be reliably stored for long times in a quantum memory experiencing Markovian noise. From a practical standpoint, we show that typically, a quantum memory with Markovian noise acting independently and identically on all qubits and a fixed time-independent global error correction mechanism becomes useless for storage after time scaling exponentially with the number of qubits. In contrast, if the error correction is local, the memory becomes useless much more quickly, i.e., after time scaling logarithmically with the number of qubits. In the setting of point-to-point communication between two spatially separated parties, our analysis provides efficiently computable bounds on the optimal rate at which bits or qubits can be reliably transmitted via long Markovian communication channels, both in the finite block-length and asymptotic regimes.
Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system's ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We also classify distinct operational criteria for preservation (e.g., ``noiseless,'' ``unitarily correctible,'' etc.) and introduce two natural criteria for measurement-stabilized and unconditionally preserved codes. Finally, for several of these operational criteria, we present efficient (polynomial in the state-space dimension) algorithms to find all of a channel's information-preserving structures.
The distribution of entangled quantum systems among the nodes of a network is a key task at the basis of the development of quantum technologies, e.g., quantum communication, quantum computation, etc. Many efforts have been devoted to identify strategies, based on pre- and post-processing operations or decoherence-free subspaces, to prevent the deterioration of such exotic correlations. However, all these approaches loose their usefulness when the noise level affecting the system surpasses a certain minimal threshold that leads to an entanglement-breaking dynamics. Here we attack this problem in the context of discrete- and continuous-time description of the system dynamics, providing some explicit examples in the context of qubit channels.
This thesis studies the capacities of quantum channels under repeated noisy evolution, with a focus on the long-time behavior of classical and quantum information transmission. The main goal is to understand how much information can still be preserved or recovered when a quantum channel is applied many times, and to develop structural, analytic, and algorithmic tools for characterizing these capacities.The core results are presented in Chapter 2. First, the thesis investigates the infinite-time regime and shows that the asymptotic capacities of quantum channels are determined by the structure of their peripheral subspace. This leads to explicit characterizations of the infinite-time classical capacity and bounds on the infinite-time quantum capacity, together with results on additivity and extensions to more general settings. The thesis then studies the finite-time behavior for GNS-symmetric channels and proves explicit exponential convergence bounds, showing that the capacities approach their infinite-time values rapidly. In addition, algorithmic methods are developed for computing the peripheral structure and the associated capacity quantities efficiently.Chapter 3 explores a related optimization-based direction motivated by Sum-of-Squares methods and semidefinite programming. Using ideas inspired by separability and SDP hierarchies, this part investigates whether machine-learning-assisted ansatz discovery can help construct more efficient SOS certificates. Although this direction is not yet complete in the specific capacity setting, it provides promising intermediate results and suggests new computational approaches for related problems in quantum information theory. Overall, this thesis provides a structural and quantitative understanding of the long-time information-carrying capabilities of noisy quantum dynamical systems, while also opening new perspectives through optimization and machine-learning-based methods.
Given a quantum Markovian noise model, we study the maximum dimension of a classical or quantum system that can be stored for arbitrarily large time. We show that, unlike the fixed time setting, in the limit of infinite time, the classical and quantum capacities are characterized by efficiently computable properties of the peripheral spectrum of the quantum channel. In addition, the capacities are additive under tensor product, which implies in the language of Shannon theory that the one-shot and the asymptotic i.i.d. capacities are the same. We also provide an improved algorithm for computing the structure of the peripheral subspace of a quantum channel, which might be of independent interest.