A general framework for analyzing the topology of quantum channels of single-particle systems is developed to find a class of genuinely dynamical topological phases that can be realized by means of discrete quantum feedback control. We provide a symmetry classification of quantum channels by identifying ten symmetry classes of discrete quantum feedback control with projective measurements. We construct various types of topological feedback control by using topological Maxwell demons that achieve robust feedback-controlled chiral or helical transport against noise and decoherence. Topological feedback control thus offers a versatile tool for creating and controlling nonequilibrium topological phases in open quantum systems that are distinct from non-Hermitian and Lindbladian systems and should provide a guiding principle for topology-based design of quantum feedback control.
A general framework for analyzing the topology of quantum channels of single-particle systems is developed to find a class of genuinely dynamical topological phases that can be realized by means of discrete quantum feedback control. We provide a symmetry classification of quantum channels by identifying ten symmetry classes of discrete quantum feedback control with projective measurements. We construct various types of topological feedback control by using topological Maxwell demons that achieve robust feedback-controlled chiral or helical transport against noise and decoherence. Topological feedback control thus offers a versatile tool for creating and controlling nonequilibrium topological phases in open quantum systems that are distinct from non-Hermitian and Lindbladian systems and should provide a guiding principle for topology-based design of quantum feedback control.
We establish a symmetry classification for a general class of quantum feedback control. For successive feedback control with a nonadaptive sequence of bare measurements (i.e., with positive Kraus operators), we prove that the symmetry classification collapses to the ten-fold AZ^{†} classes, specifying the allowed topology of CPTP maps associated with feedback control. We demonstrate that a chiral Maxwell's demon with Gaussian measurement errors exhibits quantized winding numbers. Moreover, for general (nonbare) measurements, we explicitly construct a protocol that falls outside the ten-fold classification. These results broaden and clarify the principles in engineering topological aspects of quantum control robust against disorder and imperfections.
We present topological characterization of classical stochastic processes described by discrete-time Markov chains on lattices. We point out that point-gap topology of stochastic matrices entails two distinct physical consequences that hinge on the choice of the reference point. The point-gap topology around a generic reference point is related to the direction of transport, and nontrivial topology around the origin of the complex spectrum of a stochastic matrix implies non-Markovianity caused by, e.g., feedback control. On the basis of this characterization, we identify the topological origin of directed transport in a classic experiment of Maxwell's demon [S. Toyabe et al., Nat. Phys. 6, 988 (2010)] and find the topological nature of feedback control beyond thermodynamic interpretation. We demonstrate that a topologically enforced non-Markovian classical stochastic process can be simulated by a Markovian quantum master equation, indicating a topological form of quantum advantage.
The detection of topological phases of matter has become a central issue in recent years. Conventionally, the realization of a specific topological phase in condensed matter physics relies on probing the underlying surface band dispersion or quantum transport signature of a real material, which may be imperfect or even absent. On the other hand, quantum simulation offers an alternative approach to directly measure the topological invariant on a universal quantum computer. However, experimentally demonstrating high-dimensional topological phases remains a challenge due to the technical limitations of current experimental platforms. Here, we investigate the three-dimensional topological insulators in the AIII (chiral unitary) symmetry class, which yet lack experimental realization. Using the nuclear magnetic resonance system, we experimentally demonstrate their topological properties, where a dynamical quenching approach is adopted and the dynamical bulk-boundary correspondence in the momentum space is observed. As a result, the topological invariants are measured with high precision on the band-inversion surface, exhibiting robustness to the decoherence effect. Our Letter paves the way toward the quantum simulation of topological phases of matter in higher dimensions and more complex systems through controllable quantum phases transitions.
Topological phases of matter are understood to be characterized by particular configurations of entanglement encoded within the ground states of many-body quantum systems. In this thesis, we discuss how novel phenomena associated with topological phases can arise in systems that are driven out of equilibrium or coupled to their surroundings, which are not described by ground state physics. Firstly, we consider systems undergoing unitary time evolution due to some external driving. We show how the entanglement features of the time-dependent wavefunction can be topologically characterized, and demonstrate that these topological properties are reflected in the dynamics of various quantities usually associated with topological phases in equilibrium. We introduce a new non-equilibrium topological classification scheme which can be used to predict whether or not the topological features of a given system will be preserved under time evolution. In brief, this classification captures the fact that certain symmetries (namely antiunitary symmetries) are inevitably broken once the system is driven out of equilibrium; thus any topological phenomena protected by such symmetries cannot be expected to persist in non-equilibrium scenarios. Secondly, we investigate how topologically non-trivial systems are affected by dissipative effects, i.e.~coupling to an external environment. We demonstrate that system-environment interactions facilitate processes in the system that effectively break any antiunitary symmetries, regardless of the symmetries of the microscopic Hamiltonian. Accordingly, those phases that were shown to be unstable against time-dependent driving are also fragile against coupling to an environment. To illustrate the consequences of this fragility, we consider the effects of dissipation on the coherence properties of topological bound states, as well as the conductance properties of chiral and helical topological edge modes. We find that the decoherence rate of the former and the deviation from quantized conductivity of the latter are (not) exponentially suppressed in the inverse temperature when the phase in question is protected by unitary (antiunitary) symmetries. These results regarding open systems can be connected to the same non-equilibrium classification developed in the context of isolated systems undergoing unitary dynamics. Our findings highlight a distinction between topological phases that are either robust or fragile against non-equilibrium effects and/or system-environment coupling. The ramifications for the use of such systems in quantum technologies are discussed.
The interplay between unitary dynamics and quantum measurements induces diverse phenomena in open quantum systems with no counterparts in closed quantum systems at equilibrium. Here, we generally classify Kraus operators and their effective non-Hermitian dynamical generators, thereby establishing the tenfold classification for symmetry and topology of monitored free fermions. Our classification elucidates the role of topology in measurement-induced phase transitions and identifies potential topological terms in the corresponding nonlinear sigma models. Furthermore, we establish the bulk-boundary correspondence in monitored quantum dynamics: nontrivial topology in spacetime manifests itself as topologically nontrivial steady states and gapless boundary states in Lyapunov spectra, such as Lyapunov zero modes and chiral edge modes, leading to the topologically protected slowdown of dynamical purification.
The emergence of topological entanglement, a fundamental phenomenon in condensed matter physics, is often overlooked in the context of complex, non-equilibrium systems. This research proposes a novel framework, "Phase-Steering," to quantify and control the dynamics of topological entanglement within such systems. We develop a continuous, dynamically evolving algorithm that analyzes real-time feedback to actively manipulate the topological structure of entanglement, offering a fundamentally new approach to understanding and potentially engineering topological phases. The core claim is that this method can facilitate the transition between distinct topological phases and provides a pathway for designing systems exhibiting enhanced robustness and adaptability. The work presented here details the algorithm's implementation, provides preliminary results demonstrating its effectiveness in simulating a specific non-equilibrium system, and discusses potential applications in materials science and quantum computing.