The quantum world loves order, but reality often brings disorder. In this work, physicists study one-dimensional chains of spins — tiny magnetic arrows that can point up or down. Usually, topological phases (stable states protected by symmetry) are defined for pure states, where each arrow is precisely known. But if some information is lost, the system is described by a mixed state, like a blurred photograph where details are fuzzy.
Researchers use quantum channels — mathematical operations that transform states like filters that let only part of the signal through. They introduce an order parameter that, for mixed states, takes only integer values, just as the number of turns of a thread around a spool can only be an integer. This parameter changes sharply at phase transitions, signaling a topological phase transition.
Moreover, the authors show how to generate different phases via translation (shift) and reflection transformations, even in the presence of disorder. This generalizes the famous Lieb-Schultz-Mattis theorem, which previously required pure states and the presence of an energy gap. Now the theorem works for mixed states without the notion of a gap and even without a lattice Hamiltonian.
This approach resembles trying to recognize a pattern on a fogged window: even if part of the design is hidden, the overall structure remains discernible. Similarly, even when quantum information is partially lost, topological properties can be revealed and classified. The work opens a path to studying topology in open systems that interact with their environment, and that is most real quantum devices.
Topological phases in mixed states can be compared to a Möbius strip: no matter how much you squeeze or stretch it, the property of 'twistedness' will not disappear until you tear the strip.
In Greg Egan's novel 'Schild's Ladder', quantum states are used for travel between universes, and loss of information can mean catastrophe: this work is about how to maintain order even with losses.