Mixed-state phases induced by power-law quantum channels

In the quantum world, matter usually comes in two flavors: pure states, which are perfectly isolated and predictable, and mixed states, which are messy, noisy, and constantly interacting with their surroundings. For decades, physicists have mapped out the distinct phases of pure matter, finding that even in a chaotic environment, certain patterns of order can emerge and persist. However, when a system is open to the environment, the rules change. A new kind of order has recently been proposed, one that exists only in these messy, mixed states and has no counterpart in the clean, isolated world. This phenomenon involves a specific type of symmetry breaking where a system loses a strong, rigid form of order but retains a weaker, averaged version of it. While scientists have seen this happen in systems with two or more dimensions, a long-standing belief held that in a one-dimensional line, such a transition could never occur if the noise was limited to nearby neighbors. The idea was that the noise would simply wash out any complex order before it could take hold.

A team of researchers has now challenged this limitation by introducing a new kind of noise that reaches across the entire system. Instead of just whispering to its immediate neighbors, the noise in their model talks to every other part of the line, though the strength of that conversation fades as the distance grows. By applying this long-range noise to a specific family of quantum states, they discovered that the old rules no longer apply. They found that even in a one-dimensional line, this power-law noise can trigger a dramatic shift from a state of perfect symmetry to a state where the strong symmetry breaks down, leaving only the weak version behind. This transition happens at a specific, measurable strength of noise, proving that one-dimensional systems can indeed host this exotic form of mixed-state order, provided the noise is connected enough to span the distance.

The researchers began their investigation with a family of quantum states that can be smoothly tuned between two extremes: a trivial, uninteresting state and a more complex state known as a symmetry-protected topological phase. In the pure, noise-free version of these states, the complex phase is protected by a special symmetry that keeps information safe at the edges of the system. To see what happens when the system gets noisy, the team applied a quantum channel that simulates a specific type of disturbance. This channel causes pairs of sites in the line to lose their quantum coherence, but with a twist: the probability of this happening drops off according to a power law based on the distance between the sites. If the sites are close, the disturbance is likely; if they are far apart, it is less likely, but it never completely disappears.

When they applied this long-range channel to their states, they constructed a detailed map of the resulting phases. They found that for weak noise, the system remains in a symmetric state. However, as the noise strength increases, the system crosses a sharp boundary into a new phase where the strong symmetry is spontaneously broken, leaving only the weak symmetry intact. This transition, which they call strong-to-weak spontaneous symmetry breaking, was previously thought impossible in one dimension with short-range noise. The researchers confirmed this transition by measuring specific quantities that act as thermometers for the system's order. They found that while the usual measures of order vanished, a more sophisticated measure, which looks at the system in a doubled mathematical space, clearly signaled the change. This new phase is not just a messy version of the old one; it is a distinct state of matter with its own unique properties.

Perhaps the most surprising discovery was the nature of the phase that exists just before the symmetry breaks. In the region where the noise is present but not yet strong enough to break the symmetry, the system enters a mixed-state version of the topological phase. In pure states, this phase is famous for storing quantum information at its edges, protected by the system's symmetry. In this new mixed phase, the researchers found that the system still holds onto two logical qubits of information, even though the traditional signs of this order have vanished. The standard way to detect this order, which involves looking at a specific string of correlations, fails completely in the mixed state because the noise destroys the linear signal. However, by using a different diagnostic tool that looks at the system's coherence in a non-linear way, the team showed that the information is still there, protected by a subtle symmetry that acts on both the system and its mathematical mirror image. This phase is distinct from previously studied decohered states because it retains a high degree of purity, meaning the system is not as messy as one might expect.

The study also addressed a major practical hurdle in observing these phenomena: the difficulty of measuring mixed states on real quantum computers. Because these states are inherently noisy, measuring them usually requires an enormous number of samples to get a clear signal, often making experiments impossible on current hardware. The researchers found that by choosing their initial states carefully—specifically, by tuning them to be close to a special point where the system is naturally invariant to the noise—they could create mixed states that remain surprisingly pure. This high purity dramatically reduces the number of measurements needed to detect the phase transition. They calculated that for a system of eighteen sites, probing the transition at this optimal point would require only a few hundred measurement shots, whereas probing it at a less optimal point would require tens of thousands. This suggests that the phenomenon they have described is not just a theoretical curiosity but something that can be realized and observed on near-term quantum devices.

The team's work relies on a clever mathematical trick that maps the quantum problem onto a classical statistical mechanics model, a well-understood framework for studying how particles interact. In this classical picture, the quantum noise translates into long-range interactions between spins, similar to how magnets influence each other across a distance. This mapping allowed them to use powerful computer simulations to explore the behavior of the system across a wide range of parameters. They confirmed that the transition to the symmetry-broken phase occurs at a finite noise strength for a wide range of power-law exponents, as long as the interactions are long enough. They also showed that if the interactions were short-ranged, the transition would disappear, confirming that the long-range nature of the noise is the key ingredient that allows this new phase to exist in one dimension.

Ultimately, this research opens a new window into the behavior of quantum matter in noisy environments. It demonstrates that the limitations of one-dimensional systems can be overcome by introducing long-range connections, leading to a rich phase diagram with novel states of matter. The discovery of a mixed-state topological phase that retains quantum information despite decoherence offers a new perspective on how to protect quantum data. Furthermore, the identification of a parameter regime where these states remain highly pure provides a clear path for experimentalists to test these ideas in the lab. The work suggests that the quantum world, even when messy and one-dimensional, holds more surprises and more robust forms of order than previously imagined, waiting to be unlocked by the right kind of noise.