Tsung-Cheng Lu of the University of Maryland, Yu-Jie Liu of the Massachusetts Institute of Technology, Sarang Gopalakrishnan of Princeton University, and Yizhi You of Northeastern University have established a framework linking the behavior of quantum channels to a higher-dimensional system. Their work demonstrates that a d-dimensional quantum channel can be mapped to the boundary of a (d+1)-dimensional wavefunction, revealing how strong-to-weak spontaneous symmetry breaking arises from anyon condensation on the boundary of a topological order. Researchers utilized isometric tensor network states to identify a quantum channel’s time evolution with the transfer matrix of a higher-dimensional system, enabling the construction of tunable channels exhibiting distinct mixed-state phases and transitions.

Holographic Duality Links Bulk Topological Order to Boundary States

A surprising connection between higher-dimensional topological order and the mixed states arising in quantum channels has emerged from new theoretical work. This mapping is formally defined through a principle called channel-state duality, offering a new lens for understanding complex quantum phenomena.

Built on isoTNS, they constructed continuously tunable quantum channels that exhibit distinct mixed-state phases and transitions in their steady states. This holographic approach provides a novel perspective on strong-to-weak spontaneous symmetry breaking (SWSSB), a phenomenon that arises in the steady states of these quantum channels. The researchers suggest that further exploration of this connection could unlock new insights into the behavior of complex quantum systems and the nature of mixed-state phases.

Channel-State Duality Maps Quantum Channels to Wavefunctions

The connection between a quantum channel’s behavior and higher-dimensional topological properties has been clarified through a new framework. This means that understanding the topological order in the “bulk” spacetime can reveal insights into the symmetry-breaking behavior observed on the channel’s “boundary”. The researchers utilized isometric tensor network states (isoTNS) to build this framework.

Conditional Mutual Information Reflects Bulk Entanglement

The researchers utilized channel-state duality to demonstrate this holographic mapping, showing that the steady state of a quantum channel corresponds to the reduced density matrix on the boundary of a sequentially constructed wavefunction. This isn’t merely an analogy, but a mathematically defined relationship allowing for a novel perspective on complex quantum phenomena. As the researchers write, “The spacetime bulk could be topologically ordered, and that the SW-SSB boundary arises from anyon condensation of the bulk topological order.” This condensation, a phenomenon involving exotic particles, appears to be the mechanism by which SWSSB arises.

The framework extends beyond simple scenarios, accommodating generalized symmetries including higher-form, subsystem, and fermionic symmetries. This broad applicability suggests the holographic approach could become a central tool for classifying and understanding a wide range of mixed-state phases and transitions.

The team’s findings offer a new way to analyze systems where traditional methods based on local Hamiltonians fall short, particularly those arising from nonequilibrium processes like noise and error correction. The implications of this work extend to the design of more robust quantum devices, as understanding and controlling symmetry breaking in steady states is crucial for maintaining quantum information.

Isometric Tensor Networks (isoTNS) Model Channel Evolution

The ability to finely control quantum channels, the pathways through which quantum information travels, has taken a step forward with a new framework linking their evolution to higher-dimensional topological order. Researchers have demonstrated a method for constructing and tuning these channels using isometric tensor network states, or isoTNS, revealing connections between a channel’s behavior and the properties of a related system in one higher dimension.

The construction of these tunable channels allows for precise control over the system’s behavior, opening avenues for exploring previously inaccessible quantum phenomena. The framework reveals a surprising link between spontaneous symmetry breaking in these channels and the topological order present in the higher-dimensional “bulk” system. This means that the breaking of symmetry in the lower-dimensional channel is directly connected to a physical process occurring in the higher-dimensional space, offering a new perspective on how symmetry breaking emerges in quantum systems.

Mixed-State Phases Beyond Equilibrium Hamiltonians

The conventional understanding of phases of matter relies heavily on examining ground states of systems at equilibrium, but increasingly, physicists are turning attention to the more complex realm of mixed states arising from systems far from equilibrium. The framework centers on understanding how symmetries break in mixed quantum states, specifically through a process called strong-to-weak spontaneous symmetry breaking (SWSSB). The authors state that understanding the bulk topological order is key to unlocking the properties of the boundary mixed state, offering a novel perspective on the classification of these increasingly relevant quantum states.

Toric Code Noise Illustrates Mixed-State Phase Transitions

Strong-to-weak spontaneous symmetry breaking (SWSSB) arises from anyon condensation, and the conditional mutual information (CMI) associated with SWSSB is inherited from the bulk topological entanglement entropy. This approach provides a novel perspective on strong-to-weak spontaneous symmetry breaking (SWSSB), a phenomenon that arises in the steady states of these quantum channels.

Anyon Condensation Drives Boundary SW-SSB

These channels, built upon the isoTNS formalism, provide a powerful tool for manipulating and analyzing quantum systems exhibiting SWSSB. The implications extend beyond theoretical understanding; the ability to engineer channels with specific mixed-state phases opens avenues for designing robust quantum information processing schemes. Strong-to-weak spontaneous symmetry breaking (SWSSB) in the steady state arises from the anyon condensation on the boundary of a topological order in one higher dimension.

👉 More information
🗞 Holographic Duality Between Bulk Topological Order and Boundary Mixed-State Order
✍️ Tsung-Cheng Lu, Yu-Jie Liu, Sarang Gopalakrishnan and Yizhi You
🧠 DOI: http://link.aps.org/doi/10.1103/lrqp-pnl1

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