Bader Aldossari of the Georgia Institute of Technology and colleagues have developed a new way to model intrinsically mixed-state topological phases, quantum matter exhibiting nontrivial topological phenomena even with inherent noise. The researchers constructed tensor network representations for these phases, which emerge in noisy, long-range entangled quantum systems and demonstrate the method’s applicability to scenarios like $m^{a} e^{b}$ decoherence in the $Z_{N}$ toric code.

The team’s approach reveals a tensor network representation for a mixed-state chiral semion that has no equivalent in “clean”, pure-state quantum systems, suggesting the possibility of topological order unique to noisy environments. Together, these results establish tensor networks as a natural platform for analyzing noisy intrinsically mixed-state topological matter.

Tensor Networks Capture Mixed-State Topological Order

Tensor networks now model topological phases arising from decoherence, a process where quantum information leaks from a system, in a way previously inaccessible to simulation. The framework allows for the construction of tensor network representations for a wide range of intrinsically mixed topological phases originating from strongly disordered or decohered pure-state systems. This advancement addresses a gap in the field, as prior work primarily focused on average-symmetry-protected topological phases using locally purifiable density operators, lacking a systematic tensor network approach for mixed-state topological order.

The method exploits an isomorphism between anyon decoherence within the physical Hilbert space and anyon condensation in a corresponding Choi space, a mathematical construct used to represent quantum channels. This connection enables the creation of tensor networks for systems experiencing decoherence, such as the $Z_{2}$ toric code undergoing flux-channel decoherence, where the initial pure state density matrix is denoted as $ρ_{0}$ .

The resulting topological order, while permitted in pure states, is believed to be absent from fixed-point wavefunctions with finite-dimensional Hilbert spaces, suggesting a unique tensor network representation in the mixed-state setting. A specific example of this capability is the modeling of $m^{a} e^{b}$ decoherence within the $Z_{N}$ toric code, demonstrating the method’s ability to handle complex decoherence scenarios. Researchers also successfully constructed tensor network representations for the decohered non-Abelian quantum double of $S_{3}$ , a counterintuitive result showing nontrivial topological phenomena can persist even after decoherence.

The framework’s versatility extends to pure Z/X decoherence applicable to arbitrary CSS codes, broadening its potential applications in quantum error correction and materials science. This finding signifies that the method can uncover topological order that does not exist in “clean” quantum systems.

The work suggests that long-range correlated boundaries are not necessarily required for chiral bulk behavior in mixed states, a departure from the expectations of pure-state chiral topological order. The researchers acknowledge that fundamental questions regarding bulk-boundary relations in mixed-state topological order remain open, but anticipate that the constructed tensor network representations will provide a concrete setting for extracting universal boundary data.

“Therefore, the boundary data may not be an invariant of the mixed-state phase in the sense of two-way connectivity by finite depth local channels,” the paper states, highlighting the complexity of characterizing these novel states. The team intends to further explore the scope of the method and its implications for understanding the relationship between bulk and boundary properties in intrinsically mixed topological order.

Mixed-State Semions Lack Pure-State Analogues

The ability to model quantum states beyond those achievable in pristine, isolated systems is expanding, with tensor networks now capable of representing topological order absent in pure states. Specifically, a mixed-state chiral semion, a quantum excitation, possesses a tensor network representation demonstrably lacking a counterpart in “clean” quantum systems, a finding that challenges conventional understanding of topological phases. This suggests that noise, rather than being a hindrance, can actually create novel quantum states of matter.

Choi Space Representation of Decohered States

The construction of tensor network representations begins with mapping a density matrix, denoted as ρ₀, to a state in a doubled Hilbert space, H⊗H*, via the Choi-Jamiołkowski isomorphism. This transformation allows researchers to analyze decoherence phenomena by representing the initial state and its subsequent evolution within this expanded space, facilitating the development of tensor network representations for the post-channel Choi state and, ultimately, the original density matrix.

The process relies on defining a specific tensor network structure that captures the correlations present in the initial state and how they are affected by the applied channel. Following the initial state representation, the tensor network is modified to account for decoherence, specifically through the introduction of connecting tensors that reflect the influence of noise. These tensors, derived from the application of local channels, effectively introduce correlations between different parts of the system, mimicking the effects of environmental interactions.

This approach allows for the visualization and manipulation of complex quantum states in a computationally tractable manner, enabling the study of decoherence patterns and their impact on topological order. The local channel, represented as $N_{l}$ , acts on the system by combining the original state with a modified version influenced by Pauli strings $X_{l}$ and $Z_{l - a}$ . The basis elements of the system, denoted as |x,z⟩, represent entanglements between the bra and ket layers, forming the foundation for representing the decohered Choi state.

This state is then expressed as a summation over these basis elements, incorporating the connecting tensors and a weighting factor that depends on parameters 1-p and pω^(ax_l + bz_l). The summation over α introduces a probabilistic element, reflecting the inherent uncertainty introduced by decoherence. This capability is important for understanding how topological order evolves under varying levels of noise and for predicting the behavior of quantum systems in realistic environments.

Tensor Network Construction via Kraus Operators

This advancement allows analysis of systems experiencing non-maximal strength incoherent error channels, moving past scenarios where Kraus operators act as projective measurements. The formalism readily generalizes to string-net models and states resulting from condensation, as seen in the chiral theory’s tensor network detailed elsewhere in the work. A limitation arises when commutators between different Kraus operators are described by a matrix U, preventing simultaneous diagonalization in a preferable basis and hindering the initial construction step.

This occurs when Kraus operators do not cancel out in the doubled space, and is also common with coherent noise, except in ultra-local channels where operators act on single link degrees of freedom. Despite this, the method reveals a fixed-point tensor network for the chiral semion mixed state, a surprising result indicating the existence of topological order without a corresponding pure-state analogue.

Protocol for Building Tensor Networks from Decoherence

A systematic pathway for efficiently representing noisy topological matter has been established through a protocol detailing how to build tensor networks from decoherence. This method moves beyond pure-state descriptions by constructing representations from intrinsically mixed-state topological phases, which arise in quantum systems experiencing long-range entanglement and noise. The protocol begins with a tensor network representation of the initial pure state, then uses the properties of the Choi state before decoherence.

This state is expressed as a sum over configurations, each defined by tensors that describe the relationships between different parts of the system. Ancillary legs are then attached to these tensors, extending their projective definition and enabling the incorporation of decoherence effects. Connecting tensors are added between the ancillary legs in the bra and ket layers, enforcing strong decoherence and creating a mixed-state representation.

This process yields a tensor network that is both a renormalization group fixed point, invariant under transformations and a decoherence fixed point, representing a maximally decohered state. The method’s adaptability extends beyond two-dimensional systems. It is also applicable to decohering non-Abelian anyons with trivial mutual statistics. This decomposition allows for a more efficient analysis of the system’s behavior under different scales and conditions. The procedure is not limited to specific fixed points. It can be generalized to arbitrary CSS codes, broadening its applicability to a wider range of quantum systems.

The authors note the dual nature of the resulting representation. The protocol also addresses the challenges of mapping decoherence to statistical mechanical models, which can suffer from a sign problem. By carefully constructing the tensor network and applying the appropriate transformations, researchers can circumvent this issue and obtain meaningful results.

This is achieved by expressing the tensor network in a form suitable for the renormalization group, allowing for the analysis of the system’s behavior at different length scales. The ability to construct these representations provides a powerful tool for understanding and manipulating noisy quantum systems, potentially opening the way to more robust quantum technologies.

Extending Tensor Networks to Open Quantum Systems

The ability to represent intrinsically mixed-state topological phases with tensor networks expands the scope of analysis beyond closed quantum systems. This approach moves beyond traditional studies of pure-state topological order, addressing challenges inherent in long-range entangled systems and stabilizer codes. This means the protocol reveals topological order that is fundamentally absent in “clean” quantum systems, opening the possibility of discovering entirely new phases of matter only accessible through noise and decoherence.

The protocol’s foundation lies in the ability to map decoherence to the Choi space, where incoherent noise manifests as excitations with trivial monodromy. By constructing appropriate tensor networks and applying targeted transformations, researchers can circumvent these challenges and gain insights into the system’s behavior. The work builds on established tensor network techniques, such as the tensor entanglement renormalization group (TERG), which allows for coarse-graining and identification of fixed points under renormalization.

These fixed points serve as starting points for analyzing the stability of topological phases and understanding phase transitions. The researchers first reviewed the quantum double formulation of $Z_{N}$ topological phases on a square lattice, along with standard tensor network representations of their ground states, before extending the approach to mixed-state scenarios.

The team’s protocol is applicable to a broad class of systems arising from decoherence of pure-state topological phases, establishing tensor networks as a versatile tool for exploring the intricacies of open quantum systems and their potential for quantum information processing. The results establish a pathway for further investigation into the interplay between noise, entanglement and topological order in quantum materials.

👉 More information
🗞 Tensor network representations for intrinsically mixed-state topological orders
✍️ Bader Aldossari, Sergey Blinov and Zhu-Xi Luo
🧠 DOI: http://link.aps.org/doi/10.1103/4zl5-2gw8

More like this