Duality between open systems and closed bilayer systems: Thermofield double states as quantum many-body scars

This paper establishes a duality between open many-body systems satisfying detailed balance and closed bilayer systems, revealing that the identity operator and specific eigen-operators of the Lindbladian map to quantum many-body scars in the form of thermofield double states with tunable entanglement and exponential decay dynamics.

The Gist

Imagine you are trying to understand a messy, noisy room where things are constantly changing and leaking energy (an open system). Physicists usually describe this with complex math that feels like trying to predict the weather in a hurricane.

Now, imagine you have a magic mirror. If you look at that messy room through this mirror, it transforms into a perfectly clean, silent, and stable room with two identical layers stacked on top of each other (a closed bilayer system).

This paper by Teretenkov and Lychkovskiy is about building that magic mirror and discovering what happens when you look through it.

The Magic Mirror (The Duality)

The authors found a specific mathematical trick to turn the messy, leaking room into a stable, two-layered room.

• The Condition: This trick only works if the "leakage" in the messy room follows a specific rule called "detailed balance." Think of this like a rule where the rate of things falling out of a bucket is perfectly balanced by things falling back in, based on the "temperature" of the room.
• The Result: When this rule is met, the messy math of the leaking room transforms into a clean, standard physics equation (a "self-adjoint Hamiltonian") for the two-layered room. This is huge because standard physics equations are much easier to solve and understand than the messy ones.

The "Ghost" in the Machine (The Thermofield Scar)

In the messy room, there is one very simple thing: the Identity Operator. In plain English, this is just the concept of "nothing happening" or "everything staying the same." It's the most boring, simple object you can imagine.

But when you look at this boring object through the magic mirror, it transforms into something incredibly special in the two-layered room. It becomes a Quantum Many-Body Scar.

• What is a Scar? Usually, in a complex quantum system, everything gets jumbled up and chaotic (thermalizes). A "scar" is a special, stubborn state that refuses to get chaotic. It's like a single, perfect note that keeps ringing out in a room full of static noise.
• The "Thermofield" Twist: This specific scar is called a "Thermofield Double." It's a state where the two layers of the room are perfectly entangled (linked) in a way that looks like they are at a specific temperature.
  • The Temperature Control: The "temperature" of this entanglement is directly controlled by the temperature of the reservoir in the original messy room. If the messy room is hot, the scar is highly entangled. If the room is infinitely hot, the scar becomes a famous "EPR scar" (a perfect pair of linked particles).
  • The Surprise: Even though the two layers are linked by temperature, if you look at the layers from a different angle (a "transverse" cut), they look completely unlinked and simple. This "zero entanglement" from certain angles is exactly what makes it a "scar."

The Tower of Special States

The paper doesn't just stop at the "Identity" object. The authors found that in many different types of messy rooms, there are other special "objects" (operators) that decay in a very simple, predictable way (like a ball rolling down a hill at a constant speed).

When they look at these other objects through the magic mirror, they turn into towers of scars.

• The Analogy: Imagine the messy room has a few specific levers that, when pulled, cause the system to fade away smoothly. In the two-layered room, pulling those same levers reveals a whole ladder of special, non-chaotic states (scars) hidden inside the complex system.
• The Result: This means that even in complex, chaotic two-layered systems, there are hidden pockets of order and predictability that we can find by studying the "leaky" version of the system.

Beyond the Standard Rules

Finally, the authors suggest that this magic mirror trick might work even for systems that don't follow the standard "leaking" rules (non-GKSL systems). They show an example where a very strange, non-standard math equation can still be mapped to a two-layered system with hidden special states. This suggests the mirror is even more powerful than they first thought.

Summary

In short, the paper says:

1. We found a bridge: We can turn a messy, leaking quantum system into a clean, two-layered system if the leakage follows specific rules.
2. Boring becomes special: The simplest thing in the messy system (doing nothing) becomes a special, non-chaotic "scar" in the clean system.
3. Temperature matters: The "scar" has a temperature built into it, controlled by the original system's environment.
4. Hidden order: Many messy systems have special "levers" that, when viewed through this bridge, reveal entire families of these special, non-chaotic states in the clean system.

This gives physicists a new way to find order in chaos by looking at the "leaky" side of the problem.

Technical Summary

Technical Summary: Duality between Open Systems and Closed Bilayer Systems, and Thermofield Double States as Quantum Many-Body Scars

Problem and Context
The paper addresses the theoretical challenge of relating open quantum systems, governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, to closed quantum systems. While dualities between open and closed systems are known, they typically map the Lindbladian superoperator to a non-Hermitian Hamiltonian in a doubled Hilbert space. Such non-Hermitian Hamiltonians often lack direct physical interpretation as standard quantum mechanical systems. The authors seek to establish a duality that maps the open system dynamics to a self-adjoint (Hermitian) Hamiltonian of a closed bilayer system, thereby connecting two theories with straightforward physical meanings. A specific focus is placed on identifying special eigenstates in these closed systems, known as quantum many-body scars, which are non-thermal eigenstates embedded in the bulk of the spectrum.

Methodology
The authors construct a duality map based on the following steps:

1. Heisenberg Picture Formulation: The open system is described by the GKSL equation in the Heisenberg picture: $\partial_t O_t = \mathcal{L}^\dagger O_t$, where $\mathcal{L}^\dagger$ is the adjoint Lindbladian.
2. Detailed Balance Condition: The open system is required to satisfy the detailed balance condition. This implies the dissipator can be written in a specific form involving a conserved single-body quantity $H_0$ (which commutes with the system Hamiltonian $H$) and an inverse temperature parameter $\beta$. The Lindblad operators $L_j$ act as annihilation-like operators with respect to $H_0$.
3. Vectorization and Wick Rotation: The authors define a map from operators $O$ on the open system Hilbert space $\mathcal{H}$ to vectors $|O\rangle\!\rangle_\beta$ in the doubled space $\mathcal{H} \otimes \mathcal{H}$. This map involves a similarity transformation dependent on $\beta$ and $H_0$. Crucially, they perform a Wick rotation on the dissipative coupling constant ($g = i\gamma$).
4. Dual Hamiltonian Construction: Under this map, the operator $i\mathcal{L}^\dagger$ maps to a self-adjoint Hamiltonian $H_\beta$ acting on the bilayer system. $H_\beta$ consists of the original Hamiltonian terms for each layer, a dissipative term dependent on $\beta$, and an interlayer interaction term.

Key Contributions and Results

1. The Thermofield Scar:
   The identity operator $\mathbb{1}$ on the open system side is always an eigenoperator of the Lindbladian with eigenvalue zero ($i\mathcal{L}^\dagger \mathbb{1} = 0$). Under the duality, this maps to a zero-energy eigenstate of the dual Hamiltonian $H_\beta$:
   $$|1\rangle\!\rangle_\beta = \sum_n e^{-\frac{\beta}{4}H_0}|n\rangle \otimes e^{-\frac{\beta}{4}H_0^T}|n\rangle$$
   This state is identified as a thermofield double state for the conserved quantity $H_0$.

   • Entanglement Structure: The state exhibits a tunable entanglement structure. For a bipartition separating the two layers, the entanglement entropy corresponds to the thermal equilibrium entropy of $H_0$ at temperature $\beta$. However, for "transverse" bipartitions (cutting across the layers without breaking interlayer bonds), the entanglement entropy vanishes (in the product basis limit). This zero entanglement in specific cuts, combined with its location in the bulk of the spectrum, classifies it as a quantum many-body scar.
   • Infinite Temperature Limit: At $\beta = 0$, the state reduces to a product of Einstein-Podolsky-Rosen (EPR) pairs, recovering the recently discovered "EPR scar."

2. Towers of Scars from Eigen-Operators:
   The authors identify broad classes of open systems where the adjoint Lindbladian possesses non-trivial eigen-operators $Q$ (other than the identity) that are also integrals of motion for the coherent part of the dynamics ($[H, Q]=0$).

   • These "eigen-HIoMs" (Hamiltonian Integrals of Motion) decay purely exponentially: $\langle Q \rangle_t = e^{-\Gamma t} \langle Q \rangle_0$, independent of the initial state.
   • Under the duality, these operators map to additional eigenstates of the bilayer Hamiltonian, forming "towers of scars."
   • These dual scars can be generated from the thermofield scar via the action of the operator $Q$ (appropriately transformed by $\beta$).

3. Generalization Beyond GKSL:
   The paper notes that the duality framework can be extended to superoperators that are not of the GKSL form. While such mappings do not necessarily correspond to physical open systems with standard interpretations, they can still be used to identify special eigenstates (scars) in bilayer Hamiltonians that might otherwise remain hidden. An example is provided where a non-GKSL superoperator maps to a bilayer Hamiltonian with explicitly constructible eigenstates.

Significance and Claims
The paper claims that this duality provides a powerful tool for describing complex objects in one theory by relating them to simpler objects in another. Specifically:

• It connects the simple identity operator of an open system to a non-trivial, non-thermal eigenstate (the thermofield scar) in a closed system.
• It reveals that the existence of "operator scars" (operators with pure exponential decay) in open systems is dual to the existence of quantum many-body scars in closed bilayer systems.
• It demonstrates that the entanglement entropy of these scars is directly controlled by the reservoir temperature of the dual open system.
• The authors emphasize that the detailed balance condition is crucial for obtaining a self-adjoint dual Hamiltonian; without it, the mapping generally yields non-Hermitian operators.

The work suggests that the structure of operator space fragmentation in open systems is dual to Hilbert space fragmentation in closed systems, offering a new perspective on non-ergodic behavior in quantum many-body systems. The results are presented as applicable to time-dependent Hamiltonians (leading to Floquet scars) and potential applications in quantum simulations and the study of topological effects in open systems.