⚛️ quantum physics

Nonlocality of Observables in Quasi-Hermitian Quantum Theory

This paper demonstrates that while quasi-Hermitian quantum theories with local Hamiltonians can be explicitly constructed, generic local observable algebras generally fail to exist on arbitrary lattice sites, with the specific collections of sites supporting nontrivial observables being strictly determined by the complex potential.

Original authors: Jacob Barnett

Published 2026-08-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jacob Barnett

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the standard story of quantum mechanics, the rules of the game are set by a specific kind of mathematical object called a Hamiltonian, which dictates how a system changes over time. For decades, physicists have operated under a strict rule: for a system to be physically real, this object must be "Hermitian." This requirement ensures that the results of any measurement, such as the energy of a particle, come out as real numbers rather than confusing complex values, and it guarantees that the total probability of finding a particle somewhere always adds up to one. However, in recent years, a broader class of theories has emerged where the Hamiltonian does not need to be Hermitian. These are known as quasi-Hermitian theories. They allow for systems that look non-standard on the surface but still produce real, measurable outcomes, provided one redefines how distances and angles are measured within the system's mathematical space. This redefinition is handled by a special weighting tool called a metric. While these theories are mathematically equivalent to standard ones in many ways, they offer a different perspective on what it means for a part of a system to be "local," or independent from the rest.

The question of locality is central to how we understand the universe. In a local theory, an event happening in one place should not instantly affect a distant place without a signal traveling between them. Usually, physicists assume that if a system is built from smaller parts, like a chain of atoms, the mathematics naturally splits into independent pieces for each part. This paper, however, investigates what happens to this idea of independence when we apply the rules of quasi-Hermitian quantum theory. The author, Jacob Barnett, explores whether it is possible to define "local" observables—things we can measure in a specific region without needing to know about the entire system—when the underlying mathematical rules are governed by these non-standard metrics. The investigation focuses on a simplified model of particles moving on a one-dimensional line, interacting only with their immediate neighbors, a setup that is usually considered the textbook example of a local system.

The research reveals a surprising and counterintuitive result: even though the rules governing the movement of these particles are strictly local, the things we can actually measure in a specific region are often not local at all. In a typical quantum system, if you pick a small group of sites on a lattice, you can usually find a measurement that only involves those sites. In these quasi-Hermitian models, that is frequently impossible. The ability to measure something locally depends entirely on the specific shape of the complex potential energy landscape and the arrangement of the sites. For instance, in a model where the system has a special symmetry involving a reflection and a reversal of time, the researchers found that local measurements only exist if the group of sites being measured is perfectly symmetric itself. If the group is slightly off-center, or if the complex parameters of the system are tuned to a certain value, the local observables vanish completely. In these cases, to measure anything in that region, one would effectively need to know about the entire system, rendering the concept of a local measurement meaningless for that specific collection of sites.

The study goes further to map out exactly which collections of sites allow for local measurements and which do not. By analyzing the mathematical structure of the metric, the author derives precise conditions for the existence of these local observables. It turns out that the "connectivity" of the sites matters deeply. A group of sites that is broken into separate chunks might allow for local measurements, while a single continuous chain of the same number of sites might not, depending on where the special, non-standard interactions are placed. The research shows that the space of possible local theories is much more restricted and fragile than previously thought. While the Hamiltonian driving the system remains local, the algebra of observables—the set of all possible things one can measure—becomes nonlocal. This means that the physical reality of the system, as defined by what can be measured, is entangled in a way that defies the simple intuition of independent parts.

The findings have significant implications for how we might view the structure of spacetime and quantum gravity. In theories of gravity, the evolution of the universe is expected to be local, yet the observables we can define are often nonlocal due to the nature of the theory. This paper suggests that quasi-Hermitian representations might provide a natural framework for understanding such scenarios. It demonstrates that the assumption that the physical inner product (the rule for measuring distances) must match the tensor product structure (the rule for splitting the system into parts) is unnecessarily restrictive. By relaxing this assumption, one can discover a wider variety of local quantum theories, or conversely, find that a theory which looks local in its dynamics is actually nonlocal in its observables. The work does not claim to solve the mysteries of quantum gravity, but it provides a concrete, mathematical demonstration of how the definition of "local" can shift depending on the underlying mathematical rules of the theory.

Ultimately, the paper establishes that in quasi-Hermitian quantum theories, the existence of local observables is not a guaranteed feature of a local Hamiltonian. Instead, it is a delicate property that depends on the specific details of the system's metric. The researchers proved that for a wide range of parameters, there are simply no non-trivial measurements that can be made on a subset of the system without referencing the whole. This challenges the standard view that locality is a robust feature of quantum systems and suggests that the relationship between the dynamics of a system and the observables available to an observer is far more complex than previously realized. The work serves as a cautionary tale for anyone trying to extend quantum theory to new frontiers: the tools we use to measure the world are inextricably linked to the mathematical fabric of the theory itself, and changing that fabric can fundamentally alter what it means to be local.

Technical Summary: Nonlocality of Observable Algebras in Quasi-Hermitian Quantum Theory

Problem Statement
The paper addresses a fundamental tension in quasi-Hermitian quantum theory: while such theories are mathematically equivalent to standard Hermitian quantum mechanics via a similarity transformation, the physical notion of "locality" is not preserved under this mapping. In standard quantum theory, locality is defined via a tensor product structure of the Hilbert space ( $\mathcal{H} \simeq \mathcal{H}_A \otimes \mathcal{H}_B$ ), where local observables act non-trivially only on one subsystem. In quasi-Hermitian theories, observables must satisfy $\eta O = O^\dagger \eta$ with respect to a metric operator $\eta$ . The central problem is that while the Hamiltonian may be local (acting on nearest neighbors), the metric $\eta$ often possesses non-trivial entanglement (non-factorizable structure). Consequently, the algebra of local observables in a quasi-Hermitian representation can be drastically different from, or even empty in, subsystems where local observables would exist in a standard Hermitian picture. The paper seeks to explicitly construct these local observable algebras and determine the conditions under which they exist.

Methodology
The author employs a rigorous algebraic approach to analyze the existence of local observables in two distinct frameworks:

Tensor Product Model: The paper analyzes general finite-dimensional Hilbert spaces where locality is defined by a fixed tensor product factorization. It utilizes the operator Schmidt decomposition of the metric operator $\eta = \sum_i \sqrt{\chi_i} \eta_A^i \otimes \eta_B^i$ to derive necessary and sufficient conditions for the existence of local observables.
Free Fermion Models: To bypass the exponential complexity of generic many-body problems, the analysis is applied to systems of free fermions. Here, locality is defined via the Bravyi-Kitaev criterion (operators formed by even products of creation/annihilation operators on a set of sites) rather than a strict tensor product. The problem is reduced to a first-quantized setting by analyzing the kernel of blocks of the reduced metric matrix $M$ .
Toy Models: The derived theorems are applied to a specific $PT$-symmetric toy model of free fermions on a 1D lattice. This model features nearest-neighbor hopping and complex on-site potentials (impurities) at specific sites $m$ and $\bar{m}$ . The author investigates two specific configurations: impurities at the farthest sites ( $m=1$ ) and impurities at the nearest sites ( $m=n/2$ ).

Key Contributions and Results

General Existence Theorem (Theorem 2): The paper proves that for a subsystem $B$ to contain a non-trivial local observable $O_B$ (where $O = 1_A \otimes O_B$ ), the Schmidt operators of the metric restricted to $B$ must be simultaneously reducible. Specifically, if the Schmidt number of the metric (the number of non-zero terms in its Schmidt decomposition) exceeds a bound related to the dimension of the subsystem, no local observables exist. This establishes that a non-factorizable metric generically destroys the algebra of local observables.
Free Fermion Criterion (Theorem 3): For free fermion systems, the existence of extensively local observables (observables local to a set $A$ but not to any proper subset of $A$ ) is determined by the dimension of the kernel of the off-diagonal block of the reduced metric, $K(A) = \dim(\ker M_{A'A})$ . An extensively local observable exists in $A$ if and only if $K(A) > K(S)$ for all proper subsets $S \subset A$ .
Application to $PT$-Symmetric Toy Models:
- Nearest Neighbor Impurities: When impurities are adjacent ( $m=n/2$ ), the reduced metric decomposes into parity blocks. The result shows that non-trivial local observables exist if and only if the subsystem is parity-symmetric (invariant under the exchange $i \leftrightarrow \bar{i}$ ).
- Farthest Impurities: When impurities are at the boundaries ( $m=1$ ), the existence of local observables depends on the connectivity of the subsystem. The paper provides a detailed set of conditions (Lemma 6 and Proposition 7) regarding the distance between connected components of the subsystem. For instance, isolated single sites generally do not support local observables unless they are connected to the rest of the subsystem within a specific distance (at most one site gap).
- Non-Existence: A striking result is that even for a local Hamiltonian, there exist generic collections of sites in the lattice that possess no non-trivial local observables whatsoever.
Symmetry Properties: The paper demonstrates that if a subsystem $A$ contains local observables in a $PT$-symmetric theory, its parity dual $P(A)$ also contains local observables.

Significance and Claims
The paper claims that quasi-Hermitian representations generalize local quantum theory by decoupling the definition of the physical inner product from the tensor product structure used to define locality. The author argues that assuming the physical inner product factorizes (i.e., $\eta = \eta_A \otimes \eta_B$ ) is an unnecessarily restrictive assumption.

The primary significance lies in the discovery that locality of the Hamiltonian does not guarantee the existence of local observables in quasi-Hermitian theories. The "locality profile" of a theory is dictated by the entanglement structure of the metric operator. This suggests that in contexts where the metric is non-trivial (such as certain models in quantum gravity or non-commutative field theories), the standard notion of local degrees of freedom may be emergent or entirely absent in certain regions of the system. The work provides a constructive framework to identify which subsystems retain physical observables, offering a tool for exploring broader classes of local quantum theories that are mathematically equivalent to non-local Hermitian theories but physically distinct in their observable algebras.

Technical Summary: Nonlocality of Observable Algebras in Quasi-Hermitian Quantum Theory

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