⚛️ quantum physics

On Locality of Quantum Information in the Heisenberg Picture for Arbitrary States

This paper clarifies the locality of quantum information in arbitrary states of composite systems by utilizing a modified Deutsch-Hayden framework and noncommutative values to describe quantum information as local observable values, while also addressing the challenges of retrieving entangled information through local processes.

Original authors: Otto C. W. Kong

Published 2026-07-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Otto C. W. Kong

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

Imagine the universe as a giant, cosmic video game. For decades, physicists have been arguing about how the game's "save files" work. The standard view, known as the Schrödinger picture, suggests that when two players (particles) get entangled, their save files merge into one giant, spooky file that lives everywhere at once. If you touch one player's controller, the other player's screen changes instantly, no matter how far apart they are. This feels like magic, or "non-locality," and it's been the accepted story since Einstein and Bohr started their famous debates.

But a physicist named Otto C. W. Kong is saying, "Wait a minute. What if we're looking at the game from the wrong angle?"

In this paper, Kong suggests that if we switch our perspective to the Heisenberg picture, the story changes completely. Instead of watching the "save file" (the state) move around, we watch the "controls" (the observables) evolve. And here is the twist: the information is actually local.

The Magic of the "Value"

To understand this, let's talk about what a "value" is. In our everyday world, if you measure the speed of a car, you get a number, like 60 miles per hour. That's a "real number." It's simple and commutative (60 times 2 is the same as 2 times 60).

But in the quantum world, Kong argues that the "value" of a measurement isn't a simple number. It's more like a complex, multi-layered code—a "noncommutative value." Think of it like a secret cipher. If you have a standard number, it's just a single digit. But a quantum value is a whole encrypted message that contains all the possible outcomes of a measurement, plus the relationships between them, all packed into one package.

Kong and his colleagues (building on work by Deutsch and Hayden) show that if you look at these "codes" for the local controls of a system, every single piece of information about the whole system is already sitting right there in the local controls.

The "Ghost" in the Machine

Here is the big idea: The reason entangled particles seem to talk to each other instantly isn't because information is traveling faster than light. It's because the "local" controls on Particle A already contain the full, complete description of the entire system, including Particle B.

Imagine you have two magic dice, one in New York and one in Tokyo. In the old view, rolling the New York die instantly changes the Tokyo die. In Kong's view, the New York die is actually a tiny, super-computer that already holds the entire blueprint of both dice. The "local" die doesn't need to call Tokyo to know what's happening; the information is already encoded in its own local structure.

The paper explicitly states that this description is local. If you do something to the Tokyo die (like a local process), it doesn't change the local description of the New York die. The information stays put.

What This Paper is NOT Saying

It is crucial to understand what this paper doesn't claim, because the authors are very careful about this:

It does NOT say we can send messages faster than light. The paper explicitly admits that we currently have no idea how to "read" these complex, noncommutative codes directly. We can't just look at the New York die and instantly know the Tokyo die's secret without doing some heavy statistical lifting (which takes time and involves many measurements).
It does NOT say the "spooky action at a distance" is fake. The paper agrees that if you look at the system through the lens of standard "state vectors" (the Schrödinger picture), it looks non-local. It doesn't deny the results of famous experiments like Bell's theorem. It just says, "If you change the way you describe the information, the non-locality disappears from the description."
It does NOT claim to have built a machine to do this. The authors are theorists. They are saying, "Mathematically, this makes perfect sense and solves the conceptual puzzle." They are not saying, "We built a device that reads the noncommutative value." In fact, they admit that retrieving this full information is a "huge challenge" for experimentalists and might even be impossible with current technology.

The "Collapse" Confusion

The paper also tackles the idea of "wavefunction collapse"—the moment a measurement forces a quantum system to pick a definite state. The authors suggest that what we call a "local measurement" (like checking one particle) is actually a very complex, non-local process involving the measuring device and the environment.

They use a "CNOT gate" (a quantum logic gate) as an example. They show that even a simple gate that looks like it's just touching one particle is actually a transformation that affects the whole system's description. So, when we think we are doing a "local" measurement that instantly changes a distant particle, we are actually performing a complex dance that involves the whole system. The "spooky" change isn't a violation of locality; it's just a misunderstanding of what a "local" process actually is in the quantum world.

The Bottom Line

The paper suggests that quantum information is local, but it is hidden inside "noncommutative values"—complex, code-like numbers rather than simple real numbers.

The Finding: If you describe a quantum system by the values of its local observables (using these special codes), the system is completely local. The information about the whole universe is stored locally in every part.
The Catch: We can't easily read this information yet. Our current tools (projective measurements) only give us a tiny, statistical slice of the full code.
The Confidence: The authors are very confident in the mathematical logic. They say the theory predicts this clearly. However, they are humble about the experimental reality. They suggest that while the theory allows for local retrieval of information, we don't know how to do it yet, and it might require a totally new way of thinking about experiments.

So, the next time you hear about quantum entanglement being "spooky," remember the analogy of the magic dice. Maybe the dice aren't talking to each other at all. Maybe each die is just a tiny, super-complex library that already contains the story of the whole game, and we just haven't learned how to read the books yet.

Technical Summary: On Locality of Quantum Information in the Heisenberg Picture for Arbitrary States

Problem Statement
The paper addresses the persistent conceptual difficulty regarding the "nonlocality" of quantum mechanics, particularly in the context of entangled systems. While the standard Schrödinger picture describes composite systems via non-separable state vectors (entanglement), leading to the conclusion that a local description of subsystems is impossible without losing information (via reduced density matrices), the author argues that this view is heavily dependent on the Newtonian space-time picture and the assumption that observable values must be real (commutative) numbers. The central problem is to clarify whether a complete, local description of quantum information exists for arbitrary states in a generic composite system, independent of the specific dynamics of projective measurements.

Methodology
The author employs the Heisenberg picture, where the state is fixed and observables evolve, to re-examine the nature of quantum information. The methodology involves two parallel frameworks:

Deutsch-Hayden (DH) Matrix Values: The paper revisits the DH descriptors, which represent local observables as time-evolved matrices. The author modifies this approach by decoupling the matrix values from a specific time-evolution history. Instead of viewing them as $U^\dagger(t)\beta U(t)$ , they are defined as $U_\phi^\dagger \beta U_\phi$ , where $U_\phi$ is a fixed unitary mapping a reference state $|0\rangle$ to the specific state $|\phi\rangle$ . This treats the matrix as a direct "value" of the observable for that state, stripping away redundant historical information.
Noncommutative Values: Drawing from the author's previous work on quantum mechanics as dynamics on a noncommutative geometric phase space, the paper introduces "noncommutative values" for observables. These are defined as the homomorphic image of the observable algebra under a state-dependent evaluation map. For a state $|\phi\rangle$ , the value of an observable $\beta$ is not a single real number but a set of parameters (a real expectation value and complex derivatives) that preserve the algebraic relations of the observables.

The analysis focuses on the algebraic structure of these values, demonstrating that the set of local basic observables (e.g., Pauli matrices for qubits, or position/momentum for particles) generates the full observable algebra, and their values contain the complete information of the system.

Key Contributions and Results

Redefinition of Local Quantum Information: The paper establishes that a complete local description of a composite system is possible in the Heisenberg picture. The "value" of a local observable is not a real number but a noncommutative object (a matrix or a set of complex parameters).
Strong Locality: The author demonstrates that these local values satisfy "strong locality." Specifically, the noncommutative values (or DH matrix values) of local observables in subsystem $A$ are invariant under any unitary transformation $I_A \otimes U_B$ acting solely on subsystem $B$ . This holds true even for entangled states, provided the "value" is defined as the full noncommutative object rather than a statistical distribution of measurement outcomes.
Algebraic Completeness: The values of the local basic observables form a generating set for the algebra of all observables (including nonlocal ones). Consequently, the information contained in the local basic observables is mathematically complete; the values of any global observable can be reconstructed from the local values via the algebraic homomorphism.
Clarification of Redundancy: The paper clarifies that the "dimensionality" of DH descriptors often cited as a drawback (being larger than the state vector) arises from including redundant information about the system's time-evolution history. By fixing the state and defining the value directly, the description becomes more economic and directly tied to the state's information content.
Reinterpretation of Measurement: The paper argues that the apparent nonlocality observed in projective measurements (e.g., wavefunction collapse) stems from treating measurement as a physical process that violates locality. The author contends that standard projective measurements (or CNOT-based models of measurement) are not local unitary processes ( $I_A \otimes U_B$ ) but involve global interactions or decoherence. Therefore, the change in local observables' values upon "measuring" a distant partner does not violate the locality of the quantum information itself, but rather reflects the non-local nature of the measurement interaction.

Significance and Claims
The paper claims to provide a "transparent conceptual picture" of quantum locality by shifting the focus from state vectors to the noncommutative values of observables.

Theoretical Validity: The author asserts that the dynamical theory of quantum mechanics (unitary evolution) inherently predicts a local description of information when observables are assigned noncommutative values. This challenges the prevailing view that local descriptions are impossible for entangled systems.
Nature of Quantum Information: The work posits that quantum information is fundamentally "q-number" (noncommutative) in nature, analogous to how classical information is "c-number" (commutative). The failure to recognize this has led to the misconception of nonlocality.
Experimental Limitations: The paper is modest regarding experimental implications. It explicitly states that while the theoretical description is local, the experimental retrieval of these noncommutative values via standard projective measurements is not currently possible. Projective measurements yield only classical (commutative) statistics, which are insufficient to reconstruct the full noncommutative value. The author emphasizes that the inability to retrieve this information locally experimentally does not negate the theoretical locality of the information itself.
No New Physics: The author clarifies that this is not a new physical theory or a challenge to Bell's theorem regarding local hidden variables (which rely on commutative values). Instead, it is a re-examination of the logical and mathematical implications of existing quantum dynamics, suggesting that the "nonlocality" is an artifact of the description (using real-number values or state vectors) rather than a feature of the physical reality described by the observables.

In conclusion, the paper argues that quantum information is locally encoded in the noncommutative values of local observables. The "nonlocality" of entangled states is a feature of the Schrödinger picture's state description and the limitations of projective measurements, not an inherent property of the quantum information flow in the Heisenberg picture.

The Magic of the "Value"

The "Ghost" in the Machine

What This Paper is NOT Saying

The "Collapse" Confusion

The Bottom Line

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