Locality Implies Complex Numbers in Quantum Mechanics

The paper demonstrates that under the independent source assumption, real-number quantum theories are equivalent to complex-number quantum theory only if they incorporate nonlocal hidden degrees of freedom, suggesting that complex numbers are essential for describing entanglement between independent systems.

The Gist

The Big Question: Do We Really Need "Imaginary" Numbers?

For a long time, physicists have been puzzled by a strange feature of quantum mechanics: it relies heavily on complex numbers. You know, those numbers that include "imaginary" parts (like $i$, where $i^2 = -1$).

In the real world, we measure things like probability, position, and speed using real numbers (1, 2, 3.14, etc.). We don't need imaginary numbers to build a bridge or predict the weather. So, why does the quantum world need them?

Some scientists wondered: Could we rewrite the entire rules of quantum mechanics using only real numbers? If we could, it would mean complex numbers are just a convenient math trick, not a fundamental requirement of nature.

The "Real-Number" Workaround (Stueckelberg's Rule)

The paper starts by looking at an old idea from the 1960s by a physicist named Stueckelberg. He showed that you can translate a complex quantum system into a real one.

The Analogy: The 2D Shadow vs. The 3D Object
Imagine you have a 3D object (a complex number system). If you shine a light on it, you get a 2D shadow (a real number system).

• The Trick: To make the 2D shadow act exactly like the 3D object, you have to double the size of the shadow. A single "pixel" in the complex world becomes two "pixels" in the real world.
• The Result: For a single, isolated system (like one electron in a box), this works perfectly. You can describe everything using only real numbers, provided you just use a slightly larger "canvas."

The Problem: When Two Systems Meet (The "Locality" Test)

The paper's authors, Feng, Ren, and Vedral, decided to test what happens when two independent systems interact.

The Scenario:
Imagine Alice and Bob are in different rooms (spacelike separated). They each have their own quantum system. They are "independent sources."

• Locality Principle: Alice's actions in her room should not instantly affect Bob's room, and vice versa. They are independent.
• The Goal: Can we describe what happens when their systems interact (entanglement) using only real numbers, while keeping them independent?

The Failure:
The authors found that if you try to use the standard "real-number translation" (the 2D shadow method) for two independent systems, it breaks.

• The Math Glitch: In the complex world, combining two systems is like multiplying two numbers. In the real-world translation, simply multiplying the "shadow" versions of the systems doesn't give you the right result. The math doesn't line up.
• The Consequence: Standard real-number quantum theory cannot explain how two independent systems become entangled without breaking the rules of math.

The "Fix" and Its Hidden Cost

So, can we fix the math? The paper says yes, but with a massive catch.

To make the real-number theory work for two independent systems, you have to invent a special "glue" or a new rule for how they combine. The authors call this a Modified Tensor Product.

The Analogy: The Invisible Telepathic Thread
Imagine Alice and Bob are trying to combine their puzzle pieces.

1. Complex Theory: They just snap the pieces together naturally.
2. Real Theory (The Fix): To make the pieces snap together correctly, you have to introduce a hidden, invisible thread that connects Alice's hand to Bob's hand instantly, even though they are in different rooms.

This "thread" is what the paper calls a nonlocal map.

• It means that to describe the system using only real numbers, you have to assume that Alice's local action is secretly correlated with Bob's local action through a hidden, invisible mechanism.
• This violates the principle of Locality (the idea that independent things shouldn't influence each other instantly).

The Conclusion: Why Complex Numbers Are Essential

The paper concludes with a powerful insight:

Complex numbers are not just a math trick; they are the "glue" that keeps the universe local.

• If you insist on using only real numbers, you are forced to admit that the universe has hidden, spooky, nonlocal connections (telepathy between independent systems) to make the math work.
• If you accept complex numbers, you can describe entanglement between independent systems perfectly without needing any hidden, nonlocal tricks.

The Takeaway:
Nature seems to prefer complex numbers because they allow independent systems to interact in a way that respects the speed of light and the separation of space. If we try to force the universe into a "real number only" box, we have to break the rule of locality. Therefore, complex numbers are indispensable for describing the quantum world accurately.

Summary in One Sentence

You can pretend the quantum world uses only real numbers, but if you do, you have to assume that independent systems are secretly connected by invisible, instant threads; complex numbers are the only way to describe these systems without those weird, nonlocal connections.

Technical Summary

1. Problem Statement

The paper addresses a foundational question in quantum mechanics: Are complex numbers physically indispensable, or are they merely a mathematical convenience?

• Context: While standard quantum theory (SQT) utilizes complex Hilbert spaces, alternative formulations using real or quaternionic Hilbert spaces have been proposed (e.g., by Birkhoff, von Neumann, and Stueckelberg).
• Recent Developments: Renou et al. (2021) and subsequent experiments demonstrated that real-number quantum theory (RQT) cannot simulate the results of quantum protocols involving network structures with independent sources (bi-locality) using standard tensor products.
• The Conflict: Some recent claims suggest that RQT might be consistent with the assumption of independent sources (locality) if the tensor product is redefined.
• Core Question: If RQT is modified to satisfy the independent source assumption, does it retain the property of locality, or does it implicitly require non-local mechanisms? The authors aim to determine if complex numbers are necessary specifically for describing entanglement between independent systems without invoking hidden non-locality.

2. Methodology

The authors employ a rigorous algebraic framework based on Stueckelberg's rule to map complex quantum states and operators to real-number equivalents.

• Stueckelberg's Mapping ($Map_R$):
  • A $d$-dimensional complex state $|\psi\rangle = \sum (a_j + i b_j)|j\rangle$ is mapped to a $2d$-dimensional real state:
    $$\tilde{|\psi\rangle} = \sum a_j |0\rangle|j\rangle + b_j |1\rangle|j\rangle$$
  • Complex density matrices $\rho$ and observables $A$ are mapped to real matrices $\tilde{\rho}$ and $\tilde{A}$ using the operator $XZ = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$:
    $$\tilde{A} = I \otimes \text{Re}(A) + XZ \otimes \text{Im}(A)$$
• Single System Consistency: The authors first verify that for a single system, addition, multiplication, evolution, and expectation values in the real-number formalism are mathematically equivalent to the complex formalism.
• Composite System Analysis: The core analysis focuses on composite systems formed by independent sources (e.g., Alice and Bob preparing states independently). The authors compare two approaches:
  1. Standard Tensor Product: Applying the standard tensor product to the real-mapped states ($\tilde{\rho}_1 \otimes \tilde{\rho}_2$).
  2. Modified Tensor Product: Defining a new operation ($\tilde{\otimes}$) that attempts to replicate the complex tensor product result ($Map_R(\rho_1 \otimes \rho_2)$).

3. Key Contributions and Results

A. Inconsistency of Standard Tensor Product in RQT

The authors demonstrate that for independent systems, the standard tensor product of real-mapped states does not equal the real-mapped tensor product of the original complex states:
$$Map_R(\rho_1 \otimes \rho_2) \neq \tilde{\rho}_1 \otimes \tilde{\rho}_2$$
Specifically, the dimensions and the resulting matrix structures differ. This confirms that standard RQT cannot simulate entanglement between independent sources, aligning with previous findings by Renou et al.

B. The Necessity of a Non-Local Map

To reconcile RQT with the independent source assumption, the authors analyze the "Modified Tensor Product" proposed in recent literature (e.g., Hita et al., Hoffreumon et al.).

• They show that to make the real-number description consistent with complex quantum theory for composite systems, one must introduce a nonlinear map (denoted as $\tilde{P}$).
• This map acts on the ancillary qubits (the "real/imaginary" bits introduced by Stueckelberg's rule) of the two independent systems. It effectively calculates the parity of the ancillary states:
  • $\tilde{P}(|00\rangle) = |0\rangle$
  • $\tilde{P}(|11\rangle) = -|0\rangle$
  • $\tilde{P}(|01\rangle) = \tilde{P}(|10\rangle) = |1\rangle$
• Crucial Finding: While this map $\tilde{P}$ allows the real-number theory to reproduce the predictions of complex quantum theory, it is fundamentally non-local.
  • Even if System 1 and System 2 are spacelike separated and prepared independently, the operation $\tilde{P}$ requires a joint action on the ancillary degrees of freedom of both systems.
  • This implies that a local operation on System 1 is correlated with System 2 through a "hidden nonlocal degree of freedom."

C. Equivalence to Stueckelberg's Rule

The authors argue that this modified tensor product is mathematically equivalent to the original Stueckelberg rule applied to composite systems. Both approaches require a non-local mechanism to describe the entanglement of independent sources within a real-number framework.

4. Significance and Conclusion

• Indispensability of Complex Numbers: The paper concludes that complex numbers are indispensable for describing quantum processes involving entanglement between independent systems.
• Locality vs. Real Numbers: There is a fundamental trade-off:
  1. If one insists on locality (independent sources with no hidden non-local connections), one must use complex numbers.
  2. If one insists on using only real numbers, one must accept a hidden non-local map that violates the spirit of locality, even if the sources are independent.
• Resolution of Recent Debates: The results clarify that recent claims suggesting RQT is consistent with independent sources are only valid if one accepts a non-local hidden variable structure. Just as Bell tests rule out local hidden variables but not non-local ones, the inability of standard RQT to simulate network experiments does not rule out RQT entirely; it only rules out RQT under the strict assumption of locality.
• Physical Implication: Complex numbers are not just a mathematical convenience; they are the natural language for describing local interactions and entanglement in a universe where independent sources do not share hidden non-local correlations.

In summary, the paper provides a rigorous proof that locality implies complex numbers. Any attempt to formulate quantum mechanics using only real numbers while preserving the independence of sources inevitably introduces non-local hidden degrees of freedom, thereby making the complex Hilbert space the most fundamental and local description of quantum reality.