The “No” Theorems: Physics’ Unbreakable Rules


By: Hannah Pell

Physics tells us a lot about what we can do. We can use it to predict the motions of the stars to the most fundamental constituents of matter and nearly everything in between; physics can be a powerful tool for us to realize new possibilities beyond what we’ve known before.

However, sometimes it can be just as helpful to know what we can’t do. Just as there are rules that govern our daily lives limiting what’s permissible — no talking in the library; no running by the pool; no shirt, no shoes, no service — so too there are restrictions on what’s permissible in the physical world. Many of these rules can be grouped together simply as the “no” theorems.

No-Go Theorems and Quantum Information Theory 

A “no-go” theorem in theoretical physics is, well, exactly what it sounds like — a theorem describing a situation that is not physically possible. In mathematics, the way to show that something cannot happen is to argue a proof of impossibility, which is actually more difficult than showing that something is in fact possible.

There are several theorems that are the “no-go” type within quantum information theory. One is the “no-communication” (or “no-signaling”) theorem which shows that instantaneous sharing of information between two observers is an impossibility. This theorem is an important qualifier to quantum entanglement (more popularly known as Einstein’s “spooky action at a distance”). Quantum entanglement is an effect that may characterize widely separated but potentially correlated events, and the “no-communication” theorem restricts the information that can be communicated between two far-away observers of an entangled state.

Remember our cryptographic friends Alice and Bob? The “no-communication” theorem says that if Alice and Bob are two independent observers taking simultaneous measurements on parts of a quantum-mechanical system, there is no action that Alice can perform on her part that would be detectable by Bob while observing his part (and vice versa).

There are also the “no-cloning,” “no-broadcast,” and “no-deleting” theorems. According to the “no-cloning” theorem, pure quantum states — which can be described by a single vector — cannot be identically copied. The “no-broadcast” theorem extends this to mixed quantum states, or combinations of pure quantum states. The “no-deleting” theorem states that if there are two copies of a quantum state, it is impossible to delete one of the copies. The no-cloning and no-deleting theorems together imply that quantum information is conserved — like energy, it cannot be created or destroyed (this is also described in the “no-hiding” theorem). 

 The “no-teleportation” theorem stems from “no-cloning” and states that quantum information (“qubits”) cannot be converted into classical information and then reconstructed. Although it’s impossible to make an exact copy, imperfect clones of quantum states are physically allowed. In fact, cryptographers can utilize this principle in quantum key distribution to detect eavesdropping.

No Free Lunch Theorem and Machine Learning 

“There ain’t no such thing as free lunch.” You may have heard this adage before as a reminder that you can’t get something for nothing; there are always hidden costs. The saying traces back to the late 1800s and early 1900s — saloons would offer a “free” lunch to any customer who bought at least one drink. However, since the foods offered were high in salt, customers would generally buy several drinks, so the “free” lunch wasn’t really free.

The “no free lunch” theorem in machine learning implies something similar. The theorem originated in a 1997 publication titled “No Free Lunch Theorems for Optimization” authored by physicists David H. Wolpert and William G. Macready, who wrote that no free lunch theorems “establish that for any algorithm, any elevated performance over one class of problems is offset by performance over another class.” In other words, Wolpert and Macready showed that the computational cost for finding a solution is the same for any solution method when averaged overall problems in the particular class. Even when the goal is optimization, there are no shortcuts to the desired solution — no free lunch. 

 No-Hair Theorem and Black Holes 
The “no-hair” theorem states that all black holes can be characterized by only three externally observable parameters: mass, electric charge, and angular momentum. “Hair” is a metaphor coined by theoretical physicist John Wheeler to represent all additional information about a black hole that disappears behind the black hole event horizon.

Black holes have been described as “hairless” because there are no additional “hairy” attributes to distinguish them other than mass, electric charge, and angular momentum; if two black holes have the same quantities for each parameter, then they are exactly the same. Recently, however, astrophysicists have found that one special type of black hole has “hair that can be combed.” Their study suggests that gravitational hair can be measured by gravitational-wave detectors as an additional property of a black hole, thereby violating the no-hair theorem. So rather than “no hair,” it may be more apt to say, “no hair most of the time.”

There is still no proof of a generalized no-hair theorem, and mathematicians refer to it as the “no-hair conjecture” because of this. A conjecture differs from a theorem in that it is mathematically unproven but thought to be true, whereas a theorem has been proven true. Even though the no-hair theorem lacks rigorous proof, it is in accordance with general relativity, so it’s widely accepted as valid within the physics community.

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