Why does a quantum system with many degrees of freedom imply orthogonality?

In summary, quantum decoherence refers to the loss of coherence in a quantum system when it interacts with its environment. This causes all density matrix to become diagonal after the interaction. The concept can be better understood through Dirac notation and the idea of hidden degrees of freedom in the environment, which implies approximate orthogonality of environmental states. This idea has been rigorously formulated and proved in a mathematical theorem, as discussed in the provided link.
  • #1
Erland
Science Advisor
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Quantum decoherence means that when a quantum system interacts with its environment, coherence is lost, which means that all the density matrix becomes diagonal after the interaction. I never understood why it is so, but I get a clue here:

https://en.wikipedia.org/wiki/Quantum_decoherence#Dirac_notation

It says in particular (where ##\epsilon_i## and ##\epsilon_j## are states of the environment after the interaction):

"Additionally, decoherence requires, by virtue of the large number of hidden degrees of freedom in the environment, that
bac7cc219868a877184124f84ed352815b969501
"

But why does a "large number of hidden degrees of freedom in the environment" imply approximate orthogonality of the evironmental states?

Is there any mathematical theorem where this is rigorously formulated and proved?
 
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  • #3
That was what I wanted to know. Thanks!
 

1. Why is orthogonality important in quantum systems with many degrees of freedom?

Orthogonality is important in quantum systems with many degrees of freedom because it allows for the independent description of different components of the system. This means that the evolution of one component will not affect the others, making it easier to study and understand the behavior of the system as a whole.

2. How does orthogonality relate to the superposition principle in quantum mechanics?

In quantum mechanics, the superposition principle states that a quantum system can exist in multiple states simultaneously. Orthogonality is closely related to this principle as it allows for the distinct and independent description of these different states. This is essential for understanding the behavior of quantum systems with many degrees of freedom.

3. Can orthogonality be violated in quantum systems with many degrees of freedom?

No, orthogonality is a fundamental principle in quantum mechanics and cannot be violated in systems with many degrees of freedom. Each component of the system must have a distinct and independent description, and violating orthogonality would lead to inconsistencies and contradictions in the behavior of the system.

4. How does the concept of entanglement relate to orthogonality in quantum systems?

Entanglement is a phenomenon in quantum mechanics where two or more particles become correlated in such a way that the state of one particle cannot be described independently of the others. In systems with many degrees of freedom, entanglement can arise between different components, but orthogonality ensures that each component can still be independently described.

5. Are there any practical applications of understanding the orthogonality principle in quantum systems?

Yes, understanding orthogonality in quantum systems with many degrees of freedom has many practical applications, particularly in quantum computing and cryptography. Orthogonality allows for the manipulation and control of individual components of a quantum system, which is crucial for these technologies to function properly.

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