# What does simplicial mean in the context of QM?

• I
• Jilang
In summary, simplicial and non-simplicial refer to shapes in the context of quantum mechanics. They also have different meanings outside of quantum mechanics. The concept came up in a forum discussion and was defined as polytopes in classical probabilistic theories. However, in quantum mechanics, the concept is not as straightforward and can be represented in different ways.
Jilang
What does simplicial and non simplicial mean in the context of quantum mechanics?

Jilang said:
What does simplicial and non simplicial mean in the context of quantum mechanics?
What do they mean outside quantum mechanics and what makes you think they mean anything within quantum mechanics ?

atyy said:
Thanks, but I am having some problems deciphering it. Is there a straightforward defininition?

Jilang said:
Thanks, but I am having some problems deciphering it. Is there a straightforward defininition?

Another word for simplicial is polytope - it's a shape with pointy points like a square or a triangle. In a classical probabilistic theory, every mixture is a unique mix of pure states. But this is not true for a mixed density matrix, which can be written in many ways as a mixture of pure density matrices.

Thanks. I will take that to mean of no fixed geometry.

Usually the quantum set is pictured as a circle.

## 1. What is a simplicial complex?

A simplicial complex is a mathematical structure used in topology to study geometric shapes. It is made up of simplices, which are geometric objects formed by joining vertices, edges, and higher dimensional elements.

## 2. How is a simplicial complex related to quantum mechanics?

In the context of quantum mechanics, a simplicial complex can represent the possible states of a quantum system. Each simplex in the complex corresponds to a different state, and the connections between simplices represent the transitions between states.

## 3. What is the significance of using a simplicial complex in quantum mechanics?

Using a simplicial complex can provide a more intuitive and visual representation of quantum systems, allowing for better understanding and analysis of their behavior. It also allows for the application of topological methods to study and solve quantum mechanical problems.

## 4. How is a simplicial complex constructed in quantum mechanics?

A simplicial complex can be constructed by assigning a set of basis states to the vertices of the complex and then connecting them based on the allowed transitions between states. The resulting structure is a representation of the quantum system's state space.

## 5. Can simplicial complexes be used to solve problems in quantum mechanics?

Yes, simplicial complexes have been used to analyze and solve various problems in quantum mechanics, such as determining the energy levels of a system or finding the shortest path between two states. They can also be applied to study entanglement and quantum information processing.

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