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Asked Copilot for the difference between the dotproduct and a projection:

In quantum computing, the concepts of inner product and projection are fundamental but serve different purposes:

### Inner Product

- **Definition**: The inner product (or dot product) of two vectors is a scalar value that measures the "overlap" between them. In the context of quantum mechanics, it is often used to determine the probability amplitude.

- **Notation**: If you have two quantum states, represented as kets \(|\psi\rangle\) and \(|\phi\rangle\), their inner product is written as \(\langle\psi|\phi\rangle\).

- **Purpose**: It helps in calculating probabilities and understanding the relationship between different quantum states. For example, the probability of transitioning from state \(|\psi\rangle\) to state \(|\phi\rangle\) is given by the modulus squared of the inner product \(|\langle\psi|\phi\rangle|^2\)¹.

### Projection

- **Definition**: A projection operator projects a quantum state onto a subspace. It is used to extract the component of a state that lies in a particular direction or subspace.

- **Notation**: If \(|\phi\rangle\) is a quantum state, the projection operator \(P\) onto this state is given by \(P = |\phi\rangle\langle\phi|\).

- **Purpose**: Projections are used to filter out specific components of a quantum state. For instance, if you apply the projection operator \(P\) to a state \(|\psi\rangle\), you get \(P|\psi\rangle = |\phi\rangle\langle\phi|\psi\rangle\), which is the component of \(|\psi\rangle\) in the direction of \(|\phi\rangle\)².

In summary, the inner product measures the overlap between two states, while a projection operator extracts the component of a state in a specific direction.

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Reference to a helpful site for beginners:

Qubit, An Intuition #2 — Inner Product, Outer Product, and Tensor Product in Bra-ket Notation | by Andi Sama | Medium