bra ket notation in mathematica tensor product

Contrary to the common multiplication it is not necessarily commutative as each factor corresponds to an element of different vector spaces. The tensor product is linear in both factors. Each element of V can be written as a linear combination The m-fold tensor product of a vector space is denoted by V ⊗ V ⊗ … ⊗ V = V ⊗ m. Using the bra-ket notation, the abbreviation ∣ i j⟩ = ∣ i⟩ ⊗ ∣ j⟩ is quite common. In quantum mechanics, bra-ket notation is a standard notation for describing quantum states, composed of angle brackets and vertical bars. If the vectors ϕ I, i form a base of V I and similar ϕ I I, j in V I I, we get the base vectors of V wih the tensor product ϕ i j = ϕ I, i ⊗ ϕ I I, j. If the dimensions of V I and V I I are given by dim(V I) = n I and dim(V I I) = n I I, the dimension of V is given by the product dim(V) = n I n I I. The tensor product of both vector spaces V = V I ⊗ V I I is the vector space V of the overall system. Please refer to the previous article (published in July 12 2021), Qubit, An Intuition 1 First Baby Steps in Exploring the Quantum World for a discussion on a single qubit as a computing unit for quantum computation. The common way is to introduce tensor products for vector spaces V I and V I I and their elements ψ I and ψ I I. TL DR 2 qubits inner product, outer product, and tensor product in bra-ket notation, with examples. Thus, using the bra-ket notation, the vectors ∣ ψ I⟩ and ∣ ψ I I⟩ describe the states of system I and I I with the state of the total system given by the tensor product ∣ ψ I⟩ ⊗ ∣ ψ I I⟩. For example, let us have two systems I and I I with their corresponding Hilbert spaces H I and H I I. Each subsystem is described by a vector in a vector space (Hilbert space). Tensor Products are used to describe systems consisting of multiple subsystems.

Bra ket notation in mathematica tensor product