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Essential Mathematics for Quantum Computing
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Tensor products are a way to combine vector spaces. One of the postulates of quantum mechanics is that the state of a qubit is completely described by a unit vector in a Hilbert space. The problem then becomes how to deal with more than one qubit. This is where a tensor product comes in. Each qubit has its own Hilbert space, and to describe many qubits as a system, we need to combine all their Hilbert spaces into one bigger Hilbert space.
Mathematically, that means that if we have a Hilbert space H and another Hilbert space J, we denote their tensor product as:
If H is an h dimensional space and J is a j dimensional space, then the dimension of the combined space M is h ⋅ j. In other words:
Before we go any farther, let's look at the tensor product of two vectors.
The tensor product of two vectors is denoted in the following way in bra-ket notation. You'll notice that there are four different ways...
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