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Source: Visualization of tensors – part 3A – YouTube

This videos visualizes the direct sum space and starts explaining about tensor product spaces. It shows a simple example from classical physics and a two qubit system as an example from quantum physics. We demonstrate how tensors are related to quantum entanglement.

Notes about confusing terminology: ————————————–

• The term “tensor product” can refer to the combination of two vector spaces W=U*V, creating a tensor product space. But it also sometimes refers to the combination of two vectors u and v creating a simple tensor w=u*v. The latter is sometimes also called “tensor product mapping”. In the video we used an outer product, which can perform the tensor product mapping given a coordinate system. Later in the video we refer to it as “tensor product welding”.
• The term “tensor” is sometimes defined as an element in a tensor product space W=U*V. Sometimes though the definition is more restrictive, stating tensors are only elements of a tensor product space if the vectors spaces are the same, i.e., W=V*V (allowing also for dual spaces, something we haven’t touched yet).
• The term “simple tensor” used in the video has some alternatives: “decomposable tensor”, “pure tensor”, or “elementary tensor”.
• Quantum physics uses “Hilbert spaces” which is a vector space with some additional properties, among them the existence of an inner product operation. A postulate of quantum physics says every system has a Hilbert space as its state space.
• If two systems have state spaces U and V, then the combined system’s space is their tensor product W=U*V, as explained in the video. For many purposes W is just another Hilbert space which acts as the state space of a system, hence its elements are still called “vectors”. We need their tensor-ish properties only for some purposes.
• The video shows the distinction between separable states and entangled states. There is another distinction between “pure states” and “mixed states”. A pure state corresponds to an actual physical state. The video only shows pure states. A mixed state reflects “classical uncertainty”, i.e., the fact that we may not know the physical state in full. For example, say we put in two boxes an ‘off’ qubit and an ‘on’ qubit. Then we shuffle the boxes and pick one. We know this box contains a qubit in an ‘off’ state or ‘on’ state, with 50-50% distribution. This is a mixed state: the qubit is either off or on, we just don’t know. This is a different from 50-50% superposition which is a pure state: the physical state itself contains uncertainty about the on-off property.
• Quantum physics has formalism for handling mixed states which includes an “outer product” operation, resulting with a “density matrix”. This is not what’s shown in the video. The video only refers to pure states.