Uploaded December 2025 | Updated September 2026, 2 weeks ago

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This videos visualizes the tensor product operation. It uses this visualization to continue the example of a two qubit system and entanglement.

Some notes:

Implementation of qubits:
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A common way is using quantum spin. The mathematical model of quantum spin for an electron is similar to how we show the qubit's representation. One difference though: the two states are 'up' and 'down'. This means that while in the abstract space they are 90 degrees apart, in the real world they are 180 degrees apart. So rotating the device by x degrees correspond to x/2 degrees rotation of the coordinate system in the abstract space. General 3d rotations of the device involve the imaginary part of the vector's components, which we ignored in this video.

More about entanglement:
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The following video in this channel explores the Bell theorem: youtu.be/v7jctqKsUMA
It shows how entanglement can be used to do things that classical physics doesn't allow, but not faster-than-light communication. The vector shown in that video as the spin is not really quantum spin, because quantum spin can't be fully drawn. It's related to the vector shown in the current video, except again with a double angle relation (see note above): if it's x degrees away from the vertical axis, then in the abstract space it would be x/2 degrees away. That's why you'll see theta/2 in the formulas there.

Singular value decomposition (SVD):
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SVD says every matrix can be decomposed to three R*S*T where:
* R and T are unitary, that is they perform rotations and reflections
* S is diagonal, that is it performs scaling.
Further more, S only has positive factors.
In the video, we allowed S to have negative factors, which excuses R and T from performing reflections, leaving them to do only rotations.

Degree of entanglement:
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There are several ways to measure the degree of entanglement. A common way is using "Entropy of entanglement", which measures the entropy of the distribution of the 'best' (most certain) angle of one of the qubits. It can be shown it's the same entropy for both qubits.

Music ID: ZYPKL17ZZASNJQFG