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The tensor product is not exactly like the Cartesian product because not all the elements of the tensor product of the Hilbert spaces representing each qbit can
by kgwgk 8y ago
The tensor product is not exactly like the Cartesian product because not all the elements of the tensor product of the Hilbert spaces representing each qbit can be written as the tensor product of the individual qbits.
In the cross join case, all the elements are pairs of elements from the original tables. In the Hilbert space of a composite system only the separable states can be written as the tensor product of states of the subsystems.
"Observation 3.6.5: Interesting two-qubit states. Not every 4-dimensional vector can be written as a Kronecker product of two 2-dimension vectors, e.g., you can have a two-qubit state |Ψ⟩ such that:
|Ψ⟩ = ̸= |ψ⟩|φ⟩, for any one-qubit state |ψ⟩ and |φ⟩
These types of states (called entangled states) are very intriguing and play a
fundamental role in quantum mechanics."
- lisper 8y agoInterestingly, I was just about to edit my original comment to say exactly that: there are tables which are exactly analogous to entangled states in that they cannot be written as a cartesian product of two other tables. But I have to stress that I do not intend this to be a deep insight, just an interesting (IMHO) observation.
- kgwgk 8y agoIt is interesting that the basis of the tensor product of the spaces can be constructed as the tensor product of each possible combination of the basis of the spaces. Among other things, it makes clear that the dimension of Hilbert space for the composite system is the product of the dimensions of the Hilbert spaces. But this can distract us from the most interesting part, which is that the space spanned by these basis vectors is much larger than the simple cartesian product. Focusing on superficial similarities is a two-edged sword: anchoring on a familiar concept can help or make things more difficult. And it's guaranteed to annoy people! (I'm mostly getting over it, but I still dislike the use of the word "tensor" to refer to multi-dimensional arrays. Tensor has a meaning in geometry and tensor algebra is not about doing linear algebra on 2-d slices of a larger-dimensional object.)