December 2007
Intermediate to advanced
332 pages
7h 28m
English
Content preview from Quantum Computing ExplainedBecome an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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4
TENSOR PRODUCTS
In quantum mechanics we don’t always work with single particles in isolation. In many cases, some of which are seen in the context of quantum information processing, it is necessary to work with multiparticle states. Mathematically, to understand multiparticle systems in quantum mechanics, it is necessary to be able to construct a Hilbert space H that is a composite of the independent Hilbert spaces that are associated with each individual particle. The machinery required to do this goes by the name of the Kronecker or tensor product. We consider the two-particle case.
Suppose that H1 and H2 are two Hilbert spaces of dimension N1 and N2. We can put these two Hilbert spaces together to construct a larger Hilbert space. We denote this larger space by H and use the tensor product operation symbol ⊗. So we write
The dimension of the larger Hilbert space is the product of the dimensions of H1 and H2. Once again, we assume that dim(H1) = N1 and dim(H2) = N2. Then
Next we start getting down to business and learn how to represent state vectors in the composite Hilbert space.
REPRESENTING COMPOSITE STATES IN QUANTUM MECHANICS
A state vector belonging to H is the tensor product of state vectors belonging to H1 and H2. We will show how to represent such vectors explicitly ...
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O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.Julian F.
I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.Addison B.
I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.Amir M.
I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.