# Question on tensor product

1. Jan 28, 2010

### Goldbeetle

Dear all,
why is it that the tensor product is used to describe two quantum system described by Hilbert spaces H1 and H2? What were the example systems or situations that were generalized and that led to this postulate?
Thanks.
Goldbeetle

2. Jan 28, 2010

### meopemuk

In my opinion, these two papers provide a satisfactory answer

T. Matolcsi, "Tensor product of Hilbert lattices and free orthodistributive product of orthomodular lattices", Acta Sci. Math. (Szeged), 37 (1975), 263.

D. Aerts, I. Daubechies, "Physical justification for using the tensor product to describe two quantum systems as one joint system", Helv. Phys. Acta, 51 (1978) 661.

Eugene.

3. Jan 28, 2010

### Fredrik

Staff Emeritus
I read the Aerts and Daubechies article (ok, not the whole thing, but enough to get the general idea) after the last time you posted these references. I think it's a very well-written article, but I'm not so sure that it can give you a much deeper understanding of the answer to the OP's question than the simple argument:
I had to elaborate on what the point of my calculation was the next day:
At first it seemed to me that Aerts and Daubechies made no reference to probabilities at all, but that's not entirely true. Their starting point is the quantum logic approach to QM, in which we use a mathematical structure (which is isomorphic to the lattice of closed subspaces of a complex separable Hilbert space) to represent the set of statements of the form "if I measure the observable A, the result will be in the set E with probability 1". (An arbitrary member of this set can have a different A and a different E, but the probability is always 1). So what they're doing has something to do with probabilities as well. That's why it seems to me that even if I understood the details of their argument, it wouldn't give me a much deeper understanding of why we use the tensor product. (Let me know if you disagree).

It's still a very nice paper though, and I intend to return to it if I ever get around to study quantum logic seriously. Unfortunately that probably won't be until next year.

Goldbeetle, you might find this thread useful too. It explains the definition/construction of the tensor product pretty well.

4. Jan 28, 2010

### meopemuk

Fredrik,

You have proved that tensor product is sufficient to get P(a,b)=P(a)P(b). However, you haven't proved that tensor product is necessary to get P(a,b)=P(a)P(b). In other words, after your proof there is still a possibility that some other construction (not the usual tensor product) can be in agreement with all probability postulates.

Matolcsi and Aerts&Daubechies basically filled this gap. They formulated some postulates about probabilities in compound systems (like yours P(a,b)=P(a)P(b)) and then they proved that there are only two inequivalent ways to satisfy these postulates in quantum mechanics. One is the usual tensor product of component's Hilbert spaces H = H1 x H2. The other one is the tensor product of H1 with the dual Hilbert space of H2: H = H1 x H2*. I am not quite sure about the physical meaning of the second possibility. However, it remains there as a mathematical fact.

Eugene.

5. Jan 28, 2010

### strangerep

I wonder why they make this distinction. I was under the impression that ordinary
Hilbert spaces are self-dual, i.e., H2 is isomorphic to H2*, so why bother?

6. Jan 28, 2010

### meopemuk

Apparently, the two ways of building the compound Hilbert space are not isomorphic. Both papers claim the same result independently. I don't know what is the significance of this.

Eugene.

7. Jan 29, 2010

### Goldbeetle

Thanks too all! Actually, I was looking for something else, namely some examples of compound systems (described for instance by a wave function) that can be interpreted as tensor products of hilbert spaces and that motivate the introduction of the generalization of the tensor product in conceptual framework of QM. The books I've consulted so far define more or less the tensor product and then use it, but lack any motivation.

8. Jan 29, 2010

### Fredrik

Staff Emeritus
The Aerts & Daubechies article has an example. See page 1 and the beginning of page 2.

9. Jan 29, 2010

### Goldbeetle

Fredrik,
thanks. O, it's clear now for square-integrable wave function.

10. Jan 29, 2010

### strangerep

I guess it's because even though there's a one-to-one mapping between
H and H*, the mapping is antilinear and hence not strictly isomorphic since
it doesn't preserve linearity. Cf. Ballentine's footnote on p27, and his eqn (1.8).

I.e., if $$\langle F| \leftrightarrow |F\rangle$$ , then $$\bar{c}\langle F| \leftrightarrow c|F\rangle$$ , etc.

11. Jan 29, 2010

### meopemuk

I guess you are right. The Aerts&Daubechies' proof requires use of structure-preserving maps between two quantum propositional systems. The analysis of such maps has been done in the preceding paper of the same authors:

D. Aerts and I. Daubechies, "About the structure-preserving maps of a quantum mechanical propositional system", Helv. Phys. Acta, 51 (1978), 637.

The theorem proved there is a generalization of the famous Wigner theorem about unitary/antiunitary operators. So, the two possibilities H1 x H2 (which is isomorphic to H1* x H2*) and H1 x H2* (which is isomorphic to H1* x H2) are related to two inequivalent (unitary and antiunitary) representations of structure-preserving maps.

Eugene.