The n-Category Café

Yes, it took me a while to appreciate that they were actually characterizing two categories. (We need a picture of two amigos.) So the theorem is:

Every monoidal dagger category satisfying such-and-such conditions is equivalent to either $\mathbf{Hilb}_\mathbb{C}$ or $\mathbf{Hilb}_\mathbb{R}$.

I wonder if there’s another way of distinguishing $\mathbf{Hilb}_\mathbb{C}$ from $\mathbf{Hilb}_\mathbb{R}$.

There’s a nontrivial involution of the category $\mathbf{Hilb}_\mathbb{C}$, right? Given a Hilbert space $H$, we get a new one, $\overline{H}$, which is the same as $H$ except for two things. First, scalar multiplication is twisted by complex conjugation: to scalar-multiply in $\overline{H}$, you conjugate the scalar and then scalar-multiply in $H$. Second, the order of the inner product is reversed. This should mean that the inner product on $\overline{H}$ is still sesquilinear.

But I don’t see how there could be a nontrivial involution of $\mathbf{Hilb}_\mathbb{R}$.

This is off the top of my head. I may be confused.

All very interesting! And peculiarly unrelated-seeming to the other ways that the “reconstruction of quantum theory” genre has found to select $\mathbb{C}$ instead of $\mathbb{R}$. The most common, I think, is to postulate what’s called “local tomography”; another is to posit that the generators of time evolution are conserved “observables”.

Thanks for pointing this out! I see that a paper covering an actual experiment has been published already (covered here), but I cannot find the published version of your paper.

There’s an assumption discussed in the conclusion of arXiv:2101.10873 that I found of interest, that “the independence of two or more quantum systems is captured by the tensor product structure”. To write a complex matrix as a real matrix, Renou et al. introduce a little extra dimensionality, but it’s not really a degree of freedom — it’s not dynamical, but rather only notational. They then assume that these embiggened matrices then compose by the tensor product. As they note, it’s possible to drop that assumption; what I wonder is whether there might be a positive argument to do so. In other words, does the tensor-product assumption properly reflect the physical criteria for compositing systems, like the impossibility of instantaneous signalling? Or do those physical requirements have a different implication for a theory set up using nondynamical auxiliary qubits? Is Eq. (3) of Renou et al. the right way to go even for a bipartite system, never mind three or more portions?

Suppose that Alice has two systems on opposite sides of her laboratory. Each system she treats individually using quantum theory, describing her preparation of the left-hand system by a density matrix $\rho_L$ and that of the right-hand system by $\rho_R$, modeling the measurements she might perform by POVMs, etc. Then physical constraints on signalling imply that effects combine by the tensor product while the possible joint states for the $L R$ pair are “positive on pure tensor” operators: The probability of obtaining the pair of outcomes $E, F$ on the left and right systems is $\mathrm{tr}[(E \otimes F)W_{L R}]$ for some Hermitian operator $W_{L R}$ that is unit-trace but not necessarily positive semidefinite. Moreover, we can find some other effects $\tilde{E}, \tilde{F}$ and some joint density matrix $\rho_{L R}$ such that the probability of getting that pair of outcomes is $\mathrm{tr}[(\tilde{E} \otimes \tilde{F}) \rho_{L R}]$. In other words, correlations that respect no-signalling and “local quantumness” can be emulated quantum-mechanically.

But what if we change our mathematical expression of being “locally quantum” to use embiggened real matrices? It seems to me that, from more primitive assumptions like these, one might conclude that the way to compose embiggened matrices is to extract the complex matrices, tensor-product those and embiggen the result. In other words, the way to have the theory be physically local in the way it should be is to have one generally ambient notational qubit floating about the equations. A “gambit” is somewhat like a ubit, but it doesn’t interact or have its own rotation rate; it just sits there like a cat in a sunbeam.

On the one hand, this sounds like a lengthy gloss on Renou et al.’s penultimate paragraph (“If we drop this assumption…”). On the other, there might be a little room here to explore (paraphrasing Pitowsky) the distinction between mathematical artifact and physical content.

My favorite reason to exclude real-number quantum theory is that a complete set of equiangular lines exists in $\mathbb{C}^4$ but not in $\mathbb{R}^4$. There’s some interesting number theory in that.