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In [1], the problem of simulating a Hamiltonian using repeated applications of a different set of Hamiltonians is discussed.

In particular, let $A$ and $B$ be a pair of Hermitian operators, and let $\mathcal L$ be the algebra generated from $A, B$ through repeated commutation $^{\mathbf{(\dagger)}}$.

The author then asks (first paragraph of third page) what is $\mathcal L$ for an arbitrary pair of observables $A$ and $B$, and argues that $\mathcal L$ is the space of all Hermitian matrices, unless (quoting from the paper) both $e^{iA t}$ and $e^{iB t}$ lie in an $n$-dimensional unitary representation of some Lie group other than $U(n)$.

I'm not too familiar with the theory of Lie algebras, so this statement is quite cryptic for me. How can this be shown more explicitly? Equivalently, is there a more direct way to show this fact?


$(\dagger)$: More explicitly, this is the vector space spanned by $A, B, i[A,B], [A,[A,B]], ...$

[1] Lloyd 1995, Almost Any Quantum Logic Gate is Universal, Link to PRL.

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    $\begingroup$ For anyone who likes Lie algebras more: You can just take those two symbols A, B and make the free Lie algebra $Free_2$. So no relations on the $\dagger$ besides those that make sure it is still a Lie algebra. Then let $\rho$ be the representation that goes down to actual matrices (so $\rho (A)$ is what you're calling A above). From here there are some very powerful theorems that go under the name of Kashiwara-Vergne. These are useful in understanding that long Baker-Campbell-Hausdorff formula (stronger formula than Trotter). $\endgroup$ – AHusain Jun 12 '18 at 21:51
  • $\begingroup$ @AHusain that sounds like something worthy of being an answer (not one easily understandable by me, but nonetheless..)! $\endgroup$ – glS Jun 13 '18 at 9:22
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I'm not too familiar with the theory of Lie algebras, so this statement is quite cryptic for me. How can this be shown more explicitly? Equivalently, is there a more direct way to show this fact?

At around the same time, David Deutsch et al. proved the same thing in this paper: Universality in Quantum Computation (1995), but without ever using the word "algebra" or "Lie" in the whole paper. The proof starts on page 3 and the main point is at Eq. 9, which is the same equation that appears in Seth Lloyd's paper, but here it is explained without reference to "Lie algebras". Eq. 9 is an application of what in physics we often just call the "Trotter splitting". It was written down almost 100 years earlier by Sophus Lie, but you do not need to know anything about Lie Algebras or even vector spaces in order to apply the formula as done in Eq. 9.

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  • $\begingroup$ You're welcome :) Hope it helps! $\endgroup$ – user1271772 Jun 12 '18 at 21:28
  • $\begingroup$ Why would this answer the question? In the paper, H1 and H2 are related (by a swap), so they seem exactly NOT independent as asked in the question! $\endgroup$ – Norbert Schuch Nov 19 '18 at 22:37

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